Theater-Level Stochastic Air-To-Air Engagement Modeling Via Event Occurrence Networks Using Piecewise Polynomial Approximation

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Theater-Level Stochastic Air-to-Air Engagement Modeling via Event Occurrence Networks Using Piecewise Polynomial Approximation

David R. Denhard

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THEATER-LEVEL STOCHASTIC AIR-TO- AIR ENGAGMENT MODELING VIA EVENT OCCURRENCE NETWORKS USING PIECEWISE POLYNOMIAL APPROXIMATION

Dissertation

D.R. Denhard, Major, USAF AFIT/DS/ENS/01-01

DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY

Wright-Patterson Air Force Base, Ohio

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

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Report Date Report Type Dates Covered (from... to) 01 Sep 2001 Final 01 May 1998 - 01 Sep 2001

Title and Subtitle Contract Number Theater-Level Stochastic Air-to-Air Engagement Modeling via Event Occurrence Networks Using Grant Number Piecewise Polynomial Approximation Program Element Number

Author(s) Project Number Major D. R. Denhard, USAF Task Number

Work Unit Number

Performing Organization Name(s) and Address(es) Performing Organization Report Number Air Force Institute of Technology Graduate School of AFIT/DS/ENS/01-01 Engineering and Management (AFIT/EN) 2950 P Street, Building 640 WPAFB, OH 45433-7765

Sponsoring/Monitoring Agency Name(s) and Sponsor/Monitor’s Acronym(s) Address(es) Air Force Studies and Analyses Agency 1570 Air Force Pentagon Washington DC 20330-1570 ATTN: Maj raig Sponsor/Monitor’s Report Number(s) Knierem

Distribution/Availability Statement Approved for public release, distribution unlimited

Supplementary Notes

Abstract This dissertation investigates a stochastic network formulation termed an event occurrence network (EON). EONs are graphical representations of the superposition of several terminating counting processes. An EON arc represents the occurrence of an event from a group of (sequential) events before the occurrence of events from other event groupings. Events between groups occur independently, but events within a group occur sequentially. A set of arcs leaving a node is a set of competing events, which are probabilistically resolved by order relations. An important EON metric is the probability of being at a particular node or set of nodes at time t. Such a probability is formulated as an integral expression (generally a multiple integral expression) involving event probability density functions. This integral expression involves several stochastic operators: subtraction; multiplication; convolution, and integration. For the EON probability metric, simulation is generally computationally costly to obtain accurate estimates for large EONs, transient nodes, or "rare" states. Instead, using research with probabilistic activity networks, a numerical approximation technique using piecewise polynomial functions is developed. The dissertation’s application area is air-to-air combat modeling. Subject Terms Air Combat, Event Occurrence Networks, Probabilistic Networks, Activity Networks, Piecewise Polynomial Approximation, Simulation

Report Classification Classification of this page unclassified unclassified

Classification of Abstract Limitation of Abstract unclassified UU

Number of Pages 379

The views expressed in this dissertation are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U. S. Government.

AFIT/DS/ENS/01-01

THEATER-LEVEL STOCHASTIC AIR-TO-AIR ENGAGEMENT MODELING VIA EVENT OCCURRENCE NETWORKS USING PIECEWISE POLYNOMIAL APPROXIMATION

DISSERTATION

Presented to the Faculty

Graduate School of Engineering and Management

Air Force Institute of Technology

Air University

Air Education and Training Command

in Partial Fulfillment of the Requirements for the

Degree of Doctor of Philosophy

D.R. Denhard, B.S., M.S.

Major, USAF

September 2001

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

AFIT/DS/ENS/01-01

THEATER-LEVEL STOCHASTIC AIR-TO-AIR ENGAGEMENT MODELING VIA EVENT OCCURRENCE NETWORKS USING PIECEWISE POLYNOMIAL APPROXIMATION

D.R. Denhard, B.S., M.S. Major, USAF

Approved:

Date

______Jack M. Kloeber, LTC(retired), USA (Chairman)

______Dennis W. Quinn, PE, Professor

______Dennis C. Dietz, LtCol(retired), USAF (Member)

______Raymond R. Hill, LtCol, USAF (Member)

______E. Price Smith, LtCol, USAF (Member)

Accepted:

______Robert A. Calico, Jr. Date Dean, Graduate School of Engineering and Management

Dedicated in loving memory to my father whose grin I can sense from above

Acknow ledgements

Many p eople deserve a great deal of thanks in bringing this dissertation to a

conclusion First and foremost I would like to express my sincere thanks to my

b eautiful wife Without her this dissertation would never have b een completed

Thanks bab e

Tomy committee chairman Dr Dennis Dietz Lt Col USAF Retired and Dr

Jack Klo eb er LTC USA Retired thanks for your guidance and patience Twoof

the smartest men I haveever met your critiques of the researchwere always fo cused

pro ductive and timely

To the many committee memb ers I have had over the years Col Jack Jackson

Lt Col Edward Pohl Lt Col E Price Smith and Ma j Paul Murdo ck thanks for your

guidance and participation A sp ecial thanks to Lt Col Ray Hill whose guidance over

the years and attention to detail piloted me toward graduation

DR Denhard v

Table of Contents

Page

Acknowledgements v

List of Figures x

List of Tables xiii

Abstract xvi

I Intro duction

AirtoAir Engagement

Military Mo dels

AirtoAir EngagementModels

Probabilistic Networks

Problem Description

Sequence of Presentation

II Probabilistic Network Literature

Graph Theoretic Underpinnings of Probabilistic Networks

Examples of Probabilistic Networks

Activity Networks

Petri Nets

Task Precedence SeriesParallel Graphs

Belief Networks

Complexity of Probabilistic Networks

Solution Metho ds for Probabilistic Networks

Summary vi

Page

III AirtoAir Engagement Literature

Intro duction

Analytic Mo dels

Simulation Mo dels

Summary

IV Event Occurrence Networks

Intro duction

Description

AirtoAir Engagement Examples

Underlying Probabilistic Mo del

No de State Explosion Problem

Integral Solution

State Integral Expression

EventTierIntegral Expressions

Event Tier u Integral Expressions

Bucket Analysis

Summary

V Piecewise Polynomial Approximation

Intro duction

Piecewise Polynomial Functions

Piecewise Polynomial Approximation and Interp olation

Past Use in DensityFunction Estimation and Sto chastic Op

erations

Discussion

Summary vii

Page

VI EON Solution Using Piecewise Polynomial Approximation

Intro duction

Probability DensityFunction Approximation

Integration Sto chastic Op erator Approximation

Approximation of P E e j E e

j g T g T

j T

Multiplication Sto chastic Op erator Approximation

F e f e

g T E g T

g T

c c

F e F e e f

g T g T g T E

E E

g T

j T j T

F e f e

E g T E g T

g T

c c

F e F e

g T g T

E E

i j

c c

e e F F

g T g T

E E

iT iT

c c

F e F e

g T g T

E E

j T j T

e e F F

g T g T E E

j i

n n

j i

c c c

e e F e F F

g T g T g T

E E E

j T j T

e F e F

g T g T E

c c

e e F F e F

g T g T E g T

E E

j T j T

c c c

F e e F e F

g T g T g T

E E E

j T j T

e f

g T E

g T

e f e e F F

g T E g T g T E

g T

c c

e F e F e F

g T g T g T E

E E

j T j T

f e

E g T

g T

c c c

e F e F e F e F

g T g T g T g T E

E E E

j T j T

c c c

F e F e F e F e

g T g T g T E g T

E E E

j T j T

e f

g T E

g T

Convolution Sto chastic Op erator Approximation

Simple AirtoAir Engagement Example viii

Page

Large EON AirtoAir Engagement

Summary

VI I Summary and Recommendations

Research Goals and Summary

Recommendations

App endix A AirtoAir EngagementEnvironment

A Strategy

A Missions

A AirtoAir Engagement Spatial Layout

A Aircraft

A Detection and Identication

A Weap ons

A Command Control Communications and Information

A Electronic Warfare

A Tactics and Maneuvers

App endix B THUNDERs AirtoAir Engagement Submo del

App endix C Event Occurrence Network Solution for One Versus One Air

toAir Engagement

App endix D Polynomial Approximation

D Norms

D Polynomial Interp olation and Approximation

App endix E Polynomial Co ecients and Interval Data

Bibliography

Vita ix

List of Figures

Figure Page

Military Mo del Categorization Based on Mo del Scop e

Asp ects of Mo del Resolution

Three Categories of Activities Networks DAN PAN and GAN

Six No des Typ es in a Generalized ActivityNetwork

Conversion of a Multiple Source and Sink ActivityNetwork Into a Two

Terminal Activity Network

Petri Net Example

Petri Net Example Transition Firing

Task Precedence Graph Example

Belief Network Example

Complexity Example PAN

Representation of Exp onential Polynomials by Phase Diagram

Basic Series Combination in a GERT Network

GERTNetwork of Simple Air Duel with Unlimited Passes

Backward Recurrence Time Example

Simple TwoEventEvent Occurrence Network

Event Occurrence Network for One vs One AirtoAir Engagement Ex

ample

Event Occurrence Network for One vs One AirtoAir Engagement Ex

ample with Missile Kills

Aggregated Event Occurrence Network for One vs One AirtoAir En

gagement Example with Missile Kills

Event Occurrence Network with Event Groups with Events in Each

Group

Revised Event Occurrence Network with Event Groups with Events

in Each Group with No de Truncated x

Figure Page

Graph of f t and Approximation Error for E in the Simple Airto

exp Air EngagementEvent Occurrence Network E

Graph of f t and Approximation Error for Truncated Normal Density

Function N dened on

t and Approximation Error in the Simple AirtoAir Graph of F

EngagementEvent Occurrence Network E exp

tand F Graph of the Relative Approximation Error of f tin

the Simple AirtoAir EngagementEvent Occurrence Network

Graph of P t and Error for the Simple AirtoAir EngagementEvent

Occurrence Network

Graph of the Convolution of TwoTruncated Normal DensityFunctions

Graph of the Convolution Error of TwoTruncated Normal Density

Functions Expanded Case

Graph of the Convolution Error of TwoTruncated Normal Density

Functions Reduced Case

Graph of the Convolution Error of TwoTruncated Normal Density

Functions using Lawrences algorithm

Graph of the Convolution of TwoTruncated Exp onential DensityFunc

tions

Graph of the Convolution Error of TwoTruncated Exp onential Density

Functions Expanded Case

Graph of the Convolution Error of TwoTruncated Exp onential Density

Functions Reduced Case

Graph of P t and Error for the Simple AirtoAir EngagementEvent

Occurrence Network

Graph of P t and Error for the Simple AirtoAir EngagementEvent

Occurrence Network

Graph of P t and Error for the Simple AirtoAir EngagementEvent

Occurrence Network

Graph of P t and P t Error for the Simple AirtoAir Engagement

Event Occurrence Network xi

Figure Page

Graph of P t and P t Error for the Simple AirtoAir Engagement

Event Occurrence Network

Graph of P t P t and P t Error for the Simple AirtoAir

EngagementEvent Occurrence Network

Graph of P tand P t Error for the Simple AirtoAir Engagement

Event Occurrence Network

Graphical Depiction of Blue Oensive Counter Air OCA mission

Probability of Being in a Representative Absorbing and Transient State

at a time t for Event Tier for the Large airtoair engagement

Probability of Being in a Representative Absorbing and Transient State

at a time t for Event Tier for the Large airtoair engagement

Probability of Being in a Representative Absorbing State at a time t

for Event Tier for the Large airtoair engagement

Ancker and Gafarian s sto chastic kill network for twoversus two ground

combat

Event Occurrence Network formulation of Ancker and Gafarian s sto chas

tic kill network for twoversus two ground combat

Event Occurrence Network formulation with Multiple Absorbing and

Transient States

AirtoAir Engagement Spatial Layout

Aircraft Load Factor g versus Indicated Airsp eed V

Weap ons Envelop e of AllAsp ect Missile

Weap ons Envelop e of AllAsp ect Missile under Gs

Thunder Aircraft Entity Classes

THUNDER BARCAP Patrol Area

THUNDER BARCAP SearchPattern for Perfect Command and Con

trol State

THUNDER BARCAP SearchPattern for No Command and Control

State

Interaction of Flight Group Tactics in THUNDER xii

List of Tables

Table Page

Numb er of the States k of the Markovian Mo del Given m Blue Aircraft

and n Red Aircraft

Probability of Being at a No de for the One vs One AirtoAir Engage

mentEvent Occurrence Network Example with Missile Kills

Numb er of No des for s Group Event Occurrence Networks with s

Events in Each Group

Probability of Being at a No de for the One vs One AirtoAir En

gagement Aggregated Event Occurrence Network Example with Missile

Kills

Event Sequences and Asso ciated Paths for No de for the Group

Event Occurrence Network with Events in Each Group

Error Bounds for Piecewise Polynomial Approximations of f t

Error Bounds for Piecewise Polynomial Approximations of a truncated

N density function

Approximation Error Bounds for Piecewise Polynomial Approxima

t tions of F

Error Bounds for Piecewise Polynomial Approximations of P t

Error Bounds for Piecewise Polynomial Expanded Approximations of

the Truncated Normal Convolution Example

Error Bounds for Piecewise Polynomial Reduced Approximations of

the Truncated Normal Convolution Example

Error Bounds for Piecewise Polynomial Expanded Approximations of

the Truncated Exp onential Convolution Example

Error Bounds for Piecewise Polynomial Reduced Approximations of

the Truncated Exp onential Convolution Example

Error Bounds for Piecewise Polynomial Approximations and Simulation

of P t

xiii

Table Page

Error Bounds for Piecewise Polynomial Approximations and Simulation

of P t

Error Bounds for Piecewise Polynomial Approximations and Simulation

of P t

Numb er of States and Cumulative Probability for EachEvent Tier in

the Large AirtoAir example

Numb er of States and Cumulative Probability for EachEvent Tier in

the Large AirtoAir example

Electromagnetic Sp ectrum

General Frequency Designations Bands For Radar Systems

Communication Bands

THUNDER Example Probability of Engagement Based on AEW

Control State

Probability of Engaging A Target Flight EN GFor An Intercepting

Flight in THUNDER

Probability of Engaging An Intercepting Flight EN GFor A Target

Flight in THUNDER

Weap on versus Weap on Relative Range Advantage for FC and MiG

Example

Aircraft Degree of Command and Control AC C Values for THUN

DER Example

Subintervals for Piecewise Polynomial Approximations of the exp

DensityFunction

Monic Polynomial Co ecients for exp DensityFunction Using

LU Decomp osition

Monic Polynomial Co ecients for exp DensityFunction Using

Singular Value Decomp osition

Orthogonal Co ecients and Polynomials for exp DensityFunction

Orthogonal Co ecients and Polynomials for exp DensityFunc

tion Continued xiv

Table Page

Subintervals for Piecewise Polynomial Approximations of the truncated

N DensityFunction

Monic Polynomial Co ecients for the Truncated N Density

Function Using LU Decomp osition

Monic Polynomial Co ecients for the Truncated N Density

Function Using Singular Value Decomp osition

Orthogonal Co ecients and Polynomials for the Truncated N

DensityFunction

Orthogonal Co ecients and Polynomials for the Truncated N

DensityFunction Continued xv

AFITDSENS

Abstract

This dissertation investigates a sto chastic network formulation termed an event

o ccurrence network EON EONs are graphical representations of the sup erp osition

of several terminating counting pro cesses An EON arc represents the o ccurrence

of an event from a group of sequential events b efore the o ccurrence of events

from other event groupings Events b etween groups o ccur indep endentlybutevents

within a group o ccur sequentially A set of arcs leaving a no de is a set of comp eting

events which are probabilistically resolved by order relations EONs dier from

other sto chastic networks discussed in the literature such as ActivityNetworks Petri

Nets Task Precedence Graphs and Belief Networks and are strongly inuenced by

the research of Ancker Gafarian Kress and several asso ciates to mo del m versus n

sto chastic ground combat

An imp ortant metric for an EON is the probability of b eing at a particular

no de or set of no des at time tSuch a probabilityisformulated as an integral ex

pression generally a multiple integral expression involving event probability density

functions This integral expression involves several sto chastic op erators

Subtraction

Multiplication

Convolution and

Integration

In the literature simulationbased metho ds are the dominant approximation tech

nique for obtaining metrics from sto chastic networks due to their logical simplicity

and ecient implementation However for the EON probability metric simulation is

generally computationally costly to obtain accurate estimates for large EONs tran

sient no des or rare states Instead using researchbyFergueson Shortell and xvi

Lawrence with probabilistic activity networks a numerical approximation technique

using piecewise p olynomial functions is develop ed

The dissertations application area is airtoair engagement mo deling The

dissertation contains a comprehensiveoverview of analytical and simulation based

theaterlevel air combat mo dels including an indepth review of the US Air Forces

THUNDER airtoair engagement submo del Additionally the dissertation provides

an overview of airtoair engagements and the combat environment in which these

engagements take place Factors discussed include strategy missions airtoair

spatial layout aircraft detection and identication weap ons command control

communications and information CI electronic warfare and tactics The disser

tation concludes with a large EON airtoair engagement example xvii

THEATERLEVEL

STOCHASTIC

AIRTOAIR ENGAGEMENT

MODELING

VIA

EVENT OCCURRENCE NETWORKS

USING PIECEWISE POLYNOMIAL APPROXIMATION

I Introduction

Since the intro duction of the aircraft into war the battle for control of the air

has b ecome an essential comp onentofmodernwarfare In fact since the German

attackonPoland in no nation has won a war no ma jor oensive has succeeded

and no defense has sustained itself against an opp onent who has had control of the

air page The Gulf War provided a vivid example of the devastating eects

that can o ccur when one side Iraq cedes sucientcontrol of the air to its opp onent

Coalition Forces

Tohave air sup erioritymeanshaving sucientcontrol of the air to conduct

air sea and land op erations attacks on an opp onent without serious air opp osition

as well as the abilitytoprevent eective air op erations from an opp onentpage

page The natural consequences of two or more opp onents battling

to achieve air sup eriority are combat engagements Air combat engagements may

b e divided into three categories airtoair eg aircraft versus aircraft airto

World War I was the rst war in which aircraft were used with the rst missions b eing recon

naisance page xvii

surface eg aircraft versus ground target and surfacetoair eg surfacetoair

missile SAM versus aircraft This dissertation addresses the former airtoair

engagements

AirtoAir Engagement

Since the classical stylistic dogghts of World War I airtoair engagements

have fascinated the military air community Although surfacetoair engagements

usually account for more aircraft kills than airtoair engagements surfacetoair

engagements lack the glamorous monoamono comp etitive descriptions that are

generally asso ciated with air combat page pages Historical

accounts of airtoair engagements paint these engagements as little slices out of

time which are unrep eatable the aircraft the pilots the weap ons the tactics and

the conditions all come together in an instant of time to determine a result and if

the engagementwere rep eated the outcome could easily b e reversed pages

Ten years after Wilbur and Orville Wrights rst ightina powered aircraft

unarmed reconnaissance aircraft quickly b ecame established as a vital part of war

fare The usefulness of reconnaissance aircraft led to the development of preda

tor aircraft whose sole purp ose was to destroy these reconnaissance aircraft The

rst rep orted ring of an air gun o ccurred on Aug when Lieutenant USA

Jacob Earl Fickel red his rie at a target from his single seat Curtiss biplane

Although Fickel was exploring the p otential of the aircraft as an attack platform

the concept was of obvious equal imp ortance as an airtoair means to attack other

aircraft pages and

Fickels test led to the development of a lightweight low recoil machine gun

develop ed bySamuel Neal McClean with later improvementby Colonel USA Issac

In reality air combat is far from a glamorous gentlemans comp etition Historicallyinthe

ma jority of aircraft kills over downed pilots were not even aware that they had b een red

up on

Available combat time for most combat aircraft is on the order of veminutes or less

N Lewis which proved ideal for air op eration Attached to the front righthand seat

ofaWright Mo del B yer the rst trials of the gun were made on June On

August nd Lieutenant RAF LA Strange with a Lewis machine gun

attached to his Farman biplane to oko in pursuit of a German Albatros whichwas

ying a reconnaissance mission over Maub euge and is credited with the rst use of

amachine gun in combat Until this date aircrews p ersisted with pistols and ries

despite the diculty aircrews faced in putting their aircraft into a p osition where

they could get a clear shot at the opp onent It was as late as Octob er

more than a decade after the Wright Brothers rst ight that the rst aircraft was

recorded as shot down in an airtoair engagement A French rearengined Voisin

piloted by Sergeant Joseph Franz shot down a German Avaitax By the end of

air combat had b ecome an integral part of warfare In to days timecompressed air

combat world life or death win lose or draw generally takes place in under ve

minutes page page page pages

Military Models

Before discussing airtoair engagement mo dels some background information

and terminology p ertaining to military mo dels used for combat analysis may b e help

ful to the reader Several structures exist for categorizing military mo dels page

Figure depicts one such structure for categorizing military mo dels based on

scope Scop e refers to the intented purp ose of the mo del Military mo dels range in

scop e from engineering mo dels eg a mo del of aircraft aero dynamic p erformance

to engagement mo dels eg a mo del of a one versus one airtoair engagement to

mission mo dels eg a mo del of a ight group of aircraft from takeo to attack

on target and back to the campaign or theaterlevel mo dels which represent a set

of missions op erations or battles in the pursuance of a military campaign ob jec

tive page The theaterlevel mo del in contrast to mo dels with lesser scop e

Reference lists several military mo deling taxonomies INCREASING HIGHER AGGREGATION RESOLUTION THEATER- LEVEL

MISSION

ENGAGEMENT

ENGINEERING

Figure Military Mo del Categorization Based on Mo del Scop e

shows the overall interactions of combined forces air land and naval in military

op erations

An imp ortant mo del quality is the level of detail or resolution of asp ects used

in the mo del page Figure quanties dierent asp ects of mo del resolu

tion page These asp ects include

resolution This asp ect of resolution refers to the level of ob ject mo deled Entity

Higher entity resolution might mean mo deling individual aircraft rather than

only mo deling to the level of an aircraft ight

Attribute resolution This asp ect of resolution refers to the detail of an ob jects

attributes Higher attribute resolution might mean sp ecifying the typ es of air

toair munitions carried by an aircraft rather than merely assigning a comp osite

airtoair munition to the aircraft

LogicalDep endency or Constraint resolution This asp ect of resolution refers

to the level at which constraints are enforced on the attributes of an ob ject

and their interrelationships Higher constraint resolution might mean requiring

an airtoair munition to have a captivecarry reliability rather than merely

assuming that the airtoair munition will always re page

Pro cess resolution This asp ect of resolution refers to the level at whichvarious

eects cause changes in the attribute of an ob ject Higher pro cess resolution

might mean computing airtoair attrition at the aircraft level rather than

computing airtoair attrition at the ight level and then spreading the attrition

equally across the ights aircraft

Spatial and Temp oral resolution These asp ects of resolution refer to the scales

used for space and time Higher spatial resolution might mean tracking air

craft p osition in three dimensional space rather than a two dimensional space

Higher temp oral resolution might mean tracking time advancement in steps of

seconds rather than in steps of minutes

The terms high and low resolution are used in a relativecontext here and

are not inherent prop erties of the asp ects of a mo del These terms only have meaning

when comparing two or more mo dels For example mo del As airtoair attrition

pro cess might b e considered of high resolution when compared to mo del Bs attrition

pro cess However mo del As airtoair attrition pro cess might also b e considered of

low resolution when compared to mo del Cs attrition pro cess

Military mo dels found lower on the pyramid structure in Figure tend to op

erate at a higher resolution in terms of the mo dels asp ects given ab ove than mo dels

found higher on the pyramid structure This inverse relationship b etween mo del

scop e and resolution is a direct result of current limitations in mo del complexity

size run time p erformance and cost As the scop e of a mo del increases the sheer

However the relative resolution b etween two mo dels can b e ambiguous one mo del mayhave

higher resolution than another in some resp ects but lower resolution in others page RESOLUTION

ENTITY PROCESS TEMPORAL

LOGICAL DEPENDENCY SPATIAL

ATTRIBUTE

Figure Asp ects of Mo del Resolution

numb er of combatants and weap on systems makes it imp ossible to maintain individ

ual item resolution without sacricing an increase in mo del complexity size run

time p erformance and or cost page

Closely asso ciated with the concept of resolution is the concept of aggregation

and disaggregation In general aggregation refers to the pro cess of assembling or

combining into a mass and disaggregation refers to the antithetical pro cess of dis

assembling or separating into particulars Although usually thought of in terms of

entity and pro cess asp ects of resolution aggregation and disaggregation apply to

other asp ects of resolution as well For example in the case of temp oral resolution

we can aggregate a time step of seconds into minutes and disaggregate minutes into

seconds Military mo dels found higher on the pyramid structure in Figure tend to

b e more aggregated in terms of the mo dels asp ects given ab ove than mo dels found

lower on the pyramid structure

AirtoAir Engagement Models

Todays airtoair engagement mo dels consist mainly of relatively large com

plex simulations and may b e roughly divided into two main groups according to their

intended purp ose analysis or training The airtoair engagement mo dels are used

as separate standalone mo dels or as a section of larger mo dels dep ending on the

scop e of the problem b eing investigated The level of detail in these mo dels varies

dep ending on the mo dels scop e and purp ose

In the literature I have reviewed aggregation disaggregation has b een used in reference to

entity and pro cess resolution exclusivelyFor example in the case of entity resolution aggregating

aircraft entities into a ightentity or disaggregating a ightentityinto aircraft entities in the

case of pro cess resolution aggregating sp ecic typ es of airtoair munitions on an aircraft into a

single or comp osite munition or disaggregating a comp osite airtoair munition into sp ecic typ es

of munitions

However as the relative resolution b etween two mo dels can b e ambiguous so also can the level

of aggregation For example a campaign mo del can mo del air entities to the level of individual

aircraft as can an engagement air mo del but the airtoair attrition pro cess used in the campaign

mo del might b e more aggregate than the pro cess used in the engagement mo del

The Air Forces primary theaterlevel mo del for analysis is THUNDER Op er

ational since THUNDER is a computer simulation of conventional air land

and naval warfare Volume I page The air war mo del uses a discrete event

timestepp ed sto chastic simulation whereas the ground war mo del uses a discrete

event timestepp ed deterministic simulation The naval warfare mo del is limited in

scop e and includes tasks such as carrier aircraft op erations that can b e mo deled us

ing the air war mo del The airtoair engagement algorithms used in THUNDER are

based on research p erformed by George S Fishman and Louis R Mo ore while at the

University of North Carolina These algorithms have not changed substantially

since THUNDER was rst intro duced in

In contrast to THUNDER the Air Force primary mo del for analyzing airtoair

combat at the mission engagement level is BRAWLER Op erational since the

early s BRAWLER is the Air Forces highest resolution analytical airtoair

engagement mo del BRAWLER is a discrete event timestepp ed sto chastic simula

tion that mo dels b oth within visual range and b eyond visual range engagements

Multiple ights of aircraft are explicitly mo deled through all phases of airtoair

combat BRAWLER features detailed datadriven mo dels of most of the avionics

and weap ons found on mo dern ghter aircraft Emphasis has b een placed on the

command and control asp ects of airtoair engagements including co op erative tac

tics surprise confusion and situational awareness BRAWLER is currently used to

tune parameters in THUNDERs airtoair engagement mo del page

In addition to THUNDER and BRAWLER other airtoair engagementmod

els exist in the defense community In terms of analysis mo dels several theaterlevel

The ground war mo del is based on the Armys Concept Evaluation Mo del CEM which uses

the Attrition Calibration ATCAL pro cess and the US Army Concepts and Analysis Agencys

Combat Sample Generator COSAGE mo del to attrite ground combatants

Highest as explained in the military mo dels section is a sub jective term In fact certain

asp ects of BRAWLER mightbeoflower resolution than other airtoair engagement mo dels How

ever lo oking over all asp ects BRAWLER has higher resolution than other mo dels in a ma jorityof

cases Hence the use of the term

mo dels consider airtoair engagements suchasTACWAR and the Theater Level

Campaign TLC mo del Each of these mo dels approaches the representation

of airtoair engagements dierently For example TACWAR do es not simulate

individual ights of aircraft and uses a general attrition algorithm based on re

power scores to adjudicate air combat page In contrast TLC uses

an airtoair engagement pro cess based on explicitly mo deled ights and the concept

of rounds of airtoair combat page Additionally mission engagement

level mo dels exist to analyze sp ecic problems such as force ratios Examples may

b e found in and In terms of training mo dels theater level wargames

are used to train command and sta p ersonnel Several of these mo dels consider air

combat such as the Air Forces Air Warfare Simulation AWSIM and the Armys

Corp Battle Simulation CBS Although these training mo dels are not designed for

analysis work they contain airtoair engagement algorithms that mayalsohave

application for analysis mo dels

The most signicant dierence b etween airtoair engagement mo dels of high

and low resolution is their treatment of the spatial and temp oral asp ects of mo d

eling Sp ecically higher resolution airtoair engagement mo dels tend to describ e

spatial and temp oral asp ects in greater detail than lower resolution airtoair en

gagement mo dels For example BRAWLER mo dels the spatial domain in three

dimensions longitude latitude altitude down to units of feet and the temp oral

domain in seconds page and page In contrast THUNDER

mo dels the spatial domain in two dimensions longitude latitude down to units of

meters with only a few discrete altitude settings and the temp oral domain in min

utes Volume I I page page In airtoair engagement models the

spatial and temporal aspects of a model directly determine the level of attributes con

TACWAR is a deterministic theaterlevel mo del of ground and air combat rst develop ed in

by the Institute of Defense Analysis TLC is a research prototyp e sto chastic campaign level

mo del of ground and air combat develop ed in the early s by RAND TLC serves as a test b ed

for investigating issues aimed at improving the next generation of theaterlevel mo dels

straint and process aspects that can beeectively modeled For example even though

b oth BRAWLER and THUNDER mo del airtoair engagements down to the aircraft

entitylevel the spatial and temp oral settings of THUNDER limit THUNDER from

explicitly mo deling the pro cess of few on few air combat

Probabilistic Networks

A probabilistic network is a directed graph G N A linked to an under

lying probabilistic sto chastic mo del in which probabilistic events activities are

represented by arcs or no des in GEach probabilistic event is represented by a ran

dom variable or a combination of random variables Probabilistic networks havea

wide range of applicability in the literature These networks are used to formulate

and solve problems in such diverse areas as program management computer

and communication systems and articial intelligence The app eal

and p ower of probabilistic networks in formulating and solving problems lies in the

graphical representation of the underlying probabilistic mo dels This graphical rep

resentation serves as a visual aid in conveying the structure of the probabilistic mo del

and in formulating p ossible solution techniques

This dissertation intro duces a new probabilistic network termed an event oc

currence network EON to mo del airtoair engagements An EON is a graphi

cal representation of the sup erp osition of several terminating counting pro cesses

EONs are motivated by the researchofAncker Gafarian Kress and several as

so ciates in mo deling m versus n sto chastic ground

combat Event o ccurrence networks EONs are not limited to mo deling combat

situations These networks can b e applied whenever sets of random events o ccur

THUNDER can not mo del aircraft movement in an airtoair engagement

Problem Description

Little research has b een devoted to obtaining complexity measures for proba

bilistic networks with continuous random variables This situation is a direct result

of the inability to compute exact analytical solutions for many probabilistic networks

and has led to the use of approximation metho ds Simulationbased metho ds have

b een the dominant approximation technique due to their logical simplicity and e

cient implementation When a simulation is used to solve a probabilistic network

the main complexity measure is the number of simulation runs and corresp onding

random variable draws necessary to establish a certain statistical condence in the

output

However for some probabilistic networks simulation can b ecome a compu

tationally costly undertaking to obtain accurate estimates of the output measures

As the number of random variables in the network increases so do es the number

of replications necessary in order to prop erly characterize the output measures

As an alternative to simulation numerical approximation and reduction metho ds

have b een prop osed for some probabilistic networks Network reduction involves the

rep eated application of op erators suchasmultiplication minimum maximum k

out of n convolution and conditional integration to reduce a network to a smaller

network through the aggregation of the underlying probabilistic mo del of the net

work Numerical approximation involves the use of functions such as piecewise p oly

nomial functions exp olynomial functions exp onential p olynomials functions etc

to approximate the density or distribution functions of the random variables For

probabilistic networks with several random variables numerical approximation and

reduction techniques can b e several orders of magnitude faster than simulationbased

metho ds without signicant losses in accuracy

The overall ob jective of this researchistwofold The rst goal is to develop

anetwork formulation capable of expressing sto chastic event o ccurrences and their

interactions The concept of event o ccurrence networks EONs will b e researched

EONs are graphical representations of the sup erp osition of several terminating count

ing pro cesses The second ob jective of this research is to further the investigation

of the piecewise p olynomial functions to approximate density functions and distri

butions This investigation includes the sto chastic op erations of subtraction multi

plication convolution and integration The research will expand the work of Mar

tin Fergueson and Shorttell and Lawrence The p olynomial

approximation metho d develop ed by this research will b e used to solvetwo analyti

cal airtoair engagement EONs

SequenceofPresentation

The remainder of this dissertation is divided in six chapters Chapter I I pro

vides a literature review of probabilistic networks their complexity and output mea

sures and current solution metho ds Sp ecically the review lo oks at four dierent

networks application areas activity networks sto chastic Petri nets task precedence

graphs and b elief networks Chapter I I I provides a review of airtoair engagement

literature including analytic and simulated mo dels used for theaterlevel analysis

Chapter IV intro duces event o ccurrence networks EONs and gives several examples

of airtoair engagement EONs Chapter V formally denes a piecewise p olynomial

function and provides a literature review summarizing the researchinto approximat

ing sto chastic op erators with piecewise p olynomial functions Chapter VI details the

techniquesalgorithms for solving event o ccurrence networks using piecewise p olyno

mial approximation The chapter shows two airtoair engagement EON examples

and compares the piecewise p olynomial approximation technique with simulation

Finally Chapter VI I provides a summary of the research and recommendations for

further research

II Probabilistic Network Literature

Probabilistic sto chastic networks have a wide range of applicabilityinthe

literature These networks are used to formulate and solve problems in such diverse

areas as program management computer and communication systems

and articial intelligence The app eal and p ower of probabilistic networks in

formulating and solving problems lies in the graphical representation of the un

derlying probabilistic mo dels This graphical representation serves as a visual aid

in conveying the structure of the probabilistic mo del and in formulating p ossible

solution techniques Before pro ceeding with a formal denition of a probabilistic

network a review of graph theoretic terms will b e undertaken in order to set the

framework for such a denition

Graph Theoretic Underpinnings of Probabilistic Networks

The following graph theoretic terminology may b e found in any standard graph

theory text Texts used for this review included A graph G is an

ordered pair of disjointsetsN Asuchthat A is a subset of the set of unordered

pairs of N For our purp oses the sets N and A are nite The set N is termed

the set of no des or vertices and the set A termed the set of arcs or edges An

arc fx y g that joins the no des x and y is denoted by xy or x y If xy A then

no des x and y are adjacent neighb oring no des of G and incidentwith xy Two

arcs are adjacent if they have exactly one common end no de The order of G is the

numb er of no des and is denoted by j N j The size of G is the numb er of arcs and

is denoted by j A j Gn m denotes an arbitrary graph of order n and size mThe

B C

size of a graph of order n is at least and at most A graph G N A is

a subgraph of G N AifN N and A A

Agraph G is a rpartite graph with no de classes N N N if N G

N N N and N N whenever i j r and no arc joins two

r i j

no des in the same class By denition a graph G do es not contain a lo op ie an

arc joining a no de to itself nor multiple arcs ie two or more arcs joining the same

two no des In a multigraph H multiple arcs and lo ops are allowed If the arcs of a

graph G or multigraph H are identied with ordered pairs of no des then the graph

or multigraph is considered to b e directed Otherwise G or H is undirected An

order pair a b is said to b e an arc directed from no de a to no de b and is denoted

as ab or a b

A directed path is a directed graph P of the form N P fx x x g and

AP fx x x x x x g where the elements of fx x x g are distinct

l l l

The no des x and x are the end no des of P and l j AP j is the length of P A

directed path in which x x and l is termed a cycle A directed graph G

which contains no cycles is acyclic Otherwise it is cyclic A directed acyclic graph

dag D is homeomorphic from another dag D if D can b e obtained from D by

rep eatedly inserting no des of indegree and outdegree one in the middle of arcs of

D where the indegree and outdegree of a no de x is the numb er of arcs incident

into and out of x resp ectivelyAdagD is transitive if for anytwonodesx and y

such that there is a path from x to y either x y or xy is an arc The transitive

dag of G G fN A g is referred to as the transitive closure of G

T T

For an arc x x Anodex is an immediate predecessor parent of no de

i j i

x and no de x is an immediate successor child of no de x For all x N x

j j i i i

denotes the set of immediate predecessors of x and x denotes the set of immediate

i i

successors of x A directed graph G can b e describ ed by listing either x or x

i i i

for all x N G If x thennodex is referred to as a source no de If

i i i

x thennodex is referred to as a sink no de Additionally if there is a

i i

directed path from no de x to no de x and x x then no de x is a descendantof

i j i j j

no de x Incontrast if no directed path exists from no de x to no de x thennode

i i j

x is a nondescendantofnodex

j i

Clearlythepower of graphs as analytical to ols lies in their ability to visually

represent structural relations When using directed graphs the concept of prece

dence relations is intro duced Precedence relations are sp ecied by listing x for

all x N G Having intro duced the graph theory terminology and the concept

of precedence relations wecannow pro ceed with a denition of a probabilistic net

work A probabilistic network is a directed graph G N Alinked to an underlying

probabilistic sto chastic mo del in which probabilistic events activities are repre

sented by arcs or no des in G Each probabilistic event is represented by a random

variable or a combination of random variables Since G is directed G allows for the

representation of event precedence Two p ossible mo des of graphical representation

exist

ActivityonArc AoA representation and

ActivityonNo de AoN representation

If G N A is an AoA representation the sets of arcs A denotes the events

activities and the set of no des N denotes the precedence relations among events

In contrast if G N A is an AoN representation N denotes the events activities

and A denotes the precedence relations among events Elmaghraby et al state

that a AoN graph is unique due to the onetoone corresp ondence of b oth activities

and precedence but an AoA graph is not

Examples of Probabilistic Networks

Activity Networks An activity network is a directed acyclic graph

that is used to represent activities and their interrelationships ie precedence re

lationships Figure shows the structure of three dierent categories of activity

networks Activity networks have their genesis in solving programpro ject man

agement planning and control problems and rst app eared under the names of the

critical path metho d CPM and program evaluation and review technique PERT

networks in Probabilistic or sto chastic activitynetworks are activity net

works in which some or all of the activity durations are assumed to b e random

variables These activity networks app ear under the names PERTnetworks proba

bilistic activity networks PANs and generalized activitynetworks GANs in the

literature As will b e seen shortly the name probabilistic activitynetwork has a

sp ecic and limited meaning in the literature As a result the term sto chastic ac

tivity network will b e used in place of probabilistic activitynetwork to denote an

activity network in which some or all of the activity durations are assumed to b e ran

dom variables Elmaghraby Adlakha and Kulkarni Lawrence provide

overviews of research on sto chastic activitynetworks

Activitynetworks separately dene the terms event and activityAneventisa

welldened o ccurrence in time and an activityisany undertaking that consumes

time and resources The separation of the event and activity terms claries the

concept that an eventmay dep end up on several activities with a sp ecied number of

these activities having to b e accomplished in order for the event to b e accomplished

If the principal resource of an activitynetwork is time precedence relationships

among events determine which activities must b e completed b efore other activities

can b e started and thus determine the order of o ccurrence of the activitynetworks

events

Elmaghraby makes two remarks ab out precedence in activitynetworks

Precedence is a binary relationship if an activity a precedes another activity

b denoted a b then a must b e completed b efore b is started

Precedence is a transitive relationship if a b and b c a c

This concept is implied in our denition of probabilistic network 6 2 3 2 4 5 1 6

3 2 4 5 1 Deterministic Activity Network (DAN)

Y2,3 2 3 Y Y1,2 3,6

6 1

Y 1,4 4 5 Y5,6

Y4,5 Probabilistic Activity Network (PAN)

3 (1,0) (p2,Y3,5)

(p ,Y ) 2 (1-p2,Y3,7) 1 1,2 5 (1,Y5,8)

(1,0) 4 8 1 (p3,Y4,5)

(1-p3,Y4,7) (1-p1,Y1,6) (1,Y ) 6 7 7,8

(1,Y6,7)

Generalized Activity Network (GAN)

Figure Three Categories of Activities Networks DAN PAN and GAN

Since precedence is a transitive relationship lo oping cycling is not p ermitted in

activity networks In other words if activity a precedes activity b and activity b

precedes activity c then activity c cannot precede activity a Hence activitynetworks

are directed acyclic graphs

Activitynetworks can b e mo deled using either an AoA or AoN graphical rep

resentation As will b e seen in section algorithms exist for converting a AoN

graphical representation into a AoA representation and vice versa Activitynetworks

presented in the literature generally use the AoA representation When discussing

activitynetworks an AoA representation will b e assumed for the remainder of this

dissertation unless otherwise noted Elmaghraby divided activitynetworks into

three categories

Deterministic Activity Networks DANs

Probabilistic ActivityNetworks PANs and

Generalized Activity Networks GANs

Figure illustrates the three activitynetwork categories DANs are activitynetworks

in which activity arc durations are assumed to b e constant In contrast PANs are

activitynetworks in which activity arc durations are assumed to b e random vari

ables Both DANs and PANs require that event no de realizations o ccur only when

each of the events successor activities arcs are accomplished This requirement

corresp onds to the AND logical op erator For this reason DANs and PANS are

also referred to as allAND networks In an allAND activitynetwork suchasa

DAN or PAN all activities in the network must b e accomplished in order for the

activitynetwork to b e completed

In order to relax the precedence relations among activities allAND activity

networks were extended along two lines First in DANs strict precedence among

All successor activities arc of an event no de are started when all of the events no des

predecessor activities arcs are realized

activities ie completing all preceding activities b efore starting a successor activity

was relaxed which led to DANs with generalized precedence relations GPRs

Second in DANs and PANs the nonrealization of activities was p ermitted ie

every activity in the pro ject do es not have to completed which led to generalized

activitynetworks GANs In contrast to DANs and PANs GANs expand

the no de event logical op erator set In a GAN a no de event is constructed from

the combination of an input receiver and output emitter side Three input sides

are p ossible

AND a no de event is realized when all arcs activities leading into the no de

are realized

INCLUSIVEOR a no de is realized when any one arc or combination of arcs

leading into the no de is realized and

EXCLUSIVEOR a no de is realized when one and only one arc leading into

the no de is realized

Two output sides are p ossible

DETERMINISTIC must follow a no de from which all emanating arcs are

undertaken and

PROBABILISTIC mayfollow a no de from which an emanating arc maybe

realized with probability

Since each no de has a receiver and emitter six no de typ es are p ossible in a GAN

Figure shows the p ossible combinations Additionally a GAN can accommo date

activity arc durations that are constant andor random variables Clearlya DAN

is a GAN in which all no des events are of the ANDDETERMINISTIC typ e and

all activities arc durations are constant Similarlya PAN is a GAN in which

all no des events are also of the ANDDETERMINISTIC typ e but whose activity

arc durations are random variables Output Input Deterministic Probabilistic

AND

Inclusive-OR

Exclusive-OR

Figure Six No des Typ es in a Generalized Activity Network

Since Elmaghrabys original work on GANs in the mids very lit

tle work has app eared in the literature mo difying the structure of activitynetworks

Instead this work has pro ceeded in the commercial realm due to the wide spread

adoption of simulation as the preferred analytic to ol for solving GANs Solution

metho ds for probabilistic networks will b e discussed later in section One of

the main pioneers in the use of simulation as an analytic to ol for solving GANs is

Pritsker Following the intro duction of the general evaluation and review technique

GERT to solveallEXCLUSIVEOR activity networks Pritsker devel

op ed a family of GERT simulation GERTS to ols to solve more general networking

problems Through the use of simulation event no de logical op erators and

activityarctyp es have greatly expanded Today a simulation of an activity net

work using a general purp ose simulation environmentsuch as VISUAL SLAM can

haveover twenty dierentnodetyp es and ve arc typ es

In Figure all three activitynetwork category examples are shown with a

single start no de termed a source no de and a single termination no de termed a sink

no de Referred to as a two terminal network this is the standard form for 1

Source Node 3

N-1

Sink Regular Arc Node Dummy Arc

Figure Conversion of a Multiple Source and Sink Activity Network Into a Two

Terminal Activity Network

activitynetworks found in the literature An activitynetwork mayhavemultiple

sources and sinks but it usually converted into a two terminal network through the

use of dummy no des and arcs activities of zero duration As shown in Figure this

conversion is accomplished by inserting a dummy source no de b efore the source

no des and connecting the dummy no de to the source no des via arcs of zero duration

Similarly the sink no des are connected to a single dummy sink no de through

dummy arcs of zero duration

In the literature the three primary metrics of interest in analysis of activity

networks are

Network pro ject completion time

Shortest path and

Activity and path criticalities

Clearly for sto chastic activitynetworks network completion time is expressed in the

form of a distribution Adlakha and Kulkarni Lawrence provide a compre

hensive review of research on the completion time of sto chastic activitynetworks

Analogous to network completion time the shortest path in an activitynetwork is

a useful metric for managers Alexop oulos provides a review of researchonthe

shortest path metric In addition to network completion time and shortest path

activity and paths criticalities provide measures of the activities and paths resp ec

tively that are most likely to comp ete for the longest duration in a network These

criticalities help managers identify those tasks which can b ecome b ottlenecks in a

pro jectprogram Bowman provides a review of research on activity and path

criticalities

Petri Nets APetri net is a directed bipartite graph Figure shows

an example of a Petri net Petri nets were rst used to mo del computer and com

munication systems p erformance but the use of Petri nets has expanded to include

other systems where the mo deling of concurrency parallelism and synchronization

control of parallelism are imp ortant Murata and Ciardo et al provide

overviews of researchonPetri Nets A Petri Net is generally dened by a tuple

P T I O M where P is the set of places T is the set of transitions IPT

is the input function OT P is the output function and MPZ is a

marking Z denotes the set of all nonnegativeintegers

As shown in Figure the two disjoint sets of no des of the bipartite graph are

drawn as circles which corresp ond to places and bars which corresp ond to transitions

Places contain tokens p ossibly multiple tokens which are drawn as dots A place

is input to a transition if there is an arc from the place to transition Similarlya

place is the output of a transition if there is an arc from the transition to the place

A marking of a Petri net is given bya vector whichcontains as its entries the

number of tokens in each place Tokens move based up on the enabling and ring

of transitions A transition is enabled in a marking if all of its input places contain

at least as many tokens as the multiplicity of the corresp onding input arc from

that place A transition res by removing tokens from each of its input places and

dep ositing tokens to the output places according to the output arc multiplicityEach p1 p3

p2 p4

Figure Petri Net Example

ring of a transition results in a new marking of the Petri net Each marking denes

a state of the system If the number of tokens in the Petri net is b ounded then there

are a nite numb er of markings The reachabilitysetofa Petri net is dened as

the set of all markings that are reachable from the initial marking The reachability

tree is a graphical representation of the reachability set where no des representthe

various reachable states ie markings and arcs represent p ossible paths from one

state marking to another state A reachability graph is similar to a state space

transition diagram Petri nets have b een extended for increased ease of use and

enhanced mo deling p ower For instance inhibitor arcs can b e allowed that prevent

ring of a transition when there is a token in any one of its inhibitor places

As an illustration of a ring Figure shows the result of transition t ring from

gure Initial marking b ecomes marking after t

res

AStochastic Petri net SPN is a Petri net where the transition rings are

immediate ie zero ring time or

timed ie probabilistic ring time distribution p1 p3

p2 p4

Figure Petri Net Example Transition Firing

The ring of immediate transitions is given priorityover the ring of time transi

tions Additionallyeach immediate transition is givenaweight probability which

determines its ring probability in case the transition is in conict with another

immediate transition As a consequence of the twotyp es of transition rings the

reachability set a sto chastic Petri net can b e divided into vanishing markings at

least one immediate transition is enabled and tangible markings otherwise The

tangible markings of a sto chastic Petri net corresp ond to the states of an underlying

sto chastic pro cess termed the marking pro cess

Based on the ab ove discussion of Petri nets the reader of this dissertation might

have the feeling that there is an underlying corresp ondence b etween Petri nets and

activitynetworks In fact there is corresp ondence b etween the twonetwork typ es

alb eit an informal one The parallelism b etween Petri nets and activitynetworks has

b een addressed formally by several authors in the literature Quichaud and Chre

tienne were the rst to suggest a link b etween Petri net theory and activity

networks They derived p erformance evaluation results for the transientbehavior

of bip olar synchronization schemes and also prop osed an extension of the standard

GERT analysis to networks containing AND no des recall the standard GERT anal

ysis was restricted to all EXCLUSIVEOR activitynetworks Elmaghraby

elab orated on this link citing the following similarities and dierences Similarities

between Petri nets and activity networks include

Mo deling of an entitymoving through a network ie Petri nets use tokens

while activity networks use transactions

Mo deling of activities and relationships among activities including such topics

as concurrencysynchronization etc and

Exploring similar analysis issues eg deadlo ckPetri net vs feasibility ac

tivitynetwork or reachabilityPetri net vs realizability activitynetwork

Dierences b etween Petri nets and activity networks include

Formalism from their inception Petri nets addressed issues of logical concur

rency such as reachabilitycoverabilityliveness b oundedness conversation

etc while activity networks fo cused on issues of timing economic value and

resource allo cation

Concept of State the concept of state is prominentinPetri nets and lacking

in activitynetworks and

Resource Representation it is much harder to represent resources in Petri nets

than activity networks

In conclusion Elmaghraby stated that activity networks have more p otent and gen

eral capabilities than Petri nets to mo del and analyze systems however he noted

that Petri nets retain a logical structure esp ecially when representing actions that

are conditional up on the realization of other events He prop osed researchinto a

paradigm that blended the b est features of b oth Petri nets and activitynetworks

Elmaghraby et al further discussed the blending of concepts from Petri

net and activitynetwork theory The authors prop osed a new paradigm PETAN

PETriActivity Nets as an approach to the analysis and design of software pro duc

tion pro cesses The PETAN paradigm combined the theory of generalized activity

networks GANs available simulation software SLAM I I for mo deling GANs

and the theory of Petri nets Elmaghraby et al stated that GANs p ossess un

matched analytical p ower as a result of b eing derived from the theory of sto chastic

pro cesses and software such as SLAM I I provided a means to simulate the GANs

and hence the underlying sto chastic pro cess Unfortunately as the authors stated

mo deling through either activity networks or sto chastic pro cess software is often

nonintuitive and demanding in time and resources However the authors prop osed

that the theory of Petri nets provided a highly intuitive mo deling pro cess In the

example provided by the authors a SLAM I I simulation mo del of a software pro duc

tion pro cess GAN is given and solved and a Petri Net mo del of the GAN is created

to intuitively explain the events of the pro cess

Other researchers have discussed the link b etween Petri net and activity net

work theory Cubaud showed that the computation of transition ring dates in

astochastic event graph a sto chastic Petri net for whicheach place has at most one

input transition and at most one output transition is equivalent to the computa

tion of the completion time of a PERTnetwork Lee and Murata intro duced

a b etadistributed sto chastic Petri Net BSPN whichintegrated the PERT activ

ity networks and Petri nets to solve uncertainty and concurrency problems in large

software pro jects

Task Precedence SeriesParal lel Graphs A task precedence graph

is a seriesparallel directed acyclic graph Figure shows an example of a task

precedence graph Task precedence graphs are used to mo del and evaluate paral

lel applications such as parallel programs Trogemann and Gente provide an

overview of research on task precedence graphs The no des of a task precedence

SLAM I I is a predecessor of VISUAL SLAM 2 3 4

1 5 6 11

7 8 9 10

Figure Task Precedence Graph Example

graph suchasshown in Figure represent the subtasks of the parallel applications

and the arcs represent the precedence relationships b etween the subtasks Task

precedence graphs are subsets of AoN activitynetworks As will b e shown in sec

tion the structure of these precedence graphs and seriesparallel directed graphs

in general enable ecient solution in p olynomial time Solutions to seriesparallel

directed graphs will b e discussed in greater detail in section

Belief Networks Informally a b elief network is a directed

acyclic graph linked to a joint probability distribution with certain indep endence

prop erties Figure shows two simple examples of a b elief network Belief networks

provide graphical representations of causal dep endence These networks are used

to mo del dep endencies in such diverse areas as medical diagnosis natural language

understanding circuit fault diagnosis pattern recognition machine vision nancial

auditing map learning sensor validation and forecasting

Belief networks have also b een called Bayesian networks causal networks

probabilistic inuence diagrams or knowledge maps Pearl and Jen Z X Y

X Y Z

(a) (b)

Figure Belief Network Example

zarli provide overviews of research in this area The no des of a b elief network

represent random termed chance variables whose distributions are quantied by

presence or absence of predecessor arcs into the no des Sp ecicallythechance vari

able of a no de with predecessor arcs is represented by a conditional distribution given

its immediate no de predecessors termed parents Additionallythechance variable

of a no de without predecessor arcs is represented by a marginal distribution The

joint distribution for all the chance variables in the b elief network is then obtained

bymultiplying all the conditional and marginal distributions of the no des in the

network

For example consider the twobeliefnetworks in Figure X Y and Z

representchance random variables in b oth networks In b elief network a the

joint probability distribution of X Y andZ denoted f x y z is sp ecied as

f x y z f z f x j z f y j z

where f z is the marginal probability distribution of Z and f x j z and f y j z

are the conditional probability distributions of X and Y resp ectively given Z z

In this case no de X and no de Y both have Z as a predecessor In b elief network

b f x y z is sp ecied as

f x y z f xf y f z j x y

where f x and f y are the marginal probability distributions of X and Y resp ec

tively and f z j x y is the conditional distribution of Z given X x and Y y In

this case no de X and no de Y are b oth predecessors of no de Z

Jenzarli gave the following formal denition of a b elief network a b elief

network is pair B D P where D N A is a directed acyclic graph with

variables as no des and P is a joint probability distribution for the variables in N

that can b e factored into conditional probabilities one for eachvariable given its

immediate predecessors More preciselyifX X are the random variables

represented as network no des and x denotes the set of immediate successors

of no de x then the joint distribution for the variables in N can b e represented as

follows

P x x P x j x

n i i

where P x j x P x

i i i

Shimony stated that a b elief network saves many orders of magnitude

in the size of representation Equation shows where the savings in size of repre

sentation comes from For example representing the full distribution of n binary

and thus discrete random variables directly requires probabilities Using a b elief

network one only needs

probabilities Since in most b elief networks app earing in the literature the maximum

indegree of a typical networkismuch less than n a considerable savings in size of

representation is achieved

Pearl stated the following imp ortantcharacteristics of directed acyclic

graphs and b elief networks

Criterion of dseparation if N N andN are three disjoint subsets of

no des in a directed acyclic graph D thenN is said to b e dseparate N from

N if along every path b etween a no de in N and a no de in N there is a no de x

satisfying one of the following conditions a x has converging arrows and

neither x or its descendants are in N orb x do es not haveconverging

w w

arrows and x is in N For example in Figure b elief network a no des

X and Y are dseparated bynode Z Dseparation p ermits determination of

which set of variables are indep endentofeach other given a third set in a b elief

network

Given a directed acyclic graph D and a joint probability distribution P a nec

essary and sucient condition for D to b e a b elief network of P is that each

variable X b e conditionally indep endent with resp ect to probability distribu

tion of its nondescendants variables represented by nondescendantnodes

in D given its immediate predecessor variables represented by predecessor

no des in D

Jenzarli showed a corresp ondence b etween activity and b elief networks

by demonstrating how to mo del the AoN representation of a probabilistic activity

network PAN as a b elief network Jenzarli terms these networks PERT b elief

networks In PERT b elief networks the no des represent completion times and the

arcs represent probabilistic dep endencies b etween no des The completion times of

no de i i n denoted C isgiven by

C maxC j C C D

i j j i i

where D is the duration of activity i i n The joint probabilityof

C C is determined from equation where C is the network completion time

n n

Complexity of Probabilistic Networks

Lo osely dened the complexity of a probabilistic network refers to the diculty

in analyzing a network As will b e shown later in this section the complexityofa

probabilistic network is directly related to three factors and their interactions the

top ology of the network the underlying probabilistic mo dels and the solution

metho ds used Solution metho ds for probabilistic networks will b e discussed in

more detail in section however in this section some solution metho ds will

b e briey intro duced in order to showhow these metho ds impact on the issue of

complexity in probabilistic networks Network complexity measures are quantitative

descriptions of the complexity of a network and are used to

Predict analysis pro cessing time requirements and

Compare two or more prop osed algorithms by ensuring that algorithms are

evaluated at several p oints in the range of complexity

As an example of complexity of probabilistic networks and their measures consider

the probabilistic activity networks PANs intro duced in section

Figure a and b shows two dierentPANs each with four no des and ve

arcs As stated in section a primary metric of interest in analysis of activity

networks is network pro ject completion time For PANs network completion time

is expressed in the form of a distribution Let X denote the random variable asso ci

ated with the duration of arc activity ij with density function f and distribution

function F The network completion time denoted T ofaPAN with n no des is

ij n

T T max

k n

k 2

1 4 2 3 1 4 (a)

(b)

Figure Complexity Example PAN

where k r are the paths that lead from no de to no de n and T is

k k

the completion time of path For Figure a and b the network completion

time is

T maxT T T

where in part a

T X X

T X

T X X

and in part b

T X X

T X X X

T X X

From equation and equations thru the network completion time for Figure a

T maxX X X X X

and from equation and equations thru the network completion time for Fig

ure b is

T maxX X X X X X X

In order to determine the completion times of the PANs three basic mathe

matical op erators are needed

Convolution

Maximum multiplication and

Conditional Integration

Graphically these op erators enable a PAN to b e reduced to a two terminal network

source and sink with a single arc b etween the no des The convolution op erator is

referred to as a series reduction op erator Twoarcsaresaidtobeinseries if the

arcs are separated by a single no de with only one arc incidentinto the no de and the

other arc emanating from the no de With this series condition the two arcs can b e

combined into a single arc with the convolution op erator since the distribution func

tion of the duration of the new arc can b e obtained byconvoluting the distribution

functions of two arcs in series Graphically a series reduction at no de v is p ossible

when arc u v is the unique arc into v and v w is the unique arc out of v The

series reduction then eliminates no de v and replaces arcs u v andv w with arc

u w

Similarly the maximum op erator is a parallel reduction op erator Twoarcs

are said to b e in paral lel if the arcs have the same starting and ending no des With

this parallel condition the two or more arcs can b e combined into a single arc

with the maximum op erator since the distribution function of the duration of the

new arc can b e obtained bymultiplying the distribution functions of the two or

more arcs in parallel Graphically a parallel reduction at no des v and w replaces

two or more arcs joining the no des with a single arc u v As an example of the

convolution and maximum op erators consider the PAN in Figure a Network

reduction may pro ceed as follows

Obtain Y X X a convolution op eration

Obtain Z maxY X a maximum op eration

Obtain Y X X a convolution op eration and nally

Obtain T maxZ Y a maximum op eration

Consequently twoconvolution and two maximum op erations are needed to reduce the

network in Figure a and determine the network completion time Analytically

the solution to equation is

Z Z

t t

F t F t x dF F F t x dF

T X X X X X

Note the order in which the series and parallel reductions are applied do es not aect

the nal outcome since the reduction ob eys the ChurchRosser prop erty

The PAN in Figure b can not b e completely reduced using only the convo

lution and maximum op erators since the three paths and are dep endent

Sp ecicallyarc X app ears in paths and and arc X app ears in paths and

This dep endency can b e removed by conditioning on the values of the common

arcs that are elements of more than one path The PAN in gure a may b e re

duced through two maximum and two conditional integration conditioning on X

and X op erations Analytically the solution to equation is

Z Z

t t

F t F t x F t x x F t x dF dF

T X X X X X

If one assumes that the conditional integration is the most computational time con

suming op erator followed by the convolution op erator and then the multiplication

op erator then the computing requirements of the PAN in Figure b will b e greater

than the requirements of the PAN in Figure a In other words informallyit

app ears that the PAN in Figure b is more complex than the PAN in Figure a

if one uses computing requirements as a measure of network complexity

Formally Elmaghraby and Herro elon dene network complexityinanac

tivitynetwork context as the diculty in analysis and synthesis of a given network

The authors argued that the measurementofnetwork complexity cannot b e mean

ingfully accomplished unless the use of the measure ie ob jective of the analysis is

known b eforehand For PANs Elmaghraby and Herrolen give the following measure

of network complexity MNC

MNC g v m q

where

g is a monotone increasing calibration function which is determined empiri

cally and whose form and value dep end on the computer hardware and software

and the programmers skill

v is the numb er of convolutions

m is the numb er of maximums and

q is the numb er of conditional integrations

For example in Figure a v m and q and in in Figure a v

m and q resulting in g and g resp ectively

As shown in equation the numb er of maximum op erators in a PAN is equal

to the numb er of indep endent paths in parallel In the absence of path inde

p endence the number of maximum op erators is directly related to the number of

conditioned arcs Since at the time of their writing no systematic pro cedure existed

for determining the minimum numb er of arcs to b e conditioned on Elmaghraby

and Herrolen concluded that the numb er of maximum and conditional integration

op erators must b e determined conjointly In addition the authors argued that the

measure of complexitymay b e confounded by the algorithm solution metho d em

ployed For example the MNC in equation measures the complexityofa PAN

assuming that the analytical pro cedure used is that of convoluting activities in series

and maximizing activities in parallel and conditioning on common activities

The graph in Figure b is referred in the literature as the interdictive for

bidden graph Dun showed that if a twoterminal

directed acyclic graph stdag do es not contain a subgraph homeomorphic from

the interdictive graph then the graph is seriesparallel Sp ecicallyDunproved

that a stdag is seriesparallel if and only if the graph do es not contain a subgraph

homeomorphic from the interdictive graph Seriesparallel stdags maybeeitheredge

arc seriesparallel or vertex no de seriesparallel The PAN in gure a

which has AoA graphical representation is an example of an edge seriesparallel st

dag while the task precedence graph dened in section is an example of a vertex

seriesparallel stdag Since the literature that is of relevance to this dissertation uses

edge seriesparallel terminology the term seriesparallel will b e used to mean edge

seriesparallel unless otherwise noted A stdag is seriesparallel if it can b e obtained

iteratively in the following way

A single arc is twoterminal seriesparallel with the tail b eing the source and

the head b eing the sink

If G and G are two terminal seriesparallel so is the graph obtained by

identifying the sources and sinks resp ectively parallel comp osition and

Seriesparallel graphs may also b e dened by recursive denitions denitions that sp ecify graph

characteristics and constructive denitions See

If G and G are two terminal seriesparallel so is the graph obtained by

identifying the sink of G with the source of G series comp osition

As the name implies seriesparallel stdags are completely reducible separable

to a single arc through series and parallel reductions In the case of PANs the series

and parallel reduction translates to a convolution and maximum op eration resp ec

tively in the underlying probabilistic mo del In contrast stdags with emb edded

interdictive graphs are irreducible nonseparable Seriesparallel reductions may

still b e p erformed but eventually the stdag will b e in an irreducible form For

PANs Do din proved that a PAN is not completely reducible if and only if it

contains the interdictive graph

Bein et al showed how to eliminate emb edded interdictivegraphsina

stdag by applying successive no de reductions A no de reduction is a generalization

of a series reduction and can o ccur whenever a no de v has indegree or outdegree

If v has indegree and u v is the arc into v and v w v w are the arcs

out of v then the no de reduction is accomplished by deleting no de v and replacing

the k arcs u v andv w v w withthek arcs u w v w The

k k

case where v has outdegree is symmetric A no de reduction at v on graph G is

denoted by G v

Based on the numb er of no de reductions required to reduce a stdag with em

b edded interdictive graphs Bein et al intro duced a stdag complexity measure

that describ es how nearly seriesparallel a stdag is This measure termed the re

duction complexityof G denoted G is the minimum numb er of no de reductions

sucient along with series and parallel reductions to reduce G to a single arc

Sp ecically G is smallest c for which there exists a sequence v v such that

G v v is a single arc where G denotes the graph that results when

all p ossible series and parallel reductions have b een applied to G

Bein et al proved that there exists a p olynomialtime algorithm to mini

mize no de reductions The algorithm is obtained by showing that G is equal to

the size of a minimum no de cover of its complexity transitive graph C G The

complexity graph C G of a stdag G is dened as follows The arc v wisin

C G if and only if there exits paths v w v w w and v nsuch

that v w v n fv g and w v wfw g Furthermore either

v w v wor v w v w fv wg and the arcs common to w

and v n if any form a single path The denition implies that neither no des

nor n app ears in C G Constructing C G is equivalent to computing the transitive

closure of G Since C Gisatransitive dag the minimum no de cover for C G

denoted N can then b e computed by nding the maximum matching in a bipartite

graph the complement of a minimum no de cover is a maximum indep endent set

which in a transitive dag corresp onds to a Dilworth chain decomp osition

Bein et al prop osed the following reduction sequence for a stdag G

Step Construct C G

Step Compute N

Step While N do

Perform all series and parallel reductions p ossible in G and let G G

Find v N such that the indegree or outdegree of no de v is

Let G G v

Let V V fv g

The authors showed that the rst two steps in the reduction sequence o ccur in a

time b ound of On This time estimate is based on an upp er b ound of On

for computing the transitive closure of a graph of n vertices ie step and a

time b ound of On for computing the maximum matching in a bipartite graph

Reyck and Herro elen showed an example of constructing a C G and

computing N for a stdag containing no des and arcs The authors note that

although the reduction complexity index CI intro duced by Bein et al was

dened only for AoA networks the CI can b e used for a AoN network by transferring

a AoN network into a AoA network representation Reyck and Herro elen mentioned

the work of Kawburowski et al who develop a p olynomial time algorithm

which generates an AoA network with minimal numb er of no des and CI from a

given AoN network

Although Reych and Herro elen applied the reduction CI to the multiple

resource constrained problem RCPSP in which it is assumed that an activityis

sub ject to technological precedence constraints and cannot b e interrupted once b egun

and the discrete timecost tradeo problem DTCTP in activitynetworks of the

CPM typ e using a single nonrenewable resource the reduction CI has not been

applied to stochastic networks to our know ledge Instead Lawrence dened two

interdep endent complexity measures for PANs

The rst complexity measure is the tuple c where c is the number of

terminating crossconnections in the network and is the cardinality of these

crossconnections and

The second complexity measure is the tuple c where c is the number

of starting crossconnections in the network and is the cardinality of these

crossconnections

Lawrence dened a terminating crossconnection as the ending no de x for paths

i emerging from a common no de x that is predecessor to no de x but

i i j

not an immediate predecessor x is referred to as the terminal no de for the cross

connection Similarly a starting crossconnection is the starting no de x for paths

i that end at no des x j The presence of a terminating or starting

i j

crossconnection indicates a dep endency among paths in the PAN ie the PAN

do es not have a seriesparallel top ology

Note the numb er of paths and ending no des are the same

Lawrence further showed that terminating and starting cross connections

may b e determined from the adjacency matrix M ofthePAN The adjacency matrix

M m ofagraph G or network is the n n matrix with m dened as follows

ij ij

if x x A

i j

otherwise

The numb er of terminating crossconnections c is the numb er of columns in the

adjacency matrix M whose column sums are excluding the sink no de column

Conversely is the maximum of the c column sums Similarlythenumber of

starting crossconnections c is the number of rows in the adjacency matrix M whose

row sums are excluding the source no de and is the maximum of the c row

sums

As a nal remark ab out the complexityofPANs Hagstrom obtained the

following results ab out PANs with discrete nite range activity durations

Computing a value of the distribution function of network completion time is

P complete

Computing the mean of the distribution function of network completion time

is at least as hard and

Neither of the problems in and can b e computed in p olynomial in the

numb er of p oints in the range of the pro ject duration unless P NP

The complexity of the problem relies mostly in the enumeration of the paths of the

network and manipulation of path durations which are dep endent random variables

As can b e seen by the ab ove discussion very little research has b een devoted

to obtaining complexity measures for activitynetworks with continuous random ac

tivity durations ie sto chastic activitynetworks This observation extends to

other probabilistic networks such as sto chastic Petri nets and b elief networks I

b elieve the lack of research is a direct result of the necessity to use approximation

techniques for solving probabilistic networks For example consider the convolution

and conditional integration op erators mentioned ab ove b oth these op erators re

quire an integral op eration For manycontinuous distributions such as the Gamma

with noninteger shap e parameter Normal Lognormal and Beta it is not ana

lytically p ossible to compute the integral op erator exactly ie these distribution

functions do not have a closedform This inability to compute exact analytical

solutions for many probabilistic networks has led to the use of approximation meth

o ds Simulationbased metho ds have b een the dominate approximation technique

due to their logical simplicity and ecient implementation When a simulation is

used to solve a probabilistic network the main complexity measure is the number

of simulation runs and corresp onding random variable draws necessary to establish

a certain statistical condence in the output

So o and JungMo studied the complexityofPetri nets The authors

obtained fteen structural and dynamic complexity measures and tested these mea

sures on randomly selected practical Petri nets So o and JungMo observed

that adopting a maximum ring rule for the ring of transition in a Petri net re

duces the numb er of no des places and transition in Petri net sample by four on

average The maximum ring rule is dened as follows

Let M b e the k th marking

T b e the set of transitions that are reable from M and

k k

I t b e the set of input places for t T

If t t T and I t I t thent t are red in parallel So o

n k k n

and JungMo further observed that the numb er of markings states of Petri nets

grows exp onentially with resp ect to the increasing size no de and arcs of the nets

The exp onential b ehavior of markings of Petri nets extends to sto chastic Petri

nets As with Petri nets researchinto the complexity of sto chastic Petri nets is fo

In the case of the conditional integration op erator multiple integral op erations may b e required

cused on reducing the size of the underlying net For example German presented

a mo deling paradigm whose aim is to manage the complexity of net sp ecication In

the prop osed framework termed SPNL Sto chastic Petri Net Language a sto chastic

Petri net is decomp osed into submo dels termed pro cesses whichinteract via p orts

reward and results Ports are arcs that cross pro cess b oundaries Rewards and re

sults measure the internal state and action of the pro cesses Additionally Haddad et

al intro duced ring transitions with phase distribution ring time in b ounded

generalized sto chastic Petri nets The authors stated that the transitions pro duce

increases in b oth the space and time computational complexity of the tangible mark

ing steadystate probabilities Haddad et al solved this problem by a structural

decomp osition of the underlying Petri net This decomp osition led to a reduction in

the computational resources needed to determine the steadystate probabilities As

a nal example Herman et al develop ed a pro cedure to construct large gener

alized sto chastic Petri nets from smaller comp onents using hierarchical comp osition

Additionally the authors provided a metho d to obtain p erformance indices of large

generalized sto chastic Petrinetsby stepwise comp ositional reduction

Since b elief networks are Bayesian in structure one can up date the informa

tion in the network using instantiations of some or all the random variables in the

network These instantiations are referred to as evidence Without evidence one

is solving for the joint prior distribution of the random variables in the b elief net

work With evidence one is solving for the joint posterior distribution of the b elief

network For b elief networks with discrete random variables Kim and Pearl

and Pearl showed that nding the p osterior distribution or the maximum a

p osteriori probabilityMAP of all the variables in the network given the evidence

have p olynomialtime algorithms in the sp ecial case of singly connected networks A

singly connected b elief network is a networkinwhich no more than one path exists

Also called probabilistic inference

Also called the most probable explanation MPE

between anytwo no des APERT b elief network is an example of a singly

connected network MAP is the distribution that maximizes P A j E where A is

the instantiation or value assignment that assign values to all of random variables

in the network and E is the evidence For multiple connected b elief networks with

binary discrete random variables Co op er and Shimony showed resp ec

tively that nding the p osterior distribution and MAP of all the variables in the

network given the evidence is NPhard

For b elief networks with continuous random variables simulationbased ap

proximations for nding the prior andor p osteriori given evidence distributions

of all the random variables in the network include straightsimulation

forward simulation and randomized approximation schemes suchas

Gibbs sampling Forward simulation is useful in solving

b elief networks without evidence ie prior distributions while randomized approx

imation schemes such as Gibbs sampling are useful in solving b elief networks with

evidence ie p osterior distributions

Solution Methods for Probabilistic Networks

Solution metho ds for output measures in probabilistic networks may b e cat

egorized as exact or approximateFor reasons that were alluded to in sections

and and that will b e expanded in greater detail in this section approximation

metho ds sp ecically simulation dominate the solution metho ds currently in use In

the review that follows I decided not to segregate the solution metho ds bynetwork

example opting instead to show the similarities b etween the solution metho ds of the

various network examples However when describing a solution metho d for histori

cal p ersp ective I will note the network example for which the solution metho d was

develop ed Additionally based on the research application airtoair engagement

mo deling the ma jority of the review is fo cused on solution metho ds for probabilistic

networks with continuous random variables

Exact solutions of output measures for manytypical probabilistic networks

with continuous random variables cannot b e achieved As mentioned in section

it is not analytically p ossible to compute a network reduction op eration exactly for

some distributions for example the series reduction op eration convolution for two

Beta distributions However there exist classes of functions for which reduction

op erators yield exact results for which the classes are closed under the op erators

Trogemann and Gente stated that the following items should b e taken into

account when selecting a class of function to represent random activity durations

if the total completion time of a task precedence graph is to b e computed in exact

form

The class of functions must b e closed with resp ect to multiplication dieren

tiation integration and convolution

The class of functions must b e capable of representing empirical distributions

to any required degree of accuracy and

The execution time of reduction op erations must not require excessive com

puter time or memory

These items have applicability to the reduction of non seriesparallel graphs as well

Develop ed for reducing directed acyclic networks Martin used b ounded

domain density functions represented by the class of piecewise p olynomials A

piecewise p olynomial is a function g t of the form

p t if r t r

g t

p t if r t r

n n n

where p i n are real p olynomials In order for g t in equation to

represent a density function

g t and

g xdx

Martin showed that the class of piecewise p olynomials is closed under series convo

lution and parallel maximum reduction op erations

Develop ed for seriesparallel directed graphs Cubaud used b ounded

distributions represented by the class of p ower functions PFs A PF is a function

PFt of the form

for t b PFt

where b R a and m b As equation shows the class of p ower

functions is a subset of the class of p olynomials Cubaud showed that computing a

moment of the maximum of indep endent p ower distributed random variables can

b e p erformed in time prop ortional to the number of variables

Develop ed for reducing task precedence graphs Sahner and Sahner and

Trivedi showed that distributions represented by the class of exp onential p oly

nomials EPs were closed under the op erations of multiplication convolution dif

ferentiation and integration An EP is a function F t of the form

t k

i i

e F t a t

where k N a C and C Notevery EP dened in is a valid distribution

i i i

For an EP to b e a distribution of a nonnegativerandomvariable it must satisfy the

following prop erties

F t is realvalued

F t t and

F t is monotone nondecreasing and rightcontinuous

These prop erties imply that if Rthen and if thenk

i i i i

Additionally in order for F t to b e realvalued complex numb ers must o ccur in

conjugate pairs

The class of EP distributions includes all of the Coxian phasetyp e distribu

tions Coxian distributions are dened to b e those distributions whose

Laplace transforms Ls are fractions whose numerators and denominators are p oly

nomial in s such that the degree of the numerator is less than the degree of the

denominator Sahner showed that the class of EP is exactly equivalentto

the class of Coxian distributions A Phasetyp e distribution has a rational Laplace

transform and is thus a prop er subset of the Coxian distributions The class of EP

includes exp onential hyp erexp onential Erlang and mixtures of Erlang distribu

tions EP distributions may also b e augmented with a mass at zero andor innity

to represent resp ectively the probabilitythataneventtakes no time or never n

ishes As a nal note Sahner gives analytic expressions for other paral lel reduction

multiplicative operations such as minimum k out of n and probabilistic one path

is chosen The minimum op erator is equivalent to INCLUSIVEOR receiver no de

op eration in generalized activitynetworks

Trogemann and Gente observed that the class of EPs has the disad

vantage that many parameters are required to represent distributions with small

variation and large mean value ie co ecients of variation much less than For

example an Erlang distribution with mean and variance contains stages

i i

and has distribution function exp A term EP would b e nec

essary to represent this Erlang distribution In place of EPs Trogemann and Gente

recommended the class of exp onential p olynomials EP s for task precedence

graphs EPs are derived from EPs by translation More precisely EPs H t

are functions of the following form

k t

i i i

a t e for t

i i i

H t H t with H t

i i

for t

The parameter is called the deterministic where k N a R and R

i i i i

part a the co ecient the rate and k the stage For the example ab ove a EP

i i i

can mo del an Erlang distribution mean and variance with only four terms if

Figure shows the phase diagram of EPs The parameters c where

a k

i i

can b e interpreted as branching probabilities if c Note every branch consists

of a deterministic part followed by an Erlang distribution The deterministic part

can b e used to guarantee minimum execution times Trogemann and Gente showed

that the class of EPs is closed under the op erations of multiplication convolution

integration and dierentiation

In order to mo del event distributions with b ounded domains Trogemann and

Gente truncate the right hand end of EP class to form a new class of truncated

EPs More precisely truncated EPs H are functions of the following form

for t

k t

i i i

H t H twithH t

a t e for t w

i i

i i i

for t w

k N a Rand R and i w Both EPs and where

i i i i

H w

truncated EPs can approximate empirical distributions to any required degree of

accuracy 1 k1+1 θ λ λ 1 1 1

1 k2+1 θ λ λ 2 2 2

1 kn+1 θ λ λ n n n

Deterministic Exponential

Part Phases

Figure Representation of Exp onential Polynomials by Phase Diagram

Kulkarni and Adlakha and Sahner stated that if event distributions

in a probabilistic network are exp onential then the network can b e transformed into

a continuoustime Markovchain The main drawback of this transformation is that

the numb er of states in the Markovchain increase exp onentially with the number of

no des in the graph Additionally the practical diculties of generating the states

and transitions of the Markovchain and solving a p ossibly large system of ordinary

dierential equations must b e contended with Additionally Sahner gavean

algorithm for transforming any acyclic Markovchain into a seriesparallel directed

acyclic graph dag The author noted it is more ecient to analyze a chain directly

than to translate the chain into a seriesparallel dag and reduce the graph

In sto chastic Petri nets SPNs one is generally interested in solving for the

stationary andor transient probability of b eing in a sp ecic marking or state of the

reachability set Exact solution metho ds vary dep ending on the sub class of SPNs b e

ing analyzed In other words the structure and features of a SPN generally dep end

on the sto chastic pro cess probabilistic mo del used to analyze the reachability set

Various sub classes of SPNs are obtained by imp osing restrictions on the ring distri

butions or on the eect of a transition rings on other enable transactions Ciardo

et al prop osed a hierarchy of SPN classes where mo deling p ower is reduced in

exchange for an increasingly ecient solution Included in their hierarchy are

Markov SPNs

SemiMarkov SPNs

SemiRegenerative SPNs and

Generalized SemiMarkov SPNs

The descriptions identify the underlying sto chastic pro cess used to solve for the sta

tionary andor transient probabilities of a sp ecic marking of a SPN For example

Markov SPNs are SPNs in which the probabilities of b eing in a sp ecic marking can

b e solved by a Markov pro cess suchascontinuoustime Markovchain For exact so

lutions the main obstacle is often the memory to store the reachability set Ciardo

et al gaveasaruleofthumb to markings

Ciardo et al stated that the following ring time distributions are im

p ortant in practical SPN applications constant geometric discrete exp onential

uniform p olynomial and exp olynomial The p olynomial distribution mentioned by

the authors is the same piecewise function dened by Martin in equation

The exp olynomial distribution is piecewise dened in by expressions of the form

m n

X X

a exp

ij ij

where a R and Polynomial and exp onential distributions are

ij ij

sp ecial cases of the exp olynomial distribution shown in expression Haddad et

al intro duced transitions with Coxian ring time distributions in SPN and gave

a decomp osition metho d whichallowed the derivation of a tensor expression of the

generator of the underlying Markov pro cess This tensor expression reduced the

increased complexityintro duced by the Coxian transitions keeping it of the same

order of magnitude as the complexity of the reachabilityset

Pritskers and Happs and Pritskers and Whitehouses intro duction

of the general evaluation and review technique GERT to solve all EXCLUSIVE

OR no de generalized activity networks with random activity durations motivated

an eort in the late s and early s to nd exact solution metho ds for out

put measures of sto chastic activitynetworks Such networks termed original

GERTnetworks are signal ow graph SFG representations of semiMarkov pro

cesses SMPs with one or more absorbing states In the SMP of GERTnetworks

a state represents an event the transition probability p represents the probability

that activity ij will b e realized and the holding time represents the duration of the

activity

In a GERTnetwork an activity arc is represented by the pair of parameters

p T where p is the probability of the activity b eing realized and T is a transform

of the time duration distribution of the activityTypical transforms used in GERT

networks include the moment generating function z or Laplace transforms and the

characteristic function A transform allows an additive op eration suchastimetobe

turned intoamultiplicative op eration which then can b e solved by the three basic

typ es of elementcombinations in owgraph theory The three basic typ es of element

combinations in the theory of owgraphs are

Elements in series

Elements in parallel and

Lo ops

The acronym GERT has b een expanded to cover several simulation mo dels in which the no des

are not of the EXCLUSIVEOR typ e

A SMP is a sto chastic pro cess that makes its transition from any state i to a state j whichmay

be i according to the transitionprobability matrix of a Markov pro cess but whose time b etween

transitions is a random variable that may dep end on b oth i and j (pa,Ta) (pb,Tb) 1 2 3

Before Series Combination

(papb,Ta+Tb) 1 3

After Series Combination

Figure Basic Series Combination in a GERT Network

For example supp ose the time t of an activity duration is sp ecied by its moment

generating function MGF M s where

exp f tdt if t is a continuous random variable

M s

exp f t if t is a discrete random variable

and f t is the activity duration density function For the series system shown in

Figure p p p and t t t However the MGF of t is the pro duct of

a b a b

the MGF of t and the MGF of t More precisely M M M

a b t t t

The primary outputs of the original GERT pro cedure were a the probability

of realizing a no de and its asso ciated event and b the transform eg MGF of

the time to realize a no de given that the no de is realized In order to obtain

the distribution of time to realize a no de one has to invert the transform backinto

a distribution This problem is referred to in the literature as the inverse transform

problem and is not easily solved for complex GERT networks Whitehouse

prop osed four metho ds to solve this problem

A inversion table lo okup op eration

An inversion formula

Use the rst n moments from the MGF transform to approximate the distri

bution function and

Assume the form of the distribution and estimate the parameters of the distri

bution using the moments of the MGF

Each metho d has disadvantages Whitehouse stated that the rst and second meth

o ds do not app ear practical due to the complexity of the transform derived in typ

ically sized GERT networks The third and four metho ds are approximations and

thus due not lead to exact solution

Additionally in theory AND and INCLUSIVEOR no des can b e transformed

into EXCLUSIVEOR no des at the exp ense of enlarging the number of nodes and

arcs in the network Whitehouse and Elmaghrabyprovide examples of

such expansions Quichaud and Chretienne extended GERT analysis to net

works containing b oth EXCLUSIVEOR and AND no des by searching for analyzable

EXCLUSIVEOR and AND subnetworks and reducing the subnetworks However

by the late s simulation approximation was seen as a more fruitful solu

tion metho d for GERTtyp e networks than transform metho ds By Pritsker

develop ed a GERTsimulation metho d to study sto chastic problems in inventory

reliability queuing and maintenance problems Other exact solution attempts b e

sides GERT to solve activity networks with random activity durations proved either

infeasible or met with limited success Lawrence and Adlakha and Kulkarni

provide an overview of these various attempts By the late s exact solution

metho ds for activity networks with random activity durations were abandoned

When an event distribution of a probabilistic network can not b e exactly repre

sented by the functions describ ed ab ove then the distribution can b e approximated

by the same functions For example piecewise p olynomial functions could b e t

ted to the dierentevent distributions Unfortunately an exact solution for sp ecic

output measures in the true sense can no longer b e achieved Many approxi

mation pro cedures have b een develop ed to solve output measures for probabilistic

networks These pro cedures generally fall under two categories

Simulation and

Numerical Approximation and Reduction Metho ds

Simulation discreteevent is an eective approach for approximating the out

put measures in probabilistic networks esp ecially if the network has random vari

ables with mixed distributions At the b eginning of simulation exp eriment an ana

lyst determines the numb er of replications simulations to b e made Each replication

pro duces a sample value of the output measures These sample values build a

picture of the distribution of the output measures The picture is often presented

as a frequency histogram of the sample values or set of statistics such as the mean

and variance The accuracy of the picture is a function of the numb er of replications

conducted Eective employmentofsimulation requires the user to exercise the sim

ulation through sucient replications to insure that conditions eg steady state

have b een attained with resp ect to the characterization of the output measures

Two general accepted typ es of simulations are found in the literature with

regard to output analysis

Terminating a simulation for which the o ccurrence of a natural event E

sp ecies the length of eachsimulation run and

NonTerminating a simulation for which there is no natural event E to sp ecic

the length of the simulation run

In nontermination simulations one must address steadystate and transient con

ditions of the output measures Steadystate conditions with resp ect to output

measures are dened as the limits of the measures as the length of the simulation

go es to innity In practice a large nite simulation run may provide go o d esti

mates Transient conditions with resp ect to output measures refers to uctuations

in the distribution of the output measures Law and Kelton provide a detailed

comparison of the output analysis of terminating and nonterminating simulations

Simulation of the output measures of probabilistic networks may fall into either

simulation typ e Generally probabilistic networks without cycles will b e terminat

ing eg probabilistic activitynetworks PANs while networks with cycles will b e

nonterminating eg ergo dic sto chastic Petri nets SPNs

For PANs Van Slyke was the rst to prop ose simulation as a solution

metho d for the network completion time Burt and Garman and Garman

then prop osed a conditional simulation approach based on common arcs Signal et

al develop ed a conditional simulation pro cedure similar to that of Burt and

Garman The approach conditions on the arcs in a uniformly directed cut set and

then evaluates the integral formulas numerically using simulation Both Burt and

Garman and Sigal et al solutions are time consuming and are not practical on

large PANs Following the intro duction of the general evaluation and review

technique GERT Pritsker develop ed a family of GERTsimulation GERTS to ols

for solving generalized activitynetworks GAN and other general networking and

sto chastic problems Todaysimulation of PANs and GANs can b e p er

formed on a PCbased commercial simulation package such as VISUAL SLAM

Variance reduction techniques suchasantithetic variates and control variates can b e

used to reduce the numb er of runs necessary to obtain output of a required con

dence level Elmaghraby discusses implementation of such reduction techniques

in a PAN context

For sto chastic Petri Nets Ciardo et al recommended simulation in cases

where a numerical solution is imp ossible or impractical such as the analysis of Markov

SPNs with excessively large reachability sets transient analysis of semiMarkov SPNs

or semiregenerative SPNs and analysis of a general SPN or a semiMarkov SPN

with nonMarkovian sub ordinated pro cesses As mentioned in section for b e

lief networks with continuous random variables simulationbased approximations

for nding the prior andor p osteriori given evidence distributions of all the ran

dom variables in the network include straightsimulation forward sim

ulation and randomized approximation schemes such as Gibbs sam

pling Forward simulation is useful in solving b elief

networks without evidence ie prior distributions while randomized approximation

schemes such as Gibbs sampling is useful in solving b elief networks with evidence

ie p osterior distributions

For large probabilistic networks simulation can b ecome a computationally

costly undertaking As the numb er of random variables in the network increases so

do es the numb er of replications necessary in order to prop erly characterize the output

measures The ob jective of an ecientsimulation exp eriment is to accumulate

enough sample information on the output measures so that the distribution of

the output measures is invariant with increases in the numb er of replications

Infrequent rare events can have a large impact on the numb er of replications needed

to obtain the distribution of output measures As an example consider the GAN

mentioned byLawrence This GAN has a no de with two arcs activities

emanating from it one that has a high mean activity duration when the activityis

taken but a low probability of b eing taken The other activity is just the opp osite

this activityhasa low mean activity duration but a high probability of b eing

taken The exp ected numb er of replications b efore the rst activity is taken will

b e quite large and during most simulation exp eriments the second activity will

b e rep eatedly taken b efore the rst activityistaken for the rst time A output

measure such as mean network completion time will b e initially estimated to o low

by the corresp onding sample statistic then to o high when the statistic spikes the

rst time the rst activityistaken A large numb er of replications will b e required

b efore the distribution of the mean network completion time is invariant to further

simulation runs If the simulation exp eriment is terminated b efore this invariance

is achieved then an inaccurate statistic will b e rep orted for the mean completion

time Although the ab ove example is pathological the problem of determining the

numb er of simulation runs to make for a large probabilistic network remains a hard

task If a simulation exp eriment is stopp ed to so on for cost andor time constraints

inaccurate estimates of output measures will result

As an alternativetosimulation numerical approximation and reduction meth

ods have b een prop osed for some probabilistic networks Network reduction involves

the rep eated application of op erators suchasmultiplication for example minimum

maximum k out of n convolution and conditional integration to reduce a network

to smaller network through aggregation of the underlying probabilistic mo del of the

network For probabilistic activitynetworks PANs several numerical approxima

tion and reduction metho ds have b een demonstrated Reduction pro cesses based

on arc no de duplication has b een develop ed by Martin and Do din to

approximate network completion time This reduction pro cess transforms an arbi

trary PAN Ginto a seriesparallel network G The duplications of arcs no des

is equivalent to the removal of dep endencies in G Do din showed network com

pletion time T of b oth networks are related by

T T T

G G G

where T is the b ound obtained by assuming the path durations are indep endent

Do din also develop ed an approximation of the network complete time of

PANs based on the metho d of sequential approximation Sequential approximation

constructs distributions through the no des of the network in sequential order

Do din implemented the sequential approximation and arc no de dupli

cation reduction pro cesses using discretization of continuous activity durations

In AoA networks arcs are duplicated In AoN networks no des are duplicated

Later Lawrence implemented the arc no de and sequential approximation re

duction pro cesses using p olygonal approximation of continuous activity durations

Lawrences implementation metho d expanded on the work of Martin and Fer

geson and Shortell

As noted earlier Martin used piecewise p olynomial functions reference equa

tion to approximate the density functions of random activity durations in series

parallel PANs Fergeson and Shortell observed that Martins approach created

a problem of expanding co ecients or as Lawrence terms the problem

of explo ding co ecients To illustrate this problem consider two arcs in series

Supp ose each arc has a random activity duration whose density can b e sp ecied by

piecewise p olynomial functions If the two arcs are reduced to a single arc through

a series reduction then the random variables representing the activity durations

of these arcs are summed ie convoluted If the density functions of these arcs

are represented by p olynomials with terms as high as degree m and n resp ectively

then the density function of the resulting arc can have terms as high as degree

m n A similar result holds for parallel reduction maximum op erator Thus

the seriesparallel reduction op erations of convolution and maximum increase the

numb er of degree terms in the p olynomial functions required to represent the den

sity function of the new arc The increasing numb er of terms impacts the amountof

computer storage required b ecause each new ie higher degree term requires an

additional dimension of storage in any array holding the co ecients of the p olynomial

expression For larger networks with many seriesparallel reduction op erations the

computer storage requirements for co ecients quickly explo de

Sahner also noted the explo ding cocients problem in reduction op er

ations when dealing with exp onential p olynomials EPs For example if a task

Lawrence also implemented Do dins K most critical paths approximation as part of his

research

Lawrence was Fergeson and Shortells thesis advisor

A p olynomial is completely characterized by its co ecients and hence can b e stored in array

form

precedence graph has n no des with EP distributions each with a dierent parame

ter and the no des are to b e combined with a maximum op erator then the resulting

exact EP will have terms Sahner stated that the only waytoavoid suchan

explosion is to use some symb olic approximation Sahner recommended as further

research a study of techniques to reduce the propagation of roundo and trunca

tion error due to the nite accuracy of the representation of numb ers in a digital

computer Trogemann and Gente also noted that the accuracy of analysis is

impacted by rounding errors They stated that these errors can lead to completely

false results for task precedence graphs that contain a large numb er of no des mak

ing it practically imp ossible to determine the exact distribution of the completion

time for large graphs As a result Trogemann and Gente derived approximation

formulas using extreme value theory for large task precedence graphs The authors

prove that the limiting distributions from the class of EPs and truncated EPs

for parallel maximum reduction are Gumb el extreme value typ e I and Weibull

extreme value typ e I I I distributions resp ectively

In addition to the problem of explo ding co ecients Fergeson and Shortell

observed that Martins approach created a problem of proliferating classes To

illustrate this problem consider two arcs in parallel If the density functions of these

arcs are represented by c p olynomials of b ounded domain in R then the partition

covered by the p olynomials consists of c p oints or c classes cells If the twoarcs

i i

are combined in parallel then the resulting piecewise p olynomial function obtained

by the maximum op erator may b e dened on as manyas c c classes If the two

arcs are instead combined in series then the resulting piecewise p olynomial function

obtained by the convolution op erator may b e dened on as manyas c c c c

classes Each of these new classes must b e held in computer storage creating a class

storage explosion problem for larger networks

Fergeson and Shortell solved the explo ding co ecient and proliferating

classproblems by mo deling activity duration densities as p olynomials of xed degree

piecewisedened on a xed numb er of classes This is accomplished by approximat

ing the activity duration densities of all activities in the network to p olynomial

densities of the same xed degree piecewise dened on the same xed number of

classes over their domains prior to network reduction Then when an intermediate

or nal pro duct is formed by a series or parallel reduction op eration the densityof

that pro duct is immediately transformed into p olynomials of the determined degree

piecewisedened on the determined numb er of xed classes over its domain

Fergeson and Shortell recommended using simple rst degree p olynomials

tted piecewise using least square approximation ie least squares regression over

equally spaced classes or subintervals This equates to ten linear regressions

over the b ound domain of the density function Fergeson and Shortell further rec

ommended using I cl assw idth regression tting p oints p ositioned uniformly

across each class Additionally in order to control error buildup from the p olygonal

approximation after the ten sets of regression co ecients have b een computed the

co ecients are normalized so that the probability under the approximated density

function is one

Fergeson and Shortell termed their metho d the Polygonal Approximation

and Reduction Technique PART Fergeson and Shortell examined PANs with

uniform normal and exp onential distributions and showed that the PART compared

favorably with simulation QGERT program in terms of solution accuracy and had

a faster computer pro cessing time and small computer memory requirements than

the simulation solution Fergeson and Shortell noted that the PART exp eriences

it greatest approximating error at the p eak of a distribution eg approximating

the normal density function or approximating the convolution of two normal or

exp onential distributions The authors recommended as further research adopting a

multiple linear regression approach instead of the simple linear regression approachor

increasing the numb er of classes upwards from ten Additionally at class b oundaries

two approximations of a density function exist corresp onding to the approximate

line segments of the class which terminates and the class which b egins at that class

b oundary

As mentioned earlier Lawrence adopted PART for use with Do dins

arc no de and sequential approximation reduction pro cesses Lawrence chose to use

Fergeson and Shortells results in total and justied this choice with the following

reasoning

The pro cess of tting a new reduced piecewise p olynomial function to

the higher term pro duct derived from the convolution or maximization of

two old piecewise p olynomial functions is b etter viewed as a data tting

problem rather than an interp olation problem This observation is based on

the fact that the old p olynomial functions constitute an approximation of the

underlying density functions and thus dier from the true values of densities at

p ossible p oints of interp olation although actual errors are not known Hence

a new p olynomial function is b eing tted to approximate data rather than

b eing interp olated from a given function Lawrence chose to use the customary

leastsquare approximation to measure the accuracy of the data tting

When approximating the integral or derivative of a continuous function the

higher the degree of the interp olating p olynomials the more accurate the nu

merically approximated integral or derivative but the more computationally

intensive the construction and subsequent dierentiation or integration of the

p olynomials Since the accuracy of the approximated integral or derivativecan

also b e increased by rening the partition increasing the numb er of classes by

adding additional interior p oints Lawrence adopts the strategy of Conte and

deBo or of approximating the function with p olynomial of low degree and

then rening the partition until the approximated derivativeorintegral has

the desired accuracyLawrences low degree choice is a rst degree or simple

linear regression

UnfortunatelyLawrence did not provide further analysis on rening the par

tition leaving this topic for further research Additionally the initial task of tting

a piecewise p olynomial function to a density function is an interp olation problem

Lawrence incorrectly treats this task as a data tting problem However Lawrence

did expand PART to include approximations of the triangular gamma and

b eta distributions Perhaps more imp ortantlyLawrence showed that PARTus

ing Do dins reduction algorithms were orders of magnitude faster than simulation

approximations without signicant losses in accuracy when the simulation results

are taken as true

For seriesparallel networks Cubaud showed that the class of p ower func

tions can approximate any arbitrary distribution bymatching the rst two moments

of the distribution with the two parameters of the p ower function Sahner only

partially addressed the issue of tting distribution data to the class of EP functions

Sp ecically Sahner mentioned the work of Augustin and Buscher in the t

ting of Coxian phasetyp e distributions to a given mean and co ecientofvariation

and the work of Bux and Herzog in tting a phasetyp e distribution to a given

mean variance and a number of sample values Sahner noted that Bux and Herzogs

metho d allowed more freedom in the choice of phases than Augustin and Buschers

metho d Trogemann and Gente briey reference as a guide to using EP

and truncated EP functions to approximate arbitrary distributions to any degree

of accuracy Unfortunately neither Sahner nor Trogemann and Gente presentan

example of these distribution approximating metho ds with their EP formulations

Another numerical approximation and reduction approachwas develop ed by

Mehrotra et al for approximating the network completion time in PANs The

approach is based on an estimation of the numb er of common activities across critical

paths More precisely let P denote the set of K critical paths of the network such

that V represents the sum of the common activities across the K paths and U rep

resents the sum of the noncommon activites in the ith critical path i K

The completion time of the network T is max U V Since all critical paths

iK i

in the PAN will generally not have the same common activities Mehrotra et al

prop osed using

The average numb er of common activities

The minimum numb er of common activities or

The maximum numb er of common activities

Additionally Mehrotra et al showed their approximation to have a signicant com

putational time advantage using a CPU metric over simulation without sacricing

accuracy when computing the rst two moments of the network completion time

For stated purp oses of computational ease Mehrotra et al used activity durations

with indep endent and identically distributed iid N and iid Exp distribu

tions to obtain these results The authors noted that their approximation can b e

used with noniid distributed activities as long as the common activity durations of

the network ie the U s are identically distributed

Somara jan and Lau develop ed an approachtoapproximating the net

work completion time of a PAN network based on the SchmeiserDeutsch distri

bution Sp ecically the rst four moments of network paths are calculated

andaSchmeiserDeutsch distribution is tted to each path The network com

pletion time is then computed as the maximum of n random variables represented

byeach path length Somara jan and Laus approach assumes the indep endence

of paths through the network The rst four moments of a network path are cal

culated from the rst four moments of the activity durations in the path More

preciselyify x x where x are indep endent random variables whose

n i

rst four central moments are known and x E x exp ected value of x

i i i

and x x x themth central momentof x then

m i i i i

y x

n n n

X X X

x x x y

i j i

j i i i

Equations thru are exact and valid for all distribution forms The

SchmeiserDeutsch distribution covers all distribution forms with squared skewness

and squared kurtosis and is of closed form

Summary

This chapter provided an overview of probabilistic networks including their

graph theoretic underpinnings complexity and solution metho ds for output mea

sures Sp ecically four typ es of probabilistic networks were investigated activity

networks Petri nets task precedence graphs and b elief networks and their struc

tures were compared and contrasted A probabilistic network was dened as a di

rected graph G N A linked to a underlying probabilistic sto chastic mo del in

which probabilistic events activities are represented byarcsornodesinGEach

probabilistic event is represented by a random variable or a combination of random

variables

Little research has b een devoted to obtaining complexity measures for proba

bilistic networks with continuous random variables This observation is a direct result

of the inability to compute exact analytical solutions for many probabilistic networks

and has led to the use of approximation metho ds Simulationbased metho ds have

b een the dominate approximation technique due to their logical simplicity and e

cient implementation When a simulation is used to solve a probabilistic network

the main complexity measure is the number of simulation runs and corresp onding

random variable draws necessary to establish a certain statistical condence in the

output

However for some probabilistic networks simulation can b ecome a compu

tationally costly undertaking to obtain accurate estimates of the output measures

As the number of random variables in the network increases so do es the number

of replications necessary in order to prop erly characterize the output measures

As an alternative to simulation numerical approximation and reduction metho ds

have b een prop osed for some probabilistic networks Network reduction involves

the rep eated application of op erators suchasmultiplication for example mini

mum maximum k out of n convolution and conditional integration to reduce a

network to a smaller network through the aggregation of the underlying sto chastic

pro cess of the network Numerical approximation involves the use of functions such

as piecewise p olynomial functions exp olynomial functions exp onential p olynomials

functions etc to approximate the density or distribution functions of the random

variables For probabilistic networks with several random variables numerical ap

proximation and reduction techniques can b e several orders of magnitude faster than

simulationbased metho ds without signicant losses in accuracy

III AirtoAir Engagement Literature

Introduction

In this chapter a review of airtoair engagement literature is undertaken

The review in this section deals exclusively with airtoair engagement mo deling

However in order to accurately mo del an airtoair engagement a mo deler should

have an adequate understanding of the imp ortant comp onents and variables in an air

toair engagement To this end App endix A provides a brief overview of the complex

environment of airtoair engagements and factors that determine success in this

environment Readers of this dissertation unfamiliar with the airtoair engagement

environment should read the material in App endix A

In general there is a lack of published publicdomain do cumentation on com

bat mo deling What little combat mo deling do cumentation that do es exist is usually

devoted to ground combat mo deling for example the articles in To date I

am only able to nd three journal references that sp ecically discuss airtoair en

gagement mo deling I b elievethelackofdocumentation on combat

mo deling including airtoair engagement mo dels is the result of three related fac

tors First there do es not exist a journal dedicated to combat mo deling Secondly

there is a p erception either real or imagined that access to information concerning

combat mo dels must b e kept restricted Finallytodays combat mo dels consist of

large complex simulations which are not easily explained in the page restrictions of

journal articles Congruently these simulations are created by civil contractors who

are paid to build and maintain the simulations not to write ab out them The result

of this lack of information on combat mo deling is that combat mo deling insights are

constantly having to b e relearned Hence one of the goals of this research is to pro

Articles on combat mo deling can b e found in journals suchasNaval Research Logistics Quar

terly and Military Op erations Research Neither of these journals are dedicated solely to combat

mo deling

vide a summary of the lessons learned and status of airtoair engagement mo deling

at the theaterlevel

In section it was stated that airtoair mo dels may b e divided into two main

groups according to their intended purp ose analysis or training This dissertations

fo cus is on airtoair engagement mo dels used for analysis Airtoair engagement

mo dels for analysis may b e further divided into two subgroups dep ending on the

mo deling metho d employed analytic or simulated

Analytic Models

Analytic mo dels particularly simple ones help clarify imp ortant relations that

are dicult to p erceive in a more complex mo del Though generally more abstract

than simulation mo dels analytic mo dels are characterized by transparency or the

degree to which cause and eect relationships in a mo del are apparent An analytic

mo del is usually to o simple and restricted to directly solve an op erational problem

but b ecause of its transparency its insights provide valuable guidance for simulation

mo del investigations

Analytic combat mo dels may b e traced back to the seminal works of Fisk

Osip ov Lanchester These authors were the rst to express the dynamics

of combat in mathematical terms Sp ecically they considered combat as a set of

coupled dierential equations relating the changes over time in the force level of

friendly combatants to the force level of the opp onents andor friendly combatants

and vice versa For example Lanchesters square lawofcombat hyp othesized that

under mo dern conditions that the change over time in the force level of a sides

combatants would b e prop ortional to the force level of the opp onents combatants

Helmb old and Relm provide the English translation of the ve part series of articles that

Osip ov published in the Russian journal Vo enniy Sb ornik and an overview and discussion of the

interrelationships b etween of the works of Fisk Osip ov and Lanchester

In mathematical terms wehave

y with x x

x with y y

where

xt the number of x combatants at time t

y t the number of y combatants at time t

the rate at which individual combatants in force y kill opp osing combatants

in force xand

the rate at which individual combatants in force x kill opp osing combatants

in force y

The state solution for equations and is

x x y y

showing the square law relationship

Lanchesters research started the exploration of using dierential equation for

mulations b oth deterministic and sto chastic to mo del combat attrition referred to

as LanchesterTyp e Mo dels of Warfare Taylor and Fowler provide

comprehensive reviews of LanchesterTypeModelsofWarfare Several deterministic

Lanchestertyp e airtoair engagement mo dels have b een prop osed byFrick and

Latchaw The US Air Forces THUNDER theaterlevel combat mo del uses

a set of deterministic Lanchester equations in dierence form to allo cate intercept

ing aircraft to target escort aircraft and target nonescort aircraft eg b omb ers

More precisely let

xt representthenumber of intercepting aircraft in the engagement at time t

y tand y t represent resp ectively the numb er of target escort and non

escort aircraft in engagement at time t

a represent the p ercentage of interceptors allo cated to engaging nonescort

aircraft a are allo cated to engaging escort aircraft

The resulting Lanchester equations for attrition in the airtoair engagement are

y y with y y and y y

ax with x x

a x

where

the rate at which individual escort aircraft in y kill interceptor aircraft in

the rate at which individual nonescort aircraft in y kill interceptor aircraft

in x

the rate at which individual interceptor aircraft in x kill escort and non

escort aircraft resp ectively in y and y

The state solution for equations thru is

z z x x

where

z t y t y tand

a a

Equation is used to determine the winning side either x or y through an en

gagement of total attrition If side x wins then y and y since the battle

is foughtuntil one side is annihilated y and y implies z Thus the

numb er of survivors of the engagement on side x are x z If side y wins

then x Unfortunately the number of survivors of the engagement on side y

can not b e obtained directly from equation since the expression for z involves two

unknowns y and y ie one equation and twounknown variables

However the survivors on side y can b e obtained directly from the force level

equations Sp ecically solving equations thru yields force level equations

xt x cosht sinht

x z

cosht y t y a sinht

z x

y t y a cosht sinht

Letting xt and solving for t in equation yields

t ln

The survivors on side y are then determined by equations and using t t

from equation

A measure of eectiveness MOE determines the allo cation a and a of inter

ceptors to nonescort aircraft and escort aircraft Sp ecically the winning side and

survivors on that side are calculated for a ie step i i

The p ercentage of nonescort aircraft lost A and of interceptors lost B in allo

I I

cation step i is determined The following measure of eectiveness is calculated for

each allo cation step

MOE

V B C

where

V is the measure of the value of interceptor losses and

C is the p ercentage of interceptor losses when a

The largest value of MOE in equation determines the allo cation a and a of

interceptors to nonescort aircraft and escort aircraft

Due to the general recognition that combat is a sto chastic series of events

and the limiting assumption of large force numb ers the use of deterministic Lanch

ester equations as analytic to ols have b een limited to large forceonforce engage

ments In their place sto chastic analytic mo dels have b een pursued These mo dels

are able to analyze small or intermediate numb ers of combatants the engagement

size appropriate for mo dern air combat reference App endix A

Whitehouse used an EXLUSIVEOR generalized activitynetwork GAN

or GERTnetwork to mo del a simple one versus one airtoair engagement Figure

shows the network for the airtoair engagement No de represents the event that

the interceptor engages the target aircraft Three p ossible outcome activities exist

The interceptor approaches the target aircraft res a missile and kills the tar

get aircraft This outcome activity is represented by arc with probability

p and a random o ccurrence time of t with moment generating function

MGF M s This arc terminates at no de with the event b eing a sho ot

down of the target aircraft

The interceptor approaches the target aircraft res a missile but misses the

target aircraft This outcome activity is represented byarc with proba

bility p and a random o ccurrence time of t with MGF M s This arc

terminates at no de with the event b eing a missed interceptor missile shot

The interceptor approaches the target aircraft The target aircraft res a mis

sile rst and sho ots down the interceptor This outcome activity is represented

by arc with probability p and a random o ccurrence time of t with

(p ,M (s)) 1 1 (1,3) 3

(p4,M(2,2)(s))

(p5,M(2,3)(s))

4 (p2,M(1,2)(s)) 2 (p6,M(2,4)(s))

(p7,M(2,5)(s)) 5

(p3,M(1,5)(s))

Figure GERT Network of Simple Air Duel with UnlimitedPasses

MGF M s This arc terminates at no de with the event b eing a sho ot

down of the interceptor

If the interceptor misses the target aircraft then four p ossible outcome activ

ities exist

The interceptor approaches the target aircraft res a missile but misses the

target aircraft again This outcome activity is represented by the lo op

with probability p and a random o ccurrence time of t with MGF M s

This arc terminates at no de with the event b eing a missed interceptor missile

shot

The interceptor approaches the target aircraft again res a missile and kills

the target aircraft This outcome activity is represented by arc with

probability p and a random o ccurrence time of t with MGF M s

This arc terminates at no de with the event b eing a sho ot down of the target

aircraft

The interceptor terminates the engagement This outcome activity is repre

sented by arc with probability p and a random o ccurrence time of t

with MGF M s This arc terminates at no de with the engagement b eing

terminated

The interceptor approaches the target aircraft again The target aircraft res a

missile and sho ots down the interceptor This outcome activity is represented

by arc with probability p and a random o ccurrence time of t with

MGF M s This arc terminates at no de with the event b eing a sho ot

down of the interceptor

The unconditional moment generating function of the time to sho ot down the

target aircraft is

p M sp M s

W p M s

p M s

Similar expressions can b e derived for the unconditional moment generating func

tions for the time to terminate the engagement and to sho ot down the interceptor

Whitehouse stated that more complex airtoair engagements can b e mo deled

in a similar manner A signicantdiculty of the GERTmodelsasmentioned in

section o ccurs when trying to obtain the distribution of time to realize a no de

In order to obtain the distribution of time to realize a no de one has to invert a

transform suchasgiven in equation backinto a distribution This problem is not

easily solved for complex GERTnetworks

Feigin et al used a discrete space continuous time Markovchain to an

alyze a monn airtoair engagement Sp ecically at time t an airtoair

engagement starts b etween opp osing ight group of m homogeneous blue aircraft

In terms of aircraft p erformance and weap ons characteristics

and a ightof n homogeneous red aircraft Atanytimet t an individual

aircraft is in one of four roles or states

involved in a duel with an opp onent aircraft and in the pro cess of Pursuer

acquiring a ring p osition against the opp onent aircraft

Evader involved in a duel with an opp onent aircraft and trying to minimize

the opp onents ring opp ortunity

Free not engaged in a duel with an opp onent aircraft seeking an opp onent

aircraft to attack

Downed aircraft has b een shot out of the sky by an opp onent aircraft

At t all aircraft are assumed to b e in the free role

Achange in an aircraft role is asso ciated with one of the following four events

A pursuer downs an evader with rate p where is the average rate at which

the pursuer reaches a ring p osition and res and p is the kill probabilityofa

single weap on release

A pursuer is acquired by a free aircraft from the evaders ight group with

average rate In this event the free aircraft b ecomes a pursuer the pursuer

assumes the role of an evader and the original evader b ecomes a free aircraft

Two free aircraft from the opp osing ight group engage in a duel as the result

of one of the aircraft acquiring the other with average rate ie one aircraft

b ecomes a pursuer while the other aircraft b ecomes an evader

An evader disengages a pursuer with average rate

The ob jective of eachight group blue and red is to destroyasmany aircraft in

the opp osing ight group as p ossible and each aircraft has an unlimited number of

ring opp ortunities For a given m blue and n red aircraft there are k k m n

ways based on the four events ab ove that the m n aircraft could b e assigned the

m n k

Table Numb er of the States k of the Markovian Mo del Given m Blue Aircraft

and n Red Aircraft

roles of pursuer evader free or downed Feigin et al use k states for the Markov

chain Table shows the p ossible states for a given value of m and n

Feigin et al considered the loss ratio number

N N t

B B

N N t

R R

and the attrition ratio

N t

as output measures where N tandN t refer to the numb er of aircraft on the

B R

blue and red sides resp ectively at time t The loss and attrition ratios in equations

and are the main metrics used to assess the degree of air sup eriority at the theater

level

Feigin et al stated that their mo del should b e regarded as a prototyp e of

a large family of more complete Markov mo dels The mo del can easily b e expanded

to include more detailed aircraft roles and events For example two aircraft from

the same side lead and wingman could b e allowed to engage a single opp onent

aircraft rather than the current single pursuer restriction Also states could b e

added to takeinto account the numb er of missiles remaining The payment for a

more detailed mo del is a larger state space and hence more computational time to

solve the resulting Markovchain Additionallyonemay question the validityofthe

Markovian memoryless prop erty in mo deling airtoair engagements sp ecically

do es the conditional distribution of the future state S t s given the present state

S s and the past states S u us dep end only on the present state X s and

not on the past states S u Dep ending on structure ie roles and events used

in the mo del this prop erty will not always hold For example consider an airtoair

engagement in which a ight of four aircraft loses two of its aircraft It is unlikely

that the ight will function in the same manner aggressiveness co ordination etc

given the loss of the two aircraft as they would without an aircraft loss Finallythe

fact that a homogeneous Markov pro cess is used to mo del the engagement reects

the assumption that there is no eect of time on the transition probability rates In

other words as the airtoair engagement progresses no accountistaken of factors

such as fatigue or learning on the part of the pilots

Hong et al furnish three nonMarkovian airtoair engagement mo dels

The rst mo del illustrates a one versus one airtoair engagement using the inter

ring distributions of missiles one radarguided and one infrared on each aircraft

to determine engagement outcomes The second mo del is also a one versus one air

toair engagement but unlike the rst mo del it uses the interkillingdistributions

of missiles to determine engagement outcomes Additionally the second mo del in

creases the numb er of missiles on each aircraft to four two radarguided and two

infrared and divides the engagementinto three distinct phases b eyond visual range

BVR engagement visual detection and within visual range WVR engagement

Finally the third mo del expands the state space of the second mo del to include

two aircraft on one of the sides making the airtoair engagementa twoversus one

The interring distribution describ es the distribution of time to a missile launch

In contrast to the interring distribution the interkilling distribution represents the distribution

of time to a missile kil l In other words with the interkilling distribution weareinterested only in

the missile launches that kill sho ot down an opp onent aircraft

encounter All three mo dels assume that the engagementcontinues until one side is

killed or all weap ons are exhausted

The three mo dels were develop ed under the assumption that an airtoair

engagement can b e viewed as the sup erp osition of several indep endent terminat

ing renewal pro cesses The technique of sup erp ositioning indep endent terminating

renewal pro cesses was rst prop osed byAncker for mo deling one versus one

sto chastic ground combat termed byAncker a sto chastic duel This technique

has b een used in followon researchbyAncker Gafarian Kress and several asso

ciates to mo del m versus n sto chastic ground combat

In the m versus n sto chastic combat mo del eachcombatant completes an activity

that leads to a combat event eg ring a weap on or killing a target The o ccurrence

of a combat event is mo deled as a separate renewal pro cess for each combatant with

the interarrival times of the renewal pro cess representing individual combat events

for the combatant By assuming that each combatant completes a combat event

indep endently of the other combatants eg assuming eachcombatantisshooting

at a passive xed target the renewal pro cesses are made indep endent

In order to calculate the probability of b eing in a particular state the technique

of supplementary variables is used More preciselyattimet let the supplemen

tary variable Y denote the time since the nth combat event Then the rst order

Ancker studied the oneonone sto chastic combat mo del The next attempt to solvea

sto chastic combat mo del for a larger size was the twoonone sto chastic mo del by Gafarian and

Ancker They develop ed the general solution for the state probabilities for homogeneous two

onone sto chastic combat and from these derived the winning probabilities Kress solved the

manyonone sto chastic combat problem Next Gafarian and Manion solved the homogeneous

twoontwo combat mo del Hong followed bydeveloping a solution for the state probabilities

of heterogeneous twoonone and homogeneous three ontwocombats Later Parkhideh and Ga

farian provided the general solution to manyon many heterogeneous sto chastic combat of

whichmanyonmany homogeneous combat is a subset UnfortunatelyParkhideh and Gafarians

solution is a metho d of exhaustiveenumeration and has a strong exp onential computation time

whichmakes it practically imp ossible to use for battles b eyond on As a result Yang and

Gafarian develop ed a fast approximation for the manyonmany homogeneous case Unfortu

nately as can b e seen by the references Yangs article was published b efore Parkhidehs However

Parkhideh completed his research b efore Yang UndoubtedlyParkhidehs researchled

to Yangs research t0 t

Indicates a combat event

Figure Backward Recurrence Time Example

instantaneous probability that a n th combat event will o ccur in the interval

t t dtisgiven by

f y

F y

where f and F are the density and complementary cumulative distribution

functions of the combat eventrandomvariable eg interring time or interkilling

time In renewal theory Y is referred to as the backwardrecurrencetime Figure

illustrates the concept of backward recurrence time In Figure a particular com

batant at time t has had three combat events and it has b een t time units since

the last combat event With the use of the technique of supplementary variables

the nonMarkovian combat mo dels are transformed into semiMarkovian mo dels

A ma jor diculty of the Hong et al mo del is the exp onential explosion

of the state space with an increase in the numb er of combat events This explosion

is directly related to the requirement to mo del p ossible combinations of events For

example a oneonone airtoair engagement mo del with interring distributions

where each aircraft has one radarguided and one infrared missile ie four combat

events has p ossible states In order to reduce the state space size Hong et al

recommend using interkilling distributions rather than interring distributions This

reduces the number of combat events hence reducing the state space of the mo del

Simulation Models

As stated in section to days airtoair engagement mo dels consist mainly

of relatively large complex simulations Tomany these simulated mo dels are con

sidered more credible than analytic mo dels b ecause these mo dels generally contain

more detail than analytic mo dels In simulated mo dels the pro cesses and

activities of combat are acted out In this section THUNDERs airtoair engage

ment submo del will b e discussed Although there are other theaterlevel airtoair

engagementsimulation mo dels THUNDER is the predominate mo del used bythe

US Air Force for theaterlevel analysis App endix B details the airtoair en

gagement submo del used in THUNDER for those readers who are not familiar with

this theaterlevel mo del

As can b e seen from App endices A and B air environment factors in a the

ater level airtoair engagement mo del such as those contained in THUNDER are

treated at a very aggregated level when compared to an engagementlevel mo del

suchasBRAWLER THUNDER input parameters such as probability of engage

mentEN G relative range advantage RRA probabilityofkillPK degree of

aircraft command and control AC C etc attempt to account for factors that

inuence the outcome of airtoair engagements such as tactics sensor p erformance

range advantage weap ons usage etc in a general manner In contrast the air

environment in an engagement level mo del suchasBRAWLER is treated in great

detail Sensors and weap ons are explicitly mo deled and their p erformance compared

to ight tests and lab oratory exp eriments Tactics are mo deled explicitly and are

based up on intelligence estimates of threat weap on system capabilities threat cul

tural biases and pilot prociency Blue tactics are taken from USAF air manuals

such as MultiCommand Manual MCM actual tactics development ight tests

and discussions with exp erienced pilots

As a result of the aggregated treatmentofairenvironment factors in the

THUNDER high resolution airtoair engagement submo del the submo del has

b een criticized in certain areas for not accurately reecting the actual air environ

ment Ma jor criticisms include

The single shot probability of kill SSP K calculation considers only ring

platform weap on load versus opp onent platform combinations and is ap

proximately the same regardless of whether the ring weap on is employed as

the primary munition or employed with dierent tactics or the ring plat

form encounters the same opp onent platform armed with a dierentweap ons

load For example consider an FC armed with an AIM as its

primary weap on and the AIM as its secondary weap on b eing engaged bya

threat aircraft armed with a b eyond visual range BVR missile less capable

than the AIM but with longer range than the AIM In the ma jority

of the engagements the AIM is likely to dominate all outcomes negat

ing the threat missile range advantage over the AIM In the situation that

the FC is not armed with AIMs or has exp ended them during previous

engagements this threat BVR range advantage is not reected in THUNDER

The number of weap ons red p er engagementby an aircraft is the same for all

airtoair engagements regardless of typ e of opp onent aircraft faced

Amultiple weap on salvoisalways mo deled as a SHOOTSHOOT ring do c

trine while the more typical USAF SHOOTLOOKSHOOT ring do ctrine is

not mo deled

When an airtoair engagement o ccurs weap ons are always red The mo del

do es not allow for aircraft to disengage b efore weap ons release

Although the criticisms of the THUNDER high resolution airtoair engage

ment submo del discussed ab ovedohave merit the mo del app ears to pro duce ac

ceptable results in terms of b ottomline theaterlevel air sup eriority metrics suchas

aircraft loss and exchange ratios In section it was mentioned that BRAWLER

is used to calibrate data used in THUNDER An example of this calibration pro cess

is given in reference In this calibration pro cess individual airtoair engage

ment outcomes from THUNDER were examined from an input data p ersp ective

and compared to the BRAWLER inputs and results Sp ecically four THUNDER

input variables were considered EN G RRA PKand L The result of the calibra

tion pro cess were favorable with BRAWLER and THUNDER airtoair engagement

outcomes matching within intervals attributed to the inheritant dierences of the

mo dels With as many input data factors as the THUNDER high resolution air

toair engagement submo del allows one could change other factors and lead to the

similar outcomes in terms of theaterlevel metrics

Summary

This chapter provided a review of airtoair engagement literature Todays air

toair engagement mo dels consist mainly of relatively large complex simulations and

may b e roughly divided into two main groups according to their intended purp ose

analysis or training This dissertations fo cus is on airtoair engagement mo dels

used for analysis The US Air Force predominate simulation mo del used for analy

sis is THUNDER App endix B details the airtoair engagement submo del used in

THUNDER

On the other hand analytic mo dels particularly simple ones help clarify

imp ortant relations that are dicult to p erceive in a more complex mo del Though

generally more abstract than simulated mo dels analytic mo dels are characterized

by transparency or the degree to which cause and eect relationships in a mo del

are apparent Analytic airtoair engagement mo dels discussed include Lanchester

mo dels and probabilistic mo dels

IV Event Occurrence Networks

Introduction

In this chapter a new network formulation termed an event occurrence net

work EON is intro duced to mo del the interaction of groups of event sequences

An EON is a graphical representation of the sup erp osition of several terminating

counting pro cesses EONs dier from the probabilistic networks previously discus

sion in chapter I I and are motivated by the researchofAncker Gafarian Kress and

several asso ciates in mo deling m versus n sto chas

tic ground combat An analysis metho d termed bucket analysis is intro duced to

accommo date a no de state explosion problem as the number of the events in an

EON increases The general integral solution to the probabilityofbeingatanynode

state in an EON at time t is discussed To illustrate the capabilities of EONs two

simple airtoair engagement examples are given

Description

An event o ccurrence network EON is a probabilistic network in whichan

arc represents the o ccurrence of an event from a group of sequential events b efore

the o ccurrence of events from other event groupings Events b etween groups o ccur

indep endently but events within a group o ccur sequentially A set of arcs leaving

a no de is a set of comp eting events which are probabilistically resolved by order

relations More precisely consider groups of sequential events

G G G

s s s

where G g s represent s indep endent groups of events and G represents

g g

event e in group g Let continuous random variable E represent the o ccurrence

with completion time t of event G

t density function f

t and cumulative distribution function F

t complementary cumulative distribution function F

The complementary cumulative distribution function F t represents the

t Additionally nono ccurrence of event G by time t and is equivalentto F

g E

e g

only one event can o ccur in any innitesimally small time interval dt twoormore

events cannot occur at the same instant in time

An EON is the graphical representation of the sup erp osition of several termi

nating counting pro cesses A counting pro cess fN tt g is a sto chastic pro cess

in which N t represents the total number of events that have o ccurred up to time

t Random variable E dened previously ab ove represents the interarrival

time of the eth eventforcounting pro cess N tg s If E are identically

g g

distributed for all e then counting pro cess N t is a renewal pro cess Cox and

Smith have discussed the sup erp osition of renewal pro cesses

As stated in the intro duction to this chapter EONs dier from the probabilis

tic networks previously discussion in chapter I I For example consider generalized

activitynetworks GANs Two parameters are asso ciated with the arc of a GAN

the discrete probability p that an arc is taken given the no de from whichit

emanated is realized and

the time t required to accomplish the activitywhich the arc represents

A sto chastic pro cess fX tt T g is a collection of random variables where t is an index and

X t is a random variable 1

2 3

Figure Simple TwoEventEvent Occurrence Network

For our purp oses assuming T is a continuous random variable a GAN arc

o ccurs with probability P t and has length given by the distribution of T On

the other hand an EON arc o ccurs with continuous probability pt whichis

determined by sto chastic order relations among the event random variables and has

length given by a distribution conditioned on the ordering of the event random

variables For example consider an EON comp osed of twoevent groups with one

eventineach group events G and G Figure graphically depicts the EON for

this mo del which has three states The probability that arc is chosen is equivalent

to the probabilitythat G o ccurs b efore G P t P E E with the length

t given by the conditional distribution f

E jE E

AirtoAir Engagement Examples

As an example of an EON consider the rst airtoair engagement mo del dis

cussed by Hong et al As discussed in section this mo del describ ed a one

versus one airtoair engagement in which each aircraft could re one radarguided

and one infrared missile each A constraint of the mo del was that the radarguided

missile must always b e red b efore the infrared missile Let G and G represent

g g

the events that side g g res a radarguided and infrared missile resp ectively

Further let random variables E and E represent the o ccurrence times to re the

g g

radarguided and infrared missile resp ectively for side g Figure shows the EON

for this mo del

The EON shown in Figure has no des No de represents a nonevent for

this mo del ie the nono ccurrence of the rst event neither aircraft has red a

missile No des and represent p ossible o ccurrences outcomes for the rst event

one of the aircraft has red a radar missile No de represents the eventthatthe

aircraft on side has red the radar missile ie event G while No de represents

the event that the aircraft on side has red the radar missile ie event G No des

and represent p ossible o ccurrences for the second event No de represents

the event that the aircraft on side has red an infrared missile given that the rst

eventwas a radar missile red by the same aircraft No de represents the eventof

b oth aircraft ring radar missiles Two separate paths lead to this no de dep ending

on which aircraft red the rst radar missile Finally No de represents the event

that the aircraft on side has red an infrared missile given that the rst eventwas a

radar missile red by the same aircraft In a similar manner No des and represent

p ossible o ccurrences for the third event and no de represents the o ccurrence of the

fourth and last event

Note the owdown structure of the graph in Figure Each set of no des

on the same horizontal line corresp onds to the p ossible o ccurrences of the uth event

u Six paths lead from no de to no de These paths consists of the

following sequences of events

Recall from section that E and E are referred by Hong et al as interring times

g g

1 No Event

G11 G21

2 3 1st Event

G12 G22

G21 G11 4 5 6 2nd Event

G12 G22

G21 G11

7 8 3rd Event

G22 G12

9 4th Event

Figure Event Occurrence Network for One vs One AirtoAir Engagement

Example

Number No de Path Event Sequence

G G G G

The airtoair engagement EON shown in Figure terminates with probability

one at no de as t with b oth missiles b eing exp ended byeach aircraft Clearly

this engagement is not a realistic example since the p ossibility of a missile kill has not

b een included this is the actual engagement mo del discussed by Hong et al

Supp ose missile kills are allowed in the EON More preciselylet p and p represent

g g

the resp ective probabilities that the radar and infrared missile from side g sho ots

down kills the opp onent aircraft given that the missile is red Additionally let

q p and q p represent the resp ective probabilities that the radar

g g g g

and infrared missile from side g misses the opp onent aircraft given that the missile is

red The EON for this one versus one airtoair engagementisshown in Figure

For the EON shown in Figure let G and G represent the events that

g g

side g g res a radarguided and infrared missile resp ectively that shoot down

representtheevents that side g res a radarguided and G g and let G the side

g g

and infrared missile resp ectively that misses side g The EON shown in Figure

has no des Two dierentnodetyp es are present in the graphical representation

but the graph is not bipartite since arcs join two no des from the same class eg

arc A square no de represents the o ccurrence of an event that precludes

the o ccurrence of future events Termed a concludingevent this typ e of no de

completely describ es an absorbing state in the underlying probabilistic mo del For

example no de describ es the concluding event G that the aircraft on side red

Figure Event Occurrence Network for One vs One AirtoAir EngagementEx

ample with Missile Kills

a radar missile and shot down the aircraft on side A circled no de represents the

o ccurrence of event that do es not preclude the o ccurrence of future events Termed

an intermediateevent this typ e of no de completely describ es a transient state in

the underlying probabilistic mo del For example no de represents the intermediate

event G that the aircraft on side red a radar missile missing the aircraft on

side

The EON shown in Figure has concluding events and intermediate

events Note that some of the square no des share the same concluding event For

example no des and all share the same concluding event G ie the air

craft on side res a radar missile and sho ots down the aircraft on side However

the sequence of events leading up to the concluding event for each of these no des dif

fers For no de the sequence of events is G For no de the sequence of events

is G G Finally for no de the sequence of events is G G G

Underlying Probabilistic Model

Supp ose one is interested in the probability of b eing at a particular no de or set

of no des at time t in an EON For example what is the probability that the aircraft on

side sho ots down the aircraft on side with a radar missile at time t the probability

of b eing at no de or for the EON in Figure The probabilistic model

generated by the EON structure is a nonMarkovian probabilistic model In order to

calculate the probability of b eing at a particular no de at time tweneedtoknowthe

history of all completed events The states of the underlying probabilistic mo del of

EON corresp ond on a onetoone basis with the no des of the EON absorbing states

correspond to event sequences that end with a concluding event and transient states

correspond to event sequences that end with an intermediate event Square no des

are absorbing states and circled no des are transient states App endix C details the

probability of b eing at a particular no de in Figure at time t

No de is a nonevent but will also b e classied as an intermediate event

Table gives P t the probability of b eing at a particular no de a state in

the underlying probabilistic mo del at time t t for the EON in Fig

ure assuming that missile launch times are exponential ly distributed Sp ecically

represents the completion time of event G if continuous random variable E

g g

e e

where side g res a missile of typ e e e where radarguided and

infrared at sideg that either misses or sho ots down sideg whereg

represents the opp onent side of side g then

exp E t with ie side res a radar missile

E exp t with ie side res a radar missile

E exp t with ie side res an infrared missile and

E exp t with ie side res an infrared missile

Additionally the probabilities that the radar and infrared missiles on side

and sho ot down the other side when red are p p and p

p resp ectively Exact analytic expressions for the probabilities in Table

were derived and are summarized in App endix C

From Table as t minutes the probability of b eing at a particular no de

reaches a steadystate value Note that the probability of b eing at an intermediate

no de transient state approaches zero as t The following observations can

b e made ab out the steadystate probability of b eing at a particular no de for this

airtoair engagement example

First strike radar missile kills o ccur with a probabilityof P P

Slightly over of the airtoair engagements have ended within twoevents

missile rings P P P P P P

Only slightly more than of the airtoair engagements end without a sho ot

down P t

Time t

P t

Total

Table Probability of Being at a No de for the One vs One AirtoAir Engagement

Event Occurrence Network Example with Missile Kills

In answer to the example question p osed ab ove the probability that the aircraft

on side sho ots down the aircraft on side with a radar missile is P

P P

As demonstrated in App endix C the probabilityofbeingatany no de state of

the EON in Figure at time t can b e calculated exactly when the event probability

density functions are all of exp onential form due to closure of the exp onential dis

tribution under the op eration of integration However for manyevent probability

density functions eg gamma with noninteger shap e parameter or combinations

of event probability density functions eg exp onential and normal the integral

expressions given in App endix C have no closed form solution In such cases ap

proximations of the integral expressions are needed One solution technique discussed

in the literature is p olynomial approximation Discussion of this technique is de

ferred until Chapter V

Node State Explosion Problem

Adrawback of the EON approach to mo deling the interaction of groups of

event sequences is a no de state explosion problem as the number of the events in

the network is increased For example Table shows the numb er of no des states

for s group EONs with s events in each group s This explosion problem

is directly linked to the requirement to explicitly trackevent sequences Since the

underlying probabilistic mo del of an EON is nonMarkovian past event o ccurrences

are imp ortant in determining the probability of b eing at a current no de and the time

asso ciated with reaching the no de

One approach to minimizing the network explosion problem is network aggrega

tion This approach is used byAncker Kress Gafarian and asso ciates

in mo deling m versus n sto chastic ground combat Recall from

chapter I that aggregation refers to the pro cess of assembling or combining in mass

Aggregation in EONs is pro cess aggregation This approach attempts to combine

s Number of Nodes

Table Number of Nodes for s Group Event Occurrence Networks with s Events

in EachGroup

sets of events into fewer reduced aggregated sets of events ie reduce the num

b er of states in the probabilistic mo del The cost paid for aggregation is a loss of

information detail in the mo del

For example the one versus one airtoair engagement EON shown in Figure

may b e aggregated into the EON shown in Figure This EON only has no des

concluding and intermediate asso ciated with two killing events More precisely

let G describ e the event that the aircraft from side g sho ots down the aircraft from

sideg g No de describ e the nonevent that a kill on either side has not

yet taken place and b oth sides still have missiles to launch No des and describ e

the concluding event ie single event sequence that the aircraft on side and

resp ectively sho ots down the aircraft on the other side Finally no de describ es

the concluding event that b oth sides launch all missiles without achieving a kill

engagement ends without a sho ot down

For the aggregated EON and the asso ciated probabilistic mo del in the Fig

ure the following no des and asso ciated states have b een combined from the

unaggregated EON in Figure

No de Nodes and

ag g

No de Nodes and

ag g

No de Nodes and and

ag g

No de Node

ag g 1

2 3 4

Figure Aggregated Event Occurrence Network for One vs One AirtoAir En

gagement Example with Missile Kills

ag g

where a describ es the new aggregated no des Table gives P tthe

ag g

probability of b eing at a particular no de a state in the underlying probabilistic

mo del at time t t for the aggregated EON in Figure assuming the

same parameters as used to calculate Table for the unaggregated EON

The probability of b eing a particular no de at time t as shown in Table may

b e calculated using two dierent metho ds

Time t

ag g

P t

ag g

P t

ag g

P t

ag g

P t

Total

Table Probability of Being at a No de for the One vs One AirtoAir Engagement

Aggregated Event Occurrence Network Example with Missile Kills

By summing the probabilities of b eing at the appropriate original no des in the

ag g

unaggregated EON For example P P tP tP tP tP t

P t P tP t

By dening new distributions which mo del the o ccurrence of the new aggre

gated events from the unaggregated event distributions For example one

aggregated event for the EON in Figure is the sho oting down of an air

craft The density function for suchanevent can b e dened using the density

functions of the unaggregated events

f tp f tq p f t f t q q t

g E E E

ag g

g g g

for g where denotes the convolution of the density functions and is

the Dirac Delta function

The density function in equation is a nite mixture of density functions

Note that detail has b een lost with the aggregated EON the distribution of time for

the aircraft on side g to sho ot down the aircraft on sideg is known but not which

missile typ e sho ots down the aircraft

Another approach to the no de state explosion problem is a metho d lo osely

termed by this researcher as lq bucket analysis based on the analogy of a uid

owing down through a system of buckets In this metho d intermediate nodes of

an EON may betransformedintoconcluding nodes eectively reducing the number

of no des in followon event tiers This approachallows an EON to b e truncated to

the appropriate event level For example consider the EON in Figure once again

The steadystate ndings of Table showed that slightly more than of the air

In equation the Dirac Delta function is dened such that

toair engagements end within twoevents missile rings P P P

P P P Rather than calculate the probability of b eing

at the remaining no des thru one can change no des and to

concluding no des and calculate the probability of b eing at these three no des instead

of calculating the probability of the other remaining no des in the network

The metho d of bucket analysis eliminates rare as dened by the analyst

event sequences In terms of the uid analogy the underlying probabilistic mo del

of an EON can b e viewed as a construct for the ow of probability Clearlythe

probability of b eing at no de in any EON attime t is one no events have

o ccurred by t Probability then ows to other no des in the network based

on the o ccurrence of event sequences Absorbing states can b e viewed as buckets

capturing uid probability from higher no des in the network and transient states

as leaky buckets capturing uid from higher no des but feeding this uid to lower

no des in the EON By changing an intermediate no de in an EON to a concluding

no de one has changed a transient state to an absorbing state in the underlying

probabilistic mo del

Additionally through the use of bucket analysis one do es not have to solve

for the probability of b eing in a transient state directly By viewing transient states

as absorbing states one can immediately calculate the probability of b eing in a

particular transient state For example consider calculating P t for the EON in

Figure Using conservation of probability

P tP t P tP tP tP t

bc bc bb c

where P t is the probability of b eing in transient state a when state a is viewedas

bac

an absorbing state and b is event sequence G G Although one still must

calculate the probabilities asso ciated with viewing a transient state as an absorbing

state this computation is generally simplied due to the states relationship with

an asso ciated absorbing states For example truncated state is asso ciated with

absorbing state in Figure Equation b elowgives the probability of b eing

absorbing state

q p

P t exp t

exp t

The probability of b eing in truncated state is the same as equation except p

is replaced with q

Integral Solution

In order to solve for the probability of b eing at a particular no de or set of

no des at time t in a general s group event o ccurrence network EON one of two

solution approaches may b e used simulation or integral approximation In order to

use the integral approximation solution technique the probability of b eing at any

no de in the network must b e expressed in integral form In this section the integral

solution to a general s group EON is presented

State Integral Expression As shown in App endix C the probability

ofbeingatnodeinstateattimet P t for the airtoair example in Figure

P tP E tE t

P t is equivalent to the probability of the rst event not o ccurring by time t and

can b e written in the form

c c

P tF t F t

E E

due to the indep endence of events b etween event groups For a s group EON the

expression in equation is generalized as

P t F t

Hence in order to determine the probability of b eing in state at time t of an

event o ccurrence network one needs to determine the complementary cumulative

distribution functions of the time to rst event completions in eachevent grouping

Additionally the expression in equation is equivalentto

P t F t

min

where F t is the minimum of rst event o ccurrence distributions

min

Event Tier Integral Expressions For the rst event tier those

no des states where the rst event is completed in the EON probabilistic mo del

s no des are p ossible for a s group EON These s no des consist of s pairs of con

cluding and intermediate no des A concluding no de absorbing state indicates the

completion of the rst event from event group g with success discrete probability

p eg G The corresp onding intermediate no de transient state indicates a

rst event completion with failure probability p eg G For example

consider the airtoair example in Figure Since there are twoevent groups in

this example four states are asso ciated with the rst event tier two absorbing and

two transient states States and are the corresp onding absorbing and transient

states asso ciated with the rst event completion from event group one G while

states and are asso ciated with the rst event completion from event group two

The probability of b eing in absorbing state at time t P t for the airtoair

example in Figure as shown in App endix C is

P t P E E

F e f e de

For a r group EON equation may b e generalized to

P t P E E E E

g g g

E E E E

g g s g

P E E

j g

e f e de F

g E g g

j g

for any rst tier absorbing state g g s

The probability of b eing in transient state at time t P t for the airtoair

example in Figure as shown in App endix C is

t E t E P t P E

t E t P E P E

c c

t F t e f F e de

E E

For a s group EON equation may b e generalized to

P t P E tE tE t E

g g g

E tE t

g s

P E t P E t E

j g g

j g

c c

t e f e de t F F

g E g g

E E

g j

j g

for any rst tier transient state g g s Comparing equations and

P t is similar in form to P t except that F t has b een replaced by

F t e f e de

g E g g

to reect the fact that the rst event from group g has b een completed with prob

ability p but the second event from group g has not yet o ccurred If group g

t do es not have a second event the integral in equation is replaced with F

and equation simplies to

t F P t t F

E g

j g

Event Tier u Integral Expressions The number of event tiers in

an EON is the sum of the number of events in eachevent group in the network

Sp ecicallythenumber of event tiers n ina s group EON is

n n

T g

where n is the number of events in event group g For example n fora

g T

group EON with events in each group

Section showed that s p ossible no des exist in the rst event tier for a

s group EON For event tier u u n the numb er of no des in the network

dep ends on twointerrelated factors

The number of intermediate no des in the previous event tier u and

The number of event groups that still haveevents that can o ccur

As an illustration consider the second event tier For the second event tier

between s and s no des are p ossible for a s group EON Sp ecically for each

intermediate no de in the rst event tier either s ors second tier followon

no des can o ccur representing

s events from each of the remaining s event groups that did havea

rst event o ccur in the rst event tier and

a p ossible second eventfromg th event group but only if n

Since an event o ccurrence can b e a success concluding no de or a failure

intermediate no de for each intermediate no de in the rst eventtierbetween s

and s no des are p ossible Considering there are s intermediate no des in the rst

event tier the total numb er of no des for the second event tier can range from ss

to s no des

The airtoair EON example in Figure contains twoevent groups with two

events each thus eight no des s no des are predicted for the second event tier four

concluding and four intermediate no des However there are only seven no des four

concluding and three intermediate no des present in Figure for the second event

tier The dierence is a result of combining in the EON graph event sequences that

have the same set of completed events In the airtoair example in Figure no de

accounts for the twoevent sequences G G andG G that can b e

generated bytheeventset G G

No de combining in an EON graph only happ ens for sets of event o ccurrences

that are all failures that is when combining intermediate no des transient states

Each concluding no de absorbing state in an EON describ es a unique set of events

In other words no two concluding no des describ e the same set of completed events

At the second event tier ss intermediate no des maybecombined into

no des

The probability of b eing in any concluding or intermediate no de absorbing or

transient state in event tier u denoted absor b and tr ans resp ectively at time t can

b e expressed as a singlemultidimensional integral Sp ecically for eventtier uthe

probability of b eing in absorbing state absor b is

P t P E E j E E

absor b

g T T g T

E E j E E

g T g T g T

g T

E E j E E

g T g T g T

g T

E j E E E

g T sT g T sT

P E e f e j E e de

g T E j g T g T g T

j T

g T

j g

where E j g gs represents the time of o ccurrence

j T

nono ccurrence of the rst T events in event group j and j

j j j

g g s represents the probabilistic ordering of events E and

j T

E or

j j

g T

For example consider concluding no de absorbing state in the group

by event EON shown in Figure This state o ccurs in event tier as a result

are events The four intermediate events leading to G of concluding event G

G G G and G From equation the probability of b eing in state

at time t P t is

P t P E e j E e

P E e j E e f de

E e

22 44

21 64

63 77

20 41

40 76

19 61

60 75

18 38

37 74

17 58

57 73

16 35

34 72

15 55

54 71

14 32

31 70

13 52

51 69

12 29

28 68

11 49

48 67

10 26

25 66

Figure Event Occurrence Network with Event Groups with Events in Each

Group

In the ab ove expression for random variable E and reecting that

the twoevents in event group have o ccurred and G intermediate events G

sometime b efore concluding event G Similarly for E and k

o ccurred butG o ccurred b efore G reecting that intermediate event G

b efore event G Substituting expressions for P E e j E e

P E e j E e and f de

E e

Z Z Z

t e e

T T

P t e de de e e f f

E E

T T

Z Z

f e e f e de de

E E

T T

f e e f e de de

E E

T T

whichisa multiple integral with dimensions

For event tier u the probability of b eing in intermediate no de transient state

tr ans is

t t j E t E t j E P t P E

s tr ans

sT sT T

t P E t j E

j T

where E j s represents the time of o ccurrence non

j T

o ccurrence of the rst T events in eventgroup j and j s

j j j

represents the probabilistic ordering of events E or

j j

j T

As an illustration consider intermediate no de transient state in the

group by event EON shown in Figure This state o ccurs in event tier as a

result of intermediate events G G G G andG From equation

the probability of b eing in state at time t P tis

P tP E tP E tP E tj E t

In the ab ove expression for random variable E and reecting

that all events in event group havetaken place bytimet Similarly for E

and For E and reecting that event G o ccurred

has yet to o ccur by t Substituting expressions for b efore time t but event G

t tj E t and P E t P E P E

Z Z

e t

f e e f e de de P t

E E

T T

Z Z

t e

f e e f e de de

E E

T T

Z Z

e de de e e f f

E E

T T

which is also a multiple integral in dimensions

For the last n th event tier of an EON all no des describ e concluding events

absorbing states in the network The probability of b eing at any of the no des in this

event tier at time t except one is calculated using equation The exception is the

case where all events end in failure The probability of b eing in this absorbing state

at time t is calculated using equation where and j j s

j j

Bucket Analysis

In section the concept of bucket analysis was intro duced This metho d

reduces the numb er of no des states in an EON by transforming intermediate nodes

into concluding nodes thereby eectively reducing the numb er of no des in follow

on event tiers As a further illustration consider the group EON in Figure

where each group has events This network has states Supp ose as t

P t the bucket probabilitythatevent G is completed b efore either

event G or G is rare Since the no de paths pro ceeding from no de will not

contribute signicant as dened by the analyst probability to no des in followon

event tiers no de can b e transformed from an intermediate no de to a concluding

no de Figure shows this transformation By truncating no de the number of

no des in the network are reduced from to No des and

are eliminated from Figure

In section single multidimensional integral expressions were derived for

the probability of b eing at a given concluding or intermediate no de absorbing or

transient state in an EON at time t Although many p ossible paths events se

quences can lead to a given state the probability of being in that state at time t is

independent of the path actual ly taken In other words individual paths to a given

state do not havetobetracked explicitly and the probability deliveredby these

paths calculated in order to determine the probability of b eing in the state For

example consider the probability of b eing in transient state at time t P t for

the airtoair example in Figure Using equation

Z Z

P t e de de e e f f

E E

T T

Z Z

f e e f e de de

E E

T T

Twoevent sequences lead to state G G andG G The integral in

equation can b e decomp osed into the following individual path contributions

Z Z Z Z

t t

P t f e e f e de de

E E

T T

t t e

f e e f e de de

E E

T T

Z Z Z Z

t e

f e e f e de de

E E

T T

f e e f e de de

E E

T T

22 44

21 64

63 77

20 41

40 76

19 61

60 75

18 38

37 74

17 58

57 73

16 35

34 72

15 55

54 71

14 32

31 70

13 52

51 69

12 29

28 68

11 49

48 67

10 26

25 66

Figure Revised Event Occurrence Network with Event Groups with Events

in Each Group with No de Truncated

The rst integral in the equation represents the probability of b eing in state at

while the second integral represents G time t due to the event sequence G

the probability of b eing in state due to sequence G G

In general the single integral expression representing the probability of b eing in

any state of an EON at time t can b e decomp osed into separate integrals representing

the individual path contributions of p ossible event sequences This decomp osition

can b e generated by conditioning on the random variables within one event group and

decomp osing the integral expressions of other event groups based on the conditioned

random variables For the integral expressions in equation the decomp osition

was generated by rst conditioning on the random variables in event group one and

then decomp osing the integral expression in event group two to reect the cases that

event G is completed before event G rst integral and G is completed after

event G second integral

When no des are truncated not only are other no des eliminated but event se

quences for remaining no des may b e eliminated as well As an illustration consider

the truncation of no de in the group EON shown in Figure By truncating

no de paths from no de to no des and have b een eliminated Additionally

the network paths from no de through no de to no des and have b een elimi

nated as well Ifanode truncation removes event sequences terminating at a given

state then equations and are no longer exact solutions for the probability for

being in that state at time t For example consider the probability of b eing at no de

at time t P t Twelveevent sequences terminate at no de Table shows

these sequences and the asso ciated no de paths If no de is truncated the three

event sequences with asterisks and are removed since the paths asso ciated

with these sequences contain no de Equation no longer exactly represents

P t when no de is truncated by eliminating event sequences and the

probability asso ciated with sequences has also b een eliminated Since equation

contains the probability from the three eliminated event sequences the equation can

Number Event Sequence No de Path

G G G G G

Table Event Sequences and Asso ciated Paths for No de for the Group Event

Occurrence Network with Events in Each Group

only serve as an upp er b ound on the actual value of P t However the rational

b ehind truncating a no de was that the ow of probability through the no de was

relatively small By truncating no de the analyst has made a calculated decision

that not much probabilityows through the no de and hence the event sequences

asso ciated with that no de Thus equation should b e an accurate approximation

to P t

An exact solution for the probability of b eing at any state at time t in the

network under no de truncation could b e determined and is straightforward The

following steps are necessary

For the given state determine the event sequences that have b een removed

by the no de truncation

Determine the probabilityintegral for these event sequences

Subtract this calculated integral from the appropriate nontruncated integral

equation or

As an illustration consider the three event sequences eliminated at no de by

the truncation of no de The probability of each of these sequences at time t is

Z Z Z

t e e

T T

P t e de de e e f f

path E E

T T

e e

Z Z

e de de e e f f

E E

T T

f e e f e de de

E E

T T

Z Z Z Z Z

t e e e

P t e de de e e f f

path E E

T T

e e e

e e f e f de de

E E

T T

f e e f e de de

E E

T T

Z Z Z Z Z

e e e t

e de de e e f f P t

E E path

T T

e e e

f e e f e de de

E E

T T

e de de e e f f

E E

T T

Hence the probability of b eing in any of the three event sequences at time t is

P t P tP tP t

path path path path

Z Z Z Z Z

e t e e

f e e f e de de

T T

e e

f e e f e de de

E E

T T

f e e f e de de

E E

T T

Subtracting the integral in equation from the integral in equation yields

P t P t P t

tr uncated node path

Summary

In this chapter a new network formulation termed an event occurrence network

EON was intro duced to mo del the interaction of groups of event sequences An

EON is a probabilistic networkinwhich the arcs represent the o ccurrence of an event

from a group of sequential events b efore events from other groups Events b etween

groups o ccur indep endently but events within a group o ccur sequentiallyAsetof

arcs leaving a no de is a set of comp eting events which are probabilistically resolved

by the minimum typ e op erators EONs dier from the probabilistic networks previ

ously discussion in chapter I I and are motivated by the researchofAncker Gafarian

Kress and several asso ciates in mo deling m versus n sto chastic ground combat An

analysis metho d termed bucket analysis is intro duced to accommo date a no de

state explosion problem as the number of the events in a EON is increased The

general integral solution to an EON was discussed To illustrate the capabilities of

EONs two simple airtoair engagement examples are given

V Piecewise Polynomial Approximation

Introduction

As shown in chapter IV an imp ortant metric for an Event Occurrence Network

EON is the probability of b eing at a particular no de or set of no des at time tSuch

a probabilityisformulated as an integral expression p ossibly a multiple integral

expression involving event probability density functions reference section This

integral expression involves three probabilistic op erators

Multiplication

Convolution and

Conditional Integration

For manyevent probability density functions eg gamma with noninteger shap e

parameter or combinations of event probability density functions eg exp onential

and normal these integral expressions have no closed form solution In such cases

an approximation of the integral expression is needed to obtain a solution Finding

a go o d approximation is a nontrivial problem

App endix D section D provides a review of the theoretical construct of a

norm as a measure of the accuracy of t of an approximating function g to a function

f Sp ecically section D summarizes the approximation error function e e

f g for the widely used L norm Unfortunatelysuch measurement theory can not

b e extended to EON probabilityintegral expressions In this case the approximation

error function denoted e is the dierence b etween the integral function denoted I

and the approximation integral function denoted I thatise I I When there

a I a

is no closed formed expression for I the evaluation of the error function e can not

b e undertaken Instead some form of empirical test must b e p erformed in order to

measure the accuracy of the approximation

Piecewise Polynomial Functions

An example of a piecewise p olynomial function was given in section equa

tion in conjunction with Martins solution to reducing directed acyclic

networks with continuous random variable activity durations ie probabilistic ac

tivity networks Informally a piecewise p olynomial function is a function x

dened over a closed interval obtained by dividing an interval into subintervals and

constructing a dierent polynomial over each subinterval

More precisely let the interval a b with knot sequence a x x

x b or fx g b e partitioned into n subintervals I suchthat

n i k

x x if k n

k k

x x if k n

n n

m m

and let P b e the linear space of p olynomials of degree m ie P fpx px

c x c R The spaceofpiecewise polynomial functions is dened to b e

j j

the linear space P v of functions x such that

There exist p olynomials p P k n suchthatx

p x for x I k n and

k k

d p

x x for i n where l m v

i i i

dx dx

where the multiplicityvector v v v v is dened such that v Z and

v m

A linear space over a scalar eld F is a nonempty set X that satises the following conditions

There exists a mapping of X X into X called addition and written x x and

There exists a mapping of F X into X called scalar multiplication and written x

Addition and scalar multiplication must satisfy several conditions Typical conditions are men

tioned in reference page

The denition is a mo died version of p olynomial spline denition found in reference

page

The vector v sp ecies the continuity and derivative smo othing constraints

between the piecewise segments at knots If v m ie l the two

p olynomial pieces p and p in the intervals adjoining the knot x are unrelated

i i i

to each other eg may b e a jump discontinuityat x If v m ie l

i i

then the two p olynomial pieces have their rst m v derivatives continuous across

the knot x

In most applications the setting for the approximation problems is the closed

interval a b as dened ab ove However every piecewise p olynomial has a natural

extension to the real line To accomplish this extension two additional

p olynomials p p P must b e dened such that

p x xa

p x x I

k k

p x xb

In fact this is the form of the piecewise p olynomial function g xgiven in section

equation

Piecewise Polynomial Approximation and Interpolation

In section D p olynomial interp olation and approximation is intro duced The

main drawback of p olynomials for approximation purp oses is that the class is rel

atively inexible Polynomial approximation do es an adequate job on suciently

small intervals but when larger intervals of approximation are required severe oscil

lations often app ear particularly if m the degree of the p olynomial is more than

or This observation suggests that in order to achieve a class of approximat

ing functions with greater exibilitywork should b e p erformed with p olynomials

of relatively low degree and the interval of interest should b e divided into small

subintervals The denition of piecewise p olynomial function shown in section

provides the foundation for this typ e of approximation

Schumaker proved that the nite linear space P v has dimension

m K where K v A basis for P v can b e dened with elements

i m

that have supp ort over m subintervals ie m knots In order to dene this

basis an extended partition denoted by must b e asso ciated with P v The

partition is dened as

m K

fy y y g fy g

m K

suchthat

y y a x x b y y

m n m K

m K

and

z z

y y x x x x

m m K n n

m K

The p oints fy g in are uniquely determined The rst and last m p oints

in can b e chosen arbitrary One p opular choice is y y a and

y y b

m K

A basis for P v asso ciated with the extended partition isgiven bythe

m K

set fB g where

m m m

B x y y y y x y a x b

i m i i i m

with f representing the r th divided dierence of a function f in this

i i r

m m m

case f x y andx y x y x y m with

x y

The basis elements B in equation are referred to in the literature as Bsplines

and have the following prop erties

B x for x y y

i i m

B x forx y y

i i m

m K

B x for all x a b and

i i

B x forall x y y

j j

i j m i

Prop erties and verify that Bsplines have p ositivesupportover m intervals

The term in equation provides the p ositive supp ort attribute in Prop erty

Prop erties and ab oveshow that Bsplines form a partition of unity The

factor y y in equation is a normalization factor designed to pro duce this

i m i

partition of unity

A p oint of clarication may b e helpful to the reader at this time The denition

of the space of piecewise p olynomial functions P v given in the b eginning of

this section enabled the smoothness constraints of the piecewise function x to be

built into the bases of x Such an approach reduces the approximating problem to

solely full ling the appropriate set of interpolation andor approximation constraints

Given anypoints and a function f the r th dierence over the p oints is

i i r i i r

f f

i i r i i r

i i r

i r i

where

f f

i i

For p olynomial approximation one p opular criterion in the literature is the

least squares criterion Extending this criterion to piecewise p olynomial approx

imation the L norm discussed in section D equations and is used to

construct a unique approximating p olynomial function h in eachsubinterval k k

n Sp ecically consider the seminormed linear space P v jj e jj

m xw

r r

dened on R Necessary conditions to minimize the total error function

e f h

T k

for the weighted L norm dened by equation are

min jj e jj r

R w

d p

st x x i n l m v

i i i

dx dx

Consider approximating functions h xthathave the form

k k

x k n h x c

j j

n n

Here h is a linear combination of functions

In this case equation may b e written as

T T T T

min E cy Wy y WAc cA WAc Rc

where the ab ovevectors and matrices are dened as follows

y f x

y where y k n

y f x

n kr

is a column vector of r observations on f x in the k th interval By denition

an observation z R is dened such that

z x x for k n

k k k

z x x for k n

k k k

W Diag W W isan r x r blo ck diagonal matrix where W is an

r x r matrix of the form

k k

W k n

i i

A D iag A A isan r xm n blo ck diagonal matrix where

A is an r xm matrix of the form

x x

k k

A k n

x x

kr kr

k k

c c

k n where c c

c c

is a column vector of m approximating function co ecients

R R

n n

m v xm n matrix where R is an m v x is a

i i

m matrix of the form

i i

x x

i i

d d

i i

x x

i i

dx dx

and R

mv mv

i i

d d

i i

x x

i i

dx dx

is an m v x m matrix of the form R

x x

i i

d d

x x

i i

dx dx

mv mv

i i

d d

x x

i i

dx dx

In equation the mo died error function E c is constructed as the sum

e Rc

where represents a r column vector of Lagrange multipliers Necessary conditions

to minimize E c are

E E

Hence

T T T

A Wy A WAc R

Premultiplying b oth sides of equation by RA WA

T T T T

RA WA A Wy Rc RA WA R

Using equation and solving for

T T T T

RA WA R RA WA A Wy

Using this expression for and solving for c

T T T T T

c A WA A Wy A WA RA WA R

T T

RA WA A Wy

T T

The expression A WA A Wy in equation ab ove represents the uncon

strained contribution to the co ecientvector when the multiplicityvector v

m m develop ed in App endix D equation

As discussed in App endix D for large m the matrix A WA can b e ill

conditioned making its inverse dicult to solvenumerically One way of reduc

ing the eect of illconditioning is to intro duce orthogonalityinto the matrix For

equation this can b e accomplished by requiring orthogonalitybetween functions

x within each piecewise segment h x Sp ecically for approximating func

tions h of the form in equation the orthogonality requirements are

k k

h i for i j and

i j

k k

h i for all i

i i

for all k n With these requirements the matrix A WA reduces to a

diagonal matrix B of the form Diag B B where

k k k

h i x

B k n

k k k

h i x

m m

Hence requiring orthogonalitybetween functions x within each piecewise seg

ment h x reduces equation to

T T T

c B A Wy B RB R RB A Wy

When the approximating function hx is of the form in equation a total

of m n values c co ecients must b e stored to represent hx For

example supp ose hx is a piecewise monic p olynomial approximation

n m

m m

Recall that the space of piecewise p olynomial functions P v has dimension

m K the continuity restrictions transform the m n dimensioned

m n

vector space R into the m K dimensioned vector space R

A basis for P v was intro duced in equation This Bspline representation

may b e stored in m K values Hence in general the piecewise approximating

function representation in equation is bulkier than the BSpline representation

However when the approximation has to b e evaluated at a large numb er of p oints

as with the L norm case it is advantageous to use the approximating function

representation in equation For example to evaluate the approximating function

at anypoint in the domain requires only mm op erations for the piecewise

approximation function in equation as compared to the mm op erations

required by the most ecient Bspline expansions

As discussed in App endix D an interp olating p olynomial is a p olynomial ap

proximation function in which the p olynomial approximation function matches ex

actly a nite number of points from the original function Interp olating p olynomials

provide adequate approximations over small intervals but when applied to larger

intervals severe oscillations often app ear particularly with m Toavoid these

problems an alternativeinterp olation approach divides the interval of approxima

tion into smaller intervals and uses a dierentinterp olating p olynomial over each

interval Referred to in the literature as interp olation with piecewise p olynomial

functions an interp olating piecewise p olynomial function is a piecewise p olyno

mial function hx in which the knot sequence is formed from the n interp olation

p oints

One set of interp olating piecewise p olynomial functions that has received much

discussion in the literature are spline interp olating functions Let b e a partion of

the interval a b as in denition and let m b e a p ositiveinteger the space of spline

interp olating functions S is a subspace of the space of piecewise p olynomials

P v such that

f x p x fork n and f x p x

k k k n n n

v and

m b oundary conditions

As an example of b oundary conditions consider cubic interp olating splines m

With cubic splines one of the following sets of b oundary conditions is generally

sp ecied

d p

x x natural or free b oundary or

dx dx

dp df df

x x and x x clamp ed b oundary

n n

dx dx dx dx

Earlier it was shown that Bsplines form a basis for P v Since S is

m m

a subspace of P v Bsplines also form a basis for S Hence every spline

m m

function x S has a unique representation as

m n

c B xfor x a b x

Given the set of n interp olating p oints and the representation of x in equation

the following system of equations for an approximating spline function hx is

delineated

m n

hx c B x f x k n

k i k k

The system of equations in can b e represented in matrix form as

Bc y

where the ab ovevectors and matrices are dened as follows

m m

f x c B x B x

m n

y c B

m m

f x c x B x B

n m n n n

m n

A unique solution for hx requires an additional m equations As mentioned

ab ove these equations are formed from m b oundary assumptions Additionally

the matrix B is a banded matrix as each Bspline B has nonzero supp ort over

the interval x x

i i m

As a nal note piecewise p olynomial approximation and spline interp olation

can also b e viewed from a regression prosp ectus Piecewise p olynomial regres

sion literature is divided into two main areas piecewise regression and spline regres

sion Sp ecically consider the relationship b etween y and x

y f x

obtained by piecing together dierent functions curves over dierentintervals ie

f x x

f x

f x x

n n n

In piecewise regression the n functions are referred to as phase mo dels or

regimes and the interval p oints as change p oints or join p oints In spline regres

sion the individual phase mo dels f x i n are p olynomials and

i i

more stringent conditions are imp osed on these mo dels at the changep oints

Poirier develop ed a piecewise regression scheme using cubic splines Buse and

Lim showed that cubic spline regression scheme develop ed byPoirier is a sp ecial

case of restricted leastsquares estimation

Past Use in Density Function Estimation and Stochastic Operations

Cleroux and McConalo que rst applied cubic spline interp olation to con

volution of random variables which arise in reliability problems The authors used

splines in their application b ecause maintaining the derivative of the probability den

sity functions across spline segments that makeuptheinterp olation was considered

imp ortant

Clerox and McConalogue develop ed an algorithm using the Lobatto

formula a variation of Gaussian quadature and a spline metho d to approximate

the nfold convolution of distribution functions In order for the algorithm to b e

applicable the probability density function has to b e analytic ie continuous rst

derivative on the interval The algorithm was tested on the exp onential the

truncated normal gamma and Weibull probability distributions Its application to

the gamma and Weibull distributions is restricted to the case of the shap e parameter

McConalogue generalized the algorithm to a sub class of singular probabil

ity density functions having nite singularities at the origin and innity This work

made it p ossible to approximate the nfold convolution of the gamma and Weibull

distributions with shap e parameters as small as Unfortunately implementation

of this generalized algorithm is not straightforward

Mahjo oli used piecewise p olynomial interp olation to approximate the n

fold convolution of probability density functions which are analytic over a b ounded

interval Sp ecically Mahjo oli develop ed two approximation schemes using cubic

spline functions in the form of BSplines over a uniform partition of the b ounded

interval In the rst scheme all six nonsymmetric Bsplines are discarded and only

the symmetric normalized Bsplines are considered In the second scheme only four

nonsymmetric Bsplines are eliminated and the two nonsymmetric Bsplines with

largest supp ort are included

Santoro used normalized Bsplines to representany probability density

function with nite b ounds Normalized B splines denoted M are based on

ascheme intro duced by Curry and Scho enb erg that pro duces a related basis for

P v where

m m

x a x b B x M

i i

y y

i m i

The basis elements M in equation have the additional prop erty that

Z Z

y b

i m

m m

xdx xdx M M

i i

y a

In contrast to the ab oveinterp olation schemes Fergeson and Shortell rec

ommended using simple rst degree p olynomials that were tted piecewise using

least squares regression over ten equally space subintervals to approximate maxi

mum and convolution sto chastic op erators This equates to ten linear regressions

over the b ounded domain of the density function Fergeson and Shortell recom

mended using I cl assw idth regression tting p oints p ositioned uniformly

across each class Here cl assw idth refers to the length of a subinterval Additionally

in order to control error buildup from the p olygonal approximation after the ten

sets of regression co ecients have b een computed the co ecients are normalized so

that the probability under the approximated density function is one

Fergeson an Shortell termed their metho d the Polynomial Approximtion

and Reduction Technique PART The authors used this technique with uniform

normal and exp onential distribution functions Fergeson and Shortell noted that

the PART exp eriences it greatest approximating error at the p eak of a distribution

eg approximating the normal density function or approximating the convolution

of two normal or exp onential distributions The authors recommended as further

research adopting a multiple linear regression approach instead of the simple linear

regression approach or increasing the numb er of classes upwards from ten Addi

tionally at class b oundaries twoapproximations of a density function exist corre

sp onding to the approximate line segments of the class which terminates and the

class which b egins at that class b oundary

Lawrence adopted PART for use with Do dins arc no de and se

quential approximation reduction pro cesses Lawrence also cho ose a rst degree or

simple linear regression Lawrence expanded Fergeson and Shortells technique to

include approximation of the maximum and convolution op erators for the triangular

gamma and b eta distributions Lawrence do cumented that the PART algorithms

were orders of magnitude faster than simulationapproximations without signicant

losses in accuracy when simulation results are taken as true

In contrasttoLawrence Badinelli used orthogonal p olynomials to piece

wise approximate the probability density functions of continuous random variables

and the convolution of random variables for inventory mo dels Badinelli lo oked at

ve classical orthogonal p olynomial approximations

Tchebyshev p olynomials of the rst typ e

Tchebyshev p olynomials of the second typ e

Legendre p olynomials

Laguerre p olynomials and

Hermite p olynomials

Badinelli recommended that if the density function of a random variable is de

creasing or increasing over a wide range of its domain andor changing shap e then a

more ecient metho d of approximation was to divide the domain into segments and

p erform a separate approximation over each segment ie piecewise approximation

rather than tting a single approximation over the entire domain As he stated

in this manner such a piecewise approximation metho d can achievea very close

approximation with three or four p olynomial approximations of order or instead

of a single p olynomial approximation of order or Badinelli commented that

with orthogonal p olynomials unlikeinterp olation metho ds such as splines and La

grange p olynomials one can incorp orate into the approximation as much or as little

data as needed indep endently of cho osing the number of segments in the piecewise

approximation the order of the p olynomial approximation on each segment and the

b oundaries of each segment

For b ounded domains Badinelli set upp er and lower b ounds of the piece

wise approximation of the density functions of the random variables based on the

b ounded values of the random variable For unb ounded domains Badinelli set the

b ounds of the piecewise approximation at p oints b eyond which the probability func

tion had negligble magnitude Badinelli states that in most applications such p oints

exist and can b e conservatively estimated using the Tchebyshev theorem for upp er

b ounds on tail probabilities and then tighted as density function values are computed

and found to b e to o small to b e included in the approximation Once the upp er and

lower b ounds of the approximation have b een established the interval b etween them

is divided into N adjoining segments Badinelli do es not give a criterion for selecting

Badinelli cho ose two measures for accuracy of t

the average of the absolute value of the dierence b etween true complementary

cumulative distribution function F and the tted complementary cumulative

distribution function expressed as a fraction of F and

the average of the absolute value of the dierence b etween true complemen

tary cumulative distribution function of F G and the tted complementary

cumulative distribution function of F expressed as a fraction of G

Sp ecically the measures are

j F t t j

j er r or dt j

F t

j Gt t j

j er r or j dt

where

f tdt F x

R R R

f tdtds F tdt Gx

x t x

f is the density function of the continuous random variable

d a

R R R

a d d d and

a x

is the piecewise orthogonal p olynomial that approximates f

Badinelli stated that no theoretical results on the optimal choice of the ve

classical orthogonal p olynomial typ es to minimize equations and exist in the

literature The only theoretical result is for the accuracy of t of to f reference

section D As a result Badinelli used empirical results for equations and to

indicate the b est choice of p olynomial typ e and the b est order of the approximation

when applied to a given density function In most of the empirical tests Badinelli

used a normal density function since routines for computing integrated values of the

normal accurate to the fth decimal place were available Additionally Badinelli

assumed that a pro cedure yielding an accurate approximation to various segments

of the normal density function would p erform well on other density function suchas

the gamma and b eta functions whichwere also tested

Badinelli drew the following conclusions from the empirical tests

For normal gamma and b eta density functions the Tchebyshev polynomials

of the second type obtained the b est approximation in terms of the accuracy of

t criteria

For every p olynomial typ e there is an optimal p ower of approximation When

the order of the p olynomial is to o small the approximation cannot takeon

all of the complexity in shap e of the density function When the order of the

p olynomial is to o high the approximation is volatile over the interval

Increasing the number of points used in the approximation pro cedure enhances

the accuracy of the approximation in a given segment Badinelli suggested

data p oints as typically adequate based on his empirical tests

Error terms from an orthogonal p olynomial approximation show symmetry

This has the advantageous eect of canceling p ositive and negative errors in

the tted density function under integration making the t of the integral

moreaccurate than the t of the integrand ie density function

The accuracy of the approximation of some of the ve orthogonal p olynomial

typ es deteriorates at the extreme values of a segmentsincetheweighting func

tions decrease as one approaches these values However the errors at the

fringes of a segment or domain can b e eectively truncated byttingthe

p olynomial to a wider interval than the segmentordomainover whichthe

approximation is to b e used

Accuracy of the convolution on normal density functions did not signicantly

deteriorate as the numb er of convolutions increased Badinelli used convolu

tionsfromto

Badinelli to ok the numb er of data p oints and the order of the p olynomial

approximation as proxy measures of CPU time and stated that he kept them at the

smallest values p ossible while achieving desired accuracyHowever he did not state

these measures quantitatively

Kennedy used nonclassical orthogonal p olynomials to approximate the

density functions of a random variable based on sample moments of the density func

tion the actual underlying density function is assumed to b e unknown Kennedys

metho d rst established approximate density functions based on lower order mo

ments These density functions were then rened by orthogonal p olynomial series

expansions based on higher order moments Kennedys metho d relied on the imp or

tant result that a probability distribution with a b ounded domain can b e uniquely

determined from its moments Sp ecically the second third and fourth mo

ments ab out the mean provide information ab out the spread skewness and p eaked

ness of the distribution function while higher order moments contain an increasing

amount of information ab out the tails of the distribution

Kennedy mentioned four metho ds for approximating the density function

from sample moment data

Pearson Distributions

Series Expansion

Transformation to Normality and

Principle of Maximum Entropy

Kennedy stated that classical orthogonal p olynomials form the basis for the

ma jority of series expansions As mentioned in section D Kennedy observed that

classical orthogonal p olynomials are extremely useful since they are inexp ensiveto

compute and p ossess the minimum least squares prop ertyHowever Kennedy also

noted that there are some limitations to use of classical orthogonal p olynomials to

approximate functions

Toobtainavery high degree of accuracy an excessive p olynomial order may

b e required and

Convergence is not guaranteed to b e uniform unless particular typ es of p olyno

mials are used eg p olynomials obtained after the application of the Bernstein

op erator

Kennedy stated that these p otential limitations maybeovercome by

cho osing a weighting function that resembles the sp ecic function b eing approxi

mated In section D it was mentioned that nonclassical orthogonal p olynomials

may b e generated using the GramSchmidt orthogonalization pro cess with any pos

itive weighting functionFor the approximating density functions Kennedy recom

mended selecting weighting functions of similar shap e to the density functions Since

the density functions are not know but the sample moments of the density function

In other words the nth order p olynomial approximation to a function may not necessarily b e

b etter than the n th order approximation

are known Kennedy prop osed using the extreme value Pearson typ e I and Johnson

transform distribution as the weighting functions for the p olynomial approximation

based on rst three or four sample moments These weighting functions are then used

in an orthogonal series expansion with the higher order moments In this manner

Kennedy stated an orthogonal p olynomial series expansion can b e obtained which

is stable at high order eg for th to th allowing accurate approximation of tail

probabilities

Kennedy applied his metho d to sum dierence pro duct and quotient

op erations on random variables Kennedy showed that the moments of the output of

these random variable op erations can b e expressed exactly in terms of the moments

of the input variables The density function of the output of the op erations can then

b e approximated using the nonclassical orthogonal p olynomial metho d describ ed

ab ove

Discussion

The material presented in this chapter has b een fo cused on providing the nec

essary notation and background to rationally discuss the piecewise p olynomial ap

proximation techniques to use for solving the multiple integrals presentinEvent

Occurrence Network EON formulations The author has organized this discussion

in two fo cus areas

Approximation p ower of least squares versus interp olation schemes for sto chas

tic op erations and

Advantagesdisadvantages of continuity requirements b etween piecewise p oly

nomial segments

As can b e seen in literature review of piecewise p olynomial approximation

techniques section the research trend has moved from interp olation approaches

towards least squares approaches The research of the approximation p ower of least

squares versus interp olation schemes for sto chastic op erations is scant Unser and

Daub echies investigated the approximation p ower as a function of the sampling

step h ie spacing b etween sample p oints for discrete convolutionbased signal

pro cessing algorithms The authorsanalysis summarized that for larger values of h

the least squares approximation b ehaved likeaninterp olation with twice the order

For small values of h b oth the least squares and interp olation approximation exhibit

general O h b ehavior of the error but the least squares error is usually smaller

byaknown prop ortion than the interp olation error

In terms of the exibility in approximation the least squares schemes have

a decisiveadvantage over interp olation schemes With least squares techniques as

Badinelli p oints out one can incorp orate as much or as little data as needed

indep endently of cho osing the numb er of segments in the piecewise approximation or

the order of the p olynomial approximation on each segment With the least squares

techniques the numerical integration in EON formulations can b e accomplished with

as few p oints and as small a p olynomial order as necessary on each piecewise segment

to achieve desired accuracy

In terms of the second fo cus area continuity requirements b etween piecewise

p olynomial segments the literature is again split b etween least squares and inter

p olation schemes The interp olations schemes generally involve the use of spline

functions whichby denition are a subspace of C a b while the least squares

schemes generally do not sp ecify continuity requirements However it was shown in

section that continuity requirements can also b e easily incorp orated into least

squares schemes The literature is quite uninstructive on when to imp osenot im

p ose continuity requirements For example Cleroux and McConalo que consider

continuity across segements to b e imp ortant in approximating the convolution of

random variables while Badinelli considers continuity not to b e imp ortant for

An approximation pro cedure has an Lth order of approximation if it repro duce all p olynomials

of degree m L

approximating convolutions In fact neither of the authors state the rationale for

their recommendations

It is readily apparent that continuity requirements add additional computa

tional burden to piecewise p olynomial approximations Sp ecicallyeachcontinuity

requirementbetween two segments adds an additional constraintequationtothe

approximation problem However the most convincing arguments against specify

ing continuity requirements between piecewise polynomial segments are the results

achieved without their use Neither Fergueson and Shortell Lawerence or Badnelli

maintained continuitybetween segments and achieved adequate results for sto chastic

op erators

Based on the ab ove discussion this dissertation pursues least square piecewise

p olynomial approaches without continuity requirements across segments Sp ecically

the two approaches taken are a generalized version of the FerguesonShortellLawrence

PART algorithm and a variantofthePART algorithm using Badinellis researchon

piecewise orthogonal p olynomial functions

FerguesonShortellLawrence used simple piecewise p olynomial segments of

degree one in the PART algorithm This approach approximated maximum and

convolution op erators well for random variables having linear probability density

functions such as the uniform or triangular distributions However this approach

did not work as well for nonlinear probability density functions such as the normal or

exp onential distributions Fergueson and Shortell recommended investigation of

piecewise p olynomials segments of degree greater than one but Lawrence

did not pursue this recommendation in his research A generalized PART algorithm

uses a least squares approximation with piecewise p olynomial functions of degee

m Using the notation in equation the co ecients of the piecewise p olynomial

functions are

T T

c A A A y

Note that the PART algorithm eliminates the weight matrix W and continuity

comp onentinvolving the matrix R in equation As discussed previouslythe

main problem with this approach is that for large m the matrix A A can b e ill

conditioned making its inverse dicult to solvenumerically In fact in implementing

the generalized PART algorithm in the EON solution illconditioning of the matrix

b ecomes a severe problem in the tails of distributions This problem forced a switch

of inversion engines from LU decomp osition to singular value decomp osition SVD

To eliminate p otential conditioning problems with inverting A A orthogonal

p olynomials may b e used With this approach A A reduces to a diagonal matrix

B of the form Diag B B where

k k k

h i x

B k n

k k k

h i x

m m

j i

The co ecients of the piecewise orthogonal p olynomials using the PART algorithm

are

c B A y

This approach diers from Badinellis in several resp ects First and most signi

cantly Badnelli did not use a reduction step with his algorithm he was left with

the problems of explo ding co ecients and proliferating classes Secondly Badinelli

used p olynomials satisfying a continuous orthogonality relation whichallows the re

k k

lation h i i j to hold for any p oint x in the subinterval x x For

k k

i j

discrete applications suchasthePART algorithm the ab ove orthogonality condition

holds only when x is one of the zeros of Toovercome this restriction on x a

dierent orthogonalityscheme must b e used Hayes suggested using the

threeterm recurrence relationship

k k k k k

x x b xx a

j j j j j

k n j m

b eginning with initial p olynomials

k k

x x a x and

where x and a and b are chosen to the make the discrete orthogonal

j j

relation hold

k k

h x i

j j

k k

h i

j j

k k

i h

j j

k k

h i

j j

k k

with b and a xHayess metho d is similar to the GramSchmidt orthog

onalization pro cess discussed in App endix D section D with the factor unity

instead of the factor in equation The orthogonal p olynomials generated by

Hayess metho d may b e expressed in terms of classical orthogonal p olynomials such

as Tchebyshev p olynomials via transformation However such a transformation

increases computing time

Lastly Badnelli used weight functions and expanded intervals in his approxi

mation These twotechniques provide greater p ower of approximation but must b e

used with apriori knowledge They are not well suited to a generalized algorithm

Based on Badnellis research the dissertation employs orthogonal p olynomials gen

erated using Hayess metho d in the PART algorithm At this time it needs to b e

highlighted that although the use of piecewise orthogonal p olynomials eliminates

conditioning problems with the matrix A A itintro duces the problem of maintain

ing the orthogonality of the p olynomials during the reduction pro cess

Summary

In this chapter a formal denition of a piecewise p olynomial function was

given This denition was followed byanoverview of piecewise p olynomial approx

imation techniques of least squares and interp olation Next a literature review

summarizing the researchinto approximating sto chastic op erators with piecewise

p olynomial functions was given This literature review lead to a discussion of the

most appropriate techniques for solving the multiple integrals presentinEvent Oc

currence Network formulations Two approaches were forwarded for further study

a generalized version of the PART algorithm and a variant of the PART algorithm

using piecewise orthogonal p olynomial functions

VI EON Solution Using Piecewise Polynomial Approximation

Introduction

In section equations and delineate resp ectively the probabilityof

b eing in any absorbing and transient state of an Event Occurrence Network EON

at time t These multiple integral probability expressions consist of several sto chastic

op erations In equation

P E e de e j E e f P t

g T g T j g T g T E absor b

j T

g T

j g

the integral expression de is a conditional integration op eration with multi

g T

plication op eration e The P E e f e j E

g T g T E j g T

j T

g T

j g

sub expression P E e j E e consists of s multiplica

j g T g T

j T

j g

tion op erations Additionally the sub expression P E e e j E

g T j g T

j T

consists of the following sto chastic op erations

if P E e the rst eventinevent group j has not o ccurred b efore

g T

the rst T events in event group g so an additional integration op eration is

required

e the rst T events in event group j have e j E if P E

g T j g T

j T j T

not o ccurred b efore the rst T events in event group g given that the rst

T events in event group j have o ccurred b efore the rst T events in event

j g

group g so T convolution op erations twointegrations and a subtraction

op eration are required or

if P E e allevents in event group j have o ccurred b efore the rst T

g T g

events in event group g son convolution op erations and an integration

op eration are required

Finally the sub expression f e consists of T convolution op erations

E g T

g T

Similarly for equation

P t P E t j E t

tr ans j

j T

the expression consists of s multiplication op erations Additionally the sub expres

sion P E t j E t consists of sto chastic op erations similar to those

j T

e above e j E for the absorbing state sub expression P E

g T j g T

j T

In this chapter the techniquesalgorithms for approximating these sto chastic

op erations and solving EONs are given The FORTRAN computer program that

solves for the probability of b eing at anynodeofanEONattimet is over

lines long much to o large to include in this dissertation Although the algorithms

used to pro duce the EON solution are delineated in this chapter readers desiring the

algorithms at the co de level should contact the author The chapter contains two

airtoair engagement EON examples One of the networks is the simple airtoair

engagement example created by Hong et al shown in Figure and the other

is a more complex engagementinvolving over event distributions Additionally

App endix E contains interval data and p olynomial co ecients for all the examples

given in this chapter

Probability Density Function Approximation

The rst step in solving for the probability of b eing at a no de a of an EON

at time t requires inputting eachevent o ccurrence density function f t as a

piecewise p olynomial function Sp ecically let the interval a b b e partitioned into

n subintervals I suchthat

x x if k n

k k

x x if k n

n n

and a piecewise p olynomial function of degree m h t

k k

R h t c tc

j j

approximate the event o ccurrence density function in each subinterval using the L

least squares norm

Assuming no continuitybetween segments

T T

c A A A y

where

y f x

y and y k n

y f x

n kr

is a column vector of r observations on f xinthek th interval By denition an

observation z R is dened such that

z x x for k n

k k k

z x x for k n

k k k

and A D iag A A isanr xm n blo ck diagonal matrix where

A is an r xm matrix of the form

x x

k k

A k n

x x

kr kr

k k

When using monic p olynomials as discussed in section the key op eration

in equation is the inversion of the matrix A A Two solution techniques are

available for this inversion LU decomp osition or Singular Value Decomp osition

SVD Unlike LU decomp osition SVD enables a clear diagnosis of illconditioning

situations allowing zeroing out of singularities in the matrix A A However SVD

requires that a threshold of singularity b e set apriori

When using orthogonal p olynomials the inversion of the matrix A A in equa

tion is a trivial task The more arduous eort is the calculation of the orthogonal

p olynomials for eachinterval I Hayes threeterm recurrence re

lationship shown in section equation is used to calculate the orthogonal

p olynomials

As an example of tting monic and orthogonal piecewise p olynomial functions

to a probability density function consider the completion time function of event

G in the simple airtoair engagement example created by Hong et al shown

in Figure here E exp twith The left side of Figure

t with the linear quadratic and cubic p oly shows graphs of density function f

nomial approximation techniques resp ectively For each graph two monic LUD

SVD and one orthogonal p olynomial least squares technique were used The den

sity function was truncated at of its total area the interval of approximation is

In order to maintain the integrity of the original exp onential den

sity function the area of the piecewise p olynomial approximation is broughtback

to the area of original exp onential density function one by adding in this case

to the last subinterval The implementation adds the factor toco

classw idth

ecient c Here cl assw idth refers to the length of a subinterval which in this

case is Additionally each piecewise p olynomial approximation was

determined by the using ten subintervals and eleven p oints in eachinterval based

on Fergeson and Shortells recommendation of using I cl assw idth tting

p oints Inspecting the approximations visual ly al l three piecewise polynomial tech

niques appear to t the density function adequately for the linear quadratic and

cubic polynomial cases

in the Simple Airto Figure Graph of f t and Approximation Error for E

Air EngagementEvent Occurrence Network E exp

Polynomial Approximation Error Bound

Linear Monic LUD

Linear Monic SVD

Linear Orthogonal

Quadratic Monic LUD

Quadratic Monic SVD

Quadratic Orthogonal

Cubic Monic LUD

Cubic Monic SVD

Cubic Orthogonal

Table Error Bounds for Piecewise Polynomial Approximations of f t

In order to quantitatively determine the adequacy of the t the right side of

Figure shows graphs of the p ointwise error for the three approximation tech

niques All error graphs app ear similar in shap e having a damp ed oscillatory nature

Table shows the error b ounds for these graphs Note that an additional order of

precision is gainedwitheach increase in the order of the polynomial approximation

except for the last subinterval Recall in the last subinterval an area of was

added to the approximation in order to maintain the integrity of the original exp o

nential density function This addition adversely aects the approximation of the

density function in the last subinterval but is necessary to maintain the accuracy

of other sto chastic op erators suchasmultiplication and convolution Additionally

the two monic piecewise p olynomial approximations LU Decomp osition SVD and

the orthogonal piecewise p olynomial approximation show similar accuracy for each

increase in the order of the polynomial approximation

As an additional example from Lawrences research consider tting

monic and orthogonal piecewise p olynomial functions to a truncated normal density

function N dened on Figure shows the graphs of this truncated

normal density function with the linear quadratic and cubic p olynomial approxi

mation techniques resp ectively and the graphs of the approximation error for each

of these approximation techniques For the linear case the Lawrence graphs refer

Polynomial Approximation Error Bound

Linear Monic LUD

Linear Monic SVD

Linear Orthogonal

Linear Lawrence

Quadratic Monic LUD

Quadratic Monic SVD

Quadratic Orthogonal

Cubic Monic LUD

Cubic Monic SVD

Cubic Orthogonal

Table Error Bounds for Piecewise Polynomial Approximations of a truncated

N density function

to the co ecients generated using simple regression byLawrence It should b e

noted that Lawrence did not use a matrix representation for this regression

Insp ecting the approximations visually all three piecewise p olynomial tech

niques app ear to t the density function adequately for the linear quadratic and

cubic p olynomial cases All error graphs app ear similar in shap e having a damp ed

oscillatory nature Table shows the error b ounds for these graphs As with the

exp onential example an additional order of precision is gained with each increase

in the order of the p olynomial approximation however the accuracy b etween monic

and orthogonal techniques is similar for a given p olynomial approximation order

Integration Stochastic Operator Approximation

The integration of a p olynomial expression is a straightforward task The in

tegration of a piecewise p olynomial approximation is slightly more complex but

nevertheless is also undemanding Consider the integration of the piecewise p olyno

Figure Graph of f t and Approximation Error for Truncated Normal Density

Function N dened on

mial function hx over the interval d d where

x x a

h x x I k n

k k

x x b

Sp ecically let the function H representthisintegration that is

H hxdx

The coverage of the interval d d determines the form of H Six cases exist

d x d x

d x x d x

d x x d

x d x x d x

n n

x d x x d and

n n

x d x d

n n

Cases and are trivial with H over the interval of approximation a b Case

involves integration over the interval a d while case integrates over a b Case

uses d d Finally case uses the integration interval d b

As an example of calculating H consider calculating P E e which

g T

c c

is the complementary cumulative distribution function F e Recall F e

g T g T

E E

j j

represents the nono ccurrence of event G bytimee in an EON and is equivalent

j g T

to F e Sp ecically

E g T

F e F e

g T E g T

g T

f e de

E g T g T

lje I

g T l

g T

h e de

k g T g T

lje I

g T l

X X

g T

k k

c e de

g T g T

j j

l m

X X

k k

c e de

g T g T

j j

l k

c e de for all e j e x

g T g T g T g T n

j j

For e x F e

g T n g T

The left side of Figure shows the graph of the complementary cumulative

distribution function for the rst eventfromevent group F t in the airtoair

engagement example created by Hong et al with the linear quadratic and cu

bic p olynomials approximation techniques resp ectively Again these approximations

used ten subintervals and eleven least square p oints in each subinterval Insp ect

ing the approximations visually all three piecewise p olynomial ts app ear to t the

complementary cumulative distribution function adequately

On the right side of Figure all error graphs app ear similar in shap e having

a damp ed oscillatory nature Table shows the error b ounds for these graphs As an

aside Badnelli s observation that the t of the integral is more accurate than

the t of the integrand ie density function is validated for the approximations

of F t and f t The left side of Figure displays the relative error of each

of p olynomial approximation techniques for the exp density function f t

while the right side of gure shows the relative error for the exp complementary

t For each case linear quadratic and cumulative distribution function F

cubic the complementary cumulative distribution function approximation gains an

order of magnitude of accuracy over the density function approximation

As was the case with the density function approximations the two monic

piecewise p olynomial approximations LU Decomp osition SVD and the orthogonal

Figure Graph of F t and Approximation Error in the Simple AirtoAir

EngagementEvent Occurrence Network E exp

tandF tin Figure Graph of the Relative Approximation Error of f

the Simple AirtoAir EngagementEvent Occurrence Network

Polynomial Approximation Error Bound

Linear Monic LUD

Linear Monic SVD

Linear Orthogonal

Quadratic Monic LUD

Quadratic Monic SVD

Quadratic Orthogonal

Cubic Monic LUD

Cubic Monic SVD

Cubic Orthogonal

Table Approximation Error Bounds for Piecewise Polynomial Approximations

of F t

piecewise p olynomial approximation show similar accuracy for each increase in the

order of the p olynomial approximation However the orthogonal p olynomial tech

i co ecients where as the nique requires for each degree m the storage of

monic p olynomial technique requires storage of m co ecients for each piecewise

segment For this reason further development of the orthogonal p olynomial approxi

mation technique for EON solutions is not pursued One further note since the SVD

technique allows a clear diagnosis of illconditioning situations and a x to such sit

uations monic piecewise p olynomial results will b e shown using SVD inversion only

for the remainder of this chapter

Approximation of P E e j E e

j g T g T

j T

Recall from section the expression P E e j E e

j g T g T

j T

consists of the following sto chastic op erations

P E e

g T

e or e j E P E

g T g T

j T j T

P E e

g T

The solution to P E e was given in equation in the last section Similarly

g T

the solution to P E e isabypro duct of equation as P E e is

g T g T

j j

n n

j j

e the distribution function F

g T E

l m

X X

k k

e c F e de

g T E g T g T

j j j

k l

e de for all e j e x c

g T g T g T g T n

j j

For e x F e

g T n E g T

The solution to P E e j E e can b e derived from the

g T g T

j T j T

e Conditioning on the completion of event G expression P E

g T j T

j T

e e j E e P E e j E e P E P E

g T g T g T g T g T

j T j T j T j T j T

e e P E e j E P E

g T g T g T

j T j T j T

From equation

P E e j E e P E e P E e

g T g T g T g T

j T j T j T j T

c c

F e F e

g T g T

E E

j T j T

Multiplication Stochastic Operator Approximation

Of all sto chastic op erations necessary to solve for the probability of b eing in a

particular absorbing and transient state of an EON at time tthemultiplication op

eration is the most prevalent In equation alone fourteen dierentmultiplication

op erations are p ossible

e e F e f e f P E

g T g T g T E g T E

g T g T

P E e j E e f e

g T g T E g T

j T j T

g T

c c

F e F e f e

g T g T E g T

E E

g T

j T j T

e f P E e F e f e

g T E g T E g T E g T

g T g T

c c

e F e P E P E e F e

g T g T g T g T

E E

j i

i j

e e j E e P E e j E P E

g T g T g T g T

j T j T iT iT

c c c c

F e F e F e F e

g T g T g T g T

E E E E

iT iT j T j T

P E e P E e F e F e

g T g T E g T E g T

i j

n n

i j

n n

i j

e e j E e P E P E

g T g T g T

j T j T

c c c

F e F e F e

g T g T g T

E E E

j T j T

P E e P E e F e F e

g T g T g T E g T

i j

P E e P E e j E e

g T g T g T

j T j T

c c

F e F e F e

E g T g T g T

E E

j T j T

P E e P E e j E e f e

g T g T g T E g T

j T j T

g T

c c c

e e f e F e F F

g T g T E g T g T

E E E

g T

j T j T

e P E e P E e f

g T g T g T E

j i

g T

e F e F e f

g T E g T g T E

g T

e f e j E e P E e P E

g T E g T g T g T

j T j T

g T

c c

F e F e F e f e

E g T g T g T E g T

E E

g T

j T j T

e P E e e j E P E e P E

g T g T g T g T

j T j T

k i

c c c

e F e and e F e F F

g T E g T g T g T

E E E

j T j T

e P E P E e j E e f e e P E

g T g T g T E g T g T

j T j T

i k

g T

c c c

e F e F e F F e f e

g T g T g T E g T E g T

E E E

g T

j T j T

Multiplication op erations are required to extend the solution to EONs

whose number of event groups s is three or greater Concurrently equation

requires the multiplication op erations in and ab ove

F e f e Let the complementary cumulative distri

g T E g T

g T

bution function F and density function f e b e dened as follows

E g T

g T

if e d

g T

F e if d e d

e F

g T g T

g T

if e d

g T

and

if e d

g T

f e

e if d e d f

E g T

g T g T E

g T

if e d

g T

e one rst must determine e f In order to calculate the pro duct F

g T g T E

g T

which function reaches a value of zero rst the pro duct is zero after the end of

one of the b oundary p oints d or d Additionallyiftheintervals d d and

d d are disjointoroverlap at only one b oundary p oint the domain of the

pro duct is d d the interval of the density function with two p ossible pro duct

cases

if d d the pro duct is since F e over the interval d d

g T

and

e sinceF e over this interval if d d the pro duct is f

g T g T E

g T

Assuming overlap b etween the two functions two cases are p ossible for the

domain of the pro duct

if d d thentheinterval of the pro duct is d d and

if d d then the interval of the pro duct is d d

Let F e and f e b e approximated by the piecewise p olynomial func

g T E g T

g T

tions H e andh e over the subinterval sets I and I resp ectively

g E g k k

E T T

g T

If I is partitioned into n subintervals and I is partitioned into n subin

k k

tervals then the piecewise p olynomial approximation of F e f e

g T E g T

g T

H e is partitioned into p otentially n n subintervals e h

g T g T E

g T

Hence the subinterval set of H e h e I is comp osed of a set of

g T E g T k

g T

b oundary p oints fx g fx g where k n and k n all p ossible

k k

subinterval p oints of I and I LetH e h e represent the form of

k k g E g k

E T T

g T

e h e inthek th subinterval of I the pro duct H

g E g k

T T

g T

c c

e e h e H e h H

g T k g T E g k g E

E E

T T

g T g T

j j

k k T k k T

c e c e

g T g T

where

k k

c is the m dimension p olynomial co ecientvector and is the m

dimension p olynomial basis function vector for the k st interval of I and

k k

c is the m dimension p olynomial co ecientvector and is the m

dimension p olynomial basis function vector for the k nd interval of I

Assuming a monic piecewise p olynomial approximation the ab ove expression

may b e written as

c k k T

H e h e c C x k I

g E g k k

T T

g T

where

c c

and C

c c

x e e

g T g T

Note that the order of each piecewise p olynomial segment in equation is

m In order to manage a p otential problem of explo ding co ecients once

the expanded piecewise p olynomial approximation in equation is calculated

the approximation is reduced by tting a piecewise p olynomial approximation of

degree m to the expanded approximation In order to control error buildup from

the reduced approximation the reduced approximation is then normalized to

the area of the expanded approximation This normalization is accomplished by

dividing each co ecient in each piecewise approximation by

expanded approximation area

reduced approximation area

Additionally each piecewise reduced approximation is checked to verify that it do es

not yield values less than zero if it do es the piecewise approximation is increased

by a factor c necessary to guarantee values greater than zero Nonzero values

generally app ear in the tails of a pro duct function where values of the function

approach zero

As a nal note if d d then c until d x x I

k k k

c c

e f e F F e Let the function

g T E g T g T

E E

g T

j T j T

c c

F e F e bedenedasfollows

g T g T

E E

j T j T

if e d

g T

c c

F e F e

F e F e if d e d

g T g T

g T g T g T

E E

j T j T

if e d

g T

and f e b e dened as in section

E g T

g T

If the intervals d d and d d are disjointoroverlap at only one

c c

b oundary p oint the pro duct F e F e f e is with

g T g T E g T

E E

g T

j T j T

two p ossible domain cases

if d d the domain of the pro duct is d d and

if d d the domain of the pro duct is d d

Assuming overlap b etween the two functions four cases are p ossible for the

domain of the pro duct

if d d and d d then the interval of the pro duct is d d

if d d and d d thentheinterval of the pro duct is d d

if d d and d d then the interval of the pro duct is d d and

if d d and d d then the interval of the pro duct is d d

Equations and in section are still valid for approximating H e

g T

j T

c c

e h e replacing H e with H

g T E g T k g T

E E

g T

j T

c c c c

e is the piecewise p oly e H e H e H H

g T g T g T g T

E E E E

j T j T j T j T

c c

nomial approximation of F e F e over the subinterval I

g T g T k

E E

j T j T

F e Let the distribution function F e f e be

E g T E g T E g T

g T

j j

n n

j j

dened as follows

if e d

g T

F e if d e d

F e

E g T g T

E g T

if e d

g T

and f e b e dened as in section

E g T

g T

If the intervals d d and d d are disjointoroverlap at only one

b oundary p oint the domain of the pro duct is d d the interval of the density

function with two p ossible pro duct cases

e over the interval if d d the pro duct is f e since F

g T E g T E

g T

d d and

if d d the pro duct is since F e over this interval

E g T

Assuming overlap b etween the two functions two cases are p ossible for the

domain of the pro duct

if d d thentheinterval of the pro duct is d d

if d d then the interval of the pro duct is d d

Equations and in section are still valid for approximating

H e h e replacing H e withH e H e isthe

E g T E g T k g T E g T E g T

g T

j j j

n n n

j j j

piecewise p olynomial approximation of F e over the subinterval I As a nal

E g T k

note if d d thenc in equations and for all x d x I

k k k

c c

e Let the complementary cumulative distribu e F F

g T g T

E E

j i

c c

tion functions F e and F e b e dened as follows

g T g T

E E

i j

if e d

g T

e if d e d F

F e

g T g T

g T E

if e d

g T

and

if e d

g T

F e if d e d

F e

g T g T

E g T

if e d

g T

c c

In order to calculate the pro duct F e F e one rst must determine

g T g T

E E

i j

which distribution reaches a value of zero rst the pro duct is zero after the end

of one of b oundary p oints d or d Additionally if the intervals d d and

d d are disjointoroverlap at only one b oundary p oint the pro duct of two

complementary cumulative distribution functions is simply the complementary dis

tribution function of the lower interval

Assuming overlap b etween the two complementary cumulative distribution

functions four cases are p ossible for the domain of the pro duct

If d d and d d then the interval of the pro duct is d d

If d d and d d thentheinterval of the pro duct is d d

If d d and d d thentheinterval of the pro duct is d d and

If d d and d d thentheinterval of the pro duct is d d

c c

e beapproximated by the piecewise p olynomial func e and F Let F

g T g T

E E

j i

c c

e over the subinterval sets I and I resp ectively e and H tions H

g T k k g T

E E

j i

If I is partitioned into n subintervals and I is partitioned into n

k k

c c

e e F subintervals then the piecewise p olynomial approximation of F

g T g T

E E

j i

c c

e is partitioned into p otentially n n subintervals e H H

g T g T

E E

j i

c c

e H e I is comp osed of a set of Hence the subinterval set of H

g T g T k

E E

j i

b oundary p oints fx g fx g where k n and k n all p ossible

k k

c c

e represent the form of e H subinterval p oints of I and I Let H

g T k g T k k

E E

j i

c c

e inthek th subinterval of I e H the H

g T k g T

E E

j i

c c c c

e e H e H e H H

g T k g T g T k g T

E E E E

j i j i

k k T k k T

c e c e

g T g T

where

k k

c is the m dimension p olynomial co ecientvector and is the m

dimension p olynomial basis function vector for the k st interval of I and

k k

c is the m dimension p olynomial co ecientvector and is the m

dimension p olynomial basis function vector for the k nd interval of I

Assuming a monic piecewise p olynomial approximation the ab ove expression

may b e written as

c c k k

H e H e c C x

g T g T k

E E

i j

where

k k

c c

k k

c c

C and

k k

c c

x e e

g T g T

Note that the order of each piecewise p olynomial segment in equation is

m As in section in order to manage a p otential problem of explo ding

co ecients once the expanded piecewise p olynomial approximation in equation

is calculated the approximation is reduced by tting a piecewise p olynomial ap

proximation of degree m to the expanded approximation For this op eration

an area normalization routine as outlined in section was not used as test cases

indicated no signicantimprovement in accuracy when employed However each

piecewise approximation is checked to verify that it do es not yield values less than

zero if it do es the piecewise approximation is increased by a factor c necessary to

guarantee values greater than zero As stated in section nonzero values can

app ear in the tails of the probability functions where values of the function approach

zero

As an example of the pro duct of two complementary cumulative distribution

functions consider the probabilityofbeinginstateattimet P t for the simple

airtoair engagement example created by Hong et al shown in Figure

Approximation Error Bound

Linear

Quadratic

Cubic

Table Error Bounds for Piecewise Polynomial Approximations of P t

Recall from equation in section that

c c

P t F t F t

E E

The left side of Figure shows the graph of P t and the three piecewise p olynomial

approximations using the linear quadratic and cubic monic p olynomials techniques

resp ectively The three piecewise p olynomial approximations were again determined

by the least squares criterion SVD inversion engine using ten subintervals In

sp ecting the approximations visually all three piecewise p olynomial ts app ear to

t P t adequately

In order to quantitatively determine the adequacy of the t the right side of

Figure shows graphs of the p ointwise error for the three approximations Table

shows the error b ounds for these graphs The error graphs exhibit damp ed oscillatory

b ehavior that moves towards zero except for the hump b etween As

exp ected this is caused by truncation of the exp onential functions and subsequent

area renormalization In order to fully realize the accuracy gains with the cubic

approximation truncating the event o ccurrence density functions at greater than

of the total areas would b e recommended

c c

F e F e

g T g T

E E

iT iT

c c c c

e e F e Let the functions F e F F

g T g T g T g T

E E E E

iT iT j T j T

Figure Graph of P t and Error for the Simple AirtoAir EngagementEvent

Occurrence Network

c c

and F e F e b e dened as follows

g T g T

E E

j T j T

if e d

g T

c c

F e F e

F e F e if d e d

g T g T

g T g T g T

E E

iT iT

if e d

g T

and

if e d

g T

c c

e e F F

F e F e if d e d

g T g T

g T g T g T

E E

j T j T

if e d

g T

The domain of this pro duct is identical to the domain of the pro duct

c c

F e F e f e detailed in section If the intervals

g T g T E g T

E E

g T

j T j T

d d and d d are disjointoroverlap at only one b oundary p oint the

c c c c

e is with two e F e F e F pro duct F

g T g T g T g T

E E E E

j T j T iT iT

p ossible domain cases

if d d the domain of the pro duct is d d and

if d d the domain of the pro duct is d d

Assuming overlap b etween the two functions four cases are p ossible for the

domain of the pro duct

if d d and d d then the interval of the pro duct is d d

if d d and d d thentheinterval of the pro duct is d d

if d d and d d then the interval of the pro duct is d d and

if d d and d d then the interval of the pro duct is d d

Contrary to the domain of the pro duct the pro duct

c c c c

F e F e F e F e is determined using

g T g T g T g T

E E E E

iT iT j T j T

equations and in section The approximating function is H e

g T

c c c c c

H e H e H e where H e H e

g T g T g T k g T g T

E E E E E

iT j T j T iT iT

c c

and H e H e are the piecewise p olynomial approximations of

g T g T

E E

j T j T

c c c c

F e F e and F e F e resp ectively over the

g T g T g T g T

E E E E

iT iT j T j T

subintervals I and I

k k

e and e Let the distribution functions F e F F

g T g T E g T E E

i j i

n n n

i j i

F e b e dened as follows

E g T

if e d

g T

e if d e d F

F e

g T g T E

E g T

if e d

g T

and

if e d

g T

F e if d e d

e F

E g T g T

g T E

if e d

g T

If the intervals d d and d d are disjointoroverlap at only one

b oundary p oint the domain of the pro duct and the pro duct itself havetwo p ossible

outcomes

if d d the domain of the pro duct is d d with pro duct F e

E g T

as a result of F e over d d and

E g T

if d d the domain of the pro duct is d d with pro duct F e

E g T

as a result of F e over d d

E g T

Assuming overlap b etween the two distribution functions four cases are p os

sible for the domain of the pro duct

if d d and d d then the interval of the pro duct is d d

if d d and d d thentheinterval of the pro duct is d d

if d d and d d then the interval of the pro duct is d d and

if d d and d d then the interval of the pro duct is d d

e e H Equations and in section are valid for approximating H

g T g T E E

j i

n n

j i

where H e and H e are the piecewise p olynomial approximations of

E g T E g T

i j

n n

i j

e resp ectively over the subintervals I and I As nal e and F F

g T k k g T E E

j j

n n

j j

notes if d d then c in equations and for all x d x I

k k k

If d d then c in equations and for all x d x I

k k k

c c c

F e F e F e This multiplication op

g T g T g T

E E E

j T j T

e op erator in section In fact the erator is similar to the F e f

g T g T E

g T

nooverlap and overlap domains are identical The pro ducts for the nooverlap cases

are

if d d the pro duct is since F e over the domain d d

g T

and

c c c

if d d the pro duct is F e F e since F e

g T g T g T

E E E

j T j T

over this domain

c c c

e for e H e H The pro duct approximation H

g T k g T g T

E E E

j T j T

the overlap domains d d and d d is determined from equations and

in section As a nal note if d d thenc until d x x I

k k k

e F e Let the complementary cumulative distribu F

g T E g T

tion function F and distribution function F e bedenedasfollows

E g T

if e d

g T

F e if d e d

F e

g T g T

g T E

if e d

g T

and

if e d

g T

e if d e d F

e F

g T g T E

g T E

if e d

g T

If the intervals d d and d d are disjointoroverlap at only one b oundary

p oint the two separate domain and pro duct cases are p ossible

if d d the pro duct is over the domain d d and

if d d the pro duct is

if e d

g T

e if d e d F

g T g T E

e F e F

g T E g T if d e d

g T

e if d e d F

g T g T

if e d

g T

over the domain d d

The pro duct approximation H e H e for the overlap domain

g T E g T k

d d is determined from equations and in section As nal notes if

k k

d d then c until d x x I and if d d thenc for

k k k

all x d x I

k k k

c c

F e F e F e This multiplication op er

E g T g T g T

E E

j T j T

ator is similar to the F e f e op erator in section In fact the

E g T E g T

g T

nooverlap and overlap domains are identical The pro ducts for the nooverlap cases

are

c c

if d d the pro duct is F e F e e since F

g T g T g T E

E E

j T j T

over the domain d d and

if d d the pro duct is since F e over this domain

E g T

c c

e H The pro duct approximation H e H e for

g T E g T g T k

E E

j T j T

the overlap domains d d and d d is determined from equations and

in section As a nal note if d d thenc for all x d x I

k k k

c c c

e F e F e F

g T g T g T

E E E

j T j T

c c c

f e The pro duct F e F e F e was dis

E g T g T g T g T

E E E

g T

j T j T

cussed in section The form of this pro duct is as follows

c c c

F e e F e F

g T g T g T

E E E

j T j T

if e d

g T

c c c

e if d e d e F e F F

g T g T g T g T

E E E

j T j T

if e d

g T

c c c

The domain of the pro duct F e F e F e is similar in

g T g T g T

E E E

j T j T

c c

form to the domain of the pro duct F e F e in section As

g T g T

E E

j T j T

a result the domain logic and pro duct equations from section are applicable

c c c

e e f e F e F to the calculation of F

g T g T E g T g T

E E E

g T

j T j T

F e The pro duct e F e f

g T g T E g T E

g T

F e was discussed in section The form of this pro duct is as e F

g T g T E

follows

if e d

g T

F e F e if d e d

e F e g T E g T g T F

g T E g T

if e d

g T

The domain of the pro duct F e F e is similar in form to the domain of

g T E g T

c c

the pro duct F e F e in section As a result the domain

g T g T

E E

j T j T

logic and pro duct equations from section are applicable to the calculation of

e f e F e F

g T E g T g T E

g T

c c

F e F e F e

E g T g T g T

E E

j T j T

c c

e F e F e was dis f e The pro duct F

g T g T g T E g T E

E E

g T

j T j T

cussed in section The form of this pro duct is as follows

c c

e F e F e F

g T g T g T E

E E

j T j T

if e d

g T

c c

e F e F e if d e d F

g T g T g T g T E

E E

j T j T

if e d

g T

c c

e is similar in e F e F The domain of the pro duct F

g T g T g T E

E E

j T j T

c c

form to the domain of the pro duct F e F e in section As

g T g T

E E

j T j T

a result the domain logic and pro duct equations from section are applicable

c c

e e f e F e F to the calculation of F

g T g T E g T g T E

E E

g T

j T j T

c c c

e The pro duct F e F e F e F

g T g T E g T g T

E E E

j T j T

c c c

e F e F e was discussed in sections and F

g T g T g T

E E E

j T j T

It has b een shown that the domain of this pro duct is similar in form to the domain of

c c

the pro duct F e F e in section As a result the domain

g T g T

E E

j T j T

logic and pro duct equations from section are applicable to the calculation of

c c c

e F e F e F e F

g T E g T g T g T

E E E

j T j T

c c c

e F e F e F e F

g T g T g T g T E

E E E

j T j T

f e This last EON multiplication sto chastic op erator is calculated using

E g T

g T

two previous op erators

c c c

the pro duct F e F e F e F e in section

g T g T g T E g T

E E E

j T j T

and

c c

the pro duct F e F e f e in section Here

g T g T E g T

E E

g T

j T j T

c c

replacing the pro duct F e F e with the pro duct

g T g T

E E

j T j T

c c c

e F e F e F F e

g T g T g T E g T

E E E

j T j T

Convolution Stochastic Operator Approximation

Equations and require calculation of convolutions of event o ccurrence

density functions Sp ecically consider the convoluted density function of the

o ccurrence time of the rst T events in event group g ief t This density

g T

function may b e expressed as a convolution of the density function of the o ccurrence

time of the rst T events in event group g and the event o ccurrence density

function of the T th eventinevent group g ie

f tf f t

E E E

g T g T

The rst step in solving the convolution in equation is to determine the ap

plicable approximation partition for f t Sp ecicallyletf tand f t

E E E

g T g T

b e approximated by the piecewise p olynomial functions h and h and over

E E

g T

the subinterval sets I and I resp ectivelyIfI is partitioned into n subin

k k k

is partitioned into n subintervals then the piecewise p olynomial tervals and I

approximation of f t h t is partitioned into p otentially n n subinter

E E

g T g T

vals

The subinterval set of h t I is comp osed of b oundary p oints x x

E k k k

g T

where k n and k n all p ossible combinations of I and I

k k

The domain of the convolution approximation h tis

g T

min x x max x x

k k k k

Let h t represent the form of the h tinthek th subinterval of I

E k E k

g T g T

h t h h t

E k E E k

g T g T

upper

t x h x dx h

k E k E

g T

low er

J J

k k

Here J I and J I suchthat J and J are valid sets of subintervals

k k k k k k

contributing to the k th subinterval of I iex x The conditions for the k st

k k k

subinterval of I and k nd subinterval of I contributing to the k th subinterval of

k k

I are

x x x

k k k

x x x

k k k

Conditions and may b e written for any x x x as

k k

x x x and x t x x

k k k k

x x x and t x x t x

k k k k

The ab ove expression simplies to

maxx t x x minx t x

k k k k

From equation the limits of integration in equation are

upper minx t x

k k

lower maxx t x

k k

for each subinterval combination x x J and x x J contributing

k k k k k k

to x x I

k k k

upper

In equation the sub expression h t x h x dx can b e

E k E k

low er g T

written in the following matrix form

upper

k k T k k T

c t x c x dx

low er

where

k k

c is the m dimension p olynomial co ecientvector and is the m

dimension p olynomial basis function vector for the k st interval of I and

k k

c is the m dimension p olynomial co ecientvector and is the m

dimension p olynomial basis function vector for the k nd interval of I

Assuming a monic piecewise p olynomial representation the ab ove expression

can b e written as

Z Z

upper upper

k T k T T

c x xC t dx t x h x dx h

k E k E

g T

low er low er

upper

k k T T

c XC t j

low er

where

x x

x x x

m m m

x x x

m m m

C is an upp er triangular matrix of the form

k k k

m m

m m k

c c c c

m m

k k

m k

c c mc

and t tt When either the upper or lower limits of integration are a

constant d x or x resp ectively the evaluation at that limit is obtained by

k k

simply substituting d into the matrix X in equation In this case the convolution

segment h t maintains the same p olynomial degree m as each of the convoluted

E k

g T

piecewise segments h x and h t x However when an upper or lower

E k E k

g T

limit is of the form t d t d t x or t x the degree of convolution

k k

increases to m

When an upper or lower limit is of the form x t d the matrix expression

k k T T

c XC t in equation equation may b e reformulated as

k k T

c c X T

where X is a m xm matrix of the form

m m

d d

m m

d d d

m m

m m m

d d d

m m

m m m

with I b eing a m x m identity matrix and T is an m x m

matrix of the form

m m

t t t

m m

t t t

m m m

t t t

m m m

with b eing an upp er triangular matrix of zeros

As an example of the ab oveformulation consider the convolution of two trun

cated normal density functions from Lawrence Supp ose

N truncated on f

f N truncated on

g T

The nal form of this example convolution is the truncated normal density function

truncated on f f N

E E

g T

Figure shows the graph of this convolution with the linear quadratic and

cubic monic piecewise p olynomial techniques resp ectively using Single Value De

comp osition matrix inversion Insp ecting the graphs visually the monic piecewise

Polynomial Approximation Error Bound

Linear

Quadratic

Cubic

Table Error Bounds for Piecewise Polynomial Expanded Approximations of

the Truncated Normal Convolution Example

p olynomial techniques app ear to t the truncated normal density function convolu

tion adequately

As with the results shown for multiplication sto chastic op eration the approx

imations shown in Figure consist of expanded piecewise p olynomial ts that

are reduced in degree in order to manage the problem of explo ding co ecients

Figure shows the convolution error asso ciated with the linear quadratic and cu

bic piecewise p olynomial approximations for the expanded representation Note

that the cubic piecewise p olynomial approximation starts to oscillate over the region

This is a direct result of the relatively high order of the expanded convo

lution in this case the order is m In general from equation the order of the

expanded convolution is m This oscillation problem makes applying PART

above m problematic for EON solutions Additionally as Figure and Table

show the linear piecewise p olynomial expanded approximation has less error over

regions of the convolution then the quadratic and cubic expanded approximations

eg the p eak of the distribution This result is unique to this example and is not

a general trend

In order to quantitatively determine the adequacy of the reduced represen

tation in Figure Figure shows graphs of the p ointwise error for the three

approximation techniques All error graphs app ear similar in shap e having an os

cillatory nature Table shows the error b ounds for these graphs As with the

sto chastic multiplication op eration a key concern with the convolution op eration is

the p otential for error buildup ie approximating two density functions and then

Figure Graph of the Convolution of TwoTruncated Normal DensityFunctions

Figure Graph of the Convolution Error of TwoTruncated Normal Density

Functions Expanded Case

Polynomial Approximation Error Bound

Linear

Quadratic

Cubic

Table Error Bounds for Piecewise Polynomial Reduced Approximations of the

Truncated Normal Convolution Example

using those approximations to approximate the convolution of the density functions

Note from Tables and that the order of the reduced convolution error is similar

in magnitude to the error in approximating the truncated normal density function

in Figure Although an additional order of precision is not gained with each

increase in the order of the p olynomial approximation unlike the truncated normal

density function approximation the least squares convolution approximations have

the advantageous eect of canceling p ositive and negative errors in the individual

density function approximations

As a nal observation on this example Figure shows the p ointwise error

of Lawrences single degree approximation and the authors Note that the authors

approximation has less p ointwise error on average than Lawrences This is due to

a formation error in Lawrences solution The correct representation on page

for c bf or bf should b e

i i

j j j j j j

i i i i i i

f g c f f f g g g f c f c t g g c c

Of interest to the reader and a p otential area of further investigation for sp eed

ing up the convolution algorithm is the wayinwhich algorithm identies the valid

sets of density function subintervals J and J contributing to the k th subinterval

k k

I of the convolution Martin cho ose to index his convolution algorithm by

convolution subinterval I while Lawrence indexed by density function subin

terval I Sp ecically Martin cho ose I and then searched for the rst subintervals

k k

Figure Graph of the Convolution Error of TwoTruncated Normal Density

Functions Reduced Case

Figure Graph of the Convolution Error of TwoTruncated Normal Density

Functions using Lawrences algorithm

of I and I to form I Lawrence rst found the subinterval J of the density

k k k k

function f that contributes to the I subinterval and then the corresp onding

E k

subintervals J of density function f that contribute to I The author has

k E k

g T

chosen Lawrences indexing framework based on his implementation with a reduc

tion routine with the understanding that it has not b een validated to b e the most

ecient

As another example of the convolution sto chastic op eration consider the con

volution of two shifted exp onential density functions Sp ecicallylet

exp t t f

f exp t t

g T

The nal form of this convolution is the density function

f f

E E

g T

expt expt exp

expt exp exp

expt exp expt

The ab oveformulation is based on truncating each of the exp onential density

is truncated on and f is functions at of their total area ie f

E E

g T

truncated on Figure shows the graph of this convolution with the lin

ear quadratic and cubic monic piecewise p olynomial techniques resp ectively using

SVD matrix inversion Insp ecting the graphs visually the monic piecewise p olyno

mial techniques app ear to t the convolution adequately for the quadratic and cubic

cases however the linear approximation has diculty approximating the shap e of

the convolution in the interval This is a direct result of each reduced ap

proximation using only ten subintervals The rst reduced subinterval is

which is required to span ve expanded approximation subintervals

Figure Graph of the Convolution of TwoTruncated Exp onential Density

Functions

Figure Graph of the Convolution Error of TwoTruncated Exp onential Density

Functions Expanded Case

Figure shows the convolution error asso ciated with the linear quadratic and

cubic piecewise p olynomial approximations for the expanded representation Note

that the cubic piecewise p olynomial approximations signicantly increase in error

over the region As shown with the truncated normal convolution example

this is a direct result of the relatively high order of the expanded convolution

k k

Co ecients c and c for the cubic expanded approximations are on the order of

and resp ectively Additionally as Figure and Table

show the cubic piecewise p olynomial expanded approximation has an equivalent

error b ound over the convolution region as the linear expanded approximation

In order to quantitatively determine the adequacy of the reduced represen

tation in Figure Figure shows graphs of the p ointwise error for the three

approximation techniques All error graphs app ear similar in shap e having a damp ed

Polynomial Approximation Error Bound

Linear

Quadratic

Cubic Monic SVD

Table Error Bounds for Piecewise Polynomial Expanded Approximations of

the Truncated Exp onential Convolution Example

Polynomial Approximation Error Bound

Linear

Quadratic

Cubic

Table Error Bounds for Piecewise Polynomial Reduced Approximations of the

Truncated Exp onential Convolution Example

oscillatory nature Table shows the error b ounds for these graphs From Tables

and note that the order of the reduced convolution error is similar in magni

tude to the error in approximating the exp onential density function in Figure

Although an additional order of precision is not gained with each increase in the order

of the p olynomial approximation unlike the exp onential normal density function ap

proximation the least squares convolution approximations have the advantageous

eect of canceling p ositive and negative errors in the individual density function

approximations Additionally note that the reduced cubic representation has

corrected the error asso ciated with the expanded cubic representation again this

is a result of canceling p ositive and negative errors in the expandedrepresentation

Simple AirtoAir Engagement Example

In section a simple one versus one airtoair engagementwas detailed

In this engagement each aircraft could re one radarguided and one infrared missile

at an opp onent with the ring constraint that the radarguided missile must always

b e red b efore the infrared missile Recall that random variable E represented

Figure Graph of the Convolution Error of TwoTruncated Exp onential Density

Functions Reduced Case

the completion time of event G where side g res a missile of typ e e e

where radarguided and infrared at sideg that either misses or sho ots

down sideg whereg represents the opp onent side of side g The EON for

this one versus one airtoair engagementisshown in Chapter IV Figure

For this EON sp ecic event completion functions are

exp t with ie side res a radar missile E

E exp t with ie side res an infrared missile and

E exp t with ie side res an infrared missile

Additionally the probabilities that the radar and infrared missiles on side and

sho ot down the other side when red are p p and p p

resp ectively

Implementing a generalized EON solution technique using the core concepts of

the Polynomial Approximation and Reduction Technique the probability of b eing

in any state in the EON in Figure was calculated These calculations consisted

tting linear quadratic and cubic monic piecewise p olynomials using SVD

matrix inversion to the ab ove density functions

determining the integral formation for each state probability and

solving the integral expression using the sto chastic op erations outlined in ear

lier sections of this chapter

Each of the exp onential density functions was truncated at of its total area

with each piecewise p olynomial approximation determined using ten subintervals

and I cl assw ith tting p oints In order to judge the accuracy of the EON

solution technique two comparisons where made

Approximation Error Bound

Linear

Quadratic

Cubic

Simulation

Table Error Bounds for Piecewise Polynomial Approximations and Simulation

of P t

one comparison with the exact analytic expressions for the probabilities as

summarized in App endix C and

one comparison with a simulation of the state probabilities

For this example the simulation was run for iterations with a time increment

histogram width of minutes

Consider the probability of b eing in transient state at time t P t The left

side of Figure shows the graph of P t the simulation of P t and the monic

piecewise p olynomial approximations linear quadratic and cubic resp ectively In

sp ecting the approximations and simulation visually all three piecewise p olynomial

ts and the simulation app ear to t P t adequately

In order to quantitatively determine the adequacy of the t the right side of

Figure shows graphs of the p ointwise error Table shows the error b ounds

for these graphs The error graphs exhibit damp ed oscillatory b ehavior that moves

towards zero In general the piecewise polynomial approximations provide greater

accuracy than the simulation results

For event tier one Figures and show the graphs of P t absorbing

state and P t transient state the simulation of P tandP t and the monic

piecewise p olynomial approximations as well as the p ointwise error for approximation

techniques As was the case for P t all three piecewise p olynomial ts and the

simulation app ear to t the actual solution to P t and P t adequately

Figure Graph of P t and Error for the Simple AirtoAir EngagementEvent

Occurrence Network

Figure Graph of P t and Error for the Simple AirtoAir EngagementEvent

Occurrence Network

Figure Graph of P t and Error for the Simple AirtoAir EngagementEvent

Occurrence Network

Approximation Error Bound

Linear

Quadratic

Cubic

Simulation

Table Error Bounds for Piecewise Polynomial Approximations and Simulation

of P t

Approximation Error Bound

Linear

Quadratic

Cubic

Simulation

Table Error Bounds for Piecewise Polynomial Approximations and Simulation

of P t

The right side of Figures and shows graphs of the p ointwise error

Tables and show the error b ounds for these graphs In general for P t

the simulation results provide greater accuracy then the linear piecewise p olynomial

approximation while the quadratic and cubic piecewise p olynomial approximations

provide greater accuracy than the simulation results For P t the piecewise p oly

nomial approximations provide greater accuracy than the simulation results

Figures and show the p ointwise error for various absorbing

and transient states in event tiers two three and four In general the following

observations hold

For absorbing states the simulation results provide greater accuracy then the

linear piecewise p olynomial approximation while the quadratic and cubic piece

wise p olynomial approximations provide greater accuracy than the simulation

results

For transient states the piecewise p olynomial approximations provide greater

accuracy than the simulation results

The cubic piecewise p olynomial approximation provides the b est accuracy of

all p olynomial approximations and

The greatest p ercentage increase in accuracy o ccurs when moving from the lin

ear piecewise p olynomial approximation to the quadratic piecewise p olynomial

approximation

Exceptions include

The approximations for transient state P t where the simulation results pro

vide greater accuracy then the linear piecewise p olynomial approximation

The approximation for absorbing state P twherethesimulation results pro

vide greater accuracy then b oth the linear and quadratic piecewise p olynomial

approximations and

The approximation for absorbing state P t where the piecewise p olynomial

approximations provide greater accuracy than the simulation results

As t minutes the probability of b eing at a particular no de in the airto

air engagement EON reaches a steadystate value The following observations can

b e made ab out the steadystate probability of b eing at a particular no de for this

airtoair engagement example using the cubic piecewise p olynomial approximation

First strike radar missile kills o ccur with a probabilityof P

P versus for the exact solution

Slightly over of the airtoair engagements have ended within twoevents

missile rings P P P P P P

versus for the exact solution

Only slightly more than of the airtoair engagements end without a sho ot

down P t versus for the exact solution

Figure Graph of P tand P t Error for the Simple AirtoAir Engagement

Event Occurrence Network

Figure Graph of P t and P t Error for the Simple AirtoAir Engagement

Event Occurrence Network

Figure Graph of P t P t and P t Error for the Simple AirtoAir En

gagementEvent Occurrence Network

Figure Graph of P tand P t Error for the Simple AirtoAir Engagement

Event Occurrence Network

In answer to the example question p osed ab ove the probability that the aircraft

on side sho ots down the aircraft on side with a radar missile is

P P P versus for the exact solution an agreement

to ve decimal places

Large EON AirtoAir Engagement

The large airtoair example consists of a blue oensive counter air OCA

mission For a description of the terminology used in this section refer to App endix A

As shown in gure four blue aircraft are sweeping red airspace prior to a four

ship blue strike mission Vectored by a blue airb orne early warning aircraft the four

blue sweep aircraft are attempting to engage two red defensive combat air patrols

CAPs each CAP consists of a twoship ight of red aircraft Unbeknownst to the

blue aircraft two red surfacetoair missile SAM batteries wait to engage aircraft

b ehind the red CAPs The blue aircraft are loaded out with six radar missiles and

two infrared missiles while the red aircraft are loaded out with four radar missiles

and two infrared missiles Each red SAM battery has eightSAMs

The EON for this example consists of event groups as follows

Event Groups each blue aircraft with each of these event groups having

six events

Event Groups each red aircraft with each of these event groups having six

events and

Event Groups each SAM battery with eachoftheseevent groups having

four events

for a total of events in this airtoair example Each blue aircraft is employing

sho otsho ot tactics on its rst shot opp ortunity ring two radar missiles The next

blue aircraft shot is a single radar missile followed by a single infrared missile if

necessary Additionally if needed each blue aircraft will rep eat the ab ove sequence

Figure Graphical Depiction of Blue OensiveCounter Air OCA mission

The ab ove represents the case where the blue ight engages the red ight through the

merge ring four missiles each blows through the merge turns to reengage and

engages the red aircraft again ring an additional four missiles each In contrast

the red aircraft initially employ a sho otlo oksho o t strategy employing two radar

missiles then a followon infrared missile shot This sequence is rep eated after the

blue aircraft reengage Finallyeach red SAM battery employs sho otsho ot tactics

ring two missiles p er SAM event

The event distribution and probability of success for these events are

Event Group

exp p that is the rst blue aircraft res two Event E

radar missiles with an exp onential distribution of o ccurrence

once red the missiles have a probability of success of

Event E exp p

Event E truncated N over p

Event E exp p

Event E exp p and

Event E truncated N over p

Event Group

Event E exp p

Event E truncated N over p

Event E exp p

Event E exp p and

Event E truncated N over p

Event Group

Event E exp p

exp p Event E

truncated N over p Event E

Event E exp p

exp p and Event E

truncated N over p Event E

Event Group

Event E exp p

exp p Event E

Event E truncated N over p

Event E exp p

Event E exp p and

Event E truncated N over p

Event Group

Event E truncated N over p

Event E truncated N over p and

Event E truncated N over p

Event Group

Event E truncated N over p

truncated N over p Event E

truncated N over p and Event E

Event E truncated N over p

Event Group

Event E truncated N over p

Event E truncated N over p and

Event E truncated N over p

Event Group

Event E truncated N over p

Event E truncated N over p and

Event E truncated N over p

Event Group

Event E exp p

Event E exp p and

Event Tier Numb er of States in the Tier Probability cumulative

Table Numb er of States and Cumulative Probability for EachEventTierinthe

Large AirtoAir example

Event E exp p

Event Group

Event E exp p

Event E exp p and

Event E exp p

The ab oveevent o ccurrence network EON has event tiers This EON was

rst run with the bucket probability co de engaged truncating transient no des if less

than of probabilitywas estimated to pass through the no des and using the

cubic monic piecewise p olynomial approximation With the bucket co de engaged

the EON fully truncated at event tier with total states Table shows

the total numb er of states for eachtieraswell as the steady state probabilityof

b eing in all absorbing states in that tier and all preceding tiers ie the cumulative

probability

For event tier the total steady state probability of b eing in an absorbing

state in this event tier is refer to Table This probability decomp oses

into the following steady state probabilities

Event Tier Numb er of States in the Tier

Table Numb er of States and Cumulative Probability for EachEventTierinthe

Large AirtoAir example

the steady state probability of the blue aircraft ight s rst missile shots ending

the engagementis

the steady state probability of the red aircraft ight s rst missile shots ending

the engagementis and

the steady state probability of the red SAM batteries rst missile shots ending

the engagementis

Before presenting the probability estimates over time the EON was solved

using simulation for comparison Since the network has a mixtureofexponential and

truncated normal distributions no closed form analytic solution can be derived as

was the case for the one versus one example given in the last section The initial

simulation approachwas to run the network generation program to determine the

numb er of states in the EON However the computer employed ran out of memory

during event tier or after more states where created The computer

wasaPChaving a Pentium I I I pro cessor chip running at MHz MB of RAM

and MB of virtual memoryTable shows the numb er of states for eachevent

tier through event tier

In order to pro duce simulation results a truncated version of the airtoair

EON was run Comparing tables and transient no de truncation do es not

b egin with the piecewise p olynomial approximation until event tier By taking the

ab ove airtoair engagement EON and using only the rst two events in eachofthe

ten event groups the simulation co de could b e executed This new reduced EON

will allow exact comparisons for states generated in event tiers and of the original

airtoair engagementEON

For this example the simulation was run for twomillion iterations with a time

increment histogram width of minutes Figure shows the approximation

and simulation graphs of the probability of b eing in a representative absorbing and

transient state from event tier at time t The absorbing state shown is where the

rst blue aircraft kills a red aircraft b efore another blue aircraft red aircraft and

SAM battery res a shot The transient state shown is where the rst red SAM

battery res a two missile salvo at the blue aircraft ight and misses b efore either

a blue or red aircraft res a missile The piecewise p olynomial approximation and

simulation results for the absorbing and transient state are close with a maximum

absolute error of

Figure shows the approximation and simulation graphs of the probabilityof

b eing in a representative absorbing and transient state from event tier at time t

The absorbing state shown is where the third red aircraft res a missile unsuccessfully

at the blue aircraft then the second red SAM battery kills a blue aircraft The

transient state shown is where the second blue aircraft res a two missile salvo

unsuccessfully at a red aircraft followed by the fourth blue aircraft ring a two

missile salvo unsuccessfullyFor this tier the piecewise p olynomial approximation

and simulation results are showing more dierence graphically In fact the absorbing

state did not showupinany of the simulation runs This is a general trend as the

events have less probability of o ccurring in lower tiers the simulation technique has

a dicult time picking up these rare states with the neededaccuracy

Figure Probability of Being in a Representative Absorbing and Transient State

at a time t for Event Tier for the Large airtoair engagement

Figure Probability of Being in a Representative Absorbing and Transient State

at a time t for Event Tier for the Large airtoair engagement

Figure Probability of Being in a Representative Absorbing State at a time t for

Event Tier for the Large airtoair engagement

As a nal example Figure shows the approximation graph of the probability

of all blue aircraft ring three missile each and then the rst SAM battery killing a

blue aircraft bytimet

As can b e seen by the ab ove example the currentevent o ccurrence network

formulation has a critical limitation the network concludes after only one successful

eventFor illustration if a blue aircraft sho ots down a red aircraft the engagement

is terminated Clearly this is not representative of real world air combat as the

surviving red aircraft may not get the chance to disengage nor may all blue aircraft

survive This inherentnetwork limitation can b e oset to some degree by setting

several of the success probabilities p to zero thus allowing the engagementto

continue for additional entity kills In fact when viewing such an EON formulation

Ancker and Gafarian s general renewal GR mo del is shown to b e an EON subset

For example consider twoversus two ground combat with the following rules

TwosidesA and B conduct a continuous time engagement with initial force

size on the A side and on the B side

At the b eginning of the battle each combatantonside A picks a target on

side B at random and res Eachcombatant res until he is killed or makes

a kill at which time he immediately shifts to a new target picked at random

and resumes ring

Each combatant on side B picks a target on side A at random and res Each

combatant res until he is killed or makes a kill at which time he immediately

shifts to a new target picked at random and resumes ring

All the combatants on one side are visible and within the weap on range of all

the combatants on the other side

The ammunition supply is unlimited

Every combatant res indep endently

Combat ends when one side is annihilated

Combatants have dierentinterkilling time distributions and

If a combatant s target is killed by himself he starts afresh the killing pro cess

on his next target whereas if his target is killed by another combatant from

his side his remaining time to kill is carried over to his next target

The Ancher and Gafarians GR mo del for this example is shown in Figure Here

combat starts with twocombatants on each side state and concludes when one

of four absorbing states is reached state state state or state

The identical network can b e generated via an EON formulation by setting

p p and p p and creating the following event o ccurrences

EventGroupE K tand E K t

A A

EventGroupE K tand E K t

B B

where K t is the distribution of time to the j th kill event on side i Figure

shows this EON using the state notation from Figure

In order to handle the p ossibilityofmultiple successful events the current EON

formulation can b e easily expanded to include multiple absorbing and transient states

or reactive decision making

To illustrate the concept of multiple absorbing and transient states consider

Figure The top p ortion of Figure shows the typical EON formulation de

velop ed in this dissertation Here event G has just o ccurred with probabilityof

success p State and probabilityoffailure p State The b ottom p ortion

g g

e e

of Figure shows an expanded EON formulation with two absorbing and transient

b a

and p states Here event G has just o ccurred with probabilities of success p

g g

e e

c d

States a and b and probabilities of failure q and q States c and States d

g g

e e

In order to maintain conservation of probability it is required that

d c b a

q q p p

g g g g

e e e e

To show the exibility of this expanded EON consider event G to b e the o ccurrence

of the rst event in the large airtoair engagementabove where the rst blue aircraft

res a missile salvo States a could represent the o ccurrence where blue destroys a

red aircraft and red ends the engagement while State b shows the o ccurrence where

blue destroys a red aircraft and blue ends the engagement Concurrently States c

could represent the o ccurrence where blue destroys a red aircraft and the engagement

continues while state d shows the o ccurrence where blue res but do es not destroy

a red aircraft

To illustrate reactive decision making consider the following EON formulation

Supp ose twoevent groups exist eachwithtwoevents However condition the set

of events in group two dep endent on the outcome of the rst eventinthenetwork

Figure Ancker and Gafarian s sto chastic kill network for twoversus two ground

combat

Figure Event Occurrence Network formulation of Ancker and Gafarian s

sto chastic kill network for twoversus two ground combat

Figure Event Occurrence Network formulation with Multiple Absorbing and

Transient States

If event G o ccurs rst then the set of events in group twoisfG G g else the set

of events in group twoisfG G gToshow the exibility of this expanded EON

consider event G to b e a missile ring from side in an air engagementbetween

two aircraft The set of events on side are now dep endent on whether side res

rst If side do es re rst then side could have a set of defensiveevents like

defeat side s missile shot G and disengage G If side do es not re rst

then side could have a set of oensiveevents like re a radar missile at side G

and re an infrared missile at side G

Derivation of the probabilityintegral solution for b eing in any state at time t

for these expanded event o ccurrence network formulations is left for further research

Summary

In this chapter the techniquesalgorithms for solving event o ccurrence net

works using piecewise p olynomial approximation were given The chapter showed

two airtoair engagement EON examples One of the networks is the simple airto

air engagement example created by Hong et al shown in Figure and the

other is a more complex engagementinvolving over event distributions shown in

Figure For the simple airtoair engagementinwhich an exact analytic solution

could b e derived the quadratic and cubic approximations generally showed greater

accuracy than the simulation solution For the complex airtoair engagement for

which an exact analytic solution can not b e calculated the piecewise p olynomial

and simulation solution showed similar results for probable events However as the

events had less probability of o ccurring in lower tiers the simulation technique had

a dicult time picking up these rare states with the needed accuracy

VII Summary and Recommendations

Research Goals and Summary

This goal of this researchwas to improve on the airtoair combat algorithm

used in USAFs campaign mo del THUNDER Improvement centered on creating an

airtoair combat algorithm that was eventdriven versus scripted Concurrent with

this research eort Emergent Information Technologies the develop er of THUN

DER b egan developmentonanimproved airtoair adjudication algorithm

for incorp oration into the USAFs next generation campaign mo del STORM

The work in this dissertation had a direct inuence on the creation of the new event

based airtoair adjudicator

In order to meet the ab ove research goal two ma jor research ob jectives had to

b e accomplished The rst ob jective required developing a network formulation ca

pable of expressing sto chastic event o ccurrences and their interactions Fullling this

ob jective led to the authors concept of event o ccurrence networks EONs EONs

are graphical representations of the sup erp osition of several terminating counting

pro cesses An EON arc represents the o ccurrence of an event from a group of se

quential events b efore the o ccurrence of events from other event groupings Events

between groups o ccur indep endentlybutevents within a group o ccur sequentially

A set of arcs leaving a no de is a set of comp eting events which are probabilisti

cally resolved by order relations EONs dier from probabilistic networks previous

discussed in the literature such as Activity Networks Perti Nets Task Precedence

Graphs and Belief Networks and are strongly inuenced by the research of Ancker

Gafarian Kress and several asso ciates to mo del m

versus n sto chastic ground combat

The second research ob jective required developing an Event Occurrence Net

work EON solution technique An imp ortant metric for an EON is the probability

of b eing at a particular no de or set of no des at time tSuch a probability is formu

lated as an integral expression generally a multiple integral expression involving

event probability density functions This integral expression involves several sto chas

tic op erators

Subtraction

Multiplication

Convolution and

Integration

In the literature simulationbased metho ds are the dominant approximation tech

nique for obtaining metrics from probabilistic networks due to their logical simplicity

and ecient implementation When a simulation is used to solve a probabilistic net

work the main complexity measure is the number of simulation runs and corresp ond

ing random variable draws necessary to establish a certain statistical condence in

the output

However for some probabilistic networks large networks or transient analysis

of smaller networks simulation can b ecome a computationally costly undertaking

to obtain accurate estimates of the output measures As the numb er of random

variables in the network increases so do es the numb er of replications necessary in

order to prop erly characterize the output measures As an alternative to simula

tion numerical approximation has b een prop osed for some probabilistic networks

Numerical approximation involves the use of functions such as piecewise p olyno

mial functions exp olynomial functions exp onential p olynomials functions etc to

approximate the density or distribution functions of the random variables Two

p otentially fatal problems in using numerical approximation to determine proba

bilistic network output metrics are explo ding co ecients and proliferating classes

Anumerical approximation reduction technique develop ed byFergueson and Short

tell and Lawrence using single order piecewise p olynomial approxi

mation solves these problems by taking the pro duct of a sto chastic op eration and

reducing its degree and class structure to the degree and class structure of the original

functions in the sto chastic op eration The research has expanded and op erationalized

the technique to the general order m p olynomial case as well as formulated reduc

tion techniques for subtraction multiple multiplication applications and integration

sto chastic op erators

This research eort has yielded ve ma jor contributions to the literature These

contributions are lab eled with either an applied or theoretical designator to indicate

the nature of the contribution Contributions in this dissertation include

Intro duction of event o ccurrence networks EONs as a probabilistic network

framework in whichtoinvestigate the interaction of indep endent groups of

events refer to Chapter IV Integral expressions were develop ed to deter

mine the probability of b eing in a particular no de in the network From the

prosp ectiveofstochastic pro cesses this solution is equivalent to determining

the probability of b eing in a given state at time t of a probabilistic mo del

comp osed of the sup erp osition of several terminating counting pro cesses The

oretical

Adaptation and expansion FergesonShortellLawrences least squares piece

wise p olynomial approximation and reduction technique PART to solve for

the probability of b eing in any no de of an EON at time tFergesonShortell

Lawrence develop ed PART for serial and parallel network op erations in order

to approximate the network completion distribution of a probabilistic activ

ity network Here the series op eration involved the sto chastic op eration of

convolution of two density functions and the parallel op eration involved the

sto chastic op eration of multiplication maximum of two density functions

Adapting PART concept to EONs the set of sto chastic op erations was ex

panded to include

a Subtraction op eration

b Fourteen multiplication op erations

c Convolution op eration and

d Three integration op erations

All the ab ove op erations were develop ed for general piecewise p olynomials

of degree m These op erations are nowavailable for use by programmers

Theoretical

Indepth literature review Chapter I I of probabilistic networks their com

plexity and output measures and current solution metho ds Sp ecicallythe

review lo oked at four dierentnetwork application areas activitynetworks

sto chastic Petri nets task precedence graphs and b elief networks The appli

cation areas were compared and contrasted in an eort to determine similarities

among the network typ es Applied

Literature review App endix A of the op erational airtoair engagement en

vironment Sp ecically the review provided an overview of airtoair engage

ments and the combat environmentinwhich these engagements take place

Factors discussed include strategy missions airtoair spatial layout aircraft

detection and identications weap ons command control communications

and information CI electronic warfare and tactics This review can b e used

by researchers and mo delers to mo del airtoair engagements at the theater

level Applied

Literature review of analytical and simulated theaterlevel air combat mo dels

including an indepth review of the US Air Forces THUNDER airtoair en

gagement submo del App endix B Several researchers have b eneted from an

advanced copy of this review Applied

Recommendations

Event o ccurrence networks EONs are not limited to mo deling combat situa

tions These networks can b e applied whenever sets of indep endentrandomevents

o ccur For example one promising application area is large scale simulations Here

EONs and bucket analysis could b e used to eliminate rare events and rare event

sequences By eliminating rare events and event sequences the amount of computer

simulation time needed for analysis might b e signicantly reduced as well as p erhaps

more imp ortantly an understanding of what caused the eventorevent sequence to

b e rare to b egin with

Based on this research the following is recommended for further study

Expand the set of input event distributions Currently only the exp onential

truncated rightexponential truncated left normal and truncated twoside

normal distributions have b een included This task is straightforwardasnew

distributions can b e added with ease to the EONSOLVE co de through subrou

DENSITY Fit tine POLY

Develop the probability integral solution for b eing in any state at time t for

two expanded event o ccurrence network formulations multiple absorbing and

transient states and reactive decision making These formulations would led

to the mo deling of a greater set of combat situations

Develop a piecewise p olynomial algorithm to solve for the completion time

of general activity networks using the work of Beinet al Agrawal et

al Lawrence and this dissertation The solution would have a direct

commercial application

Appendix A AirtoAir Engagement Environment

The following summary is intended to illuminate what the exp erts sayare

the imp ortant comp onents and variables in airtoair engagements and to distill

this information where p ossible to aid in the mo deling of these engagements The

material b elow has b een taken from unclassied airtoair engagement literature

This summary is not intended to b e an exhaustive literature survey but to provide

an overview of the complex environment of airtoair engagements and the factors

that determine success in this environment

An airtoair engagement b egins when opp onent aircraft are detected and ends

when the opp onent aircraft are either killed or neutralized or the attack is termi

nated The time from b eginning of the engagement to end may b e a matter of sec

onds page v The outcome of an airtoair engagement dep ends on manyinter

woven factors None of these factors should b e lo oked at in isolation the interaction

eects are much more imp ortant in determining the results of an engagement The

following factors of the airtoair engagementenvironment are discussed strategy

missions airtoair spatial layout aircraft detection and identication weap ons

command control communications and information CI electronic warfare and

tactics and maneuvers

A Strategy

As with land and sea engagements when twocombatants meet in the air

environment they have several engagementchoices For instance b oth combatants

could continuously engage each other until one combatant is destroyed ie a war

of attrition Another p ossibility is for one combatanttomove in strikea blow and

retire b efore the other combatant can resp ond Yet another approachinvolves doing

something unexp ected like using a decoy combatant pulling a third combatant

into the engagement or striking indirectly at the combatantby destroying his base

of op eration or an air space controller page The engagementchoices

exercised by a combatant dep end on the air strategy employed bythatcombatant

An imp ortant comp onent of an eective air strategy is an analysis of the nature

of the opp onent intheairenvironment Manyways exist to categorize an opp onent

For example he may b e rational irrational fanatic rigid exible indep endent

innovative determined or do ctrinaire To the extent an opp onent can b e assigned

to a category his airtoair engagement plans can b e anticipated and his reactions

in the air environment can b e predicted Related to analyzing the nature of the

opp onent intheairenvironment is knowing the nature of your own side in the air

environment As you are analyzing the strengths and weaknesses of your opp onent

in the air environment so is your opp onent if he is comp etent analyzing yours

Your opp onent will try to exploit your weaknesses and avoid your strengths By

knowing the nature of your side in the air environment you can predict opp onent

air strategy page

Air strategy evolves into rules of engagementROE and general instructions

for airtoair engagements It has a signicant eect on how airtoair engagements

will b e fought For example consider these three ROE

BVR ROE Aircraft have p ermission to re at any unknown

ID ROE In order for aircraft to re at a target the target must b e rst

conrmed as hostile byany electronic or visual means or

Visual ROE In order for aircraft to re at a target the target must b e

conrmed as a hostile by visual means or through a hostile act

Clearly aircraft with BVR ROE haveadvantage over aircraft with Visual ROE

in terms of rst sho ot capability BVR ROE aircraft do not have to identify an

aircraft visually typically within nautical miles nm range whereas the Visual

ROE aircraft do Hence the BVR ROE aircraft can launchweap ons if the aircraft

is equipp ed with suchweap ons at a longer range than the Visual ROE aircraft

As section A shows rst sho ot advantage can b e a decisive factor in airtoair

engagements page

A Missions

Airtoair engagements are a bypro duct of the interactions b etween opp onents

executing separate air missions There are manyways to classify an air mission I

will use the mission classication scheme used in United States Air Force THUNDER

simulation THUNDER classies twentytwo dierent air missions Volume

I pages Alphab etically by acronym the air missions are

Airb orne Refueling AAR

Airb orne Early Warning AEW

AirtoAir Escort AIRESC

Battleeld Air Interdiction BAI

Barrier Combat Air Patrol BARCAP

Close Air Supp ort CAS

Closein NonLethal Air Defense Jamming CJAM

Closein Lethal Air Defense Suppression CSUP

DefensiveCounterAir DCA

Lethal Direct Air Defense Suppression CSUP

Escort NonLethal Air Defense Jamming EJAM

Escort Lethal Air Defense Suppression ESUP

Fighter Sweep FSWP

High Value Asset Attack HVAA

Long Range Air Interdiction INT

Oensive Counter Air OCA

Over FLOT Defense Counter Air ODCA

Reconnaissance RECCE

Stando NonLethal Air Defense Jamming SJAM

Stando Reconnaissance SREC

Stando Lethal Air Defense Suppression SSUP

Strategic Target Interdiction STI

An airtoair engagement o ccurs when a friendly or own aircraft group inter

cepts an opp onent group of aircraft or vice versa For illustration purp oses assume

that a friendly aircraft group intercepts an opp onent aircraft group An intercept

can o ccur one of twoways

previous airb orne friendly aircraft vector to the opp onent aircraft or

friendly aircraft launch and vector to the opp onent aircraft

In the THUNDER mission classication scheme the previous airb orne friendly

aircraft are executing a BARCAP mission and the launched friendly aircraft are ex

ecuting either a DCA ODCA or HVAA mission ABARCAP mission involves

friendly aircraft patrolling a designated area in friendly territory p ositioned so as

to facilitate interception of opp onent aircraft b efore the opp onent aircraft strikea

friendly target A BARCAP formation generally consists of a rotating racetrack or

bit ie a circular route with one aircraft or elementpointing toward the exp ected

threat axis at all times The BARCAP mission is b est suited to situations when

the direction of the approach or route of the opp onent aircraft is known with some

degree of certainty The BARCAP mission provides the ight with certain tactical

advantages These include

aircraft are deployed in the air

sucient time to use radar to search for detect and monitor all targets along

the threat axis and

at least one aircraft is in p osition to monitor the threat axis

An imp ortant consideration for the BARCAP mission is orbiting altitude Orbiting

altitude is usually chosen according to the exp ected altitude of the opp onent but

weap ons and fuel considerations also play imp ortantrolesVolume I I page

pages page

DCA ODCA and HVAA missions wait on the ground until opp onent air

craft are detected and then are scrambled to intercept The DCA mission involves

friendly aircrafts sitting on runway alert and waiting to intercept opp onent aircraft

detected in friend ly territory while an ODCA mission involves a friendly aircraft sit

ting on runway alert waiting to intercept an opp onent aircraft detected in opponent

territory Finallya HVAA mission involves a friendly aircraft sitting on runway

alert waiting to intercept a AEW AAR or SREC mission opp onent aircraft DCA

ODCA and HVAA mission aircraft havetheadvantage of countering attacks from

any direction with equal ease Volume I I page pages

The primary goal of a BARCAP DCA ODCA or HVAA mission is to protect a

friendly target Although destruction of opp onent strikeorhighvalue asset HVA

aircraft is ideal destruction is not the only way to accomplish the mission goal

Merely threatening opp onent strikeorHVA aircraft is often sucient to force aircraft

ocourse or to defend themselves eg drop b ombs in the case of the strike aircraft

leading to a mission ab ort In either case the friendly aircraft haveachieved a

mission kill page

Additionally the opp onent aircraft group maycontain an air escort element

THUNDER classies escort missions as either FSWP or AIRESC A FSWP mission

involves friendly ights ying over the opp onents territory for the purp ose of en

gaging and destroying opp onent aircraft The FSWP mission denies the opp onent

In the case of the HVAA mission protecting the target means preventing a high value asset

HVA aircraft from p erforming its mission For example preventing a reconnaissance aircraft from

ying over a given area

use of the airspace and makes the airspace safer for friendly strike aircraft FSWP

missions are oensive in nature and own as a precursor to strike missions such

as CAS BAI OCA INT and STI FSWP mission ob jectives are achieved when

opp onent air combat aircraft in the strike aircraft packages ingress andor egress

route are shot down or driven o Often the opp onents air defense network will

b e sup erior to friendly capabilities in the area so the unexp ected attack should b e

guarded against Command control communication and information CI are crit

ical elements in the success of the FSWP The ability of the friendly ight to nd

identify and engage opp onent aircraft while maintaining an acceptable chance of

survival rest in great measure on relative CI capabilities CI capabilities will b e

discussed in greater detail in section A Volume I I page pages

page

Although FSWP generally provides the most eective means of clearing a safe

path through opp onent territory for a friendly strikepackage sometimes the neces

sary of protecting other typ es of aircraft such as transp orts in a hostile environment

requires an air escort AIRESC mission In general there are four typ es of escort

missions

Reception Escort meet the escorted aircraft as they return from hostile terri

tory and guard its retreat from pursuing opp onent aircraft

Remote Escort p osition ahead of the escorted aircraft taking the form of a

sweep

Detached Escort p osition strategically around the escorted aircraft normally

within visual range to engage opp onent aircraft trying to attack the escort

aircraft and

Close Escort p osition around the escorted aircraft for terminal defense

Todays USAF do ctrine stresses indep endentFSWPmissionsover AIRESC missions

Volume I I page pages

The goal of any air mission whether its an airtoair mission or an airtoground

mission is to accomplish the mission while maintaining adequate survivabilityIn

many cases one must weigh the probability of mission success against the chances

of survival Except for very critical missions where success is absolutely essential

when the probabilityofsurvival b ecomes unacceptable aircraft will ab ort their mis

sion page

A AirtoAir Engagement Spatial Layout

There are an innite numb er of starting conditions for any airtoair engage

ment engagements have unique spatial layouts or setups Figure shows one

suchlayout In this gure two aircraft are in the pro cess of entering an airtoair

engagement This layout can b e easily extended to multiple air combat by picturing

each aircraft in Figure as a ight Each aircraft passes through ve combat zones

Outside Sensor Zone In this zone an aircraft can not detect the presence

of an opp onent with his own sensor system eg radar Detection requires

the intervention of a third party such as a dedicated airb orne or groundbased

radar platform eg AWACs

Beyond Visual Range BVR Intercept Zone In this zone an aircrafts sensor

system is able to detect the presence of an opp onent The pilot of the aircraft

is p ositioning the aircraft for weap ons employment

BVR Engagement Zone Aircraft is within the maximum launch range of its

BVR weap onry

BVRWVR Transition Engagement Zone Aircraft is well within range of its

BVR weap onry and is transitioning to a visual contact with the opp onent

Shortrange weap onry b ecomes available for launchtowards the end of this

zone

Or as is the case with the ma jority of engagements one of the aircraft is engaging the other

undetected

Within Visual Range WVR Engagement Zone Pilot of the aircraft is able

to visually identify the opp onent page

Eachofthecombat zones in Figure dep ends on the unique spatial and

environment factors of the engagement For example given the RCS of the aircraft

on side B and the sensor system capabilities of the aircraft on side A the pilot on

side A is able to theoretically detect the aircraft on side B at nautical miles nm

transition from the Outside Sensor Zone to BVR Intercept Zone This detection

distance can easily change based on the relative p osition of the two aircraft it is

harder to detect an aircraft straight on then on the its b eam side As another

example consider the transition to the WVR engagement zone the pilot on side B

is able to detect the aircraft on side A at nm while the pilot on side A is able to

detect the aircraft on side B at nm These visual detection distances can clearly

change based on environmental factors such as the p osition of sun and human factor

capabilities such as the eyesight of the pilots page

As a nal comment ab out the spatial layout of airtoair engagements Fig

ure includes two sets of dashed lines that denote the transition from an oensive

to neutral p osture and the transition from a neutral to defensive p osture for a side

For example consider the oensiveneutral p osture line for the aircraft on side B

This line coincides with the maximum launch range of the BVR weap on of the air

craft on side A The aircraft on side B is no longer in an oensive p osture if it is

in the weap ons envelop e of the aircraft on side A Oensive neutral and defensive

p ostures will b e discussed in greater detail in section A

A Aircraft

In airtoair engagements aircraft p erformance is imp ortant esp ecially when

approaching the within visual range WVR transition zone shown in Figure ie

less than nm In order to employa weap on sho ot at an opp onent it is necessary out y t Spatial La

Side A Range to Target (nm) Merge 80 70 60 50 40 30 20 15 10 5 WVR Figure AirtoAir Engagemen Neutral/Defensive Zone Line 5 Side BRangetoTarget( BVR/WVR Transition Engagement Zone Outside Side A’s 10 Sensor Zone BVR Engagement Zone 15

20 BVR Intercept 30 Zone Side B nm Offensive/Neutral

Line 40 ) Outside Side B’s 50 Sensor Zone Side A 60

to p oint the aircraft at the opp onent or in the opp onents near vicinity dep ending

on the weap on b eing employed The p erformance measures of most interest are

Turn Performance instantaneous and sustained and

Energy Performance climb and acceleration

Turn p erformance or maneuverability is the ability of an aircraft to change the direc

tion of its motion in ight while energy p erformance is the abilitytochange energy

state ie to climb andor to accelerate Other asp ects of aircraft p erformance such

as takeo landing range and endurance although critical are concerned more with

how the air combat aircraft gets to and from the airtoair engagement area rather

than how it p erforms within that area page page

Turn p erformance is classied as either instantaneous or sustained Instanta

neous turn p erformance refers to the aircrafts turn capabilities at anygiven moment

under existing ight conditions such as altitude and sp eed Sustained turn p erfor

mance refers to the aircrafts turn capabilities for an extended length of time under

existing ight conditions Aircraft turns are measured in three interrelated ways

Load Factor g indicates the centrifugal acceleration generated by the turn

and is usually expressed in units of G

Turn Radius R indicates howtight the aircraft is turning and is generally

expressed in feet or nautical miles nm and

Turn Rate TR indicates how fast the aircraft moves around the turn radius

and is usually expressed in degrees p er second

Well dened physical relationships exist b etween the three parameters ab ove For

fairly high load factors

V g

R and TR

g V

One G unit is equivalent of the nominal acceleration of gravityf tsec

If lo oking down on an aircraft from ab ove the turn radius is the distance from the center of

the turn circle to the aircraft

where V is the true airsp eed sp eed in relation to the ground of the aircraft As can

b e seen from equation turn radius R and turn rate TRare inversely propor

tional at a given load factor G and airspeed The greater the turn rate the smaller

the turn radius and vice versa Equation also shows that turn radius is mini

mized and turn rate is maximized at a high load factor and low airsp eed pages

page

Based on the discussion ab ove in order to obtain a maximum load p erformance

minimum turn radius R and maximum turn rate TR the b est course of action

would app ear to b e a high load factor g at minimum airsp eed V However there

is a limit to the load factor that can b e obtained at anygiven airsp eed Figure

shows the relationship b etween load factor and indicated airsp eed sp eed shown on

the aircrafts airsp eed indicator The left side of Figure lab eled lift limit in

dicates the maximum load factor that an aircraft can generate at a given airsp eed

The curvature of the b oundary reects the variation of lift capabilityL with the

square of airsp eed Lift capability is the aero dynamic force generated by the aircraft

p erp endicular to its direction of motion and represents the force available to turn the

aircraft Lift capability is mainly pro duced by an aircrafts wings The sp ecic lift

capability that can b e pro duced by a given wing is dep endent on the airsp eed and

altitude of the aircraft and is roughly prop ortional to the square of airsp eed The

upp er and lower b oundaries of Figure depict the structuralstrength limits of the

aircraft The intersection of the lift limit and structural limit dene a crucial aircraft

sp eed in ghter p erformance known as the corner velo cityV At the corner ve

lo city an aircraft attains its maximum instantaneous turn p erformance pages

page

To see that an aircraft attains maximum instantaneous turn p erformance at

corner velo cityV a closer lo ok at Figure is necessary In the case of turn rate

TR the rapid rise in load factor g above aircraft sp eed V leads to improved

instantaneous turn p erformance culminating at the corner velo city as evidenced by V ersus Indicated Airsp eed v g actor

Figure Aircraft Load F 8 Maximum Structural G Limit 6

4 Limit Speed

Lift Limit 2

Factor (g) 1 Aircraft Load 0

-2 Stall Speed @ 1G -4

Vs Vc VD Indicated Airspeed

equation Since the load factor is limited by the aircrafts structure ab ove corner

velo city further increases in airsp eed degrade decrease the instantaneous turn rate

of the aircraft In the case of turn radius R the rapid rise in the load factor

ab ove stall sp eed leads to improved instantaneous turn p erformance as turn radius

decreases as evidenced by equation until after corner velo city is obtained In most

mo dern ghters V is b etween to knots of indicated airsp eed The aircraft

with the greatest useable load factor at a given airspeed has superior instantaneous

turn performance ie faster turn rate and smal ler radius at that speed pages

Sustained turn p erformance is a little more complex In Figure the b ound

aries of the gure are dep endent on the aircrafts weight altitude conguration and

power setting No aircraft can sustain corner velo city while pulling a maximum load

factor for long Pulling to o great a load factor will cause airsp eed to bleed o b e

low corner velo city due to aircraft drag eects This sp eed lost can only b e regained

by reducing the load factor further reducing turn p erformance until the aircraft

accelerates back to corner velo cityby reducing altitude gravityprovides needed ac

celeration or increasing engine thrust Since maintaining the capabilitytostayat

corner velo city and altitude in a turn is an imp ortant consideration sustained turn

p erformance dep ends on the ability of engine thrust to overcome increased drag A

imp ortant measure of engine thrust is the thrusttoweight ratio at a given altitude

and airsp eed As the name implies the thrusttoweight TW ratio compares the

amount of thrust pro duced by an aircrafts engines to the weight of the aircraft

The higher the ratio the quicker an aircraft can gain or regain airsp eed For most

combat aircraft the TW ratio is around to however for high p erformance

aircraft such as the F and F the ratio is greater than in some altitude

and airsp eed regions allowing acceleration in the vertical direction TW is a critical

comp onent in sustained turn p erformance pages page page

page

As stated ab ove energy p erformance is the abilitytochange energy state ie

to climb andor accelerate The energy state of an aircraft is expressed through a

quantity known as sp ecic energy E and is dened mathematicallyas

M V

M A E

where M is the mass of the aircraft V is the true sp eed of the aircraft G is the

acceleration of gravity ftsec and A is the altitude of the aircraft The rst

term in equation is termed kinetic energy and represents the energy asso ciated

with the sp eed of the aircraft The second term in equation is potential energy and

represents the energy stored in the altitude of the aircraft page page

The energy state of an aircraft can b e changed through the through the ap

plication of p ower thrust and drag The rate of change in sp ecic energy E is

known as sp ecic excess p ower P and is expressed mathematicallyas

T D

P V

where T is total engine thrust D is total aircraft drag W is aircraft weight and V

is true airsp eed Equation shows that whenever thrust is greater than drag the

rate of change in sp ecic energy is p ositive resulting in an increase in energy state

Conversely if drag exceeds thrust at any time the energy state will decrease The

sp ecic excess p ower P of an aircraft under given conditions of weight congura

tion engine thrust sp eed altitude and load factor determines the available energy

p erformance or energy maneuverability Higher sp ecic excess p ower can givean

aircraft the ability to out zo om and out maneuver a less energyecient opp onent

page page

Presently all high p erformance combat aircraft are subsonic aircraft with the

ability to make short sup ersonic dashes through the use of afterburners Although

aircraft mayenter airtoair engagements at sp eeds greater than the sp eed of sound

results from Southeast Asia and the Middle East suggest that most airtoair engage

ments take place at sp eeds less than the sp eed of sound Higher airsp eeds particu

larly ab ove the sp eed of sound typically mean reduced maneuverability pages

page page

One imp ortant area of research in aircraft maneuvering is thrustvectored en

gines Thrustvector engines enable pivoting of the exhaust nozzles This pivoting

can provide suchcharacteristics as turn without bank and increased load factor with

increased pitch For example at the September Farnb orough England Air

Show the Soviet Sukhoi Su was able to p erform sup ercobras pullbacks

from the horizontal and kulbits or somersaults Both maneuvers are imp ossi

ble in nonthrust vectoring aircraft Although many defense analysts criticize the

combat value of sup ermaneuvering aircraft in a short range airtoair engagement

ie the loss of energy or sp eed in a short range engagement is usually fatal these

maneuvers can causes AWACS and air intercept radars to hiccup and temp orar

ily lose track of a target for several seconds The US will not haveanequivalent

maneuvering aircraft until the F is op erational pages

Most air combat aircraft generally op erate or cruise at an altitude of

to ft and an airsp eed of knots Many air combat aircraft exp erience a

marked changed in p erformance when passing an altitude of to ft

combat aircraft are much more maneuverable in the midtwenties however engine

Afterburner refers to a series of engine settings where additional fuel is sprayed directly into the

exhaust gases of the nal combustion chamb er The highest afterburner setting can provide a

increase in thrust of to days turb ofan combat aircraft engines Unfortunately using afterburner

settings consumes fuel at roughly three to four times the rate of nonafterburner dry engine

settings The highest maximum p ower setting without afterburner is termed mil military p ower

The USAFs nextgeneration Advanced Tactical Fighter ATF is required to sustain sup ersonic

cruise sp eeds sp eeds ab oveMach at altitude in mil p ower ie without the use of afterburners

p erformance drops o signicantly in the upp er twenties and turn p erformance is

thus degraded The lower twenties often oers the right balance of high true airsp eed

and endurance low fuel consumption while providing adequate indicated airsp eed

for maneuvering For greater range endurance and b etter maneuverability most

of to days combat aircraft cruise at high subsonic sp eed on the order of Mach to

In terms of the sp eeds at an airtoair engagement USAF combat aircraft such

as the FC will pursue the intercept phase of an engagement at ab out Mach

accelerating to Mach to launchbeyond visual range BVR weap ons The increase

in sp eed serves two purp oses rst the increased sp eed adds to BVR missile sp eed

thus increasing BVR missile reach second if the BVR missiles miss then the sp eed

can b e converted into a turn or energy advantage The Eagles will generally stayat

this sp eed Mach unless entering a turning engagement where opp onent sp eed

and own ship cornering velo citycomeinto play pages page

A Detection and Identication

Analogous to a ground battle in order to destroy an opp onent in the air

he must rst b e found The range at which an opp onent aircraft is detected the

detection range is a critical factor in airtoair engagements Simply put detection

range equates to reaction time The greater the detection range the larger the

time to react to the opp onent aircraft and more reaction time means more time

to setup sp ecic engagement tactics In the air environment aircraft are detected

by radar infrared sensors electronic countermeasures such as a radar and missile

warning receivers communication intercepts and visual observation In all these

cases detection dep ends on extracting and interpreting information provided bythe

electromagnetic sp ectrum page page

The electromagnetic sp ectrum is an ordered arrayofknown electromagnetic

EM radiations or energies The sp ectrum extends from short in terms of wave

length cosmic rays through long electric current The two principal ways in which

EM radiation is describ ed are byits frequency f and wavelength The frequency

of a particular EM waveisthenumb er of oscillations in the wave during a one second

time p erio d while wavelength is the distance through which the wave propagates in

one complete cycle The wavelength of an EM wave is related to the frequency of

the waveby the equation

where c is equal to the sp eed of light and of all EM radiation through space

ms pages F pages thru approximately x

Table details the electromagnetic sp ectrum Frequencies b etween ab out

Hz to kHz are commonly referred to as audio frequencies b ecause the human ear

can detect oscillations in the air pressure at these frequencies Radio communica

tions typically takes place at frequencies from kHz to GHz and overlap radar

frequencies whichtake place b etween GHz and Hz Infrared radiation o ccupies

a region from to x Hz while visible light o ccupies a very narrow p ortion

of the sp ectrum with frequencies in the order of Hz page page

page

By far the main metho d for aircraft detection involves extracting and inter

preting energy from the radar p ortion of the electromagnetic sp ectrum The term

radar is derived from the original name given to the technique Radio Detection And

Ranging invented by the British during World War I I At its most basic level a

radar system op erates by radiating EM energy and detecting the echo returned from

reecting ob jects targets The nature of the echo signal provides information ab out

the target such as range asp ect elevation velo city etc page page

Radars can b e classied by their use and generally fall into three categories

early warning acquisition or guidance Earlywarning radars are usually low fre

quency longwavelength sets requiring large antennas Their size generally precludes

Frequency Hz Wavelength m Typ e

x Cosmic Photons

x Gamma Rays

x Gamma Rays X Rays

x XRays

x Soft X Rays

x Ultraviolet XRays

x Ultraviolet

x Visible Sp ectrum

x Infrared

x Farinfrared

x Microwaves

x Microwaves Radar

x Radar

Television FM Radio

x ShortWaveRadio

x AM Radio

x Long WaveRadio

x Induction Heating

x Power

Direct Current

Table Electromagnetic Sp ectrum

their use on air combat aircraft Earlywarning radars are characterized by relatively

long range and p o or resolution and are used primarily for airb orne or groundbased

intercept control An aircraft detected byanearlywarning radar may app ear on a

radar display as a blip representing several miles in width and many closely spaced

aircraft may app ear as a single target Air combat aircraft are generally equipp ed

with an acquisition radar which has a higher frequency smaller antenna shorter

range and b etter resolution than early warning radars Acquisition radars usually

have the capabilitytotrack a target in order to gain more detailed information on

its relative p osition sp eed altitude etc Air combat aircraft radars also are gener

ally capable of guiding airtoair weap ons to a target thus also serving as guidance

radars Given sucient resolution a radar can discern something ab out the tar

gets size and shap e refer to identication techniques explained later in this section

Table shows the general frequency designations bands at which radar systems

op erate Each frequency band has particular characteristics that make it b etter for

certain applications than for others pages pages

Aircraft may b e detected by either groundbased or airb orne radar USAF

air sup eriority do ctrine relies mainly on the use of airb orne radar platforms suchas

the E Sentry Airb orne Warning and Control System AWACS and E Hawkeye

platforms as well as the radars of air combat aircraft The E Sentrys radar APY

op erates in the S or EF band meaning that it generates radar waves in the

to GigaHertz GHz range with a wavelength of from to cm The APY

radar uses the pulseDoppler principle relying on precise measurement of the tiny

frequency shift in waves reected from moving ob jects to distinguish ying aircraft

from background ground clutter This gives the radar the ability to lo ok down and

detect lowying targets as long as they are moving faster than knots pages

The APY radar has a scanning capability and can op erate in seven

mo des

Standard Radar Bands Frequency Wavelength

HF MHz cm cm

VHF MHz cm

UHF GHz cm

L GHz cm

S GHz cm

C GHz cm

X GHz cm

Ku GHz cm

K GHz cm

Ka GHz cm

MM GHz cm

NATO Bands Frequency Wavelength

A Hz

B Hz cm

C GHz cm

D GHz cm

E GHz cm

F GHz cm

G GHz cm

H GHz cm

I GHz cm

J GHz cm

K GHz cm

L GHz cm

M GHz cm

Table General Frequency Designations Bands For Radar Systems

PulseDoppler NonElevation Scan PDNES This high pulse rep etition fre

quency PRF mo de provides lo ok down surveillance to the surface using

pulseDoppler with narrow Doppler lters and a sharp b eam to eliminate

ground clutter The PDNES mo de is optimized for detection out to the radar

horizon however this mo de do es not measure target elevation The PDNES

mo de is used when range information is paramount to elevation data Atan

altitude of m ft this mo de is able to detect targets out to a range

of approximately nautical miles nm

PulseDoppler Elevation Scan PDES This mo de is similar to the PDNES

mo de but includes a electronic vertical radar b eam scan to provide target

elevation with a slight decrease in maximum detection range

BeyondtheHorizon BTHThislow PRF mo de provides longrange surveil

lance of medium and high altitude aircraft without Doppler This mo de ob

tains target range and azimuth information only no target elevation data is

provided

Passive In this mo de the radar transmitter is turned o and the receiver left

on to obtain electronic countermeasures ECM information such as enemy

jamming lo cation

Maritime This mo de provides a maritime surveillance capabilityThisinvolves

the use of a very short radar pulse to provide the resolution required to detect

moving or anchored surface ships

InterleavedOne The PDNES and Maritime mo des can b e interleaved

InterleavedTwo The PDES and BTH mo des can b e interleaved

PRF refers to the numb er of radar pulses sent out in a given time p erio d PRF will b e discussed

in greater detail in FCs radar system APG description

Since the radar b eam is ab ove the horizon there is no ground clutter and a Doppler b eam is

not needed

The PDNESMaritime and PDESBTH interleaved mo des are noted as providing

the necessary p erformance for most op erational situations Further exibilityis

generated by the APY radar b eing able to divide the surveillance volume of azimuth

scans into to a maximum of sectors eachofwhich can b e congured with its own

op erating mo de page page page

Most air combat aircraft radar systems op erate in the Xband to GHz

This frequency band oers the b est compromise in terms of radar system design

and p erformance By far the most critical factor in radar system design for an air

combat aircraft is antenna size For an air combat aircraft the most convenient

p osition for the antenna is the aircrafts nose where limited space due to aero dy

namic considerations constrains allowable antenna size An X band radar system

pro duces a go o d compromise antenna size while maintaining adequate system p er

formance Dropping frequency to a higher p erformance Sband radar system such

as on the E Sentry demands a larger antenna while increasing to a Kband radar

oers a smaller antenna size but a radar signal whichisadversely aected by me

teorological conditions suchasrain USAF air combat aircraft such as the FC

Eagle FE Strike Eagle FC Falcon and F Raptor contain sophisticated

radar systems page page

The FC Eagle uses the APG radar system while the FE Strike Eagle

uses the APG radar system Each system is an allweather programmable multi

mo de pulseDoppler radar op erating with a numb er of selectable frequencies in

the IJbands to GHz With a detection range of greater than nm

nm for the APG against a large radar cross section aircraft eg aTu

Bear b omb er b oth the APG and APG radar combine long range with

features such as automatic detection and lo ckon within nm nm for APG

In addition the Xband generates adequate gain the outputinput p ower ratio to pro duce a

nelycontrolled b eam without unwanted big sidelob es Sidelob es giveaway the aircrafts presence

on opp onent radar warning receivers without contributing to detection capability

Some versions of the Eagle are equipp ed with the APG radar system

These systems contain a programmable signal pro cessor whichgives the ability

to rapidly change the radar through software rather than a hard wired circuit design

to accommo date new tactics mo des and weap ons Both the APG and APG

can simultaneously track and engage multiple targets page page

page page

Three pulse rep etition frequency PRF ranges are used in the APG and

APG radar systems low medium and high PRF refers to the numb er of radar

pulses transmitted in a given time p erio d Each of the three PRF ranges has its

strengths and weaknesses LowPRFtypically to pulses p er second radar

mo des transmit radar pulses at sucientintervals so that each pulse has enough

time to travel out to the target b e reected and return to the radar b efore the next

pulse is transmitted Low PRF radar transmissions provide excellent lo okup long

range capabilityHowever low PRF radar transmissions tend to pro duce massive

ground clutter when searching in a lo okdown capacity making targets harder to

dierentiate In contrast high PRF typically to pulses p er second

radar mo des transmit radar pulses so oner than reected pulses are returned In

high PRF radar mo des the radar is able to measure the Doppler shift in pulses

reected by the target Thus high PRF unlikelow PRF radar transmissions

can acquire and discriminate targets from ground clutter when searching in a lo ok

down capacityHowever high PRF radar transmissions are p o or at providing target

distance due to the high reected pulse return rate and are generally useless in

b eam side attacks where there is relatively little Doppler shift pages

pages pages page

Recall Doppler shift is the tiny shift in frequency of a reected pulse caused by the relative

motion b etween the target and the radar system As the target approaches the radar system more

reected pulses p er second reach the system but as the target recedes fewer reected pulses p er

second reach the system Hence the Doppler shift is providing velo city information only

In order to improve distance determination lower frequencymo dulation FM or twosimulta

neous closelyspaced frequencies are transmitted But this is at the exp ense of reduced detection

range due to the complex pro cessing required of the returning signals In addition FM ranging has

an error of ve p ercentorso

Medium PRF typically to pulses p er second radar mo des oers

a go o d compromise to low and high PRF mo des at distances less than nm or

so Medium PRF transmissions provide target distances while acquiring and dis

criminating targets from ground clutter when searching in a lo okdown capacity

The three PRF ranges used in the APG and APG radar systems eectively

lter out ground clutter giving the radar a lo okdown sho ot down capability ie

the radar can track and engage an aircraft ying at low altitude while retaining an

excellent lo okup sho ot up capability pages pages pages

page

The APG radars horizontal or azimuth scan has three selectable arcs

or centered directly in front of the aircraft The vertical or elevation

scan has three selectable bars a bar is a slice of airspace with a vertical depth of

perbarbar bar or bar forvarying vertical coverage

With an antenna sweep sp eed of secbar the largest search pattern a

bar scan can take up to fourteen seconds to complete page

Additionally the APG radar system has a low probabilityofintercept mo de

LPI mo de designed to defeat the radar warning receivers and electronic supp ort

measure ESM detectors reference section A on opp onent aircraft by using

techniques like frequencyhopping and p ower regulation This means an Eagle or

Strike Eagle can conduct an active radar search and RWRESMequipp ed aircraft

will not detect the search Conventional radars lo cate aircraft by transmitting high

energy pulses in a narrow frequency band and nding highenergy returns However

a good RWRESMequipp ed aircraft can pick up these highenergy pulses at over

two times the radars eectiverange On the other hand LPI radars transmit

lowenergy pulses over a wide band of frequencies When the multiple returns are

received by the aircraft the radars signal pro cessor integrates all the individual

pulses back together Even though the amount of returning energy is ab out the

Scan p erformance statistics could not b e found in the literature for the APG radar system

same as a conventional radars highenergy pulse most warning systems can not

detect the LPI radar since each individual LPI pulse has signicantly less energy

than a conventional radar pulse Additionally the LPI radar frequencies used do

not necessarily correlate to the normal frequency search pattern of these warning

systems This gives the Eagle or Strike Eagle a tremendous advantage in anylong

range engagement as the pilot do es not have to establish a lo ckon when ring a

mediumrange missile such as the AIM AMRAAM The rst realistic indication

that an opp onent aircraft will have of an attackby an Eagle or Strike Eagle maybe

the screams from his radarwarning receiver telling him that the AMRAAM s radar

has lit o lo cked on and is in the nal stages of intercept page

The FC Falcon uses the APG radar which is an Xband GHz

pulse Doppler radar The radar has range out to can scan a arc horizon

tally and has elevation setting of or bar As with the F the range at which

airtoair targets are detected is increased by using a high PRF mo de Once detected

a medium PRF mo de can b e employed to get additional range and angle informa

tion The APG radar is capable of tracking ten targets simultaneously page

page

The radar system to b e used on the F the APG diers considerably

from the radar systems describ ed ab ove For instance the antenna of the APG

is a xed elliptical active arraycontaining ab out individual radar Trans

mitReceive TR mo dules Each TR mo dule is ab out the size of an adults nger

and is a complete radar system in its own right The APG can sweep a

multiplebar search pattern However instead of taking fourteen seconds to sweep

a sixbar pattern like the APG the APG will search the equivalentvol

ume almost instantaneously This is b ecause the active array can form multiple radar

b eams to rapidly scan an area and do es not need motors or mechanical linkages to

movetheantenna for a scan The APG will also have the critical LPI search

mo de page

Recallitwas stated as the b eginning of this section that USAF air sup eriority

do ctrine relies mainly on the use of airb orne radar platforms This reliance is a

direct result of the oensive nature of USAF air sup eriority do ctrine which seeks to

obtain control of opp onent airspace Since the ground under an opp onents airspace

is by denition controlled by the opp onent airb orne radar platforms in conjunction

with spacebased assets such as satellites provide the most eective means to moni

tor opp onent airspace In contrast a ma jority of the nations in the world including

several of the former Soviet Union republics and asso ciated satellite nations do not

have the adequate airb orne assets b oth in numb er and quality to conduct a pro

longed oensive air campaign The air sup eriority do ctrine of these nations centers

on maintaining an adequate air defense capability Detection of opp onent aircraft

in an air defense system relies not only on the use of airb orne radar platforms but

also on groundbased radar systems

Groundbased radar systems are quite diverse in nature Systems maybe

mobile or xed twodimensional provide b earing and range information only or

three dimensional b earing range and height information and provide high or low

altitude coverage or b oth To provide an illustrative example of the capabilities of

such systems consider the groundbased radar systems of the Peoples Republic of

China page

Two dimensional air defense radar systems are generally characterized by rel

atively long range however these systems can only provide b earing and range infor

mation Chinese two dimensional radar systems include the C HNR and

HN mobile long range air defense warning radar systems The C ground

based radar system can op erate simultaneously in two band regions within the VHF

frequency band to MHz considered the low band region and to

I tried rst to nd an op en unclassied reference do cumenting the p erformance characteristics

of the groundbased radar systems of the former Soviet Union since these systems are prevalentin

many of regions of the world where conict can b e exp ected to take place UnfortunatelyIwas

not able to lo cate a reference As a substitute I cho ose the Chinese groundbased radar systems

where information was readily available

MHz high band region The low band radar wave has a horizontal b eamwidth of

while the high band radar wave has a horizontal b eamwidth of The C

radar has a pulse length of microseconds m and a PRF of Hz The

HNR groundbased radar system op erates in the UHF band to GHz The

radar has a detection range of approximately nm against a square meter tar

get The HNR radar has a range resolution of m and an azimuth resolution

of Lastly the HN groundbased radar system op erates in the VHF frequency

band to MHz The radar has a pulse length of microseconds km

and a PRF of Hz This radar has longest detection range of any of Chinas air

defense radars with a range of approximately nm The Typ e HN radar has

a range resolution of m and an azimuth resolution of pages

In contrast to two dimensional air defense radar systems three dimensional air

defense radar systems are characterized as generally b eing of shorter range than their

two dimensional counterparts however three dimensional systems provide altitude

information in addition to b earing and range Chinese three dimensional radar

systems include the JY the JYA JY JY air surveillance radars The JY

radar is a mobile vanmounted air surveillance radar that op erates in the GH bands

to GHz This radar can b e employed as the main radar sensor for an automated

tactical defense system or can b e used as an indep endent radar The JY radar has

a horizontal b eamwidth of a vertical b eamwidth of and a pulse length

of microseconds m This radar also has three PRFs

and Hz The JY radar has a detection range of approximately nm and

can cover altitudes up to nm Additionally the JY has a range resolution

of m and an azimuth resolution of and can simultaneously trackupto

targets A variant of the JY the JYA mobile air surveillance radar can carry out

re control of missiles andor antiaircraft artillery guns This radar has a reduced

detection range of approximately nm and a reduced altitude coverage up to

nm pages

The Chinese JY radar is a mobile low altitude three dimensional air surveil

lance radar designed for air defense gap lling airp ort surveillance and coastal de

fense This radar op erates in Eband to GHz with a microsecond

m pulse length The radar rotates at rpm providing azimuth coverage

and vertical coverage The JY radar has a horizontal b eamwidth of a

vertical b eamwidth of and a PRF of Hz This radar has a detection range

of approximately nm Additionally the JY has a range resolution of

m and an azimuth resolution of and can simultaneously track up to targets

The JY is a xed long range three dimensional air surveillance radar op erating

in the Fband to GHz The radar rotates at rpm providing azimuth

coverage and vertical coverage The JY had a detection range of approxi

mately nm and can cover altitudes up to nm Additionally the JY has

a range resolution of m an azimuth resolution of and an altitude resolution

of pages

In addition to the two three dimensional groundbased radar systems the

Peoples Republic of China has twooverthehorizon OTH radar systems in North

China and the South China Sea As there name implies OTH radar systems can

lo ok past the radar horizon the line of sight limited byradarby the curvature of the

earth These systems generally use the ionosphere as a mirror in the sky to b ounce

transmitted radar waves o the upp er atmosphere and use the same path for reected

returns Unfortunately OTH radar warning systems have no closein capabilityThe

Chinese OTH radar systems op erate in HFband to MHz have a rep orted

detection range from to nm with a azimuth coverage pages

page

In order to direct surfacetoair weap ons such as AA guns or surfacetoair

missiles an air defense system needs re control radars that can precisely trackthe

ightofanopponent aircraft The Peoples Republic of China has several surface

With a probability of detection of and Swerling I of sq m

toair re control radar systems In addition to the JYA radar mentioned ab ove

other surfacetoair re control radar systems include the and The mobile

surfacetoair re control radar system is used with antiaircraft AA guns

usually and mm and comes in three versions A B and C All versions

of this radar op erate in the IJband to GHz and is capable of b oth search

and track functions and is generally used with a computer and optical range nder

The JY radar has a horizontal b eamwidth of a vertical b eamwidth of

and two pulse lengths narrow micro seconds m PRF of Hz and

broad microseconds m PRF of Hz The A has a detection range

on a ghter size target of approximately nm in search mo de with a maximum

tracking range of nm The minimum reliable tracking range is ab out m

The B intro duced a identication friend or fo e IFF systems and increased the

frequency coverage and maximum detection range search mo de to nm The

C has a maximum detection range of nm The Chinese surfacetoair re

control radar system is a mobile all weather system consisting of radar optical unit

and computer and op erates with various calib er AA guns and surfacetoair missiles

The radar op erates in the GHIbands to GHz and is used primarily against

aircraft attacking from low and medium altitude The has a maximum detection

range of nm pages

Once detected by the groundbased radars of an air defense system air defense

aircraft will b e vectored to intercept opp onent aircraft either from defensive ground

alert DCA OCA or HVAA missions or combat air patrol BARCAP mission p o

sitions Several formidable p otential opp onent aircraft presentchallenges to USAF

air sup eriority op erations two of these include the Russian MIG Fulcrum and the

Sukhoi Su FlankerB Both the Fulcrum and the Flanker contain sophisticated

radar systems The SuSK FlankerB K is the exp ort version of the Su has

the NI IP S Myech N track while scan coherent pulseDoppler radar system

The N radar op erates in the Ilow Jband to GHz This radar has lo ok

down sho ot down capability and can track up to ten targets simultaneously while

engaging two The N radar has a search range up to nm and tracking range

up to nm against large size targets and a search range of nm and tracking rage

of nm against MIG size targets in the forward hemisphere The searchrange

against MIG size targets is up to nm in the rear hemisphere The N

radar has three dierent elds of vision by by and by and a

search eld of by page page page

The MIGC Fulcrum has the RLS RP N Sapr coherent pulse

Doppler radar system The N radar op erates in the Ilow Jband to GHz

This radar has lo ok down sho ot down capability and can track up to ten targets

simultaneously while engaging one The N radar has a search range up to nm

and tracking range up to nm against large size targets and a search range of

nm against MIG size targets in the forward hemisphere The searchrangevaries

from nm against MIG size targets to nm against large size targets in the

rear hemisphere The N radar tracking limits are and in elevation

and in azimuth page page page

As stated earlier in this section a radar system op erates by radiating EM

energy in this case radar waves and detecting the echo returned from reecting

ob jects targets As is the case with EM energy such as lightwaves a radar wave

can b e deected diused or absorb ed Radar reectivity is aected by the physical

prop erties of the reected ob ject and the asp ect of the ob ject presented to radar

The amount of reected radar radiation provide by an aircraft can b e expressed by

an area metric known as radar cross section RCS RCS is a complex measure that

dep ends on the crosssectional area of that radiated aircraft geometric cross section

howwell the aircraft reects radar radiation material reectivity and howmuch

of the reected radar radiation travels backtoward the radar system directivity

For example of the aects of geometric cross section take an aircraft ying directly

toward a radar wave The physical dimensions presented to the radar system would

b e fairly small the leading edge of the wings and tail plus the crosssection of the

fuselage and engines are probably all that would b e seen If the same aircraft now

turns through the radar system would now see the whole length of the fuselage

and the slab section of the wings and tail Clearlythe case would presenta

larger radar image or radar cross section b ecause there is more of the target for the

radar waves to hit and reect back page

AdditionallyRCS is dened as the cross sectional area of a p erfectly conducting

sphere of radius a RC S a that reects the same amount of energy as the

aircraft This conducting sphere is the only radar ob ject with a geometric cross

sectional area equal to its RCS page

Factors that determine the detection range of a sp ecic radar system include

maximum range of the radar aircraft including munitions radar signature and

jamming techniques used App endix B provides a general discussion of these factors

In recentyears the concept of stealth or low observable technology has gar

nered a great deal of discussion Two stealth techniques exist to reduce a radar

system eective detection range

External shaping and

Coating the aircraft with radarabsorbing materials RAMs

Both these techniques reduce the eective detection range by reducing the amount

of reected radar radiation provided to the radar system ie RCS of the target

aircraft By carefully shaping the external surfaces of an aircraft one can reduce

the RCS of the aircraft The F incorp orates such shaping in its design For

existing nonstealth aircraft designs such as the F or F RAM coatings can

cut the RCS Both shaping and coating are eective against b oth the typical short

This is true only when the circumference of the sphere is at least an order of magnitude larger

than the radar wavelength under consideration

wavelengths of conventional air combat aircraft and the longer wavelengths of air

and groundbased defense radars pages pages

Although radar is the main sensor used to detect aircraft in the air infrared

IR sensors are b ecoming an increasingly imp ortant and capable means for aircraft

detection IR sensors are passive these sensors detect infrared energy pro duced byan

aircraft The ma jority of infrared energy pro duced by an aircraft is absorb ed bywater

vap or and carb on dioxide gas in the atmosphere however two aircraft detection

windows exist in the infrared band The rst window termed midIR o ccurs at a

frequency of to x Hz wavelength of to x meters Infrared energy from

the heat of an aircrafts engine parts and exhaust falls into this midIR region The

second window termed long IR o ccurs at a frequency of to x Hz wavelength

of to x meters Infrared energy caused by solar heating or by air friction

on the fuselage of the aircraft falls into this longIR region Mo dern dualband

infrared search and track IRST systems can lo ok for aircraft in b oth IR windows

IRSTs are comp osed of wide eldofview sensors that use automated detection and

tracking routines to nd targets in highly cluttered backgrounds pages

page page

Russian air combat aircraft such as the MIG and Su contain sophisticated

IRST sensors The SuSK is equipp ed with the OLS IRST that provides az

imuth and elevation information This IRST can scan large areas and detect aircraft

at ranges out to to nm although to nm is a more reasonable range esti

mate against a nonafterburning nonIR stealthly aircraft The OLS is combined

with a laserrange nder to provide distance information The laserrange nder is

capable of tracking a target out to a range of nm in the rearward hemisphere and

nm in the forward hemisphere The MIGC is equipp ed with UOMZ OEPS

IRST which can detect ghtersized targets out to nm in the forward hemisphere

and nm in the rearward hemisphere pages pages

On the other hand the USAF has chosen not to install such systems on cur

rent air combat aircraft such as the F and F Apparently this p osition is

b eginning to change due to the fact that at least two USAF aircraft participating

in the Gulf War were probably shot down by Iraqi MIGs using a combination of

that aircrafts IRST helmetmounted sight and the R Archer infrared airtoair

missile page

Clearly current IRST systems do not provide greater detection ranges than

radar systems however IRST systems are passive ie do not generate the EM

radiation needed for detection where as radar systems are active Thus the use of

IRST systems can not b e detected whereas the use of radar systems can Stealth

countermeasures to IR sensors only address the midIR detection band Little can

b e done ab out the long IR band some IR emission from solar or friction heating of

an aircrafts outer skin will always remain present MidIR stealth countermeasures

include co oling engine exhaust where the most infrared energy is generated page

page

Once an aircraft is detected in order to engage the opp onent esp ecially from

beyond visual range BVR under most common rules of engagementonemust

b e certain he is rst an opp onent Timely aircraft identication is probably the

toughest problem in the air environment This problem is not unique to airtoair

engagements as it applies to surfacetoair engagements as well Aircraft can b e

identied visually or electronicallyTwotyp es of electronic identication pro cedures

exist

co op erative identication and

nonco op erativeidentication

Co op erative identication dep ends on transp onders carried by friendly aircraft In

co op erativeidentication an unknown aircraft is interrogated with a co ded radio

signal from a friendly aircrafts Identication Friend or Fo e IFF electronic identi

cation system Friendly aircraft resp ond with a co ded reply and are classied as

friendly aircraft Opp onent aircraft do not reply Unfortunately neither do friendly

aircraft with inop erative equipment Despite advances in IFF systems one basic

problem has remained determining whether the lack of a resp onse from an air

craft means that the aircraft is hostile or whether the aircraft is friendly with a

nonresp onsive transp onder The relatively low reliability of the co op erative iden

tication pro cedure has led to restrictive rule of engagementROE that require

several indep endent means of verifying that a nonreplying aircraft is reallytruly

an opp onent b efore the aircraft can b e red at The tragic sho ot down of two

Army helicopters in Northern Iraq in by FCs provides a vivid illustration

of shortcoming of the co op erativeidentication pro cedure page page

page page

In contrast the nonco op erative electronic identication pro cedure relies on var

ious radar metho ds of identifying aircraft without a resp onse and is able to identify

most friendly and opp onent aircraft How this is done is highly classied Never

theless one p ossible means discussed in op en sources is to fo cus a highresolution

radar b eam on a target and with advanced signal pro cessing determine a return pat

tern that is unique to the aircraft and can b e compared to the known parameters

of friendly and hostile aircraft For example with a highresolution radar b eam you

might b e able count the numb er of blades in the aircrafts engine fan or compres

sor Knowing the blade count tells you the typ e of engine and can giveyou a good

idea as to whether the aircraft is hostile page pages page

page

Most USAF air combat aircrafts are equipp ed with co op erative and nonco op

erative electronic identication capabilityFor example the E Sentry is equipp ed

with a highly sophisticated IFF system capable of interrogating virtually any IFF

transp onder in the world within nm The system rep ortedly has some nonco op

APX IFFTactical Digital Link IFFTADILC system

erative capabilities as well The FC Eagle and FE Strike Eagle APG radar

contains nonco op erativeidentication capabilities and the FC Falcon mayhave

nonco op erativeidentication capabilities as well page

The F will have an improved capability to conduct nonco op erativeidenti

cations The Fs APG radar will b e able to form incredibly ne b eams allowing

the signal pro cessor to generate a highresolution radar image of an aircraft through

Inverse Synthetic Ap erture Radar ISAR mo de pro cessing An ISARcapable radar

uses the Doppler shifts caused by rotational changes in the aircrafts p osition with

resp ect to the radar antenna to create a D map of the aircraft With a go o d D

radar image an integrated aircraftcombat system might conceivably identify the

target by comparing the image to a stored database page

A Weapons

The aircraft gun ushered in the era of airtoair engagement and is by far

the most widely used and imp ortant airtoair weap on in history However the

development and employment of capable airtoair missiles has expanded the combat

domain of airtoair engagements Since the June ArabIsraeli War all ma jor

airtoair engagements have included the use of airtoair missiles The advent

of airtoair missiles enabled aircraft to sho ot and kill opp onents quicker and from

greater distances than guns alone haveallowed Aircraft no longer have to close to

within hundreds of feet of an opp onent and maneuver into a small cone b ehind the

opp onent in order to enable the aircraft to hold guns on the opp onent long enough

to re sucient bullets to destroy the opp onent page page

Airtoair missiles may b e divided into two main categories radarguided and

non radarguided or passively guided Radarguided missiles are further divided

into two categories semiactive and active Active radarguided airtoair missiles

such as the US Advanced Medium Range AirtoAir Missile AMRAAM carry their

own radar where as semiactive radarguided airtoair missiles such as the Sparrow

AIM have a radar receiver but no transmitter and pick up radar signals transmit

ted by the aircraft and reected from the target Semiactive radarguided airtoair

missiles require the launching aircraft to lo ck on with radar a single opp onent and

to remain lo cked on until the missile impacts in order to provide targeting informa

tion for the missile If radar lo ck is lost from the launching aircraft the missile will

not track the opp onent In contrast active radarguided airtoair missiles can pro

vide their own targeting information An aircraft launching an active radarguided

missile can break o after missile launch and leavetoght another opp onent or exit

the combat area page page page

In the USAF airtoair missile inventory the AMRAAM has replaced the nal

version of the Sparrow the AIMM Unfortunately the Sparrowwas plagued by

signicanttechnological aws during its ve decades of the service The AIM was

over four times as eective in Desert Storm as it was in Vietnam however the missile

only achieve a kill rate in Desert Storm in Vietnam Additionally almost

half of the AIMs launched in Desert Storm failed to function prop erly page

The current deployed version of the AMRAAM is the AIMA Termed the

Slammer the AIMA is inches in diameter and inches long with a center

stabilizing n of inches and a rear guidance n span of inches The AIM

A weights lb The AIMA had a complete air intercept radar system The

radar of the ring aircraft sends the AMRAAM the three dimensional heading and

sp eed of the target aircraft then the missile ies out to a p oint where it switches to

its own radar If the target aircraft is within the radar cone of the AIMAs

seeker the missile lo cks up the target aircraft and then pro ceeds to intercept Addi

AMRAAM may b e red in a numb er of dierent mo des At longer range the missile may

b e continually up dated with fresh target information from the launching aircrafts radar while in

ightuntil the missiles radar takes over Alternatively if the aircraft survivability is an issue the

launching aircraft can let the missile go without midcourse up dates and trust the AMRAAMs

active radar to work unaided Finally for shorter engagement distances lo ckon of the missiles

seeker can b e accomplished prior to launch

tionally the AIMA is equipp ed with a proximity fuse that detects and detonates

when a target aircraft is within lethal range The imp etus for the AMRAAMs intro

duction into service was the gradual implementation of groundbased and airb orne

sensors that provided a detailed picture of the air environment at ranges imp ossible

a decade ago Information from assets suchasAWACs pip ed via datalink to the

aircraft combined with aircraft sensors allow the identication and targeting of op

p onent aircraft at very long ranges with a high degree of accuracy page

As an illustrative example consider launching an AIMA at a target aircraft

from an FC In the F C the pilot selects an airtoair mo de for the radar suchas

BORE Boresight ie where the radar is sighted down the centerline of the aircraft

or TWS DOGFIGHT where the radar is in a mo de useful for closein dogghting

The pilot then selects the AIMA from the stores control panel and cho oses either

SLAVE or BORE to program the AIMs radar to accept commands from the

Fs onb oard APG radar Once a radar contact is established with the target

aircraft the Fs onboard weap ons computer establishes a re control solution

including elapsed time from AIMA launchuntil the AIMAs radar go es

active Atthispoint the Fs Head Up Display HUD will b egin to give the pilot

steering cues to bring the F into range to re Once the HUD gives the pilot an IN

RANGE indication the pilot press the weap ons release switch on the control stick

The AIMA is launched and will accept up dates from the Fs radar if pilot

has selected a FIRE AND UPDATE mo de until either pilot breaks radar contact

from maneuvering or the AIMA hits the target At this p oint the ring aircraft

is free to re another missile seek another target aircraft or leave the engagement

area page

The SLAVE option lo cks the seeker onto whatever target the Fs radar is currently tracking

while BORE p oints the Fs radar straight ahead along the line of ight where the rst target

the radar sees is lo cked

Other capable radarguided missiles include the semiactive Russian Vym

p el R NATO co de name AA Alamo Future developments include the

French MICA UK FMRAAM and Russias rst active radarguided missile the

R NATO co de name AA AMRAAMski

The ma jority of non radarguided passively guided airtoair missiles are heat

seeking or infrared IR missiles such as the US Sidewinder AIM The frequency

of the IR p ortion of the electromagnetic sp ectrum is just b elow that of visible light

and well ab ove that of radar These missiles guide on the IR energy pro duced bythe

opp onent aircraft All IR missiles are launch and leave or re and forget ie the

missile is autonomous after launch tracking on the opp onent at which the missile

was directed For example to launch an AIM at a target aircraft in the FC all

that is required is to select AAM from the stores control panel The seeker in the

AIM then lo oks for the target aircraft in front of the FC When the seeker is

lo cked on to the target aircraft an audio growl tone in the pilots headset alerts

the pilot that the AIM is ready to launch All the pilot of the FC has to do to

launch the AIM is squeeze the trigger Once an IR missile is in ight the ring

aircraft is free to re another missile seek another target or leave the engagement

area page page page

The currently deployed third generation version of the Sidewinder is the AIM

M Termed the Mike the AIMM is inches in diameter and inches long

with a forward canard wingspan of inches and a rear stabilizer wingspan of

inches The AIMM weights lb The AIMM has an ob oresight capability

of and scans in b oth short and middle wavelength infrared lightaswell as the

Also IR versions of this missile

When the rst test launches of what would b ecome the Aerial Intercept Missile Nine AIM

were conducted in the missiles snakelikeight path towards the test targets provided the

name it would carry for the next half century of service the Sidewinder page

Or if the radar is already lo cked onto the target aircraft the seeker head can b e slaved to the

radar and the seeker will also lo ckonto the target aircraft

The abilitytolock up a target o the ring aircrafts centerline

long wavelength ultraviolet sp ectrum Additionally the AIMM is equipp ed with

a proximity fuse like the AIMA If the missile should miss the target a rare

event due to the accuracy of the guidance system the warhead will still detonate

sp ewing its fragmentation pattern at the target aircraft page

Other capable IR missiles include the Russian Vymp el R NATO co de name

AA Archer and Israel Python Many defense exp erts b elieve the AA

has provided its combat aircraft the Sukhoi Su Flanker and the MiG Ful

crum with a signicant advantage in short range combat scenarios such as bugout

the moment that an friendly aircraft breaks out of the engagement and rundown

the subsequent tail chase over NATO aircraft The AA has an ob oresight

capabilityof expanded to immediately after launch and a deectorbased

thrustvectoring system that provides for increased agility Additionally the AA

is the rst airtoair missile to b e coupled with an op erational helmet mounted

sight HMS The Python has an ob oresight capabilityofapproximately and

maneuvers using aero dynamic control The AA and Python have the led the

US to pursue an upgrade to the AIMM the AIM X AIMX program require

ments call for a missile with a ob oresight capability using a HMS page

page

One metho d for visualizing the capabilities and limitations of an airtoair

weap on is to study weap on envelop es A weap ons envelop e is the area around the

target aircraft where missiles or gun can b e eective The weap ons envelop e is

dened by

Angleo the dierence measured in degrees b etween the ring aircrafts

heading and the target aircrafts

Range the distance b etween the ring aircraft and the target aircraft and

This is based on an op erational assessment conducted when German reunication broughta

squadron of AA equipp ed MIG s into NATO service The assessment indicated that Western

aircraft equipp ed with the AIMM would b e at a distinct disadvantage in airtoair engagements

at ranges b elow nm

Asp ect Angle the dierence measured in degrees b etween the target aircrafts

tail and the ring aircraft

The weap ons envelop e of allasp ect missiles such as the AIMM and the AIM

lo ok like a doughnut with the outside ring b eing the maximum range of the missile

and the inside ring b eing the minimum range Figure shows a typical weap on

envelop e for an allasp ect missile Theweap on envelop e in Figure has an oval

shap e with more of the envelop e area in front of the target aircraft than b ehind This

disp ortionality is due to the fact that a missile red at a high asp ect from in front

of the target aircraft has a greater eective range than a missile red at low asp ect

from b ehind When a missile is red at a target aircraft head on the fact that

the target aircraft is ying towards the missile will help the missile reach its target

by reducing the distance the missile needs to travel to reach the target aircraft

This dierence mayvary byasmuch as to from a rear asp ect missile

shot Although shown in two dimensions in Figure an actual airtoair weap ons

envelop e is three dimensional however the envelop e would probably only slightly

vary in the third dimensional dep ending on the ring aircrafts p osition relativeto

the target aircrafts plane of symmetry pages pages

When a target aircraft turns the weap ons envelop e shifts Figure shows a

typical weap on envelop e for an allasp ect missile for a target aircraft making a G

turn Generally the limits of the maximum and minimum ranges in frontofthe

target aircraft move in the direction of the turn while the ranges b ehind the target

aircraft move in on the b elly side of the turn pages

In addition to changing with target aircraft maneuvers the weap ons envelop e

of an airtoair missile also changes with ring aircrafttarget aircraft sp eeds and

altitude Clearly in order to get a full picture of the capabilities of a particular

airtoair missile pilots must consider manyweap on envelop e charts covering a wide

Figure is taken from page

Figure is taken from page Note: Weapon Zone is taken at a fixed Angle-Off

180

RMAX

RMIN

Bogey

90 90

Figure Weap ons Envelop e of AllAsp ect Missile RMAX

RMIN

Figure Weap ons Envelop e of AllAsp ect Missile under Gs

range of target aircraft maneuvers ring aircrafttarget sp eeds and altitudes Even

if pilots could draweachweap on envelop e from memory they still must determine

the parameters necessary to recognize whichenvelop e they are in and their p osition

range and asp ect within the envelop e In order to solve the hard problem of

envelop e recognition most combat aircraft are equipp ed with a tracking radar system

and recontrol computer These systems can accurately assess and displaythe

missile envelop e capabilities as well as other limits that might b e deemed necessary

In order to make inputs to the radar computer it is necessary to maintain a radar

track rather than just detect the target aircraft The transition from radar detection

to track is referred to target lo ck Dep ending on the sophistication of the radar this

may b e an automatic pro cess or a manual pro cess pages

As shown in Figures and the range of an airtoair missile can b e sig

nicantly aected by whether the airtoair engagement takes place headon on the

b eam or in a tailchase conguration For more capable airtoair missiles the b est

engagement range is usually achieved with the target o nose of the ring air

craft degrees asp ect As mentioned the range of an airtoair missile can b e

signicantly aected by b oth the sp eed and altitude of the ring and target aircraft

The greater the sp eed of the ring aircraft the greater the range of the missile since

aircraft sp eed adds to the missile sp eed Also missile range increases with alti

tude since the range of the missiles engine increases with altitude The range

capabilities of airtoair missiles are generally divided into three groups

Short to indicate a missile with a maximum range of nm km or less

Note As discussed in section A there is a reduction in aircraft maneuverability at sp eeds

ab oveMach Hence there is a practical limit on aircraft ring sp eeds if one desires to re

tain maneuverabilityFor example F Eagle generally launch BVR missiles at sp eeds around

Mach This ring sp eed strikes a go o d balance b etween increased missile range and adequate

maneuverability

Note However missile maneuverability decreases with altitude Furthermore missile range is

greater when ring from a higher altitude to a target at a lower altitude than ring from a lower

altitude to a higher altitude This is the result of p otential energy eects describ ed in section A

Medium to indicate a missile with a maximum range greater than nm

km and less than nm km and

Long to indicate a missile with a maximum range greater than nm km

These range groupings are in a constant state of ux as new higher range missiles

enter national inventories For example the AIMM is consider a short range

missile even though it can theoretically y out to a range of nm against a non

maneuvering target page pages pages page

Typically IR non radarguided missiles have a shorter range capability than

radarguided missiles For example the IR AIMM Sidewinder has a maximum

engagement range approaching a headon target of nm while the radarguided

AIM AMRAAM has a maximum engagement range of approximately nm

In the past IR missiles were generally only useful at visual ranges that is when

pilots of opp osing aircraft could see each other distances of several nautical miles or

less This limitation placed IR missiles rmly in the short range missile category

However more capable IRST sensors reference section A esp ecially those on

Russian aircraft such as the Su and MIG have help ed push the launchenvelop e

for IR missiles well past the short range missile category into the medium range

missile categoryFor example the Russian AA Archer has an engagementrange

from nm to nm dep ending of the engagement scenario AdditionallyIR

versions of the Russian A Alamo have a maximum engagement range of nm

for medium and high altitude and nm for low altitude targets using high IR

emitting afterburner engines

Range data is for newer RM version The older RM version had an engagement range

of nm to nm

Referenced distances are for the D version of the AA missile Russian designation R

ET The B version of the AA missiles Russian designation RT has a maximum

engagement range of nm for medium and high altitude and nm for low altitude targets using

high IR emitting afterburner engines The AA D version has a longer range solid prop ellant

ro cket than the AA B version

In contrast to IR missiles radar guided missiles can usually b e launched at

ranges much greater than a few nautical miles placing them in the medium and

sometimes long range missile category The United States has emphasized ghting

airtoair engagements from longer ranges and has fo cused development funding on

radar guided missiles that can op erate in the heart of the medium range missile

envelop e to nm or more For example the AIM AMRAAM can kill

an approaching headon target aircraft out to approximately nm while in a

tail chase engagement which requires the missile to overtake the target this range

drops to approximately nm The minimum engagement range for the AIM is

approximately to nm Although Russia has emphasized development of shorter

range IR missiles the Russians still have develop ed a capable semiactive radar

guided missiles in the AA Alamo The Russian AA Alamo has a maximum

engagement range of nm for medium and high altitude and nm for low

altitude targets page page page

The sp eed and op erating altitude capabilityoftheabove IR and radar guided

missiles varies The AIMM Sidewinder has a sp eed of Mach with a maximum

ight time of one minute and an unrep orted op erating altitude The AA Archer

has a sp eed of knots and an op erating altitude capability of ft m to

ft km The AA Alamo missiles IR and radar versions havean

unrep orted sp eed with an op erating altitude capability of sea level to ft

km for the A and B versions with the C and D versions p ossibly having some

limited capability to ft km Finally the AIM AMRAAM had a

rep orted sp eed of Mach and an op erating altitude capability of sea level to

ft km

The maximum engagement range of the AIM AMRAAM is quite op en to debate in the

literature Jane rep orts a maximum engagement range of nm while Brassey rep orts a range of

nm Clancy a range of nm and Missile Forecast a range of nm

Referenced distances are for the C version of the AA missile Russian designation RER

The A version of the AA missiles Russian designation RR has a maximum engagement

range of nm for medium and high altitude and nm for low altitude targets The AA D

version has a longer range solid prop ellantrocket than the AA A version

Typical measures of an airtoair missiles eectiveness are the noescap e zone

and the singleshot kill probability The noescap e zone refers to the volume of

airspace around the ring aircraft where a target aircraft will b e unable to get away

from a launched missile no matter how hard and fast the target aircraft maneuvers

The probability of the missile achieving a singleshot kill P isdenedasthe

ssk

pro duct of ve factors

P P P P P P

ssk e l g f k

where

P whether an aircraft successfully forces an opp onentinto an engagement

P whether the pilot res a missile

P whether the missile guides successfully

P whether the fuse works and

P whether the warhead kills the target

Assuming indep endent missile launches the probability of killing a single aircraft

can b e increased by ring multiple missiles For example consider these two missile

ring strategies

Sho ot In this strategy an aircraft res one missile at the target aircraft and

waits for the outcome of the missile shot b efore deciding to ring another

missile and

Sho otSho ot In this strategy an aircraft res a second missile at the target

b efore knowing the outcome of the rst missile

The probabilityofkillis P and P P P P for the Sho ot and

ssk ssk ssk ssk ssk

Sho otSho ot strategies resp ectively If P this results in for the

ssk

There are higher order cases such as Sho otSho otSho ot however the increase in probability

of kill is marginal when comparing current missile P s

ssk

Sho ot strategy and a for the Sho otSho ot strategy page page

page

Radarguided airtoair missiles unlike IR airtoair missiles are generally un

aected byenvironmental factors suchasweather or time of day This is a direct

result of the fact that radarguided airtoair missiles provide their own source of

illuminating energy required for guidance and homing where as IR missiles dep end

on the illuminating energy provided by the target aircraft IR missiles do not func

tion well in clouds or rain b ecause IR energy is absorb ed and dissipated bywater

vap or Additionally the greater the environmental background energy or radiation

eg sun reections o of water snow clouds or terrain the stronger the illu

minating energy provided by the target aircraft must b e in order for an IR missile

to see guidehome on the target aircraft Although radarguided airtoair mis

siles are aected byenvironmental factors suchasweather or time of daytheyare

susceptible to countermeasures eg jamming that aect the illuminating energy

generated by the missile Missile countermeasures will b e discussed in greater detail

in section A An additional disadvantage of an active radarguided missile is its

p otential for fratricide Once an active radarguided missile is separated from the

ring aircraft and lo cked onto a target aircraft the ring aircraft has no further role

in the missiles op eration In a close crowded engagement the active radarguided

missile may not b e able to discriminate b etween opp onent and friendly aircraft and

fratricide could o ccur page pages page

In contrast to presentday airtoair missiles to days aircraft guns app ear rel

atively lowtech However this app earance do es not imply that the aircraft gun

serves no useful purp ose in to days air engagementenvironment in fact quite the

opp osite is true Aircraft guns unlike airtoair missiles can b e used against surface

targets and have no minimum range requirements The aircraft gun is allasp ect

weap on with a circular weap on envelop e around the target aircraft depicting the

In the case of semiactive radar missiles the aircraft provides the illuminating source

guns maximum range Additionally a gun pro jectile is not vulnerable to counter

measures once it leaves the barrel The MIG and SU come equipp ed with the

GSH mm cannon The aircraft gun used on USAF ghters is the MMA

Vulcan mm cannon The version used in the FC Eagle holds rounds which

yields ab out seconds of ring time Ammunition used in the Vulcan includes the

PGU which has armor piercing explosive fragmentation and incendiary capa

bilities all in a single round page page page

A Command Control Communications and Information

Command control communications and information CI play a signicant

role in airtoair engagements As an example consider the rst night of the air

battle in the Gulf War During the rst nightofthewar USAF aircraft destroyed

the ma jority of the Iraqi air defense network Without groundbased command

and control Iraqi interceptors blasted around trying to intercept coalition strike

aircraft they could not detect Another example from the Gulf War involves a four

ship of FCs patrolling the skies over Baghdad An E AWACS warned the ight

of an Iraqi Mirage F which had just taken o and was closing on the four FCs

from astern With the AWACs warning one of the FCs broke hard turned b ehind

the Mirage and destroyed the F b efore it could inict damage on the ight page

At the heart of USAF CI is the E AWACs As mentioned in section A the

E AWACS ies on the edge of the air battle area scanning airspace for unidentied

aircraft referred to as b ogies and opp onent aircraft referred to as bandits

TypicallyanAWAC is organized into separate surveillance and control sections

In the surveillance section three to vetechnicians monitor the air trac in the

assigned airspace and pass on information to the control section Comp osed of twoto

veweap ons controllers sitting at multipurp ose consoles the control section guides

friendly aircraft to intercept bandits or b ogies Dep ending on its particular mission

Band Frequency MHz Wavelength m

VHF

UHF

Table Communication Bands

an AWACs also may carry senior sta ocers radar technicians radio op erators a

communications technician and a computer technician page page

An imp ortant asp ect of the CI is the communications equipmentavailable to

participants in airtoair engagements Voice communication generally takes place

in three frequency bands of the electromagnetic sp ectrum HF VHF and UHF

Table A shows the frequency and wavelengths of these communication bands The

ma jorityofvoice communication o ccurs in the VHF and UHF bands VHFUHF

communication systems are essentially lineofsight systems where the range of the

equipment is directly prop ortional to the aircrafts altitude In order to provide the

necessary range for long distance ights the HF band is used The FC Eagle

is equipp ed with a ANARC VHFUHF auxiliary transceiver and VHFUHF

auxiliary transceiver with cryptographic capability The basic ARC covers the

VHFUHF band providing channels over the range to MHz in KHz

increments In addition the FC is equipp ed with ANARC HF transmit

terreceiver This system provides channels over the range to MHz in

kHz increments page page page

In order to provide voice communication capabilityina heavy electronic cout

nermeasure ECM environment many USAF communication system including the

FCs ANARC are equipp ed with HaveQuick capabilityHaveQuick pro

vides the user with a jamresistant VHFUHF capabilityby giving the communi

cation systems a frequencyhopping technique The frequency hopping technique is

implemented by storing a pattern of frequencies to b e used for a given day and utiliz

ing the pattern according to the time of day The strength of HaveQuickliesinthe

apparently random manner in which frequencies are jump ed page page

Data communication is another imp ortant asp ect of any airb orne communica

tions system Airb orne communication systems such as the FC Eagles ALARC

provide for general data transmission Also USAF air combat aircraft suchas

the FC and FC Falcon are equipp ed with tactical air navigation TACAN

radios which allow pilots to determine the separation distances of aircraft within

the ight group to a resolution of a tenth of a mile In addition to the standard

communication and TACAN systems the new JointTactical Information Data Sys

tem JTIDS provides for expanded data communication b etween aircraft JTIDS

terminals allow the linking of any aircraft so equipp ed to an aerial lo cal network

The secure ie unjammable and untappable data link allows the sharing of in

formation from a planes sensors and other systems with other aircraft ships and

ground units equipp ed with JTIDS terminals JTID terminals are able to transmit

a full situational rep ort including radar contacts sending aircraft p osition altitude

and heading and even fuel and armament status counting gun b omb and mis

sile rounds onb oard JTIDS terminals are currently available on the E Sentry

AWACs as well as new E JointSTARS ground surveillance aircraft and select

FC aircraft page page page

JTIDS op erates on frequencies in the to MHz frequency band

sharing with TACAN but avoids the IFF transp onder frequencies which are also

in this band Future datafusion eorts b eyond JTIDS include collecting infrared

and ultraviolet scans from satellites or a RC Cobra Ball which can pickup

rapid movement suchasarocket launch and electronic intelligence ELINT from

platforms suchastheRCVW Rivet Joints which can plot opp onent radars and

their status Clearly systems like JTIDs are the only quiet form of communication

in strict emissions control situations page pages

A Electronic Warfare

Electronic warfare EW is dened as an action to determine exploit reduce

or prevent use of the electromagnetic sp ectrum and is organized into two ma jor cat

egories EW supp ort measures ESM and electronic countermeasures ECM ESM

involves actions taken to searchforintercept lo cate record and analyze radiated

electromagnetic energy while ECM involves actions taken to prevent or reduce use

of the electromagnetic sp ectrum page page page

ESM uses passive radar intercept or warning systems which detect radar emis

sions from aircraft radar missiles and surfacetoair missile installations ESM

systems are preset to cover the p ortion of electromagnetic sp ectrum used byex

p ected threat radars The simplest system is the radar warning receiver RWR

which supplies the relative b earing of the emission on an aircraft display Most

combat aircraft haveRWRs which are tuned to provide a warning only when an op

p onents re control radar has established a lo ck on the aircraft ESM systems then

increase in complexity through tactical ESM to full ELINT intelligencegathering

capability These systems are designed to nd and analyze radar emissions and then

classify the typ e of radar pro ducing the emissions ELINT capable combat aircraft

include sp ecialized EW aircraft such as the EFA Raven The range at whicha

radar emission is detected by a radar warning system is inversely prop ortional to the

squared radar range rather than the quaded radar range need for detection For this

reason radar warning systems can detect a radiating radar at far greater distances

then those of radars own target detection capability page page

With its Blo ck up date the E Sentry will b e equipp ed with four passive

ESM frequency receivers that are capabilityofintercepting radar emissions from

more than nm These receivers are also capable of detecting the mo de in which

the radar is op erating eg widescan narrowscan etc This ESM capability

can b e used to provide raw data as to the targets height and sp eed as well as to

classify the radar system and hence the aircraft using the system For example

search emissions characteristic of an NO Slot Back radar system would indicate

the target to b e a MIG page

The FC Eagle and FE Strike Eagle are equipp ed with the Loral AN

ALRC radar warning receiver RWR This RWR has a digitally controlled dual

channel receiver covering the E and Jbands to GHz and is capable of sorting

and identifying threat aircraft The ANALRC is part of the Tactical Electronic

Warfare System TEWS that provides control of all selfprotection systems The

TEWS also includes the Magnavox ANALQ electronic warfare system This

equipment probably covers the area b eyond the HIJ band of the ANALR

The SuSK Flanker and MiGC Fulcrum have the SPO Beryoza L

RWR On the SuSK the RWR sensors are mounted on the sides of the air

intakes and on the tailb o om while on the MiGC the sensors are on the wingro ot

extensions wing tips and p ort n pages

In addition to radar warning systems ESM also include dedicated missile

warning systems for b oth infrared and radar missiles Historically of all

aircraft shot in an airtoair engagementnever detected the opp onent that killed

them Missilewarning systems can provide spherical coverage around the air

craft enabling a pilot to know when a missile has b een red page

In contrast to a ESM systems ob jective to obtain information the ob jective

of an ECM system is to deny the opp onent information ie detection p osition

track and classication ECM systems may b e classied in a number of ways

main purp ose active or passive etc The most common typ e of ECM is radar

jamming Radar jamming denies the opp onent the use of sp ecic wavelengths or

p ortions of the electromagnetic sp ectrum by transmitting signals that degrade

the receptive capability of radar systems Radar jamming may b e activepowered

signals or passive no p ower required Twotypical examples of radar jamming are

noise jamming and deceptive jamming Noise jamming is an attempt to pro duce

a strong signal that will overp ower a target aircrafts return when it is received

by an opp onent radar Since reected target energy is more sensitive to target

range than received noise noise jamming is eective at longer distances but will

eventually b ecome ineective ie burnthrough at closer distances Deceptive

jamming involves techniques that generate false targets on an opp onent aircrafts

radar page page page

Radar Jamming is deployedemployed in ve ma jor ways stando escort

mutual supp ort co op erative selfscreening and standforward In the stando

jamming SOJ case the jamming platform remains close but outside the range of

the opp onents weap ons SOJ platforms typically employ highp ower jamming in

order to disrupt radar systems at the long ranges required to stay out of opp onents

weap ons range In the escort jamming case the jamming platform accompanies the

strikepackage jamming radars to protect the strike aircraft In the mutual supp ort

case the jamming capabilities of individual jamming platforms are co ordinated as

combat elements against several radar systems In the selfscreening case a jammer

is used to protect the carrying aircraft For example FC carries the internally

mounted Northrop ALQV radar jammer which op erates automatically re

quiring only that pilot turn it on Finally in the standforward jamming case the

jamming platform is lo cated b etween the strikepackage and the opp onent radars

and jams the radars to protect the strikepackage page page

In addition to radar jamming one can jam the data and voice communications

of opp onent aircraft degrading the command and control For example in the

Lebanon air war the Syrians launched waves of aircraft to intercept the Israelis

but the Israelis jammed the data and voice communications on which the Syrian

pilots were dep endent As a result the Syrian ghters were reduced to uncontrolled

The standforward jammer is usually within the range of opp onentweap ons and may suer a

high attrition rate Thus only the use of relativelowcost remotely piloted vehicles is practical for

this mission

singles trying to op erate against a sup erior number of wellcontrolled opp onent air

craft page

ECM is also used to defeat airtoair missiles For radarguided airtoair mis

siles several classes of ECM exist The rst class are devices like gate stealers that

attack the tracking circuitry of missile without angle deception The next class is ex

p endables Passive exp endables likecha and blivets and more advanced forms like

illuminated cha and active exp endables The next class is high p ower systems like

Cross Eye and Cross Pole that disrupt or break angle track For IR airtoair missiles

the conventional ECM like ares and ash lamps are designed to defeat reticle and

pseudo imaging seekers The ANALE chaare disp enser in the F Eagle

automatically programs the ejection of chaare cartridges in resp onse to inputs

from the ESM systems eg in resp onse to evasive aircraft maneuvers Full automa

tion takes the guesswork out of timing the go o d discharge of chaare page

page

In order to counter opp onent ECM many aircraft and missiles are equipp ed

with electronic countercountermeasures ECCM to ensure the eective use of the

electromagnetic sp ectrum despite opp onentECMFor example many radars are

frequencyagile over a wide band If the rate of frequency agility is fast enough a

jammer may not b e able to follow the frequency changes and the radar can counter

the jamming eects Akey role is also played by those ECCM techniques which

cannot b e classied as electronic suchashuman factors metho ds of radar op era

tion and radar deployment tactics For example emission control the appropriate

assignment of op erating frequencies to various radars can impact ECM page

This typ e of ECCM is eective against sp ot jamming where the jamming frequency is centered

around a narrow bandwidth Barrage or broadband jamming radiated across the entire band of

the radar sp ectrum could still jam frequency agile systems

As a nal note many ESM systems control the deployment and op eration of

ECM and ECCM with the link b etween ESM ECM ECCM often automatic For

example a missilewarning system could automatically deploy exp endable counter

measures cha and ares or a radar system may automatically change frequencies

when jammed Automated ESM systems reduce pilot workload and react in real time

to threats such as a missile launch page cite page Cla page

A Tactics and Maneuvers

The sole purp ose of air combat aircraft is to destroy other aircraft The aircraft

itself is only a weap ons platform designed to bring weap ons into a ring p osition

against an opp onent aircraft The goal of all air combat aircraft maneuvers and

tactics is to p osition the aircraft into weap on ring parameters while simultaneously

frustrating those of the opp onent An aircraft maneuver p ositions an aircraft in the

three dimensional combat arena by actuating its control vectors of roll pitch and

thrust An aircraft tactic on the other hand is a series of these maneuvers designed to

accomplish a sp ecic ob jective in the airtoair engagement For example supp ose

the mission for a particular group of friendly aircraft is to intercept a group of

opp onentstrike aircraft ie a BARCAP mission The friendly aircraft will rst

assess the initial geometry of the engagement and select an air tactic to establish

contact with the opp onent aircraft After acquiring the opp onent aircraft with their

sensors the friendly aircraft will attempt to place the opp onent aircraft within the

lethal ring envelop e of their weap ons by a series of appropriate maneuvers As

the friendly aircraft close in on the opp onent aircraft the opp onent aircraft can b e

exp ected to resp ond with a series of their own tactics Hence a sequence of interactive

maneuvers and tactics will develop b etween the two groups of aircrafts In general

the actual maneuvers and tactics selected in a given airtoair engagement dep end

on each pilots understanding or p erception of the combat situation Tactics and

the corresp onding maneuvers are designed to takeadvantage of the strengths of

the aircraft and weap ons and exploit anyweaknesses in the opp onent Each tactic

has strong and weak p oints and can b e anticipated and defeated byanopponentif

applied to o often with the same routine pages page pages

Perception is p erhaps the most imp ortant factor in the air environmentin

determining tactics and maneuvers The task of the pilots in the ight is to obtain

as much tactical information electronic visual communicated as p ossible and then

to lter and analyze the information The abilityofeachmemb er of the ight

to gather and to pass critical information to other memb ers of the ightplays a

signicant role in success in the air environment An imp ortant concept in the

employment of the tactics and maneuvers is the organization of the ightinto aircraft

or ship elements USAF aircraft typically engage opp onent aircraft in two or four

ship elements For fuel economy as well as tactical reasons a tactical spread

out formation is preferable to a close formation In a tactical formation aircraft

are p osition abreast ab out to nm dep ending on air environment conditions

and are usually stacked high or low ab out nm Aircraft elements co ordinate

tactics and maneuver and share workloads Once engaged ights usually attempt

to maintain only twoship elements This size has b een found to b e most eective

in maintaining mutual supp ort while minimizing co ordination problems page

pages

In the ma jority of cases the side taking the rst shot in an airtoair engage

ment seizes the initiativeby forcing opp onent aircraft to react against the missiles

to prevent a kill A missile in the air generally attracts the attention of the pilot

in the target aircraft often causing him to forget ab out the ring aircraft or ring

his own weap ons For this reason rst shot can b e critical even if not successful A

Two fuel states are generally considered in mission planning Bingo fuel and Joker fuel Bingo

fuel is the fuel state that allows an air combat aircraft to return to a tanker or home base with an

acceptable safety reserve Joker fuel is the fuel state that the pilot of an air combat aircraft should

think ab out disengaging if involved in an airtoair engagement

missile in the air establishes a psychological set b etween combatants that places the

pilot of the target aircraft in a defensive frame of mind This reaction maymakethe

opp onent more vulnerable to other weap ons as well Getting the rst shot requires

longrange weap ons and identication systems As discussed in section A allasp ect

airtoair missiles can b e red at a target aircraft from any direction Although all

asp ect capable most of these missiles are b etter in some situations than in other

with b eam asp ects often causing the most diculty reference Figure One way

to increase a missiles eective range is to re at a signicantly higher altitude than

the opp onent aircraft This gives the missile a reserve of p otential energy that can

b e converted into kinetic energy page pages pages

As shown in section A a typical airtoair engagement starts with nm

plus range separation b etween friendly and opp onentights An airspace controller

AWACS or GCI may p oint the friendly ight in the right direction but the ight

must eventually nd the opp onent aircraft on their sensor systems In order to

get into weap ons parameters the friendly ightmust intercept reduce the range

between the opp onent ight The choice of intercept tactics dep ends on the en

gagement situation Many factors contribute to this choice for example mission

goals rules of engagement the geometry of the engagement numb er of friendly and

opp onent aircraft friend and opp onent aircraft and weap on capabilities capability

of the pilots oensive and defensive p otential and so on In addition to placing

the friendly ight within weap ons parameters of the opp onentight the intercept

tactics chosen should keep the opp onentight from getting into weap ons param

eters Intercept tactics have names such as singleside oset trail sweep pincer

and drag The basics of almost all of these intercepts involve frontquarter and

The choice b etween optimizing oensive or defensive p otential often dep ends on the likeliho o d

and results of b eing attacked by the opp onent ight If the ightiscondent they can detect

and neutralize an opp onentight attack and either defeat or escap e the opp onent then optimizing

oensive p otential is a reasonable choice Otherwise a defensive p osture should b e taken

Refer to chapter of reference for other tactics

sternconversion intercepts As the name implies the frontquarter intercept is one

in which the intercepting ight approaches the target ight from the target ights

front quarter The stern conversion converts an initial front quarter intercept into

a nal rearhemisphere p osition for the ight The b ottom line on intercept tactics

is that there are many tactics to cho ose from and choice is highly scenario dep en

dent page page pages page

During the intercept p ortion of airtoair engagements most of the friendly

ights knowledge of opp onentisprovided by onb oard sensors mainly radar and

communications Throughout the course of the intercept the friendly ight attempts

to comprehend the tactical situation in terms of the spatial three dimensional lay

out and temp oral relationships b etween the ight other friendly ights and the

opp onent The pro cess of developing and up dating a spatial layout of the opp onent

is called radar sorting Sorting requires the ight to distinguish all p otential targets

in terms of azimuth elevation range and velo city The goal of sorting is to de

cide which opp onent aircraft to attack After deciding which aircraft to attack the

friendly ightmust b ecome more sp ecic in controlling the intercept geometryUntil

this p oint general rules of thumbwere sucient eg oset left or right stay at high

altitude Beyond Visual Range BVR weap ons employment is the culmination

of the intercept if the rules of engagementallow The friendly ight compares its

tactics to the opp onentights tactics and ies within the weap ons envelop e of the

missile pages pages

If all has gone according to plan and BVR weap ons are launched the ightmay

pro ceed to within visual range WVR op erations IR missiles and guns Conversely

Sorting answers such questions as Howmany threat aircraft are out thereWhat formation

are they inWhat are they doingThe goal is to sort the opp onen t b efore entering opp onent BVR

missile range

An altitude oset can prevent or delay acquisition of the friendly ightby the opp onent ight

if radar sorting pro cedures did not dierentiate each individual aircraft or the

ROE did not allowaBVRweap ons employment eg visual identication required

or insucient identication obtained the ightmayhave to continue into closer

range of the opp onent ight As a result the ightmay opt to forego BVR weap ons

employment and pro ceed directly to short range op erations Information provided

by the electronic supp ort measures ESM eg RWR helps the ight decide whether

to continue oensive actions or to execute defensive actions for survival If defensive

the ightmay b e forced to ab ort the engagement page

By nm or less from the target the ighthasentered the WVR transition

zone At this zone each pilot in the ightmust decide whether his aircraft is in

an oensive defensive or neutral p osition and whether he should attack evade and

reengage or disengage Factors that determine p osition include radar and ESM

situational awareness for example a pilot might consider his aircraft to b e in

An oensive p osition if he has high radar SA and a blank RWR scop e

A neutral p osition if he has a radar lo ck and a single RWR spike and

A defensive p osition if low radar SA and manyRWR spikes

If deciding to attack the pilot must execute short range op erations which rely on air

combat maneuvering tactics infrared missiles and guns and visual contact rather

than sensors On the other hand an evade and reengage action may b e attempted if

the pilot elects to pursue a followon BVR attack If deciding to disengage the pilot

should p erform defensive maneuvering to separate from the engagement page

When pressing to WVR typically nm or less two basic tactical approaches

are available the angles ght and the energy ght In the angles approach the

pilot rst seeks to gain a p osition advantage angles even at the exp ense of relative

Sorting is dicult within nm of the threat b ecause of rapid changes in threat azimuth and

altitude

energy and then maintain or improve this advantage in order to get within weap on

ring parameters In the energy approach the pilot rst seeks to gain an energy ad

vantage without yielding a decisive p osition disadvantage Once an energy advantage

has b een gained the pilot converts this advantage into a lethal p osition disadvan

tage without giving up the entire energy advantage Each of these approaches has

advantages and disadvantages dep ending on the weap ons involved page

If friendly aircraft enters a turning engagement termed dogght with an

opp onent aircraft he allows the other aircraft in the opp onentight an opp ortunity

to sho ot USAF tactics emphasize early shots causing disruption keeping airsp eed

up and avoiding getting drawn into a dogghttyp e engagement Pilots must realize

their escap e window or safe path out of an engagement Factors that the drivethe

p osition of the escap e window include

range from opp onent ight the greater the range the more op en the escap e

window

energy relative to opp onent aircraft the greater the energy the more op en the

escap e window and

angleo and asp ect with opp onentight headon pass typically gives the b est

chance for an op en escap e window

A turning engagement among several aircraft can quickly degenerate into a fur

ball typ e of ght A furball is not desired since each pilot is op erating indep en

dently and giving mutual supp ort only by presence page page vii

page

In a turning engagement maneuver capability is paramount In this closein

environment basic ghter maneuvers BFM describ e how aircraft maneuver against

each other in oneversusone combat BFM describ es sp ecic concepts of air combat

aircraft turns turning ro om and turn circles and is generally group ed in three cate

gories dep ending on the p osition of aircraft relative to the opp onent in the airtoair

engagement

Oensive BFM Oensive BFM implies a p osition andor sp eed advantage

over the opp onent The goal of oensive BFM is to shot down the opp onent

in the minimum amount of time Hence maneuvers center around placing the

opp onentinweap on envelop es while at the same time denying the opp onent

weap on launch opp ortunities Anytime a weap on shot can b e taken it should

b e done

Defensive BFM Defense BFM implies a p osition andor sp eed disadvantage

over the opp onent The goal of defensive BFM is to stayalive and separate

from the opp onentifgiven the opp ortunity Maneuvers center around creating

weap on launch problems for the opp onentthus buying time and surviving a

little longer By extending the engagement the opp onentmay b e forced into

an error If the opp onent res a missile the b est way to ght the missile is with

asp ect by placing the oncoming missile on the b eam asp ect Missiles

generally havetheworst lineofsight and rate problems to solve at this asp ect

The practice of b eaming in combination with electronic countermeasures such

as cha and are disrupts and frequently defeats the tracking capabilityofthe

missile and causes the missile to miss

Neutral BFM Neutral BFM implies a null p osition andor sp eed over the

opp onent and vice versa Two options exists to the pilot he can separate or

he can stay and attempt to get an oensive p osition on the opp onent If the

pilot choices to engage he must b e aware of his p osition in regard to the escap e

window

Time considerations are imp ortant in turning engagements Then though a pilot

may b e winning the engagement he started with one opp onent he may b e losing an

engagement with a second opp onent he do es not see page

As a nal note most airtoair kills are against aircraft that have no idea that

they are ab out to b e red up on The further away an aircraft can re a missile at

an opp onent and still have the missile b e eective the b etter Of particular concern

to USAF planners are opp onent air combat aircraft attacking high value assets such

as AWACs and tankers using radar quiet intercept RQI techniques initially relying

solely on passive detection systems such as radar warning receivers and infrared

search and track IRST systems page pages

Appendix B THUNDERs AirtoAir Engagement Submodel

THUNDERs airtoair engagement submo del consists of two separate engage

ment mo dels describ ed by the mo del do cumentation as low resolution and high

resolution Volume I I page In order to describ e the high resolution airto

air engagement submo del some background information on THUNDERs aircraft

entity mo deling structure is appropriate Three hierarchical aircraft entity classes

exist in THUNDER Flight Group Flight and Aircraft Figure illustrates these

entity classes and their hierarchy

The basic aircraft entity class mo deled in THUNDER is the aircraft An

aircraft entity consists of a platform and weap on loads eg a FC with four

AIMs AMRAAM and four AIMs Sidewinder The next aircraft entity

class in the hierarchy is the ight A ightentity consists of a group of homogeneous

aircraft entities eg a four ship of FCs each carrying four AIMs and four

AIMs The last aircraft entity class in the hierarchy is the ight group A ight

group entity consists of a group of homogeneous andor heterogeneous ights eg

twoights of four FCs or a ight of four FCs escorting a ightoftwo FCs

to a ground target

As noted in Figure when an airtoair engagement o ccurs b etween two air

craft ight groups one is termed the intercepting ight group and the other the target

ight group An airtoair engagement in THUNDER is triggered by an intercepting

ight group detecting a target ight group Detection o ccurs in two manners

The target ight group is detected while ying through an air zone patrolled

bytheintercepting ight group The intercepting ight group is p erforming a

BARCAP mission The BARCAP mission is describ ed in section A or

The description of the THUNDER airtoair engagement submo del will b e limited to the high

resolution submo del onlyThelow resolution mo del is trivial in comparison to the high resolu

tion mo del For further information concerning the low resolution mo del the reader may refer

to Flight Flight Group Group

Flight Flight

Aircraft Aircraft

Intercepting Target

Figure Thunder Aircraft Entity Classes

The target ight group is detected by the intercepting ight group after the in

tercepting ight group has own to an intercept p oint to engage the target ight

group The intercepting ight group is p erforming a DCA ODCA or HVAA

mission DCA ODCA or HVAA missions are describ ed in section A

Before explaining the detection pro cess in THUNDER it might b e helpful to

understand howight paths are mo deled Thunder uses a horizontal x y square

grid system to mo del ight paths Each grid intersection denes a p ossible p oint

a ight path may pass through Mission planning mo dules in THUNDER pro duce

a ight path for every ight group Flight paths are either direct or nondirect in

nature dep ending on a departure aireld setting If a ight group is given a direct

ight path the ight group pro ceeds on a direct route to the target In nondirect

routing the ight group pro ceeds to a p oint p erp endicular to the target and a setback

distance on the friendly side of the forward line of tro ops FLOT and then pro ceeds

directly to that target Nondirect ight paths are intended to minimize the time

and distance sp entover opp onent territory A few discrete altitude z settings may

b e used for eachight path p oint Volume I I page

The air zones patrolled byanintercepting ight group p erforming a BARCAP

mission consist of a rectangular patrol area whose size is determined bythe type

number and ight size of aircraft ying the BARCAP mission Sp ecically for each

aircraft typ e input variables width and depth dene a rectangular area that can b e

patrolled by that aircraft typ e and the input variable ight size denes the standard

patrol unit size for that aircraft typ e The width of the patrol area is calculated

as the pro duct of the width variable and greatest integer value of the ratio of the

numb er of aircraft in the intercepting ight group to the ight size The depth of

the patrol area corresp onds to the depth variable Figure shows a patrol area

for a four ship of MiGs consisting of twoights ie ight size variable for the

MiGs is two Each MiG can cover an area of width nm km and

depth nm km Hence the patrol area of the fourship of MiGs has a

width of nm and a depth of nm Volume I I pages

If the ight path of the target ight group enters the patrol area of the inter

cepting ight group the target ight group is sub ject to detection The probability

of detection at search time t P t is based on a random search algorithm

SW t

P t exp

where

S the sp eed of the target ight groups path through the patrol area

W the diameter of the searchsweep

t the time the intercepting ight group has sp ent searching for the target

ight group and

In the case of the BARCAP mission the intercepting ight group is comp osed of only one

aircraft typ e

The ight size variable allows for element formations For example consider a fourship of

aircraft split as two ship elements In this scenario the eective search width would b e doubled

with each element acting as a separate search unit

Figure is taken from Volume I I page Depth (108 nm)

Aircraft Width (65 nm) Flot

Target Flight

Flight Group Width (130 nm) Group Path

Figure THUNDER BARCAP Patrol Area

A the search area the intercepting ight group must cover page

This search algorithm places a searchpathoflengthL St into the search area

A in a random fashion This means that the lo cation and course of the search path

at anygiven time is indep endent of the lo cation and course at other times not to o

close to the rst The random search mo del in equation yields a lower b ound on

search eectiveness pages

The variables W and A in equation are dep endentonanumerical ex

pression of the degreeofcommand and control among aircraft in the intercepting

ight group The concept of command and control will b e addressed in greater de

tail later in this section For illustration purp oses consider the intercepting ight

groups command and control value FGCC tobeagiven value b etween and

The search strategies are

If the intercepting ight group was in a p erfect command and control state

FGCC aircraft in the intercepting ight group would conduct individ

ual ight size searches in specic strips of the search area or

If the intercepting ightgroupwas in a no command and control state

FGCC aircraft in the intercepting ight group would combine the

diameter of individual aircraft searchsweeps over the entire patrol area

Figure shows the search pattern for the p erfect command and control state

for the four ship of MiGs In this state eachight element searchs a dierent

subsection of the patrol area dened by the aircraft patrol area parameters width

sear ch

W is dened as the diameter of the searchsweep or circle whichis where A

sear ch

is the searchsweep area For a single aircraft A is sw eep detect where sw eep is the

sear ch

sweep angle of the radar in degrees and detect is the distance at which the aircrafts radar detects

the target

Dened by the ight size variable

and depth For this state denoted

w idth depth sw eep

A W detect fsize

fsize

where

detect the distance at whichanintercepting aircrafts radar detects the target

ight group

fsize intercepting aircraft ight element size

n thenumber of intercepting aircraft in the patrol area

sw eep intercepting aircrafts radar sweep angle

w idth intercepting aircraft patrol area width and

depth intercepting aircraft patrol area depth

Figure shows the search pattern for no command and control state for the four

ship of MiGs In this state each aircraft combines aircraft searchsweeps to search

the entire aircraft patrol area For this state denoted

sw eep

Aw idth depth n W detect

page

Letting Pt and Pt represent the probability of detection at search time

t under search state and resp ectively and weighting the probabilities in search

state and by FGCC and FGCC resp ectively yields

S W t S W t

A A

P t FGCC exp FGCC exp

Equation is evaluated at

t mint t

tar g et sear ch Depth (108 nm)

detect Flight Group Width (130 nm)

fsize * sweep

Figure THUNDER BARCAP SearchPattern for Perfect Command and Con trol State

Depth (108 nm)

n * sweep

detect

Flight Group Width (130 nm)

Figure THUNDER BARCAP SearchPattern for No Command and Control

State

where

t the time the target ight group will b e in the patrol area and

target

t the time since the intercepting ight groups last airtoair engagement

sear ch

ie search time page

Once the value of P t is determined a random drawismadefroma U

to determined if the intercepting ight group detects the target ight group If

the intercepting ight group detects the target ight group an airtoair engagement

takes place else the target ight group passes unhindered through the patrol area

In order to illustrate equation consider the fourship of MiGs packaged

as two elements mentioned ab ovecovering a patrol area of width of nm and a

depth of nm Assume that a four ship of FCs enters the patrol area traveling

at knots parallel to the depth of the patrol area ie t hrs in the

patrol area Further assume that the MiGs have a radar detection range of

nm against the FCs a radar sweep angle of and a FGCC of For this

situtation

W nm nm

nm nm

A nm

S W t

exp P exp

and

W nm nm

A nm nm nm

S W t

P exp exp

A detection takes place if the value of P t is greater than or equal to the random draw

The concept of radar detection range will b e detailed shortly

Thus the detection probability for this example is P whichis

Intercepting ight groups p erforming a DCA ODCA or HVAA missions are

rst alerted to target ight group by a GCI detection event A GCI detection event

o ccurs when target ight group is detected by a groundbased or airb orne early

warning AEW platform Groundbased detection o ccurs when the target ight

group crosses a groundbased detection line The lo cation of this line is dep endent

on several factors including the altitude high or low of the target ight group

Sp ecically the detection distance dist from the groundbased radar lo cation is

computed as

dist mindist dist RC S

GB max i

where

dist the groundbased radars maximum detection range

max

dist the groundbased radars one square meter target detection range de

p ending on the target ight groups altitude i low high and the given radar

band and

RC S the target ight groups radar cross section in the groundbased radar

band

Once within the groundbasedradars detection distance THUNDER considers the

target ight group to be detectedAnintercepting ight group is then senttointercept

the target ight group at the earliest p ossible p oint Volume I I page

Similarly airb orne early warning AEW detection o ccurs when the target

ight group enters the radar detection circle around the AEW platforms orbit p oint

The AEW detection distance dist is computed as

AE W

dist mindist dist RC S los

AE W max tar g et

where

dist the AEW platform radars maximum detection range

max

dist the AEW platform radars one square meter target dectection range

tar g et

for the given radar band

RC S the target ight groups radar cross section in AEW platforms radar

band and

los line of sightbetween the AEW platform and the target ight group

los is computed as

r r

los r cos cos

r al t r al t

tar g et AE W

where

r radius of the earth km

al t altitude of the target ight group and

target

al t altitude of the AEW platform

AE W

Equation assumes a round cueball earth Once within the AEW radars

detection distance THUNDER considers the target ight group to bedetectedAn

intercepting ight group is then senttointercept the target ight group at the earliest

p ossible p oint Volume I I page

For DCA ODCA and HVA missions once the intercepting ight group arrives

at its target p oint ie the p oint where the target ight group is rep orted to b e

THUNDER computes the radar detection range detect at which the intercepting

ight group radar can detect the target ight group based up on the intercepting

Equation uses the angle from the center of the earth to the AEW platform and the target

ight group in place of the tangent of the angle THUNDER do cumentation claims this approxi

mation holds since aircraft altitudes are small with resp ect to the radius of the earth

ight groups radar and target ight groups radar cross section RCS and use of

jammers This radar detection range is the same range used to calculate W in

equation Detection o ccurs if closest distance b etween the target ight groups

ight path and the target p oint dist is less than the detection range detect

closest

ie if

dist detect

closest

Volume I I page

The detection range detect is calculated using a horizontal x y detection

range hdetect and the altitude z separation distance al tbetween the intercepting

and target ight groups Based on the Pythagorean relation

hdetect al t if al t detect

detect

if al t detect

The THUNDER horizontal x y detection range algorithm uses the fact that

the radar cross section RC S of a ight group in a given radar band is prop ortional to

the horizontal radar detection range hdetect raised to the fourth p ower page

This is expressed mathematically as

RC S hdetect

Eachight group in THUNDER has a radar cross section RCS RCS is an

area measure of the ight groups ability to reect search radar energy The RCS of

the ight group is radar band or frequency sp ecic ie the RCS of the ight group

varies dep ending of the band frequency of the search radar For each applicable

radar band the RCS of a ight group is determined by nding the maximum radar

cross section of the ights in the ight group For example supp ose a ight group is

comp osed of three ights with the following radar cross sections for a given radar

band

FlightARCS square meters

FlightBRCS square meters

FlightCRCS square meters and

The RCS of this ight group is square meters The RCS of a ight is calculated

minfsizen RC S

where

fsize aircraft ight element size

n is the numb er of aircraft in the ight and

RC S aircraft RCS

THUNDER assumes that the maximum numb er of aircraft that would b e in the

resolution cell of the search radar is the ight size sp ecied for particular aircraft

this assumes aircraft in a ight are ying suciently close to each other that they

do not app ear as separate targets The RCS of an aircraft for a given radar band

is a function of the aircraft fuel and weap ons conguration and is calculated as the

sum of

basic aircraft RCS

aircraft conguration RCS delta eg FC vs FE

fuel conguration RCS delta eg drop tanks and

weap ons conguration RCS delta eg missiles

THUNDER has historically based the basic aircraft RCSona onose asp ect at the given

radar band frequency of the search radar

Volume I I page

Using the relationship in equation the nonjammed ie assuming no

radar jamming horizontal detection range hdetect for a ight group of radar

noj am

cross section RC S at a given radar band is

hdetect hdetect RC S

noj am sq noj am

where hdetect is the nonjammed horizontal detection distance for a RCS

sq noj am

of unit area Volume I I page

Jamming eects in THUNDER consists of escort main lob e and stando

side lob e These jamming eects are assumed to b e used only when the search

radar is detected This assumption allows the jammed horizontal radar detection

range hdetect to b e related to unjammed horizontal detection range hdetect

jam

The jammed horizontal radar detection range is also referred to as the burnthough

range the range at which a jammed search radar can burnthough the jamming

Using the relationship in equation the jammed horizontal detection range

hdetect for a ight group of radar cross section RC S at a given radar band is

jam

hdetect hdetect RC S

jam sq jam

where hdetect is the jammed horizontal detection range for a RCS of unit

sq jam

area The hdetect value for escort jamming is determined using a lo okup table

sq jam

of ranges by jammer typ e and radar typ e The hdetect value for stando

sq jam

jamming is set in an input curve describing the range as a function of stando

jammer range jammer typ e and radar typ e When multiple jammers are presentthe

As the name implies escort jamming is an electronic countermeasure ECM tactic in whichthe

jamming platform accompanies the strike package jamming radars to protect the strike package

In contrast in stando jamming the jamming platform remains close but outside the range of

opp onent air defense systems jamming these systems to protect the strikepackage page

jammed horizontal detection range is taken to b e the minimum jammed horizontal

detection range jamming eects are not combinedVolume I I page

The THUNDER high resolution airtoair engagement submo del hence re

ferred to as the THUNDER airtoair engagement mo del adjudicates combat via

three pro cesses

A pro cess that calculates a single shot probability of kill SSPK for each

aircraft against p ossible opp onent aircraft

A pro cess that aggregates aircraft versus aircraft SSPK calculations to attrition

rates for ightversus ight and ight group versus ight group levels

A pro cess that assesses engagement losses using draws from binomial distribu

tions

In order to illustrate the pro cesses as they are describ ed let us consider the case

where a target ight group comp osed of a single ight ies through an air zone

patrolled byanintercepting ight group comp osed of a single ight Sp ecically

consider an intercepting ight group comp osed of a ight of four MiGs each

carrying two AAs two AAs and two GSH and a target ight group comp osed

of a ight of four FCs each carrying four AIMs and four AIMs ying a

FSWP mission reference section A The intercepting ight group MiGs will

b e referred to as side A and the target ight group FCs will b e referred to side

A single shot probability of kill SSPK is calculated as follows for each aircraft

versus opp onent aircraft combination

SSP K EN G LC H PK

where

SSP K single shot probability of kill of an aircraft against an opp onent air

craft

EN G probability of engaging the opp onent aircraft

LC H probability of ring a weap on at the opp onent aircraft and

PK probability of killing an opp onent aircraft given a weap on ring

Since a ight is comp osed of homogeneous aircraft in THUNDER the SSPK calcula

tion can also b e viewed as a ightversus ightentitylevel pro cess The discussion of

the SSPK calculation highlights whichentitylevel aircraft or ight the engagement

pro cess is b eing viewed from

An engagement wil l always occur in THUNDER if the intercepting ight group

detects the target ight group The probability of an aircraft or ight engaging an

opp onent aircraft or ight addresses those air environmentvariables that aect the

ability to engage after initial detection The probability of engaging the opp onent

EN G at the aircraft or ight level is mo deled in THUNDER as the pro duct of

two factors

A general engagement probability EN G based on whether or not either

AE W

ight group is under airborne or groundbasedcontrol termed in THUNDER

Airb orne Early Warning AEW control and

A mo dication comp onent TAC reecting the ight group tactics own by

each side

A ight group is consider to b e in AEW control if the controller is in b oth com

munication range of the friendly ight group and radar range of the opp onent ight

group Four separate AEW control states are p ossible

Neither side is in AEW control

The own ight group is in AEW control only

The opp onent ight group is in AEW control only and

EN G EN G EN G EN G

Side A MiGs

Side B FCs

Table THUNDER Example Probability of Engagement Based on AEW Con

trol State

Both the own and opp onentight groups are AEW control

For the current case let EN G AE W i A B refer to the probability of en

gagement conditioned on AEW control state AE W Table shows the

probabilities for our example The probability of engagement conditioned on AEW

control state is opponent platform dependentFor example in our case the FCs

would have a dierent set of engagement probabilityvaluesiftheywere to engage

MiGs rather than the MiGs For illustration purp oses let us assume that b oth

ight groups are in AEW control thus from Table

EN G AE W EN G and

EN G AE W EN G

As mentioned ab ove the next contributing factor to a sides EN G is the sides

ight group tactics Flight group tactics are implemented at the ight level For the

intercepting ight group the ight group can cho ose to employ one of these three

ight group tactics

The intercepting ight group engages the target ight groups escort aircraft

only or

The intercepting ight group engages the target ight groups other non

escort aircraft only or

The intercepting ight group engages the target ight groups escort aircraft

with a p ercentage a and other nonescort aircraft with a p ercentage a

Escort aircraft and hence escort ights are those entities that have b een given an

AirtoAir Escort AIRESC mission reference section A In THUNDER a is

referred to as the allo cation ratio The allo cation ratio is calculated using Lanchester

dierence equations and an allo cation measure of eectiveness Section details

this calculation Since the target ight group FCs in our example do es not

contain a ight with an AIRESC mission recall the FCs are ying an FSWP

mission the intercepting ight group allo cation ratio is a

For the target ight group the ight group can cho ose to employ one of these

three ight group tactics

All escorts split from the target ight group to engage the intercepting ight

group or

All escorts stay with the target ight group or

A p ercentage m of the escorts stay with the target ight group and a p ercentage

m of the escorts split from the target ight group to engage the intercepting

ight group

In THUNDER m is referred to as the mutual supp ort defender ratio The mutual

supp ort defender ratio is a user inputted variable For our case let m

Figure shows the interaction of the intercepting and target ight group

tactics Four mo dication comp onents TAC exist dep ending on the typ e of target

ight group involved escort or other

If a ight from the intercepting ight group is engaging an escort ight from the

target ight group TAC a The term a represents the allo cation of

aircraft in the intercepting ight to engaging target ight group escort aircraft

If a ight from the intercepting ight group is engaging an other nonescort

ight from the target ight group TAC a The term a represents the

allo cation of aircraft in the intercepting ight to engaging target ight group

other aircraft

Escort Target Flight Other NonEscort Target Flight

Intercepting Flight EN G a EN G a

AE W AE W

Table ProbabilityofEngagingATarget Flight EN GFor An Intercepting

Flight in THUNDER

If an escort ight from the target ight group is engaging a ightfromthe

intercepting ight group TAC a m m The term a

m represents the p ortion of the escort ight that engages a p ortion of an

intercepting ight engaging escort aircraft The term m represents the p ortion

of the escort ight protecting the other nonescort aircraft in the target ight

group

If an other nonescort ight from the target ight group is engaging a ight

from the intercepting ight group TAC a mm The term a m

represent the p ortion of the other nonescort ight that engages a p ortion

of an intercepting ight engaging the other nonescort aircraft The term

m represents the p ortion of the escort ight protecting the other nonescort

aircraft in the target ight group

Table and Table show the probabilities of engaging the opp onentEN G

at the aircraft or ight level when for the intercepting and target ights resp ectively

For our example with a and m

A A

EN G EN G AE W a and

B B

EN G EN G AE W a m m

In this example ight group tactics do not aect the engagement probabilities since

there are no escort aircraft involved

The next variable in equation the probabilityofringaweap on at the

opp onent aircraft LC H is a combination of several factors INTERCEPTORS a 1-a

m ESCORTS 1-m

NON-ESCORTS

Figure Interaction of Flight Group Tactics in THUNDER

Intercepting Flight

Escort Target Flight EN G a m m

AE W

Other NonEscort Flight EN G a mm

AE W

Table ProbabilityofEngagingAnIntercepting Flight EN GFor A Target

Flight in THUNDER

Relative Range Advantage variable indicating the number of weap ons of a

sp ecic weap on typ e an aircraft can re b efore an opp osing aircraft can re

a single weap on of a sp ecic weap on typ e

Opp onents Probability of Engagement ie Opp onents ENG

Number of Friendly and Opp onents Aircraft in Each Flight

Opp onents Probability of Kill ie Opp onents PK and

Opp onents Force Multiplier

For each aircraft in an airtoair engagement THUNDER determines the impact

of weap on rings using the concept of a comp osite or aggregate weap on Recall

each aircraft in THUNDER consists of a platform and weap on loads eg a FC

with four AIMs AMRAAM and four AIMs Sidewinder For each airto

air engagement THUNDER sp ecies a userinputted maximum numb er of launches

for each aircraft For our example the maximum number of weap on rings p er

engagement for the FCs and MiGs is set at Additionally for each aircraft

THUNDER sp ecies a userinputted prioritized list of weap ons rings For our

example the prioritized list of weap ons rings for the MiGs and FCs are

Firing Sequence MiG FC

st AA AIM

nd AA AIM

rd AA AIM

th AA AIM

th GS AIM

th No Ammo AIM

If the FC were to use its maximum number of weap on rings during an

engagement the F would re an AIM and an AIM THUNDERs airtoair

engagement mo del assumes that an aircraft wil l always take its maximum number

of weapon rings in an engagement unless the aircraft has an insucient number of

weapons remaining For our example the FC and MiG aircraft entities will

always re weap ons p er engagement unless they have only remaining weap on

to re Anaggregate weapon is created from the weap ons to b e red in the air

toair engagement For our example the aggregate weap on for the FC will b e a

combination of an AIM and an AIM and for the MiG a combination of an

AA and AA The sp ecics of these combinations will b e addressed shortly

In order to re an aggregate weap on at an opp onent aircraft one of these three

scenarios must exist for the ring aircraft These scenarios are

The ring aircraft has an overall relative range advantage and can re its

aggregate weap on at the opp onent aircraft without threat of b eing killed by

the opp onent aircrafts aggregate weap on or

This situation is not p ossible in our case since the total number of weap ons for each aircraft

divided by the maximum numberofweap on rings in an engagementareintegervalued for the

FC and for the MiGs

AIM AIM

AA AA GSH

AIM

GSH

Table Weap on versus Weap on Relative Range Advantage for FC and MiG

Example

The ring aircraft do es not haveanoverall relative range advantage but the

opp onent aircraft can not engage the ring aircraft or

The ring aircraft do es not haveanoverall relative range advantage but the

ring aircraft survives an sp ecied numb er of rings of an aggregate weap on

by the opp onent aircraft

Each of these mutually exclusive scenarios has a probability of o ccurrence These

probabilities are then added together to determine LC H the probability of ring

aweap on at an opp onent aircraft entity Let these probabilities of o ccurrence b e

designated by LC H i resp ectivelyThus LC H LC H LC H LC H

In order to compute LC H the probability of o ccurrence of scenario one

the concept of relative range advantage needs to b e addressed THUNDER denes

relative range advantage as the number of weap ons of a sp ecic weap on typ e an

aircraft can re b efore an opp osing aircraft can re a single weap on of a sp ecic

weap on typ e Table shows the weap on versus weap on relative range advantages

for our example As can b e seen from Table an FC armed with AIM s

has a single relative range advantage against a MiG armed with either AAs

AAs or GSH s In other words the FC can re an AIM b efore a MiG

armed with either AAs AAs or GSHs re a weap on Additionallya

MiG armed with AAs has a single relative range advantage against an FC

armed with only AIMs

Scenario one equates to the situation were the ring aircraft can re a weap on

at the opp onent aircraft b efore the opp onent aircraft res a weap on at it In our

example there are four p ossible combinations of rst weap ons to b e red b etween a

FC and a MiG These combinations are

AIM versus AA

In weap on combinations and the FC has a single relative range advantage

over the MiG In weap on combination the MiG has a single relativerange

advantage over the FC In combination neither aircraft has a relativerange

advantage Assuming that weap on combinations are random the FC will havea

single relative range advantage over the MiG of the time the MiG will

have a single relative range advantage over the FC of the time and neither

aircraft will have a relative range advantage over the other of the time Hence

B A

and LC H LC H

A B

LC H and LC H may also b e derived from the aggregate weap on concept

In our example assuming the maximum number of weap ons rings each the

FC will re an AIM and AIM and the MiG will re an AA and AA

resp ectively each engagement The aggregate weap on for the FC will b e an equal

combination of an AIM and an AIM and for the MiG an equal combination

of an AA and AA In other words the aggregate weap on for the FC will b e

have of the characteristics of the AIM and of the characteristics of the

AIM THUNDER terms these characteristics fractional launches Let FL denote

the fractional launches of weap on typ e j for side i i A B For the MiGs

i A let j represent the AA and j represent the AA resp ectively

For the FCs i B let j represent the AIM and j representthe

AIM resp ectively Hence for our example

Now

A B A

LC H FL FL

B A B A B

LC H FL FL FL FL

LC H the probability of o ccurrence of scenario two dep ends on the o ccurrence

of two indep endentevents

The ring aircraft do es not have a relative range advantage on the opp onent

aircraft The probabilityofthiseventis LC H

The opp onent aircraft opp do es not engage the ring aircraft The probability

of this eventis EN G

opp

Hence LC H LC H EN G For our example

opp

A A B

LC H LC H EN G

B B A

LC H LC H EN G

LC H the probability of o ccurrence of scenario three dep ends on the o ccur

rence of three indep endentevents

The ring aircraft do es not have a relative range advantage on the opp onent

aircraft The probabilityofthiseventis LC H

The opp onent aircraft opp engages the ring aircraft The probability of this

eventis EN G

opp

The ring aircraft survives n weap on rings from the opp onent aircraft opp

b efore his rst weap on ring The probabilityofthiseventis PK

opp

where PK is the probabilityofkillgiven the ring of an opp onent aircraft

opp

entitys aggregate weap on

The probability of kill given the ring of an aggregate weap on is calculated in manner

similar to the calculation of LC H PK values in THUNDER areweapon versus

platform dependent Let PK denote the probabilityofkillofweap on typ e j for side

i i A B Again for our example the MiGs i A let j representthe

AA and j represent the AA resp ectively Fpr the FCs i B let j

represent the AIM and j represent the AIM resp ectively Hence for our

example

Now

A A A A A

PK FL PK FL FL

B B B B B

PK FL PK FL FL

The number of weap ons rings from the opp onent aircraft n is determined

by considering the pro duct of two factors

A force multipliervariable FMthatallows particular weap ons to act as force

multipliers reducing the attrition of the aircraft entity carrying them

The ratio of numb er of aircraft entities in b oth engaging ightentities

The force multipliervariable FM is calculated via the following equation

P P

fac opp opp opp

opp

FL FL L FL FM minRRA

opp

i j

i j j ij

where

i represents p ossible weap on typ es of the ring aircraft fac

j represents p ossible weap on typ es of the opp onent aircraft opp

RRA is the relative range advantage of the opp onent aircraft and weap on typ e

j against the ring aircraft and weap on typ e iand

L representthenumber of weap on rings the opp onent aircraft maytake

opp opp

The rst term in the ab ove equation minRRA L FL represents the

opp

ij j

advantage the opp onent aircraft has in ring weap on typ e j against the ring aircraft

ring weap on typ e iThisadvantage takes into accountthenumber of weap ons of

typ e j that can b e red by the opp onent aircraft in an airtoair engagement For

example the opp onent aircraft with weap on typ e j mayhavea two shot advantage

opp

RRA over the ring aircraft with weap on typ e i but it mayhaveonlyone

opp

Hence the opp onent aircraft with weap on weap on of typ e j to re L FL

opp

typ e j only has a one shot advantage min over the ring aircraft with weap on

fac opp

represents the fraction of rst weap on FL typ e i The second term FL

i j

rings by the opp onent aircraft ring weap on typ e j and the ring aircraft ring

weap on typ e iFor our example letting the AIM and AA have a subscript

designation of and the AIM and AA have a subscript designation of the force

multipliervariables are

X X

A A A A B

minRRA FM L FL FL FL

ij j j i

i i

min min

X X

B B B B A

FM minRRA L FL FL FL

ij j j i

i i

min min

Letting x i A B representthenumb er of aircraft on side i for our example

the number of weap ons rings from the opp onent aircraft n is

A A

n FM

B B

n FM

Thus the probabilityoftheevent that the ring aircraft survives n weap on rings

from the opp onent aircraft b efore his rst weap on ring for our example is

For our example LC H the probability of o ccurrence of scenario three is

A A B B n

LC H LC H EN G PK

A A n B B

EN G PK LC H LC H

The probability of ring an aggregate weap on LC H is the sum of the prob

abilities of o ccurrence for the three scenarios given ab ove Sp ecically LC H

LC H LC H LC H For our example LC H is

A A A A

LC H LC H LC H LC H

B B B B

LC H LC H LC H LC H

For our example the last variable in equation the probability of killing an

opp onent given ring a weap on PKwere shown in the discussion of scenario three

LC H tobe

Hence for our example the single shot probability of kill SSPK using equa

tion are

A A A A

SSP K EN G LC H PK

B B B B

SSP K EN G LC H PK

The second pro cess in the THUNDER airtoair engagement mo del aggregates

the aircraft versus aircraft SSPK calculations to attrition rates for the ightversus

ight and ight group versus ight group levels The pro cess is conducted in the

following manner for each ring ightversus opp onentightcombination

The attrition rate of a single aircraft in the opp onentight when engaged by

the entire ring ight ring one weapon each is calculated

The attrition rate of the single aircraft in the opp onent ight when engaged

by the entire ring ight ring al l available weapons is calculated

First for each ring ightversus opp onentightcombination the attrition

rate of a single aircraft in the opp onentight when engaged bytheentire ring

ight ring one weapon each is calculated This attrition rate calculation determines

two discrete probabilities

The probabilityof i aircraft from the ring ighti fac where fac is

the numb er of aircraft in the ring ight engaging the opp onent aircraft

The probabilityof j out of i engaging ring aircraft kill an opp onent aircraft

AEW Control No AEW Control

MiG AC C Value

FC AC C Value

Table Aircraft Degree of Command and Control AC C Values for THUNDER

Example

The pro cess used to calculated the probabilitythat i aircraft from the ring ight

will engage the opp onent aircraft is based on a numerical expression of the degree

of command and control among aircraft in the ring ight The concept of com

mand and control was intro duced with the probability of detection in equation

For each aircraft a userinputted variable termed aircraft degree of command and

control AC C AC C is provided An AC C corresp onds to p er

fect command and control and an AC C corresp onds to no command and

controlVolume I I page AC C is a sub jective measure of the degree of com

mand and control given an aircrafts sensor suite and weap ons conguration ACC

values are also dep endent on whether or not the ring ight group is under airb orne

or groundbased control termed in THUNDER Airb orne Early Warning AEW

control Table shows the ACC values for the ights in our example Earlier we

assumed that b oth ight groups were in AEW control thus

AC C

A ights degree of command and control FCCvalue is assumed to b e the average

of all the AC C values of the aircraft in the ight Considering that a ightentity

by denition is comp osed of homogenous aircraft entities FCC AC C

THUNDER assumes a weighted combination of two ring ight engagement

strategies based on the degree of command and control of the ight These strategies

are

If the ring ightwas in a p erfect command and control state FCC

aircraft in the ring ightwould divide themselves evenly over the aircraft in

the opp onentight or

If the ring ightwas in a no command and control state FCC

aircraft in the ring ightwould divide themselvesrandomlyover the aircraft

in the opp onent ight

Letting S ifac represent the probability that i aircraft from the ring

ight engage an aircraft from the opp onent ight under the rst strategy

fac

m if i m

opp

fac

m if i m

opp

otherwise

fac

suchthatxistheinteger p ortion of the value x and opp is the num where m

opp

b er of aircraft in the opp onentight Under the second strategy I binfac

opp

Letting S ifac represent the probability that i aircraft from the ring

ight engage an aircraft from the opp onent ight under the second strategy

fac

C B

i faci

if i fac

opp opp

otherwise

Let P i fac represent the probability that i aircraft from the ring

ight engage an aircraft from the opp onentight Weighting the probabilities in

strategy one S and strategy twoS by FCC and FCC resp ectively yields

i i

P FCC S FCC S

i i i

For our example the MiG and FC ights b oth have ring aircraft entities

For the MiG ight

P FCC S FCC S

For the FC ight

P FCC S FCC S

Al lowing for the possibility of multiple kil ls of an opponent aircraft by dierent

ring aircraft the probabilityof j outofthei engaging ring aircraft killing an

opp onent aircraft is

B C

ij j

SSP K SSP K

Hence for each ring ightversus opp onent ightcombination the attrition rate

probability of kill of a single aircraft in the opp onentightentity when engaged

by the entire ring ight is ring one weapon each

fac

X X

C B

ij j

AT T R P SSP K SSP K

i j

or equivalently

fac

P SSP K AT T R

For our example

ATTR

Next for each ring ightversus opp onentight combination the attrition rate

of a single aircraft in the opp onentight when engaged bytheentire ring ight

ring al l available weapons is calculated This attrition rate calculation is similar to

that shown for the ring ight ring one weap on each Two discrete probabilities

are determined by the calculation

The probability of that the opp onentight will b e engaged i times by the ring

ightentityi flwherefl is the maximum number of weap on rings

of the ring ight

The probabilitythat k out of the i engagements by the ring ight kill an

opp onent aircraft

THUNDER assumes a weighted combination of two ring ight group engage

ment strategies based on a numerical expression of the degree of command and con

trol among ights in the ring ight group and on the maximum numb er of launches

any aircraft in the ight group can makemax The strategies are

If the ring ight group is in a p erfect command and control state FGCC

ights in the friendly ight group would divide themselves evenly over the

ights in the opp onent ight group or

If the ring ight group is in a no command and control state FGCC

ights in the friendly ight group would divide themselves randomly over the

ights in the opp onent ight group

A ight groups degree of command and control FGCCvalue is assumed to b e the

average of all FCC values of the ights in the ight group

Letting S jmax represent the probability that j ights from the

ring ight group engage a ight from the opp onent ight group entity under the

rst strategy

max

m if j m

fli

max

m if j m

fli

otherwise

max

such that xistheinteger p ortion of the value x and fli is the num where m

fli

b er of ights in the opp onentight Under the second strategy J binmax

fli

Letting S jmax represent the probability that j ights from the ring

ight group engage a ight from the opp onentight under the second strategy

fac

C B

j maxj

if j max

fli fli

otherwise

Let P jmax represent the probabilitythat j ights from the

ring ight group engage a ight from the opp onentight group Weighting the

FG FG

probabilities in strategy one S and strategy twoS by FGCC and

j j

FGCC resp ectively yields

FCC S FCC S P

j j

A note of clarication may b e helpful to the reader at this time THUNDER assumes

that an opp onent ight in the opp onentight group will b e engaged at most max

times by ights in the ring ight group When calculating the probability of that

the opp onent ight will b e engaged i times bya specic ring ighti fl

where fl is the maximum number of weap on rings of the ring ight THUNDER

uses the appropriate values of P For example consider a ring ight group

comp osed of two ights Assume ight one and two can re and weap ons p er

engagement resp ectively The most an opp onentight in the opp onent ight group

can b e engaged is times max The probability that ight one will engage

FG FG

the opp onent ightonceandtwice is P and P resp ectively

For our example eachight group has only one ight Thus each ight will

engage the other with the maximum numb er of ring opp ortunities The MiG

and FC ights b oth havetwo ring opp ortunities each Hence for the MiG

FG FG

and FC ights P and P

Al lowing for the possibility of multiple kil ls of an opponent aircraft by the

dierent engagements the probabilitythat k out of the i engagements by the ring

ight kill an opp onent aircraft is

B C

ik k

ATTR AT T R

Hence for each ring ightversus opp onent ightcombination the attrition rate

probability of kill of a single aircraft in the opp onentight when engaged bythe

entire ring ight is ring al l available weapons

X X

B C

FG ij j FG

P AT T R AT T R ATTR

i j

or equivalently

FG FG i

AT T R P AT T R

For our example

ATTR

In order to ensure that the attrition rates AT T R that a ring ight inicts

on opp onent ights are less than or equal to one THUNDER normalizes the attrition

rates for each ring ightbymultiplying the attrition rate by the fraction of the

numb er of rings by the ring ight and the numb er of total rings by the ring

ight group or

nor m FG

AT T R AT T R

where fl designates ight rings and tl designates total ight group rings For our

nor m FG

example fl tl thus AT T R AT T R

The last pro cess in the THUNDER airtoair engagement mo del is the loss

assessment The loss assessment pro ceeds as follows

For each opp onentight in the opp onentight group the numb er of aircraft

lost due to the engagement with each ring ight in the ring ight group is

nor m

calculated using a draw from a binomial distribution binopp AT T R

The actual amount of aircraft loss is the minimum of the binomial draw and

the number of weap on rings designated for the ring ight

The total amount of aircraft loss from the opp onentight is the minimum of

the sum of actual aircraft losses for each ring ight in the ring ight group

and the numb er of aircraft in the opp onentight

This assessment is p erformed for each opp onentight in the opp onent ight

group

Clearly actual aircraft losses can not exceed weap on rings

Clearly aircraft losses can not exceed the numb er of aircraft in the opp onent ight

Appendix C Event Occurrence Network Solution for One Versus

One AirtoAir Engagement

Figure shows the EON for this one vs one airtoair engagement from Hong

et al Recall for this engagement that each side aircraft has two missiles

one radarguided and one infrared The radarguided missile must always b e red

b efore the infrared missile Let G representtheevent where side i res a

missile of typ e j j where radarguided and infrared at side i that

either misses or sho ots down side i where i represents the opp onent

side of side iFor example the event G represents the concluding event that

the aircraft on side red a radar missile and shot down the aircraft on side

For each no de in gure has the following sequence of events o ccur

No de no events havetaken place

No de G

No de G G

No de

G G or

G G

No de G G

G No de G

G G No de G

No de

G G G or

or G G G

G G G

No de

G G G or

G G G

No de

G G G or

G G G

No de

or G G G

G G G or

G G G

No de G G G

No de

G G G G or

G G G G

No de

G G G G or

or G G G G

G G G G or

G G G G

No de

G G G G or

G G G G

No des marked with an denote a sequence of intermediate events that end with

a concluding event Hence no des and end with

concluding events while no de ends the engagement since no events are left to

o ccur ie b oth aircraft are out of weap ons

Let continuous random variable E represent the o ccurrence time of event

G with

density function f t

distribution function F t and

complementary distribution function F t

Also let p and q p represent the probabilities that missile j from side i

i i i

j j j

sho ot downs and misses resp ectively side i given that the missile j is red For the

example shown in chapter IV

E exp t with ie side res a radar missile

E exp t with ie side res a heat missile and

exp t with ie side res a heat missile E

Additionally the probabilities that the radar and infrared missiles on side and

sho ot down the other side when red are p p and p p

resp ectively

Exact expressions for the probabilities in Tablewere derived and are sum

marized b elow

P t P E tE t

Z Z

y dy x dx f f

E E

t t

Z Z

exp x dx exp y dy

t t

exp t

P t P E E

F x f x dx

exp x p exp x dx

p exp t

P t P E t E E t

c c

x dx t x f F F

E E

exp t exp t x q exp x dx

q exp t exp t

P t P E tE t E

c c

F F t y f y dy

E E

exp exp t t y q exp y dy

q exp t exp t

E P t P E

F y dy y f

y dy exp y p exp

p exp t

E E P t P E

Z Z

x t

dx x dx x x f f x F

T T E E T

exp x

p exp x x q exp x dx dx

T T

p q exp t

exp t

P t P E E t E t

Z Z

t x

t f x x f F x dx dx

E T E T

exp t

Z Z

t x

q exp x x q exp x dx dx

T T

q q exp t

exp t exp t

P t P E E E

Z Z

t y

f y F y x f x dx dy

E E

p exp y

dy exp y x q exp x dx

q p

exp t exp t

E tE E tE E P t P E

E tE E tE E P E

Z Z

t y

c c

dy F t y f y F t x f x dx

E E

Z Z

x t

c c

dx y dy t y f F x t x f F

E E

exp t y q exp y

exp t x q exp x dx dy

exp t x q exp x

exp t y q exp y dy dx

q q exp t

exp t exp t exp t

P t P E E E

Z Z

x t

dx x f y dy x y f F

E E

p exp x

dx exp x y q exp y dy

q p

exp t exp t

P t P E tE E t

Z Z

t y

f F t y y f y dx dy

E T E T

exp t

Z Z

t y

q exp y y q exp y dy dy

q q exp t

exp t exp t

E E P t P E

Z Z

y t

dy f y y f y dy y F

T E T E T

exp y

y dy exp y y q dy exp p

T T

exp t p q

exp t

P t P E E E

Z Z Z

y x t

dy f x x f x dx dx f y

E T E T E

p exp y

Z Z

y x

dy q exp x x q exp x dx dx

T T

q q p

exp t exp t

exp t

P t P E E tE E E E t

t E E E tE E P E

P E E tE E E t

Z Z Z

t x y

F t y f f y x x f x dx dy dx

E E T E T

Z Z Z

t x x

f x x f x F t y f y dy dx dx

E T E E T

Z Z Z

t y x

F dy x dx dx y x x f t y f f

T T E E E

Z Z

t x

exp t y q exp y

q exp x x q exp dy dx x dx

T T

Z Z

t x

q exp x x q exp x

y dy dx dx exp t y q exp

exp t y q exp y

Z Z

y x

dy q exp x x q exp x dx dx

T T

q q q

exp t

exp t exp t

exp t

E E E E E P t P E

P E E E E E E

Z Z Z

t x y

x dx dy dx x x f y f F x y f

T T E E T E

Z Z Z

t x x

f x x f x x y f y dy dx dx F

E T E T E T

Z Z

t x

exp x y q exp y

dy dx p exp x x q exp x dx

T T

Z Z

t x

p exp x x q exp x

dx dx exp x y q exp y dy

T T

q p q

exp t

exp t exp t

P t P E E E E E E

P E E E E E E

Z Z Z

t y y

f y x f y y f F x dx y dy dy

E T E T E T

Z Z Z

t y x

dx dy y dy y y f F x f y x f

T T E E T E

Z Z

t y

p exp y y q exp y

dy dy exp y x q exp x dx

T T

Z Z

t y

exp y x q exp x

p exp y y q exp y dy dx dy

T T

q q p

exp t

exp t exp t

t E E E tE E P t P E

P E E tE E E E t

t E E tE E P E

Z Z Z

t y x

f y y f y dy dx dy F t x f x

E T E T E

Z Z Z

y t y

F x dx dy dy y t x f y y f f

T E T E E

Z Z Z

x y t

dx F t x f x f y y f y dy dy

E E T E T

Z Z

t y

exp t x q exp x

q exp y y q exp y dy dx dy

T T

Z Z

t y

q exp y y q exp y

exp t x q exp x dx dy dy

exp t x q exp x

Z Z

x y

q exp y y q exp y dy dy dx

T T

q q q

exp t

exp t exp t

exp t

P t P E E E

Z Z Z

y t x

dx y dy dy y y f f x f

T T E E E

p exp x

Z Z

y x

dx q exp y y q exp y dy dy

T T

q q p

exp t exp t

exp t

P t P E E E E E E t

P E E E E E t

P E E E E E E t

Z Z Z Z

t y y x

T T

x dx dy dx dy y x x f y y f f f

T T T E T E E E

Z Z Z Z

t y y x

T T

f y y f y f x x f x dx dx dy dy

E T E E T E T T

Z Z Z Z

x t y x

T T

f y y f y dy dx dx dy f x x f x

E T E T T E T E

Z Z Z

t y x

T T

p exp y y q exp y

q exp x x q exp x dx dy dx dy

T T T

Z Z

t y

p exp y y q exp y

Z Z

y x

q exp x x q exp x dx dx dy dy

T T T

Z Z Z

t y x

T T

q exp x x q exp x

dx dx dy p exp y y q exp y dy

T T T

q q q p

exp t

exp t exp t

exp t

exp t exp t

P t P E E E E E E t

t E E E E P E

t E E E E E P E

P E E E E E t

P E E E E E E t

Z Z Z Z

y t y x

T T

f y y f y f x x f x dx dy dx dy

E T E E T E T T

Z Z Z Z

t y y x

T T

f y y f f y x x f x dx dx dy dy

E T E E T E T T

Z Z Z Z

x t y x

T T

x x x f f dx dx dy y dy y y f f

T E E T T T E E

Z Z Z Z

t x y x

T T

f x x f x f y y f y dy dx dy dx

E T E E T E T T

Z Z Z Z

t x x y

T T

f x x f x f y y f y dy dy dx dx

E T E E T E T T

Z Z Z Z

y y x t

T T

f y y f f y x x f x dx dy dy dx

E T E E T E T T

Z Z Z

t y x

T T

q exp y y q exp y

q exp x x q exp dy dx dy x dx

T T T

Z Z

t y

q exp y y q exp y

Z Z

y x

x dx dx dy dy exp x x q exp q

T T T

Z Z Z

x y t

T T

q exp x x q exp x

dx dx dy q exp y y q exp y dy

T T T

Z Z Z

t x y

T T

q exp x x q exp x

q exp y y q exp y dy dx dy dx

T T T

Z Z

t x

q exp x x q exp x

Z Z

x y

dx dx q exp y y q exp y dy dy

T T T

Z Z Z

t x y

T T

q exp y y q exp y

dy dy dx q exp x x q exp x dx

T T T

q q q q

exp t

exp t exp t

exp t

exp t exp t

q q q q

exp t

exp t exp t

exp t

exp t exp t

t E E E E E P t P E

t E E E E P E

t E E E E E P E

Z Z Z Z

x t x y

T T

dx dy dx y dy y y f f x x x f f

T T T E E T E E

Z Z Z Z

t x x y

T T

f x x f x f y y f y dy dy dx dx

E T E E T E T T

Z Z Z Z

y y t x

T T

x dx dy dy dx x x f f y y y f f

T T T E E T E E

Z Z Z

t x y

T T

p exp x x q exp x

dx dy dx q exp y y q exp y dy

T T T

Z Z

t x

p exp x x q exp x

Z Z

y x

dx dx q exp y y q exp y dy dy

T T T

Z Z Z

t x y

T T

q exp y y q exp y

dy dy dx p exp x x q exp x dx

T T T

q p q q

exp t

exp t exp t

exp t

exp t exp t

The no des of EON corresp ond on a oneonone basis with states of the underlying

sto chastic pro cess absorbing states corresp ond to event sequences that end with a

concluding event and transient states corresp ond to event sequences that end with

an intermediate event

Appendix D Polynomial Approximation

D Norms

One of the most basic problems in the numerical metho ds literature is the

problem of approximating a known function f with a approximation function h

In order that h b e a go o d approximation to a given function f we require the

error function e ie f h to b e small in some sense

More precisely the criterion of a small error function is quantied with a

metric known as a norm A norm is a realvalued function jj x jj dened on a linear

space X suchthat

jj x jj Positivity

jj x jj if and only if x Positive Deniteness

jj x jj j j jjx jj for any arbitrary scalar Homogeneity and

jj x y jjjjx jj jj y jjTriangle Inequality

The pair X jj jj is referred to as a normed linear space The norm of the error

function jj e jj or jj f h jj determines the adequacy of the approximation The b est

approximation h istheapproximation such that the condition

jjjjf h jj jj f h

The framework of metric spaces provides a metho d of measuring the go o dness of an approxi

mation since one of the basic prop erties of a metric space is a distance function Sp ecically a pair

X d is a metric space if X is a nonempty set of ob jects called p oints and d is a realvalued

function from X X to R called the metric of the space satisfying the following four prop erties

for all p oints x y z in X

Positive dx y and dx x

Strictly Positive if dx y then x y

Symmetry dx y dy x and

Triangle Inequality dx y dx z dz y

for all h in the approximation set AHowever unless A is a compact setabest

approximation may not exist

Generally many norms can b e dened on a given linear space X Clearly

the choice of norm plays an imp ortant role in the adequacy of the approximation

what may b e a go o d approximation when measured in one norm maybea very

p o or approximation in another norm One of the most widely used norms

in approximation theory is the L norm For the normed linear space C

of continuous realvalued functions dened on the interval the L norm is

dened using the error function as

p jj e jj j ex j dx

jj e jj sup j ex j

For the ndimensional normed linear vector space R of real column vectors xthe

L norm is dened using the error function as

p jj e jj j ex j

px i

jj e jj max j ex j

x i

The norms dened in equations and are referred to in the literature as semi

norms Seminorms satisfy prop erties and of a norm but not necessarily

prop erty b ecause jj e jj may not imply ex

D Polynomial Interpolation and Approximation

Interp olation refers to the pro cess of approximating a function f xwithhx

using a nite numb er of sp ecied data p oints More preciselygiven the n paired

values x fx i naninterp olating approximating function hxis

i i

Closed and b ounded

dened such that

hx f x i n

i i

The criterion in equation minimizes the error function e for the L semi

norm given in equation

An interp olating function hx can b e used to estimate the value of f at any

arbitrary p oint xIfx lies in the interval x x then the estimate is termed inter

p olation else the estimate is extrap olation Many dierenttyp es of interp olating

functions can b e used eg p olynomials rational functions trigometric functions

For p olynomial interp olation there is a unique p olynomial of degree n or less

that satises the n constraints in equation

As the number of interp olation p oints increases the degree of the interp olat

ing p olynomial generally must b e increased to t these sp ecied p oints However

high degree p olynomials typically have an oscillatory nature that can lead to in

accurate approximations within intervals dened by the interp olation p oints Ad

ditionally for functions that change their shap e over dierentintervals a single

interp olation p olynomial is generally not a go o d approximation For these reasons

an alternativeinterp olation approach uses a dierent p olynomial over eachsubinter

val This approach is referred to in the literature as interp olation with piecewise

p olynomial functions and will b e discussed further in section

In many cases it may b e unreasonable to require that the approximating func

tion h exactly t the function f at sp ecied p oints ie hx f x Two cases

i i

when an exact t may not b e desired are

If some question exists as to the accuracy of the individual values f x i

n eg f x f where the errors are unknown or

i i i i

These functions change their character to o much to b e approximated by a p olynomial of

low degree and using a high degree p olynomial can cause signicantapproximation error within

intervals due to the oscillatory nature of the high degree p olynomial

If a simple function is desired eg as is our case f is approximated bya

p olynomial function h thatiseasiertointegrate

The rst case falls under the area of regression in the literature Although this

dissertation do es not view approximation from a regression p ersp ective ie errors

in the sp ecied p oints results from the regression literature when viewed from the

second p ersp ective are applicable

For the two cases mentioned ab ove another approximation criterion is required

One p opular criterion in the literature is the least squares criterion The least

squares criterion is the L norm shown in equations and with p The

L norm gives equal weight to the error function for each p ointin C and

R For some purp oses it is appropriate to require the approximation to b e b etter

have less error over some parts of the interval than over other parts To this end

weighted generalized versions of the L norm dened in equations and are

jj e jj p w x j ex j dx

jj e jj p w j ex j

pxw i i

resp ectively where w x and w are weight functions In equation wx is de

ned such that w x for all x p ositive function and w xdx For

equation the w are p ositive If w x and w the L norm in equa

i i p

tions and results

Necessary conditions to minimize the error function e ie f h for the

weighted L norm dened by equations and are

jj e jj

where c j m is the set of parameters for hFor C and R the

solution to equation is resp ectively

Z Z

w xf xh xdx w xhxh xdx

j j

r r

X X

w f x h x w hx h x

i i j i i i j i

i i

j m Equations and are referred to in the literature where h

as normal equations Both these equations contain scalar inner pro duct terms

Equation contains the scalar pro duct

f f w xf xf xdx

while equation contains the scalar pro duct

hu v i w u v

i i i

where f and f are arbitrary functions and u and v are arbitrary vectors

The smallness of jj e jj is strongly dep ends on the choice of approximating

function h One class of choices are those that have the form

hx c x

j j

Here h is a linear combination of basis functions selected a priori

In this case equations and reduce to a system of m linear equations for

determining the m co ecients unknowns c note eachofthem equations

contains all m co ecients For R equation is of the form

T T

A WAc A Wy

where

x x f x c

A y c W

x x f x c

r m r r m

j m m

c x the When h is a p olynomial function p of degree m ie p

set of basis functions f g in equation are the monic p olynomials

For this set of basis functions the matrix A in equation is of the xx

form

x x

For large m the matrix A WA can b e illconditioned making the linear system in

equation dicult to solvenumerically

One way of reducing the eect of illconditioning is to intro duce orthogonality

into the linear system Two functions or vectors a and b are orthogonal if ha bi

reference equations and Additionally a sequence of functions or vectors

is an orthogonal system if h i fori j and h i for all i

i j i i

If in addition jj jj for all i the sequence is called an orthonormal system Func

tions or vectors in an orthogonal system are linearly indep endent

When the approximating function h is comp osed of a nite orthogonal system

of basis functions ie

h c

j j

the b est approximation of h is

h fi

jj jj

where jj jj h i In this case orthogonality reduces equations and

j j

from a system of m equations in m unknowns c s to m equations each

of whichcontains only one unknown one c For R equation reduces to

Bc A Wy

where

h i x

m m j r

The matrix B is diagonal and do es not p ose conditioning problems

Orthogonal basis p olynomials on an interval a bmay b e generated using the

following recursive relation based on equation

Let b e the constant function a x b

Let b e the linear function x x x a x b where

h x i

j j

jj jj

Let b e dened by the threeterm recurrence relation

x x a x b and j

j j j j j

where

jj jj

This pro cess is referred to as the GramSchmidt orthogonalization pro cess

Two ma jor classes of orthogonal p olynomials are distinguished in the literature

classical and nonclassical The three systems of classical orthogonal p olynomi

als are the Jacobi Tchebyshe Gegenbauer and Legendre Laguerre and Hermite

These classical p olynomial systems are distinguished from nonclassical orthogonal

p olynomials by their asso ciation with a generalized Ro drigue formula from whichthe

classical orthogonal p olynomials may b e derived by successive dierentiation of func

tions of their weighting functions Additionally classical orthogonal p olynomials also

satisfy a SturmLiouville secondorder dierential equation The major advantage of

the classical systems is that they possess simple recurrencerelations which relate any

threesuccessive polynomials Recurrence formulae also exist for nonclassical p oly

nomials but they are more dicult to obtain However nonclassical orthogonal

p olynomials with any p ositiveweighting function w x may b e generated on

any part of the real line using the GramSchmidt orthogonalization pro cess detailed

ab ove

Appendix E Polynomial Coecients and Interval Data

Table shows the ten subintervals used in the piecewise p olynomial approxi

mation for the exp density function example shown in Section Figure

Tables and detail resp ectively the monic p olynomial co ecients c for the

exp density function example using LU Decomp osition and Singular Value De

comp osition for matrix inversion Here the p olynomial function in subinterval k is

of the form h t c t where m is the degree of the p olynomial

Tables and show the p olynomial co ecients c and orthogonal p oly

nomials t for the exp density function example Here the p olynomial

k k

twhere m is the function in subinterval k is of the form h t c

j j

numb er of orthogonal p olynomials in the approximation

Table shows the ten subintervals used in the piecewise p olynomial approxi

mation for the truncated N density function example shown in Section

Figure Tables and detail resp ectively the monic p olynomial co ecients

c for this density function example using LU Decomp osition and Singular Value

Decomp osition for matrix inversion Here the p olynomial function in subinterval k

c is of the form h t t where m is the degree of the p olynomial

and orthogonal p oly Tables and show the p olynomial co ecients c

nomials t for the truncated N density function example Here the

k k

p olynomial function in subinterval k is of the form h t c t where m

j j

is the numb er of orthogonal p olynomials in the approximation

For the p olynomial co ecients and interval data for the other examples in this

dissertation please contact the author

Subinterval Number Subinterval

Table Subintervals for Piecewise Polynomial Approximations of the exp

DensityFunction

For m

k k

k c c

For m

k k k

c c k c

For m

k k k k

k c c c c

Table Monic Polynomial Co ecients for exp DensityFunction Using LU

Decomp osition

For m

k k

k c c

For m

k k k

c c k c

For m

k k k k

k c c c c

Table Monic Polynomial Co ecients for exp DensityFunction Using Sin

gular Value Decomp osition

Polynomial co ecients

k k k k

k c c c c

Orthogonal Polynomials

k t Polynomial

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

Table Orthogonal Co ecients and Polynomials for exp DensityFunction

Orthogonal Polynomials

k t Polynomial

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

Table Orthogonal Co ecients and Polynomials for exp DensityFunction

Continued

Subinterval Number Subinterval

Table Subintervals for Piecewise Polynomial Approximations of the truncated

N DensityFunction

For m

k k

k c c

For m

k k k

c c k c

For m

k k k k

k c c c c

Table Monic Polynomial Co ecients for the Truncated N Density

Function Using LU Decomp osition

For m

k k

k c c

For m

k k k

c c k c

For m

k k k k

k c c c c

Table Monic Polynomial Co ecients for the Truncated N Density

Function Using Singular Value Decomp osition

Polynomial co ecients

k k k k

k c c c c

Orthogonal Polynomials

k t Polynomial

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

Table Orthogonal Co ecients and Polynomials for the Truncated N

DensityFunction

Orthogonal Polynomials

k t Polynomial

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

t t

t t t

t t t t

Table Orthogonal Co ecients and Polynomials for the Truncated N

DensityFunction Continued

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Vita

Ma jor Denhard is currently a joint air op erations analyst at USCENTCOM s

Combat Analysis Division at MacDill AFB Up on completing a Bachelor of Science

degree in mechanical engineering from Carnegie Mellon Ma jor Denhard was com

missioned in the United States Air Force through the Reserve Ocer Training Corp

program in May Assigned to the Training System Program Oce at the Aero

nautical Systems Center in January Ma jor Denhard managed several aircraft

simulator programs including the F and F unit training device and F B

and B improved visual system upgrade In March Ma jor Denhard earned a

Master of Science in Op erations ResearchattheAirForce Institute of Technology

AFIT

Up on completion of his Masters Ma jor Denhard was selected to followon in

the Department of Op erational Sciences PhD program Ma jor Denhard arrived at

the Air Force Studies and Analysis Agency AFSAA on the Air Sta in and

was assigned to the Air and Space Sup eriority branches While at AFSAA Ma jor

Denhard conducted numerous studies including F and JSF air to air and air to

ground mission eectiveness as well as assessments of strategic missile defense and

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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER

THEATER-LEVEL STOCHASTIC AIR-TO-AIR ENGAGEMENT MODELING 5b. GRANT NUMBER VIA EVENT OCCURRENCE NETWORKS USING PIECEWISE POLYNOMIAL APPROXIMATION 5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) 5d. PROJECT NUMBER

D. R. Denhard, Major, USAF 5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAMES(S) AND ADDRESS(S) 8. PERFORMING ORGANIZATION Air Force Institute of Technology REPORT NUMBER Graduate School of Engineering and Management (AFIT/EN) AFIT/DS/ENS/01-01 2950 P Street, Building 640 WPAFB OH 45433-7765 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) Air Force Studies and Analyses Agency 1570 Air Force Pentagon Maj Craig Knierem Washington DC 20330-1570 [email protected] 11. SPONSOR/MONITOR’S REPORT NUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED.

13. SUPPLEMENTARY NOTES

14. ABSTRACT This dissertation investigates a stochastic network formulation termed an event occurrence network (EON). EONs are graphical representations of the superposition of several terminating counting processes. An EON arc represents the occurrence of an event from a group of (sequential) events before the occurrence of events from other event groupings. Events between groups occur independently, but events within a group occur sequentially. A set of arcs leaving a node is a set of competing events, which are probabilistically resolved by order relations. An important EON metric is the probability of being at a particular node or set of nodes at time t. Such a probability is formulated as an integral expression (generally a multiple integral expression) involving event probability density functions. This integral expression involves several stochastic operators: subtraction; multiplication; convolution, and integration. For the EON probability metric, simulation is generally computationally costly to obtain accurate estimates for large EONs, transient nodes, or "rare" states. Instead, using research with probabilistic activity networks, a numerical approximation technique using piecewise polynomial functions is developed. The dissertation's application area is air-to-air combat modeling. 15. SUBJECT TERMS Air Combat, Event Occurrence Networks, Probabilistic Networks, Activity Networks, Piecewise Polynomial Approximation, Simulation

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON ABSTRACT OF Raymond R. Hill, Lt Col, USAF (AFIT/ENS) a. REPORT b. ABSTRACT c. THIS PAGE PAGES 19b. TELEPHONE NUMBER (Include area code)

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