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

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

Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 9-2001 Theater-Level Stochastic Air-to-Air Engagement Modeling via Event Occurrence Networks Using Piecewise Polynomial Approximation David R. Denhard Follow this and additional works at: https://scholar.afit.edu/etd Part of the Operations Research, Systems Engineering and Industrial Engineering Commons, and the Statistical Models Commons Recommended Citation Denhard, David R., "Theater-Level Stochastic Air-to-Air Engagement Modeling via Event Occurrence Networks Using Piecewise Polynomial Approximation" (2001). Theses and Dissertations. 4545. https://scholar.afit.edu/etd/4545 This Dissertation is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. g 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. Report Documentation Page 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 j T j T Multiplication Sto chastic Op erator Approximation c F e f e g T E g T E g T j 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 j n j 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 i c e F e F g T g T E E j i n j c c e e F F e F g T g T E g T E E j T j T i n i c c c F e e F e F g T g T g T E E E j T j T i e f g T E g T c e f e e F F g T E g T g T E E g T j i n j c c e F e F e F g T g T g T E E E j T j T i n i 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 k i n k 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 k i n k 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

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