QUEUEING THEORY WITH APPLICATIONS TO PACKET TELECOMMUNICATION QUEUEING THEORY WITH APPLICATIONS TO PACKET TELECOMMUNICATION JOHN N. DAIGLE Prof. of Electrical Engineering The University of Mississippi University, MS 38677 Springer ISBN 978-0-387-22859-4 (eBook) Print ISBN 978-0-387-22857-0 Additional material to this book can be downloaded from http://extras.springer.com ©2005 Springer Science + Business Media, Inc. Print ©2005 Springer Science + Business Media, Inc. Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: http://ebooks.kluweronline.com and the Springer Global Website Online at: http://www.springeronline.com NOTE TO INSTRUCTORS A complete solution manual has been prepared for use by those interested in using this book as the primary text in a course or for independent study. Inter- ested persons should please contact the publisher or the author at http://www.olemiss.edu/~wcdaigle/QueueingText to obtain an electronic copy of the solution manual as well as other support materials, such as computer programs that implement many of the computational procedures described in this book. Contents List of Figures xi List of Tables xv Preface xvii Acknowledgments xxiii 1. TERMINOLOGY AND EXAMPLES 1 1.1 The Terminology of Queueing Systems 2 1.2 Examples of Application to System Design 9 1.2.1 Cellular Telephony 9 1.2.2 Multiplexing Packets 11 1.2.3 CDMA-Based Cellular Data 14 1.3 Summary 17 2. REVIEW OF RANDOM PROCESSES 19 2.1 Statistical Experiments and Probability 20 2.1.1 Statistical Experiments 20 2.1.2 Conditioning Experiments 22 2.2 Random Variables 27 2.3 Exponential Distribution 33 2.4 Poisson Process 39 2.5 Markov Chains 45 3. ELEMENTARY CTMC-BASED QUEUEING MODELS 57 3.1 M/M/1 Queueing System 58 3.1.1 Time-Dependent M/M/1 Occupancy Distribution 58 3.1.2 Stochastic Equilibrium M/M/1 Distributions 60 3.1.3 Busy Period for M/M/1 Queueing System 76 viii QUEUEING THEORY FOR TELECOMMUNICATIONS 3.2 Dynamical Equations for General Birth-Death Process 81 3.3 Time-Dependent Probabilities for Finite-State Systems 83 3.3.1 Classical Approach 84 3.3.2 Jensen’s Method 88 3.4 Balance Equations Approach for Systems in Equilibrium 91 3.5 Probability Generating Function Approach 98 3.6 Supplementary Problems 101 4. ADVANCED CTMC-BASED QUEUEING MODELS 107 4.1 Networks 108 4.1.1 Feedforward Networks: Fixed Routing 109 4.1.2 Arbitrary Open Networks 110 4.1.3 Closed Networks of Single Servers 111 4.2 Phase-Dependent Arrivals and Service 122 4.2.1 Probability Generating Function Approach 124 4.2.2 Matrix Geometric Method 138 4.2.3 Rate Matrix Computation via Eigenanalysis 143 4.2.4 Generalized State-Space Methods 146 4.3 Phase-Type Distributions 152 4.4 Supplementary Problems 156 5. THE BASIC M/G/1 QUEUEING SYSTEM 159 5.1 M/G/1 Transform Equations 161 5.1.1 Sojourn Time for M/G/1 165 5.1.2 Waiting Time for M/G/1 167 5.1.3 Busy Period for M/G/1 167 5.2 Ergodic Occupancy Distribution for M/G/1 170 5.2.1 Discrete Fourier Transform Approach 170 5.2.2 Recursive Approach 180 5.2.3 Generalized State-Space Approach 183 5.3 Expected Values Via Renewal Theory 210 5.3.1 Expected Waiting and Renewal Theory 210 5.3.2 Busy Periods and Alternating Renewal Theory 216 5.4 Supplementary Problems 219 6. THE M/G/1 QUEUEING SYSTEM WITH PRIORITY 225 6.1 M/G/1 Under LCFS-PR Discipline 226 6.2 M/G/1 System Exceptional First Service 229 6.3 M/G/1 under HOL Priority 236 Contents ix 6.3.1 Higher Priority Customers 238 6.3.2 Lower Priority Customers 241 6.4 Ergodic Occupancy Probabilities for Priority Queues 244 6.5 Expected Waiting Times under HOL Priority 246 6.5.1 HOL Discipline 248 6.5.2 HOL-PR Discipline 249 7. VECTOR MARKOV CHAINS ANALYSIS 253 7.1 The M/G/1 and G/M/1 Paradigms 254 7.2 G/M/1 Solution Methodology 259 7.3 M/G/1 Solution Methodology 261 7.4 Application to Statistical Multiplexing 265 7.5 Generalized State Space Approach: Complex Boundaries 278 7.6 Summary 290 7.7 Supplementary Problems 294 8. CLOSING REMARKS 297 References 301 Index 309 About the Author 315 List of Figures 1.1 Schematic diagram of a single-server queueing system. 2 1.2 Sequence of events for first customer. 3 1.3 Sequence of events for general customer. 4 1.4 Typical realization for unfinished work. 6 1.5 Blocking probability as a function of population size at a load of 11 1.6 Queue length survivor function for an N-to-1 multi- plexing system at a traffic intensity of 0.9 with N as a parameter and with independent, identically distributed arrivals. 14 1.7 Queue length survivor function for an 8-to-1 multiplex- ing system at a traffic intensity of 0.9 with average run length as a parameter. 15 1.8 Comparison between a system serving a fixed number of 16 units per frame and a system serving a binomial number of units with an average of 16 at a traffic inten- sity of 0.9. 16 2.1 Distribution function for the random variable defined in Example 2.4. 29 3.1 Survivor function for system occupancy for several val- ues of 63 3.2 Schematic diagram of a single-server queueing system. 65 3.3 Schematic diagram of a simple network of queues. 76 3.4 Sequence of busy and idle periods. 76 3.5 Sequence of service times during a generic busy period. 77 xii QUEUEING THEORY FOR TELECOMMUNICATIONS 3.6 Busy period decompositions depending upon interar- rival versus service times. 79 3.7 Time-dependent state probabilities corresponding to Ex- ample 3.1. 86 3.8 Steps involved in randomization. 90 3.9 State diagram for M/M/1 System. 92 3.10 State diagram for general birth-death process. 92 3.11 State diagram for general birth-death process. 95 3.12 State diagram illustrating local balance. 98 4.1 Block diagram for window flow control network. 117 4.2 State diagram for phase process. 122 4.3 State diagram for system having phase-dependent ar- rival and service rates. 123 5.1 Survivor functions with deterministic, Erlang-2, expo- nential, branching Erlang and gamma service-time dis- tributions at 176 5.2 Survivor functions for system occupancy with message lengths drawn from truncated geometric distributions at 178 5.3 Survivor functions for system having exponential ordi- nary and exceptional first service and as a parameter. 198 5.4 Survivor functions with unit deterministic service and binomially distributed arrivals with N as a parameter at 199 5.5 Survivor functions with unit-mean Erlang-10 service and Poisson arrivals with C as a parameter at a traffic load of 0.9. 201 5.6 Survivor functions with unit-mean Erlang-K service and Poisson arrivals with K as a parameter at a traf- fic load of 0.9. 203 5.7 Survivorfunctions with and Pade(2, 2) ser- vice, Poisson arrivals, and a traffic load of 0.9. 205 5.8 Survivor functions for deterministic (16) batch sizes with Pade deterministic service and Poisson arrivals at a traffic load of 0.9 for various choices of 206 List of Figures xiii 5.9 Survivor functions for deterministic (16) batch sizes with Erlang-K-approximated deterministic service and Poisson arrivals at a traffic load of 0.9 for various choices of K. 207 5.10 Survivor functions for binomial (64,0.25) and deter- ministic (16) batch sizes with deterministic service ap- proximated by a Pade(32, 32) approximation and Pois- son arrivals at a traffic load of 0.9. 208 5.11 A sample of service times. 211 5.12 An observed interval of a renewal process. 212 6.1 HOL service discipline. 237 7.1 Survivor functions for occupancy distributions for sta- tistical multiplexing system with 0.5 to 1.0 speed con- version at 273 7.2 Survivor functions for occupancy distributions for sta- tistical multiplexing system with equal line and trunk capacities at 277 7.3 Survivor functions for occupancy distributions for sta- tistical multiplexing system with and without line-speed conversion at 278 7.4 Survivor functions for occupancy distributions for wire- less communication link with on time as a parameter. 291 List of Tables 5.1 Blocking probabilities versus occupancy probabilities for various service time distributions. 180 5.2 Parameters for Example 5.5. 193 5.3 Formulae to compute parameter values for Example 5.6. 197 5.4 Parameter values for Example 5.8. 200 5.5 Occupancy values as a function of the number of units served for the system of Example 5.8. 202 5.6 Comparison of values of survivor function computed using various Pade approximations for service time in Example 5.10. 204 5.7 Possible data structure for representing the input pa- rameters in a program to implement the scalar case of the generalized state space approach. 209 5.8 Possible data structure for representing the output pa- rameters in a program to implement the scalar case of the generalized state space approach. 209 7.1 Definition of the phases for the problem solved in Ex- ample 7.1.