J. MEDHI STOCHASTIC MODELS IN QUEUEING THEORY

Transcription

1 J. MEDHI STOCHASTIC MODELS IN QUEUEING THEORY SECOND EDITION ACADEMIC PRESS An imprint of Elsevier Science Amsterdam Boston London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

2 Contents Preface xv CHAPTER 1 Stochastic Processes Introduction Markov Chains Basic ideas Classification of states and chains Continuous-Time Markov Chains Sojourn time Transition density matrix or infinitesimal generator Limiting behavior: ergodicity Transient solution Alternative definition Birth-and-Death Processes Special case: M/M/1 queue Pure birth process: Yule-Furry process Poisson Process Properties of the Poisson process Generalization of the Poisson process Role of the Poisson process in probability models Randomization: Derived Markov Chains Markov chain an an underlying Poisson process (or subordinated to a Poisson process) 33

3 viii Contents Equivalence of the two limiting forms Numerical method Renewal Processes Introduction Residual and excess lifetimes Regenerative Processes Application in queueing theory Markov Renewal Processes and Semi-Markov Processes 39 Problems 41 References and Further Reading 46 CHAPTER 2 Queueing Systems: General Concepts Introduction Basic characteristics The Input or arrival pattern of customers The pattern of service The number of servers The capacity of the system The queue discipline Queueing Processes Notation Transient and Steady-State Behavior Limitations of the Steady-State Distribution Some General Relationships in Queueing Theory Poisson Arrival Process and Its Characteristics PASTA: Poisson arrivals see time averages ASTA: arrivals see time averages 62 References and Further Reading 62 CHAPTER 3 Birth-and-Death Queueing Systems: Exponential Models Introduction The Simple M/11111 Queue Steady-state solution of M/A4/ Waiting-time distributions The output process Semi-Markov process analysis System with Limited Waiting Space: The MIM11/K Model Steady-state solution Expected number in the system L K Equivalence of an M/M/1/K model with a two-stage cyclic model 80

4 Contents ix 3.4 Birth-and-Death Processes: Exponential Models The MPti/o0 Model: Exponential Model with an Infinite Number of Servers The Model M/M/c Steady-state distribution Expected number of busy and idle servers Waiting-time distributions The output process The M/M/c/c System: Erlang Loss Model Erlang loss (blocking) formula: Recursive algorithm Relation between Erlang's B and Cformulas Model with Finite Input Source Steady-state distribution: M/M/c/ Irn(m>c). Engset delay model Engset loss model MIMIcIlm/(rn > c) The mode! M/M/c//m (m < c) Transient Behavior Introduction Difference-equation technique Method of generating function Busy-period analysis Waiting-time process: Virtual waitingtime Transient-State Distribution of the M/M/c Model Solution of the differential-difference equations Busy period of an M/M/c queue Transient-state distribution of the output of an M/M/c queue Multichannel Queue with Ordered Entry Two-channel model with ordered entry (with finite capacity) The case M 1, N=N Particular case: M N 1 (overflow system) Output process 144 Problems and Complements 145 References and Further Reading 159 CHAPTER 4 Non- Birth -and- Death Queueing Systems: Markovian Models Introduction, The system M/Ek/ The system Ek/ M/1 170

5 X Contents 4.2 Bulk Queues Markovian bulk-arrival system: WM/ Equivalence of M s. /M/1 and M/E,./1 systems Waiting-time distribution in an WM/1 queue Transient-state behavior The system MVM/ Queueing Models with Bulk (Batch) Service The system M/M(a, b)/ Distribution of the waiting-time for the system M/M(a, b)/ Service batch-size distribution M/M(a, b)/1: Transient-State Distribution Steady-state solution Busy-period distribution Two-Server Model: M/M(a, b)/ Particular case: M/M(1, b)/ The M /M(1, b)/c Model Steady-state resultsm/m(1, b)/c 208 Problems and Complements 210 References and Further Reading 217 CHAPTER 5 Network of Queues Network of Markovian Queues Channels in Series or Tandem Queues Queues in series with multiple channels at each phase Jackson Network Closed Markovian Network (Gordon and Newell Network) Cyclic Queue BCMP Networks Concluding Remarks Loss networks 241 Problems and Complements 242 References and Further Reading 249 CHAPTER 6 Non - Markovian Queueing Systems Introduction Embedded-Markov-Chain Technique for the System with Poisson Input 256

6 Contents Xi 6.3 The MIGIl Model: Pollaczek-Khinchin Formula Steady-state distribution of departure epoch system size Waiting-time distribution General time system size distribution of an M/G/1 queue: supplementary variable technique Semi-Markov process approach Approach via martingale Busy Period Introduction Busy-period distribution: Taketcs integral equation Further discussion of the busy period Delay busy period Delay busy period under N-policy Queues with Finite Input Source: MIG111 IN System System with Limited Waiting Space: M/G/1/K System The M7G/1 Model with Bulk Arrival The number in the system at departure epochs in steady state (Pollaczek-Khinchin formula) Waiting-time distribution Feedback queues The M /et, b)/1 Model with General Bulk Service The G/M/1 Model Steady-state arrival epoch system size General time system size in steady state Waiting-time distribution Expected duration of busy period and idle period Multiserver Model The M/G/00 model: transient-state distribution The model GIMIc The model M/G/c Queues with Markovian Arrival Process 324 Problems and Complements 326 References and Further Reading 334

7 Xii Contents CHAPTER 7 Queues with General Arrival Time and Service-Time Distributions The G/G/1 Queue with General Arrival Time and Service-Time Distributions Lindley's integral equation Laplace transform of VJ Generalization of the Pollaczek-Khinchin transform formula Mean and Variance of Waiting Time W Mean of W (single-server queue) Variance of IN Multiserver queues: approximation of mean waiting time Queues with Batch Arrivals G('`) IGI The Output Process of a G/G/1 System Particular case Output process of a G/G/c system Some Bounds for the G/G/1 System Bound for e) Bounds for E(W) 360 Problems and Complements 368 References and Further Reading 371 CHAPTER 8 Miscellaneous Topics Heavy-Traffic Approximation for Waiting-Time Distribution Kingman's heavy-traffic approximation for a G/G/1 queue Empirical extension of the AVG/1 heavy-traffic approximation GIM1c queueinheavy traffic Brownian Motion Process Introduction Asymptotic queue-length distribution Diffusion approximation for a G/G/1 queue Virtual delay for th'e G/G/1 system Approach through an absorbing barrier with instantaneous return Diffusion approximation for a G/G/c queue: state-`dependent diffusion equation 395

8 Contents xiii Diffusion approximation for an M/G/c model Concluding remarks Queueing Systems with Vacations Introduction Stochastic decomposition Poisson Input queue with vacations: [exhaustive-service] queue-iength distribution Poisson input queue with vacations: waiting-time distribution M/G/1 system with vacations: nonexhaustive service Limited service system: MIG11-1/a, model Gated service system: MIG11-14 model MIG111K queue with multiple vacations Mean value analysis through heuristic treatment Design and Control of Queues Retrial Queueing System Retrial queues: model description Single-server model: MIM /1 retrial queue M/G/1 retrial queue Multiserver models Model with finite orbit size Other retrial queue models Emergence of a New Trend in Teletraffic Theory Introduction Heavy-tail distributions M/G/1 with heavy-tailed service time Pareto mixture of exponential (PME) distribution Gamma mixture of Pareto (GMP) distribution Beta mixture of exponential (BME) distribution A dass of heavy-tail distributions Long-range dependence 454 Problems and Complements 455 References and Further Reading 461 Appendix 469 Index 477