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An Introduction to Queueing Theory Modeling and Analysis in Applications

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An Introduction to Queueing Theory Modeling and Analysis in Applications U. Narayan Bhat An Introduction to Queueing Theory Modeling and Analysis in Applications Birkhauser Boston • Basel • Berlin Contents Preface xi 1 Introduction 1 . 1.1 Basic System Elements 1 1.2 Problems in a Queueing System 2 1.3 A Historical Perspective 4 1.4 Modeling Exercises 11 2 System Element Models 13 2.1 Probability Distributions as Models 13 2.1.1 Deterministic Distribution (D) 14 2.1.2 Exponential distribution; Poisson process (M) 14 , 2.2 Identification of Models 18 2.2.1 Collection of Data 18 2.2.2 Tests for Stationarity 18 2.2.3 Tests for Independence 19 2.2.4 Distribution Selection 19 2.3 Review Exercises 20 3 , Basic Concepts in Stochastic Processes 23 3.1 Stochastic Process 23 3.2 Point, Regenerative, and Renewal Processes 23 3.3 Markov Process 24 4 Simple Markovian Queueing Systems 29 4.1 A General Birth-and-Death Queueing Model 29 4.2 The Queue M/M/l 34 - 4.2.1 Departure Process 40 4.3 The Queue M/M/s 43 4.4 The Finite Queue M/M/s/K 51 4.5 The Infinite-Server Queue M/M/oo 58 Contents 4.6 Finite-Source Queues 59 4.7 Other Models 62 4.7.1 The M/M/l/1 System 62 4.7.2 Markovian Queues with Balking 64 4.7.3 Markovian Queues with Reneging 66 4.7.4 Phase-Type Machine Repair 66 4.8 Remarks 68 4.9 Exercises 68 Imbedded Markov Chain Models 75 5.1 Imbedded Markov Chains 75 5.2 The Queue M/G/l 76 5.3 The Queue G/M/l 98 5.4 Exercises 112 Extended Markov Models 115 6.1 The Bulk Queue M(X)/M/\ 115 6.2 The Bulk Queue Af/AfW/1 118 6.3 The Queues M/Ek/\ and Ek/M/\ 120 6.4 The Bulk Queues M/GK/l and GK/M/\ 123 6.5 The Queues Ek/G/\ and G/Ek/\ 126 6.6 The Queue M/D/s 126 6.7 The Queue M/M/X with Priority Disciplines 127 6.8 Exercises 138 Queueing Networks 141 7.1 Introduction 141 7.2 The Markovian Node Network 142 7.3 Queues in Series 144 7.4 Queues with Blocking 147 7.5 Open Jackson Networks 150 7.6 Closed Jackson Networks 152 7.7 Cyclic Queues 154 7.8 Operational Laws for Performance Analysis 155 7.9 Remarks 157 7.10 Exercises 158 Renewal Process Models 161 8.1 Renewal Process 161 8.2 Renewal Process Models for Queueing Systems 166 The General Queue G/G/l and Approximations 169 9.1 The General Queue G/G/l 169 9.2 Little's Law L = XW 173 9.3 Approximations 175 9.4 Diffusion Approximation 178 Contents ix 9.5 Fluid Approximation 180 9.6 Remarks 183 10 Statistical Inference for Queueing Models 185 10.1 Introduction 185 10.2 Birth-and-Death Process Models 187 10.3 Imbedded Markov Chain Models for M/G/l and G/M/l 191 10.4 The Queue G/G/l 193 10.5 Other Methods of Estimation 194 10.6 Tests of Hypotheses 197 10.7 Control of Traffic Intensity in M/G/l and G/M/l 197 10.8 Remarks 199 11 Decision Problems in Queueing Theory 201 11.1 Introduction 201 11.2 Performance Measures for Decision Making 202 11.3 Design Problems in Decision Making 202 11.4 Control Problems in Decision Making 205 12 Modeling and Analysis Using Computational Tools 207 12.1 Mean Value Analysis 207 12.2 The Convolution Algorithm 211 12.2.1 Computing Other Performance Measures 213 12.3 Simulation 214 12.4 MATLAB 217 12.5 Exercises 223 Appendices A Poisson Process: Properties and Related Distributions 229 A.I Properties of the Poisson Process 229 A.2 Variants of the Poisson Process 231 A.3 Hyperexponential (HE) Distribution 233 A.4 Erlang Distribution (Ek) 234 A.5 Mixed Erlang Distributions 234 A.6 Coxian Distributions; Phase-Type Distribution 235 A.7 A General Distribution 236 A.8 Some Discrete Distributions 236 B Markov Process 239 B.I Kolmogorov Equations 239 B.2 The Poisson Process 240 B.3 Classification of States 242 B.4 Phase-Type Distributions 243 x Contents C Results from Mathematics 247 C.I Riemann-Stieltjes Integral 247 C.2 Laplace Transforms 248 C.3 Generating Functions 250 References 253 Index 265.
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