Contents Preface... v 1 The Exponential Distribution and the Poisson Process... 1 1.1 Introduction... 1 1.2 The Density, the Distribution, the Tail, and the Hazard Functions... 2 1.2.1 The Hazard Function and the Memoryless Property (Version 1)... 3 1.2.2 The Memoryless Property (Version 2)... 4 1.2.3 The Memoryless Property (Version 3)... 5 1.2.4 The Least Among Exponential Random Variables... 6 1.2.5 The Erlang Distribution... 8 1.2.6 The Hyperexponential Distribution... 9 1.2.7 A Mixture of Erlang Distributions... 9 1.3 The Poisson Process... 10 1.3.1 When Have They Actually Arrived?... 11 1.3.2 Thinning and Superpositioning of Poisson Processes... 13 1.4 Transforms... 14 1.4.1 The z-transform... 14 1.4.2 The Laplace-Stieltjes Transform... 16 1.5 Exercises... 17 2 Introduction to Renewal Theory... 21 2.1 Introduction... 21 2.2 Main Renewal Results... 22 2.2.1 The Length Bias Distribution and the Inspection Paradox... 22 2.2.2 The Age and the Residual Distributions... 24 2.2.3 The Memoryless Property (Versions 4 and 5)... 28 2.3 An Alternative Approach... 31 2.4 A Note on the Discrete Version... 32 2.5 Exercises... 34 ix
x Contents 3 Introduction to Markov Chains... 37 3.1 Introduction... 37 3.2 Some Properties of Markov Chains... 38 3.3 Time Homogeneity... 39 3.4 State Classification... 40 3.5 Transient and Recurrent Classes... 41 3.6 Periodicity... 42 3.7 Limit Probabilities and the Ergodic Theory... 42 3.7.1 Computing the Limit Probabilities... 44 3.8 The Time-Reversed Process and Reversible Processes... 44 3.9 Discrete Renewal Processes Revisited... 46 3.10 Transient Matrices... 47 3.11 Short-Circuiting States... 48 3.12 Exercises... 50 4 From Single Server Queues to M/G/1... 51 4.1 Introduction... 51 4.2 Why Do Queues Exist at All?... 52 4.3 Why Queues Are Long?... 52 4.4 Queueing Disciplines... 53 4.5 Basics in Single Server Queues... 54 4.5.1 The Utilization Level... 54 4.5.2 Little s Law... 54 4.5.3 Residual Service Times... 56 4.5.4 The Virtual Waiting Time... 57 4.5.5 Arrival and Departure Instants... 59 4.6 ASTA and the Khintchine Pollaczek Formula... 59 4.7 The M/G/1 Model... 60 4.7.1 Examples... 61 4.7.2 The Busy Period of an M/G/1 Queue... 62 4.7.3 Stand-By Customers and Externalities... 64 4.7.4 M/G/1 Queues with Vacations... 65 4.8 The G/G/1 Queue... 66 4.8.1 Lindley s Equation... 67 4.9 Exercises... 68 5 Priorities and Scheduling in M/G/1... 71 5.1 An M/G/1 Queue with Priorities... 71 5.1.1 Conservation Laws... 71 5.1.2 The Optimality of the C Rule... 72 5.1.3 Waiting Times in Priority Queues... 73 5.1.4 Shortest Job First (SJF)... 75 5.1.5 Preemptive Priority... 76 5.2 Exercises... 78
Contents xi 6 M/G/1 Queues Using Markov Chains and LSTs... 81 6.1 Introduction... 81 6.2 The Markov Chain Underlying the Departure Process... 82 6.2.1 The Limit Probabilities... 83 6.3 The Distribution of Time in the System... 86 6.3.1 Arrival, Departure, and Random Instants... 88 6.3.2 Observable Queues... 89 6.4 Busy Period in an M/G/1 Queue Revisited... 90 6.5 A Final Word... 92 6.6 Exercises... 93 7 The G/M/1 Queueing System... 99 7.1 Introduction and Modeling... 99 7.2 The Stationary Distribution at Arrival Instants... 100 7.2.1 The Balance Equations and Their Solution... 100 7.2.2 Exponential Waiting Times... 103 7.2.3 The Queue Length at Random Times... 103 7.3 Exercises... 104 8 Continuous-Time Markov Chains and Memoryless Queues... 107 8.1 The Model... 107 8.2 Examples... 108 8.3 The Limit Probabilities... 118 8.3.1 The Limit Probabilities and the Balance Equations... 119 8.3.2 The Embedded Process... 124 8.3.3 Uniformization... 124 8.3.4 The Cut Balancing Theorem... 124 8.4 The Time-Reversed Process... 128 8.4.1 A Condition on the Limit Probabilities... 128 8.4.2 The Time-Reversed Process Is Markovian... 130 8.4.3 Time-Reversible Processes... 131 8.4.4 Poisson Processes Stemming from Markov Processes... 132 8.5 Exercises... 133 9 Open Networks of Exponential Queues... 139 9.1 Open Networks of Exponential Queues: Model and Limit Probabilities... 139 9.1.1 Partial Balancedness... 143 9.2 Processes in Open Networks of Memoryless Queues... 143 9.3 Sojourn Times in Open Network of Queues... 147 9.3.1 The Unconditional Mean Waiting Time... 147 9.3.2 The Arrival Theorem... 147 9.4 Generalizations... 149 9.5 Exercises... 149
xii Contents 10 Closed Networks of Exponential Queues... 151 10.1 The Model and the Limit Probabilities... 151 10.1.1 Partial Balancedness... 154 10.2 The Convolution Algorithm... 155 10.3 Short-Circuiting Stations... 158 10.4 The Arrival Theorem... 159 10.5 Mean Value Analysis (MVA) for Closed Networks of Queues... 160 10.5.1 The Mean Value Analysis (MVA) Algorithm... 161 10.6 Generalizations... 162 10.7 Exercises... 163 11 Insensitivity and Product-Form Queueing Models... 165 11.1 Introduction... 165 11.2 Symmetric Queues... 165 11.2.1 Examples... 166 11.2.2 Product Form and Insensitivity... 167 11.3 One-Chance Queues... 172 11.3.1 Examples... 173 11.3.2 Product Form and Insensitivity... 173 11.3.3 Proof of Theorem 11.5... 175 11.4 BCMP Network of Queues... 178 11.5 Exercises... 178 12 Two-Dimensional Markov Processes and Their Applications to Memoryless Queues... 181 12.1 Model Description... 181 12.2 Examples... 182 12.2.1 Example 1: The M=Er=1 Model Revisited... 182 12.2.2 Example 2: The Two Shortest Truncated Symmetric Queues... 185 12.2.3 Example 3: Two Servers: When Customers Join the First When Its Queue Is Not Too Long... 193 12.2.4 Example 4: Game with Tokens... 195 12.3 The Balance Equations... 197 12.3.1 Example 1 (Cont.)... 198 12.3.2 Example 2 (Cont.)... 199 12.3.3 Example 3 (Cont.)... 199 12.3.4 Example 4 (Cont.)... 200 12.4 Solving for the Limit Probabilities... 201 12.4.1 Example 1 (Cont.)... 203 12.4.2 Example 3 (Cont.)... 204 12.4.3 Example 4 (Cont.)... 210 12.5 The General Case... 214 12.6 Exercises... 215 References... 217 Index... 219
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