Page 5 - A First Course In Stochastic Models

Contents

Preface ix

1 The Poisson Process and Related Processes 1
1.0 Introduction 1
1.1 The Poisson Process 1
1.1.1 The Memoryless Property 2
1.1.2 Merging and Splitting of Poisson Processes 6
1.1.3 The M/G/∞ Queue 9
1.1.4 The Poisson Process and the Uniform Distribution 15
1.2 Compound Poisson Processes 18
1.3 Non-Stationary Poisson Processes 22
1.4 Markov Modulated Batch Poisson Processes 24
Exercises 28
Bibliographic Notes 32
References 32
2 Renewal-Reward Processes 33

2.0 Introduction 33
2.1 Renewal Theory 34
2.1.1 The Renewal Function 35
2.1.2 The Excess Variable 37
2.2 Renewal-Reward Processes 39
2.3 The Formula of Little 50
2.4 Poisson Arrivals See Time Averages 53
2.5 The Pollaczek–Khintchine Formula 58
2.6 A Controlled Queue with Removable Server 66
2.7 An Up- And Downcrossing Technique 69
Exercises 71
Bibliographic Notes 78
References 78
3 Discrete-Time Markov Chains 81

3.0 Introduction 81
3.1 The Model 82