Introduction to Stochastic Models

Preface ix

Chapter 1. Introduction to Stochastic Processes 1

1.1. Sequences of random variables 1

1.2. The notion of stochastic process 10

1.3. Martingales 13

1.4. Markov chains 17

1.5. State classification 24

1.6. Continuous-time Markov processes 27

1.7. Semi-Markov processes 33

Chapter 2. Simple Stochastic Models 37

2.1. Urn models 37

2.2. Random walks 39

2.3. Brownian motion 44

2.4. Poisson processes 50

2.5. Birth and death processes 59

Chapter 3. Elements of Markov Modeling 61

3.1. Markov models: ideas, history, applications 61

3.2. The discrete-time Ehrenfest model 63

3.3. Markov models in genetics 79

3.4. Markov storage models 110

3.5. Reliability of Markov models 124

Chapter 4. Renewal Models 149

4.1. Fundamental concepts and examples 149

4.2. Waiting times 155

4.3. Modified renewal processes 159

4.4. Replacement models 161

4.5. Renewal reward processes 165

4.6. The risk problem of an insurance company 168

4.7. Counter models 171

4.8. Alternating renewal processes 180

4.9. Superposition of renewal processes 182

4.10. Regenerative processes 186

Chapter 5. Semi-Markov Models 189

5.1. Introduction 189

5.2. Markov renewal processes 190

5.3. First-passage times and state classification 196

5.4. Reliability 200

5.5. Reservoir models 207

5.6. Queues 218

5.7. Digital communication channels 222

Chapter 6. Branching Models 227

6.1. The Bienaymé-Galton-Watson model 227

6.2. Generalizations of the B-G-W model 271

6.3. Continuous-time models 302

Chapter 7. Optimal Stopping Models 315

7.1. The classic optimal stopping problem 315

7.2. Renewal with binary decision 333

Bibliography 343

Notation 367

Index 369