Introduction to Probability Models (eBook)

Front Cover 1
Introduction to Probability Models 4
Copyright Page 5
Table of Contents 6
Preface 12
Chapter 1. Introduction to Probability Theory 14
1.1. Introduction 14
1.2. Sample Space and Events 14
1.3. Probabilities Defined on Events 17
1.4. Conditional Probabilities 20
1.5. Independent Events 23
1.6. Bayes' Formula 25
Exercises 28
References 33
Chapter 2. Random Variables 34
2.1. Random Variables 34
2.2. Discrete Random Variables 38
2.3. Continuous Random Variables 44
2.4. Expectation of a Random Variable 49
2.5. Jointly Distributed Random Variables 57
2.6. Moment Generating Functions 71
2.7. Limit Theorems 79
2.8. Stochastic Processes 83
Exercises 85
References 95
Chapter 3. Conditional Probability and Conditional Expectation 96
3.1. Introduction 96
3.2. The Discrete Case 96
3.3. The Continuous Case 101
3.4. Computing Expectations by Conditioning 104
3.5. Computing Probabilities by Conditioning 113
3.6. Some Applications 120
Exercises 138
Chapter 4. Markov Chains 150
4.1. Introduction 150
4.2. Chapman-Kolmogorov Equations 153
4.3. Classification of States 156
4.4. Limiting Probabilities 164
4.5. Some Applications 174
4.6. Branching Processes 181
4.7. Time Reversible Markov Chains 184
4.8. Markov Decision Processes 195
Exercises 199
References 211
Chapter 5. The Exponential Distribution and the Poisson Process 212
5.1. Introduction 212
5.2. The Exponential Distribution 213
5.3. The Poisson Process 221
5.4. Generalizations of the Poisson Process 248
Exercises 256
References 267
Chapter 6. Continuous-Time Markov Chains 268
6.1. Introduction 268
6.2. Continuous-Time Markov Chains 269
6.3. Birth and Death Processes 271
6.4. The Kolmogorov Differential Equations 278
6.5. Limiting Probabilities 285
6.6. Time Reversibility 293
6.7. Uniformization 299
6.8. Computing the Transition Probabilities 302
Exercises 305
References 314
Chapter 7. Renewal Theory and Its Applications 316
7.1. Introduction 316
7.2. Distribution of N(t) 318
7.3. Limit Theorems and Their Applications 322
7.4. Renewal Reward Processes 331
7.5. Regenerative Processes 338
7.6. Semi-Markov Processes 344
7.7. The Inspection Paradox 347
7.8. Computing the Renewal Function 349
Exercises 352
References 362
Chapter 8. Queueing Theory 364
8.1. Introduction 364
8.2. Preliminaries 365
8.3. Exponential Models 369
8.4. Network of Queues 385
8.5. The System M/G/1 394
8.6. Variations on the M/G/1 398
8.7. The Model G/M/1 403
8.8. Multiserver Queues 408
Exercises 414
References 423
Chapter 9. Reliability Theory 424
9.1. Introduction 424
9.2. Structure Functions 425
9.3. Reliability of Systems of Independent Components 431
9.4. Bounds on the Reliability Function 435
9.5. System Life as a Function of Component Lives 446
9.6. Expected System Lifetime 454
9.7. Systems with Repair 458
Exercises 462
References 469
Chapter 10. Brownian Motion and Stationary Processes 470
10.1. Brownian Motion 470
10.2. Hitting Times, Maximum Variable, and the Gambler's Ruin Problem 473
10.3. Variations on Brownian Motion 475
10.4. Pricing Stock Options 476
10.5. White Noise 487
10.6. Gaussian Processes 489
10.7. Stationary and Weakly Stationary Processes 492
10.8. Harmonic Analysis of Weakly Stationary Processes 497
Exercises 499
References 504
Chapter 11. Simulation 506
11.1. Introduction 506
11.2. General Techniques for Simulating Continuous Random Variables 511
11.3. Special Techniques for Simulating Continuous Random Variables 519
11.4. Simulating from Discrete Distributions 527
11.5. Stochastic Processes 534
11.6. Variance Reduction Techniques 545
11.7. Determining the Number of Runs 555
Exercises 555
References 564
Index 566