Probability and Statistics with Reliability, Queuing, and Computer Science Applications

Kishor S. Trivedi, PhD, is the Hudson Professor of Electrical and Computer Engineering at Duke University, Durham, North Carolina. His research interests are in reliability and performance assessment of computer and communication systems. Dr. Trivedi has published extensively in these fields, with more than 600 articles and three books to his name. Dr. Trivedi is a Fellow of the IEEE and a Golden Core Member of the IEEE Computer Society.

Preface to the Paperback Edition ix

Preface to the Second Edition xi

Preface to the First Edition xiii

Acronyms xv

About the Companion Website xix

1 Introduction 1

1.1 Motivation 1

1.2 Probability Models 2

1.3 Sample Space 3

1.4 Events 6

1.5 Algebra of Events 7

1.6 Graphical Methods of Representing Events 11

1.7 Probability Axioms 13

1.8 Combinatorial Problems 19

1.9 Conditional Probability 24

1.10 Independence of Events 26

1.11 Bayes’ Rule 38

1.12 Bernoulli Trials 47

2 Discrete Random Variables 65

2.1 Introduction 65

2.2 Random Variables and Their Event Spaces 66

2.3 The Probability Mass Function 68

2.4 Distribution Functions 70

2.5 Special Discrete Distributions 72

2.6 Analysis of Program MAX 97

2.7 The Probability Generating Function 101

2.8 Discrete Random Vectors 104

2.9 Independent Random Variables 110

3 Continuous Random Variables 121

3.1 Introduction 121

3.2 The Exponential Distribution 125

3.3 The Reliability and Failure Rate 130

3.4 Some Important Distributions 135

3.5 Functions of a Random Variable 154

3.6 Jointly Distributed Random Variables 159

3.7 Order Statistics 163

3.8 Distribution of Sums 174

3.9 Functions of Normal Random Variables 190

4 Expectation 201

4.1 Introduction 201

4.2 Moments 205

4.3 Expectation Based on Multiple Random Variables 209

4.4 Transform Methods 216

4.5 Moments and Transforms of Some Distributions 226

4.6 Computation of Mean Time to Failure 238

4.7 Inequalities and Limit Theorems 247

5 Conditional Distribution and Expectation 257

5.1 Introduction 257

5.2 Mixture Distributions 266

5.3 Conditional Expectation 273

5.4 Imperfect Fault Coverage and Reliability 280

5.5 Random Sums 290

6 Stochastic Processes 301

6.1 Introduction 301

6.2 Classification of Stochastic Processes 307

6.3 The Bernoulli Process 313

6.4 The Poisson Process 317

6.5 Renewal Processes 327

6.6 Availability Analysis 332

6.7 Random Incidence 342

6.8 Renewal Model of Program Behavior 346

7 Discrete-Time Markov Chains 351

7.1 Introduction 351

7.2 Computation of n-step Transition Probabilities 356

7.3 State Classification and Limiting Probabilities 362

7.4 Distribution of Times Between State Changes 371

7.5 Markov Modulated Bernoulli Process 373

7.6 Irreducible Finite Chains with Aperiodic States 376

7.7 The M/G/ 1 Queuing System 391

7.8 Discrete-Time Birth–Death Processes 400

7.9 Finite Markov Chains with Absorbing States 407

8 Continuous-Time Markov Chains 421

8.1 Introduction 421

8.2 The Birth–Death Process 428

8.3 Other Special Cases of the Birth–Death Model 465

8.4 Non-Birth–Death Processes 474

8.5 Markov Chains with Absorbing States 519

8.6 Solution Techniques 541

8.7 Automated Generation 552

9 Networks of Queues 577

9.1 Introduction 577

9.2 Open Queuing Networks 582

9.3 Closed Queuing Networks 590

9.4 General Service Distribution and Multiple Job Types 620

9.5 Non-product-form Networks 628

9.6 Computing Response Time Distribution 641

9.7 Summary 654

10 Statistical Inference 661

10.1 Introduction 661

10.2 Parameter Estimation 663

10.3 Hypothesis Testing 718

11 Regression and Analysis of Variance 753

11.1 Introduction 753

11.2 Least-squares Curve Fitting 758

11.3 The Coefficients of Determination 762

11.4 Confidence Intervals in Linear Regression 765

11.5 Trend Detection and Slope Estimation 768

11.6 Correlation Analysis 771

11.7 Simple Nonlinear Regression 774

11.8 Higher-dimensional Least-squares Fit 775

11.9 Analysis of Variance 778

A Bibliography 791

A.1 Theory 791

A.2 Applications 796

B Properties of Distributions 804

C Statistical Tables 807

D Laplace Transforms 828

E Program Performance Analysis 835

Author Index 837

Subject Index 845