Probability and Statistics with Reliability, Queuing, and Computer Science Applications (Paperback, 2)

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