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

Provides a comprehensive introduction to probability, stochastic processes and statistics

This book covers fundamental concepts in probability and statistics, and relates these concepts to computer science and engineering. The author begins with a five–chapter–long coverage of probability theory designed for a one–semester introductory course on applied probability. Real–world examples and problems are used to help readers understand applied probability concepts. Chapters six though nine in turn are designed to be the core of an introductory course on stochastic processes and their applications. The remaining two chapters discuss statistical interference and regression that can form a core of a course on statistics.

Theory and applications of Markov chains to model reliability, availability, performance and performability of computer systems and networks are extensively discussed Numerical solution techniques for Markov chains and stochastic Petri nets as a means of automatically generating large Markov chains are discussed Applications include, fault tolerant and dependable computing, real–time systems, cellular wireless systems, and software reliability Includes over 200 in–text examples as well as self–study exercises for each section Provides access to a book companion website with an instructor s manual and power point slides

Probability and Statistics with Reliability, Queuing and Computer Science Applications, 2nd Edition is written for senior undergraduate and graduate students interested in electrical and computer engineering, reliability engineering, and applied mathematics. This book will also be of interest to practicing engineers and researchers in these areas.

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 vii

Preface to the Second Edition ix

Preface to the First Edition xi

Acronyms xiii

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 23

1.10 Independence of Events 25

1.11 Bayes Rule 37

1.12 Bernoulli Trials 45

2 Discrete Random Variables 61

2.1 Introduction 61

2.2 Random Variables and Their Event Spaces 62

2.3 The Probability Mass Function 64

2.4 Distribution Functions 66

2.5 Special Discrete Distributions 68

2.6 Analysis of Program MAX 92

2.7 The Probability Generating Function 96

2.8 Discrete Random Vectors 99

2.9 Independent Random Variables 104

3 Continuous Random Variables 115

3.1 Introduction 115

3.2 The Exponential Distribution 119

3.3 The Reliability and Failure Rate 124

3.4 Some Important Distributions 129

3.5 Functions of a Random Variable 148

3.6 Jointly Distributed Random Variables 153

3.7 Order Statistics 157

3.8 Distribution of Sums 167

3.9 Functions of Normal Random Variables 182

4 Expectation 193

4.1 Introduction 193

4.2 Moments 197

4.3 Expectation Based on Multiple Random Variables 200

4.4 Transform Methods 207

4.5 Moments and Transforms of Some Distributions 217

4.6 Computation of Mean Time to Failure 228

4.7 Inequalities and Limit Theorems 237

5 Conditional Distribution and Expectation 247

5.1 Introduction 247

5.2 Mixture Distributions 255

5.3 Conditional Expectation 262

5.4 Imperfect Fault Coverage and Reliability 269

5.5 Random Sums 279

6 Stochastic Processes 289

6.1 Introduction 289

6.2 Classification of Stochastic Processes 294

6.3 The Bernoulli Process 300

6.4 The Poisson Process 304

6.5 Renewal Processes 314

6.6 Availability Analysis 319

6.7 Random Incidence 328

6.8 Renewal Model of Program Behavior 332

7 Discrete–Time Markov Chains 337

7.1 Introduction 337

7.2 Computation of n–step Transition Probabilities 341

7.3 State Classification and Limiting Probabilities 347

7.4 Distribution of Times Between State Changes 356

7.5 Markov Modulated Bernoulli Process 358

7.6 Irreducible Finite Chains with Aperiodic States 361

7.7 The M/G/ 1 Queuing System 377

7.8 Discrete–Time Birth Death Processes 385

7.9 Finite Markov Chains with Absorbing States 392

8 Continuous–Time Markov Chains 405

8.1 Introduction 405

8.2 The Birth Death Process 412

8.3 Other Special Cases of the Birth Death Model 446

8.4 Non–Birth Death Processes 454

8.5 Markov Chains with Absorbing States 496

8.6 Solution Techniques 520

8.7 Automated Generation 530

9 Networks of Queues 555

9.1 Introduction 555

9.2 Open Queuing Networks 560

9.3 Closed Queuing Networks 568

9.4 General Service Distribution and Multiple Job Types 596

9.5 Non–product–form Networks 604

9.6 Computing Response Time Distribution 617

9.7 Summary 630

10 Statistical Inference 637

10.1 Introduction 637

10.2 Parameter Estimation 639

10.3 Hypothesis Testing 692

11 Regression and Analysis of Variance 727

11.1 Introduction 727

11.2 Least–squares Curve Fitting 732

11.3 The Coefficients of Determination 735

11.4 Confidence Intervals in Linear Regression 738

11.5 Trend Detection and Slope Estimation 742

11.6 Correlation Analysis 745

11.7 Simple Nonlinear Regression 748

11.8 Higher–dimensional Least–squares Fit 749

11.9 Analysis of Variance 751

A Bibliography 765

A.1 Theory 765

A.2 Applications 770

B Properties of Distributions 777

C Statistical Tables 780

D Laplace Transforms 801

E Program Performance Analysis 808

Author Index 811

Subject Index 819