# Probability and Statistics with Reliability, Queuing, and Computer Science Applications, 2nd Edition

ISBN: 978-1-119-28542-7
880 pages
July 2016
For Instructors

## Description

An accessible introduction to probability, stochastic processes, and statistics for computer science and engineering applications

Second edition now also available in Paperback. This updated and revised edition of the popular classic first edition relates fundamental concepts in probability and statistics to the computer sciences and engineering. The author uses Markov chains and other statistical tools to illustrate processes in reliability of computer systems and networks, fault tolerance, and performance.

This edition features an entirely new section on stochastic Petri nets—as well as new sections on system availability modeling, wireless system modeling, numerical solution techniques for Markov chains, and software reliability modeling, among other subjects. Extensive revisions take new developments in solution techniques and applications into account and bring this work totally up to date. It includes more than 200 worked examples and self-study exercises for each section.

Probability and Statistics with Reliability, Queuing and Computer Science Applications, Second Edition offers a comprehensive introduction to probability, stochastic processes, and statistics for students of computer science, electrical and computer engineering, and applied mathematics. Its wealth of practical examples and up-to-date information makes it an excellent resource for practitioners as well.

An Instructor's Manual presenting detailed solutions to all the problems in the book is available from the Wiley editorial department.

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Preface to the Paperback Edition ix

Preface to the Second Edition xi

Preface to the First Edition xiii

Acronyms xv

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

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## Author Information

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.

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## Reviews

"The book offers a comprehensive introduction to probability, stochastic processes, and statistics for students of computer science, electrical and computer engineering, and applied mathematics. Its wealth of practical examples and up-to-date information makes it an excellent resource for practitioners as well." (Zentralblatt MATH, 2016)

"I highly recommend this book for academics for use as a textbook and for researchers and professionals in the field as a useful reference." (Interfaces, September/ October 2004)

"This introduction...uses Markov chains and other statistical tools to illustrate process in reliability of computer systems, fault tolerance, and performance." (SciTech Book News, Vol. 26, No. 2, June 2002)

"...an excellent self-contained book.... I recommend the book to beginners and veterans in the field..." (Computer Journal, Vol.45, No.6, 2002)

"This book is a tour de force of clear, virtually error-free exposition of probability as it is applied in a host of up-to-date contexts.... It will richly reward the...reader.... Read this book cover to cover. It’s worth the effort." (Technometrics, Vol. 45, No. 1, February 2003)

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## Related Websites / Extra

Author's web site for InstructorsA Solutions Manual and Power Point Slides are available at the associated web site.
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Instructors Resources
Wiley Instructor Companion Site
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