2nd Edition

# Reliability Engineering Probabilistic Models and Maintenance Methods, Second Edition

By

## Joel A. Nachlas

ISBN 9781498752473
Published December 1, 2016 by CRC Press
378 Pages 148 B/W Illustrations

USD \$145.00

Prices & shipping based on shipping country

## Book Description

Without proper reliability and maintenance planning, even the most efficient and seemingly cost-effective designs can incur enormous expenses due to repeated or catastrophic failure and subsequent search for the cause. Today’s engineering students face increasing pressure from employers, customers, and regulators to produce cost-efficient designs that are less prone to failure and that are safe and easy to use.

The second edition of Reliability Engineering aims to provide an understanding of reliability principles and maintenance planning to help accomplish these goals. This edition expands the treatment of several topics while maintaining an integrated introductory resource for the study of reliability evaluation and maintenance planning. The focus across all of the topics treated is the use of analytical methods to support the design of dependable and efficient equipment and the planning for the servicing of that equipment. The argument is made that probability models provide an effective vehicle for portraying and evaluating the variability that is inherent in the performance and longevity of equipment.

With a blend of mathematical rigor and readability, this book is the ideal introductory textbook for graduate students and a useful resource for practising engineers.

Preface.................................................................................................................... xiii

Author......................................................................................................................xv

1 Introduction......................................................................................................1

2 System Structures............................................................................................5

2.1 Status Functions.....................................................................................5

2.2 System Structures and Status Functions............................................7

2.2.1 Series Systems...........................................................................7

2.2.2 Parallel System..........................................................................8

2.2.3 k-out-of-n Systems................................................................... 10

2.2.4 Equivalent Structures............................................................. 12

2.3 Modules of Systems............................................................................. 17

2.4 Multistate Components and Systems................................................ 18

Exercises........................................................................................................... 19

3 Reliability of System Structures................................................................23

3.1 Probability Elements............................................................................23

3.2 Reliability of System Structures......................................................... 24

3.2.1 Series Systems......................................................................... 24

3.2.2 Parallel Systems.......................................................................25

3.2.3 k-out-of-n Systems...................................................................25

3.2.4 Equivalent Structures.............................................................26

3.3 Modules................................................................................................. 31

3.4 Reliability Importance......................................................................... 32

3.5 Reliability Allocation...........................................................................35

3.6 Conclusion.............................................................................................36

Exercises........................................................................................................... 37

4 Reliability over Time.................................................................................... 39

4.1 Reliability Measures............................................................................ 39

4.2 Life Distributions.................................................................................44

4.2.1 Exponential Distribution.......................................................45

4.2.2 Weibull Distribution...............................................................46

4.2.3 Normal Distribution...............................................................49

4.2.4 Lognormal Distribution......................................................... 51

4.2.5 Gamma Distribution.............................................................. 52

4.2.6 Other Distributions................................................................ 52

4.3 System-Level Models...........................................................................54

Exercises...........................................................................................................58

viii Contents

5 Failure Processes............................................................................................ 61

5.1 Mechanical Failure Models................................................................ 62

5.1.1 Stress–Strength Interference................................................. 62

5.1.2 Shock and Cumulative Damage...........................................64

5.2 Electronic Failure Models...................................................................71

5.2.1 Arrhenius Model.....................................................................71

5.2.2 Eyring Model...........................................................................72

5.2.3 Power Law Model...................................................................72

5.2.4 Defect Model...........................................................................72

5.3 Other Failure Models..........................................................................73

5.3.1 Diffusion Process Model........................................................73

5.3.2 Proportional Hazards............................................................ 78

5.3.3 Competing Risks.....................................................................80

Exercises...........................................................................................................83

6 Age Acceleration............................................................................................85

6.1 Age Acceleration for Electronic Devices........................................... 87

6.2 Age Acceleration for Mechanical Devices........................................89

6.3 Step Stress Strategies...........................................................................92

6.4 Concluding Comment.........................................................................93

Exercises...........................................................................................................93

7 Nonparametric Statistical Methods...........................................................95

7.1 Data Set Notation and Censoring......................................................96

7.2 Estimates Based on Order Statistics..................................................98

7.3 Estimates and Confidence Intervals..................................................99

7.4 Kaplan–Meier Estimates................................................................... 102

7.4.1 Continuous Monitoring of Test Unit Status...................... 102

7.4.2 Periodic Monitoring of Test Unit Status............................ 105

7.5 Tolerance Bounds............................................................................... 107

7.6 TTT Transforms.................................................................................. 109

7.6.1 Theoretical Construction..................................................... 109

7.6.2 Application to Complete Data Sets..................................... 113

7.6.3 Application to Censored Data Sets..................................... 118

7.7 Nelson Cumulative Hazard Estimation Method..........................122

Exercises......................................................................................................... 124

8 Parametric Statistical Methods................................................................. 129

8.1 Graphical Methods............................................................................ 129

8.2 Method of Moments.......................................................................... 135

8.2.1 Estimation Expressions........................................................ 136

8.2.2 Confidence Intervals for the Estimates.............................. 139

8.3 Method of Maximum Likelihood.................................................... 143

8.4 Maximum Likelihood Method with Data Censoring.................. 159

Contents ix

8.5 Special Topics...................................................................................... 161

8.5.1 Method of Moments with Censored Data......................... 161

8.5.2 Data Analysis under Step Stress Testing........................... 164

Exercises......................................................................................................... 167

9 Repairable Systems I: Renewal and Instantaneous Repair................ 173

9.1 Renewal Processes............................................................................. 174

9.2 Classification of Distributions and Bounds on Renewal

Measures............................................................................................. 181

9.3 Residual Life Distribution................................................................ 186

9.4 Conclusion........................................................................................... 189

Exercises......................................................................................................... 190

10 Repairable Systems II: Nonrenewal and Instantaneous Repair........ 193

10.1 Minimal Repair Models.................................................................... 194

10.2 Imperfect Repair Models..................................................................200

10.3 Equivalent Age Models.....................................................................203

10.3.1 Kijima Models.......................................................................203

10.3.2 Quasi-Renewal Process........................................................ 210

10.4 Conclusion........................................................................................... 214

Exercises......................................................................................................... 214

11 Availability Analysis.................................................................................. 217

11.1 Availability Measures........................................................................220

11.2 Example Computations.....................................................................223

11.2.1 Exponential Case..................................................................223

11.2.2 Numerical Case.....................................................................225

11.3 System-Level Availability.................................................................227

11.4 Nonrenewal Cases............................................................................. 232

11.4.1 Availability under Imperfect Repair..................................233

11.4.2 Availability Analysis for the Quasi-Renewal Model.......235

11.5 Markov Models.................................................................................. 239

Exercises......................................................................................................... 245

12 Preventive Maintenance............................................................................. 247

12.1 Replacement Policies......................................................................... 248

12.1.1 Elementary Models............................................................... 248

12.1.2 Availability Model for Age Replacement..........................253

12.1.3 Availability Model for Block Replacement........................255

12.1.4 Availability Model for Opportunistic Age

Replacement.................................................................... 257

12.1.4.1 Failure Model......................................................... 262

12.1.4.2 Opportunistic Failure Replacement Policy.......265

x Contents

12.1.4.3 Partial Opportunistic Age Replacement

Policy................................................................. 268

12.1.4.4 Full Opportunistic Age Replacement Policy..... 271

12.1.4.5 Analysis of the Opportunistic Replacement

Models..................................................................... 271

12.2 Nonrenewal Models.......................................................................... 274

12.2.1 Imperfect PM Models........................................................... 275

12.2.2 Models Based on the Quasi-Renewal Process..................277

12.2.3 Models Based on the Kijima Model................................... 281

12.3 Conclusion...........................................................................................283

Exercises.........................................................................................................284

13 Predictive Maintenance.............................................................................. 287

13.1 System Deterioration.........................................................................288

13.2 Inspection Scheduling....................................................................... 289

13.3 More Complete Policy Analysis.......................................................290

13.4 Models and Analysis Based on Continuous Process

Monitoring.......................................................................................... 294

13.4.2.1 Time Series Methods............................................ 298

13.4.2.2 Conditional Probability Methods.......................300

13.5 Conclusion...........................................................................................304

Exercises.........................................................................................................305

14 Special Topics...............................................................................................307

14.1 Statistical Analysis of Repairable System Data..............................307

14.1.1 Data from a Single System...................................................307

14.1.2 Data from Multiple Identical Systems............................... 310

14.2 Warranties........................................................................................... 314

14.2.1 Full Replacement Warranties.............................................. 315

14.2.2 Pro Rata Warranties.............................................................. 317

14.3 Reliability Growth............................................................................. 319

14.4 Dependent Components................................................................... 323

14.5 Bivariate Reliability........................................................................... 325

14.5.1 Collapsible Models................................................................ 326

14.5.2 Bivariate Models................................................................... 327

14.5.2.1 Stochastic Functions............................................. 327

14.5.2.2 Correlation Models...............................................330

14.5.2.3 Probability Analysis............................................. 331

14.5.2.4 Failure and Renewal Models...............................335

Exercises......................................................................................................... 341

Contents xi

Appendix A: Numerical Approximations.....................................................343

Appendix B: Numerical Evaluation of the Weibull Renewal

Functions..........................................................................................................347

Appendix C: Laplace Transform for the Key Renewal Theorem..............353

Appendix D: Probability Tables......................................................................355

References............................................................................................................ 359

Index......................................................................................................................365

...

## Author(s)

### Biography

Joel A. Nachlas received his B. E. S. from the Johns Hopkins University in 1970, his M. S. in 1972 and his Ph. D. in 1976 both from the University of Pittsburgh. He served on the faculty of the Grado Department of Industrial and Systems Engineering at Virginia Tech for 41 years and retired in March, 2016. His research interests are in the applications of probability and statistics to problems in reliability and quality control. In addition to his normal teaching activities during his time at Virginia Tech, he served as the coordinator for the department’s graduate program in Operations Research and for their dual master’s degree that is operated with Ecole des Mines de Nantes in France. From 1992 through 2011, he regularly taught Reliability Theory at the Ecole Polytechnique de Nice-Sophia Antipolis. He is the co-author of over fifty refereed articles, has served in numerous editorial and referee capacities and has lectured on reliability and maintenance topics throughout North America and Europe.

## Reviews

"The text presents material in a clear and logical progression. Concepts, covered in the early chapters, are developed through to systems level prediction techniques in the final chapters. The text is appropriate for both graduate level courses and also practitioners."
— John Andrews, University of Nottingham, United Kingdom

"Overall, the text offers a well-written, cohesive, in-depth treatment of descriptive and prescriptive concepts in reliability evaluation and maintenance planning. It should prove a valuable resource for both graduate students and practicing engineers."
— Lisa Maillart, University of Pittsburgh, USA

"The book discusses reliability engineering with a combination of statistical rigor and good readability. It covers both reliability models and analysis of failure data. The text is well organized and methodological and contains examples and exercises as well. Students will certainly find it an excellent introduction to reliability concepts and modeling approaches, and practitioners will find it a great reference source."
— Bengt Klefsjö, Luleå University of Technology, Sweden

"Professor Nachlas’ book is a great contribution to regular and advanced topics in textbooks on reliability and maintenance. It covers several areas with up-to-date material, particularly in statistical analysis and preventive and predictive maintenance."
Dragan Banjevic, University of Toronto, Canada