Reliability engineering : Probabilistic models and maintenance methods

RELIABILITY

ENGINEERING

Probabilistic Models and

Maintenance Methods

Second Edition

RELIABILITY

ENGINEERING

JOEL A. NACHLAS

Probabilistic Models and

Maintenance Methods

Second Edition

Boca Raton London New York

CRC Press is an imprint of the

Taylor & Francis Group, an informa business

CRC Press

Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300

Boca Raton, FL 33487-2742

© 2017 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works

Printed on acid-free paper

Version Date: 20161019

International Standard Book Number-13: 978-1-4987-5247-3 (Pack - Book and Ebook)

This book contains information obtained from authentic and highly regarded sources. Reasonable

efforts have been made to publish reliable data and information, but the author and publisher cannot

assume responsibility for the validity of all materials or the consequences of their use. The authors and

publishers have attempted to trace the copyright holders of all material reproduced in this publication

and apologize to copyright holders if permission to publish in this form has not been obtained. If any

copyright material has not been acknowledged please write and let us know so we may rectify in any

future reprint.

Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,

transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or

hereafter invented, including photocopying, microfilming, and recording, or in any information stor￾age or retrieval system, without written permission from the publishers.

For permission to photocopy or use material electronically from this work, please access www.copy￾right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222

Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro￾vides licenses and registration for a variety of users. For organizations that have been granted a photo￾copy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are

used only for identification and explanation without intent to infringe.

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com

Dedicated to the memory of Betty Nachlas

vii

Contents

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

13.4.1 Observable Degradation Processes ................................... 294

13.4.2 Unobservable Degradation Processes............................... 297

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

xiii

Preface

The motivation for the preparation of a second edition was my wish to expand

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 orientation of the topical develop￾ment is that probability models provide an effective vehicle for portraying

and evaluating the variability that is inherent in the performance and lon￾gevity of equipment.

The book is intended to support either an introductory graduate course

in reliability theory and preventive maintenance planning or a sequence of

courses that address these topics. A fairly comprehensive coverage of the

basic models and of various methods of analysis is provided. An under￾standing of the topics discussed should permit the reader to comprehend

the literature describing new and advanced models and methods.

Notwithstanding the emphasis upon initial study, the text should also

serve well as a resource for practicing engineers. Engineers who are involved

in the design process should find a coherent explanation of the reliability

and maintenance issues that will influence the success of the devices they

create. Similarly, engineers responsible for the analysis and verification of

product reliability or for the planning of maintenance support of fielded

equipment should find the material presented here to be relevant and easy

to access and use.

In preparing this second edition, the treatment of statistical methods for

evaluating reliability has been expanded substantially. Several methods for

constructing confidence intervals as part of the parametric estimation effort

are described and methods for treating data derived from operating repair￾able devices have also been added. In addition, the analysis of nonstation￾ary models of repairable equipment maintenance has been updated and

expanded. These expansions along with numerous other minor improve￾ments to the text should make this book an even more useful resource for

both students and practitioners.

The background required of the reader is a sound understanding of prob￾ability. This subsumes capability with calculus. More specifically, the reader

should have an understanding of distribution theory, Laplace transforms,

convolutions, stochastic processes, and Markov processes. It is also worth

mentioning that the use of the methods discussed in this book often involves

substantial computational effort, so facility with numerical methods and

access to efficient mathematical software is desirable.

xiv Preface

One caveat concerning the coverage here is that the treatment is strictly

limited to hardware. Reliability and maintenance models have been devel￾oped for applications to software, humans, and services systems. No criti￾cism of those efforts is intended but the focus here is simply hardware.

The organization of the text is reasonably straightforward. The elemen￾tary concepts of reliability theory are presented sequentially in Chapters 1

through 6. Following this, the commonly used statistical methods for eval￾uating component reliability are described in Chapters 7 and 8. Chapters

9 through 13 treat repairable systems and maintenance planning models.

Here again the presentation is sequential in that simple failure models pre￾cede those that include preventive actions and the renewal cases are treated

before the more realistic nonrenewal cases. In the final chapter, four inter￾esting special topics, including warranties, are discussed. It is worth noting

that four appendices that address aspects of numerical computation are pro￾vided. These should be quite useful to the reader.

Naturally, many people have contributed to the preparation of this text.

The principal factor in the completion of this book was the support and

encouragement of my wife Beverley. An important practical component of

my success was the support of Virginia Tech, especially during sabbaticals

when progress with writing is so much easier.

I acknowledge the significant computational capability provided to me by

the Mathematica software. Many of the analyses included in this text would

have been much more taxing or even impossible without the strength and

efficiency the Wolfram software provides.

I also wish to extend my thanks directly to three of my students, each of

whom contributed to my efforts. Edvin Beqari stimulated my increased inter￾est in and analysis of the diffusion models of degradation. He also directed

much of my analysis of that topic. Elliott Mitchell-Colgan helped to expand

the sets of exercises included at the end of the chapters. Paul D’Agostino

invested very many hours in verifying a majority of the complicated numeri￾cal analyses used for examples or for exercise solutions.

I express my profound gratitude to all of my graduate students who have

taught me so much about these topics over the years. May we all continue to

learn and grow and to enjoy the study of this important subject.