2nd Edition
Reliability Engineering Probabilistic Models and Maintenance Methods, Second Edition
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
13.4.1 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
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.
"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
