Reliability engineering probabilistic models and maintenance methods

RELIABILITY ENGINEERING Probabilistic Models and Maintenance Methods Second Edition RELIABILITY ENGINEERING Probabilistic Models and Maintenance Methods Second Edition JOEL A NACHLAS 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 storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.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 provides licenses and registration for a variety of users For organizations that have been granted a photocopy 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 Contents Preface xiii Author .xv Introduction .1 System Structures 2.1 Status Functions 2.2 System Structures and Status Functions 2.2.1 Series Systems 2.2.2 Parallel System 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 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 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 vii viii Contents 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 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 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 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 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 Index A Age acceleration, 97, 166 accelerated life testing, 85 density functions, 86–87 distribution functions, 86–87 for electronic devices activation energy function, 89 Arrhenius model, 87–88 effectiveness, 86 Eyring model, 89 Nachlas model, 88 power law model, 89 representative thermal cycle, 88 for mechanical devices cumulative damage model, 90 cumulative distribution function, 92 deterioration process, 90 instantaneous rate of aging, 91 killing function, 92 proportional hazards models, 91 rate of actuation, 90 stress–strength models, 91 stress testing, 90 step stress strategies, 92–93 stress screening, 85–86 timescale compression, 86 Age replacement policy, 289 age, 258, 262, 273 availability function, 255–256 availability measures, 253 convenient notation, 248–249 cost per cycle, 251 expected cost function, 252–253 modified policy, 282 operation, 248 preventive vs corrective maintenance, 253 renewal function, 254–255 renewal intervals, 253–255 repair time distribution, 254 Weibull life distribution, 255–256 Army material systems analysis activity (AMSAA) model, 321 Arrhenius model acceleration equation, 166 age acceleration, 87–88 chemical conversion reactions, 71 hazard functions, 72 reaction rate, 71–72 Weibull life distribution, 72 Availability analysis, 278 availability measures availability function, 222 average availability, 220–221 limiting availability, 220 limiting average availability, 221 point availability, 220 exponential case exponential life distribution, 223 exponential repair time distribution, 223 limiting availability, 225 limiting value, 224 Markov models availability function, 243 birth–death process, 244 Chapman–Kolmogorov forward differential equations, 240, 242 expected first passage time to system failure, 244 failure intensity, 244 limiting availability, 243–245 parallel configuration, 241 state transition diagram, 241 system availability, 242 two-state transition diagram, 239–240 nonrenewal cases cost models, 232–233 under imperfect repair, 233–235 quasi-renewal model, 235–240 numerical case, 225–227 365 366 system-level availability availability bounds, 229 availability function, 227 average availability, 230 average system uptime per cycle, 231 average time down per cycle, 231 component replacement rates, 231 individual failure and repair times, 230 limiting availabilities, 228, 230 limiting average system availability, 232 point availabilities, 228 series system, 228 system status, 227 B Backward recurrence time, 176, 189, 266 Birnbaum–Saunders distribution, 52, 70–71 Bivariate reliability collapsible models, 326–327 correlation models, 330–331 device life, 326 probability analysis bivariate (partial) differential equation, 334 cumulative failure probability, 331–332 failure and renewal models, 335–340 hazard function, 333 life distribution function, 334 longevity vectors, 331–332 marginal distributions, 334 MIFR, 333–334 moment-generating function, 334–335 reliability function, 332 stochastic functions analytical approach, 329 bivariate device failure hazard function, 329 bivariate life distribution, 329–330 conditional density, 329 definition of failure models, 327 marginal density, 328 marginal distribution, 330 probability distribution, 328 Index Blackwell’s theorem, 181 Block replacement policy availability function, 256–257 convenient notation, 248–249 delayed renewal process, 256 device replacement, 255 Kijima model, 282 operation, 248 policy time, 250–251 with quasi-renewals, cost rate, 277 renewal function, 256 total cost per unit time, 250 Weibull life distribution, 257 C Catastrophic equipment failures, Chapman–Kolmogorov backward differential equation, 74 Chapman–Kolmogorov forward differential equations, 240, 242 Chi-square coordinates, 357 Coherent systems, 7, 32 Competing risk models crude life distribution, 81 device reliability function, 81 hazard function, 83 joint survivor function, 80 life length of device, 80 marginal (net) survival function, 81 survival probabilities, 82 Conditional probability methods, degradation process bivariate normal distribution, 300 conditional density, 300, 302 estimating equations, 302 failure and survival probabilities, 301 failure time, 301 censored, 303 simulated, 303 joint density, 301 logarithm of likelihood function, 302 marker variable, 301–303 residual life density, 304 Wiener process, 300 Condition-based maintenance, see Predictive maintenance Constant failure rate (CFR) distribution, 44, 54 367 Index Crow’s model, 320–321 Cumulative distribution function (CDF), 346 Cut vector, 15 survival function, 74 system state change, 74 Weibull hazard function, 75 Digamma function, see Psi function Duane’s model, 320–321 D Decreasing failure rate (DFR), 43–44, 183–184, 201–202 Defect model, 72–73 Degradation processes, 287 killing function, 75 mean value and diffusion functions, 75 mechanical and chemical, 76 observable density function, 294 disadvantage, 297 distribution function, 295 estimation equations, 295 failure threshold, 294, 296 likelihood function, 295 regression model, 297 residual life length, 296 residual reliability vs time, 297 unobservable conditional probability methods, 300–304 marker variable, 297 time series methods, 298–299 Dependent components bivariate exponential survivor function, 325 joint survival probability, 325 mutual interactions, 323 shared load system, 324 shock models, 324–325 Development process, 319 Diffusion process models degradation process, 294, 296–297 discretized approximations, 75 Fokker–Planck equation, 74 gamma process, 73 killing function, 92 killing process, 75 life distribution models, 76 Markov process, 74 Poisson process, 73–74 state boundary conditions, 76–78 E Electronic failure models Arrhenius model, 71–72 defect model, 72–73 Eyring model, 72 power law model, 72 Equivalent infinite series expansion, 186 Equivalent structures algebraic effort, 27 computed values, 28 cut vector, 15 inequality, 29 lower bound, 28, 30 minimax bounds, 30–31 minimum cut set, 15–17 minimum paths and minimum cuts, 26, 30 minimum path set, 13 minimum path vector, 13 path vector, 12 reliability values, 26–27 system reliability function, 27 upper bounds, 28–29 Wheatstone bridge minimum cut equivalent structure, 16–17 minimum path equivalent structure, 14–15 reliability block diagram, 12 system status values, 12–13 Erlang distribution, see Gamma distribution Estimates, reliability based on order statistics, 98–99 and confidence intervals beta distribution, 101 binomial distribution, 100 density function, 101 life distribution, 102 number of survivors, 100 point estimates, 99–101 tolerance bounds, 107–109 368 Kaplan–Meier estimates continuous monitoring, 102–105 periodic monitoring, 105–107 Nelson cumulative hazard estimation method, 122–123 Exponential distribution bivariate, 334, 337, 339 conditional survival probability, 45 confidence intervals, 139–141 constant hazard function, 45 density function, 45 life distribution, 46 life lengths of components, 130 likelihood function, 145–146 method of moments with censored data, 161 population mean, 136 renewal process, 177, 179–180 Weibull distribution, 159 Eyring model, 72, 89 F Failure and renewal models, bivariate reliability bivariate exponential distribution, 337 bivariate normal distribution, 338 bivariate normal models, 340 Hunter’s bivariate exponential distribution, 339 inverse transform, 338–339 joint distribution, 335, 340 key integral renewal equation, 336–337 Laplace transform, 337 marginal distributions, 340 noninstantaneous repair cases, 338 number of renewals, 335 renewal density, 336, 338 renewal function, 336–337 renewal vector, 335–336 repair time density, 339 time–frequency duality relation, 335 Failure processes bivariate, 330 competing risk models, 80–83 diffusion process models, 73–78 Index electronic failure models Arrhenius model, 71–72 defect model, 72–73 Eyring model, 72 power law model, 72 excessive stress level, 164 homogeneous, 311 mechanical failure models shock and cumulative damage, 64–71 stress–strength interference, 62–66 nonhomogeneous, 308, 320 principal models, 61 proportional hazards models, 78–80 Failure/survival probability, 98 Fisher information matrix, 147–148, 150, 153, 156, 160 Fokker–Planck equation, 74 Forward recurrence time, 176, 189 Full opportunistic age replacement policy, 271 Full replacement warranties expected profit function, 316–317 expected profit model, 315–316 fixed duration, 317 hazard function identity, 316 key feature, 314 optimal policy time, 317 product generation cost, 315 vs pro rata warranties, 317–318 G Gamma distribution advantage and disadvantage, 52 Cox’s result, 163 cumulative damage model, 68 cumulative intensity function, 197–198 density function, 52, 199 gamma function, 344–345 method of maximum likelihood, 152–153, 155 method of moments, 138, 142–143 renewal process, 177 system deterioration, 288 Gompertz distribution, 53 369 Index H operating intervals, 206–207 random variable, 205, 209 residual life distribution, 283 revised recursion, 209 successive mean residual life lengths, 209 time and frequency domains, 205 transition probability function, 207 virtual age stochastic process, 205 Hessian matrix, 147, 160 Homogeneous Poisson process (HPP), 198–199, 309 HPP, see Homogeneous Poisson process I IFR, see Increasing failure rate IFRA, see Increasing failure rate on average distribution Increasing, decreasing, bathtub (IDB) distribution, 53 Increasing failure rate (IFR), 43, 183–184, 250 Increasing failure rate on average (IFRA) distribution, 54 definition, 44 imperfect repair models, 201–202 shock and cumulative damage models, 67, 69 Inverse power law model, 89 K Kaplan–Meier estimates continuous monitoring, 102–105 periodic monitoring, 105–107 Key integral renewal equation, 336–337 Key renewal equation, 337 Key renewal theorem, 178–179, 188, 219, 282 availability function, 222 Laplace transform, 353 Kijima models block replacement policy, 282 cost per replacement cycle, 282 density function, 206, 209 expectations on measures, 205 generalized renewal density, 282 generalized renewal function, 283 hazard function, 281 Kijima I model, 203–204 Kijima II model, 204, 209 mean residual life under minimal repair, 208, 210 minimal repair as upper bounds, 208, 210 L Lognormal distribution, 51–52 Lomnicki’s method, 186 M Magnuson–Moss Act, 314 Makeham distribution, 52–53 Markov models availability function, 243 birth–death process, 244 Chapman–Kolmogorov forward differential equations, 240, 242 expected first passage time to system failure, 244 failure intensity, 244 limiting availability, 243–245 parallel configuration, 241 state transition diagram, 241 system availability, 242 two-state transition diagram, 239–240 Maximum likelihood method binomial probability, 143–144 confidence interval for quantile, 155 with data censoring, 159–160 density function, 152, 156 derivative expressions, 150 exponential distribution, 145 extreme value distribution, 150–151 Fisher information matrix, 147–148, 150, 153 gamma distributions, 157 gamma function, 152 Hessian matrix, 147 joint distribution, 144 life test data, 156–157 marginal distributions, 145 370 maximum likelihood estimation equation, 144 normal distribution, 157–159 partial derivatives, 152 1–α probability confidence intervals, 149, 157 psi function, 154 variance and covariance values, 150 variance–covariance matrix, 147–148, 154 variance term, 149 Weibull distribution, 146–147, 157 Mechanical failure models shock and cumulative damage models Birnbaum–Saunders distribution, 70–71 conceptual duality, 67 description, 64–65 failure threshold, 68 fatigue failure, Miner’s rule, 69 IFRA distribution, 67 life distribution model, 66 life length distribution, 66 reliability function, 67–68 state-dependent arrival rate, 68 survival function, 69 stress–strength interference model basic concept, 63 device reliability, 62 interference at time values, 64–65 random dispersion, 62 reliability function, 64–66 time evolution of strength distribution, 63–64 Weibull distribution, 64 Method of moments with censored data Cox’s result, 162–163 exponential distribution, 161 gamma distribution, 163 sample variance, 162–163 surrogate sample mean, 161 Weibull distribution, 162 confidence intervals chi-square variate, 139 distribution parameter, 140 exponential distribution, 140 gamma distribution, 142–143 Index for mean, 141 Nelson’s method, 140 normal distribution, 140–141 for reliability, 140 sampling distribution, 139–140, 142 for scale parameter, 142 for standard deviation, 142 for θ, 139 for Weibull parameters, 140 distribution moments, 135 estimation expressions coefficient of variation, 137–138 disadvantage, 138–139 exponential distribution, 136 gamma distributed life test data, 138–139 gamma distribution, 138 normal distribution, 137 numerical search values, 137 variance, 136 Weibull distribution, 136, 138 Miner’s rule, fatigue failure, 69 Minimum cut set, 15–17 Minimum path set, 13 Minimum path vector, 13 Modules of systems, 17–18 Multivariate increasing failure rate (MIFR), 333–334 N Nelson cumulative hazard estimation method, 122–123 New better than used in expectation (NBUE), 182–184 New better than used (NBU) life distributions, 182, 184, 249 New worse than used in expectation (NWUE) life distribution, 182–184 New worse than used (NWU) life distribution, 182 Nonhomogeneous Poisson process (NHPP) construction, 194 cumulative intensity function, 197–198 failure intensity function, 309, 320 Index frequency probabilities, 197–198 gamma distribution, 197–200 hazard function, 194–195 imperfect PM models, 276 mapping, 199 minimal repair, 200 Weibull distribution, 199 Nonparametric statistical methods advantage, 95 data set notation and censoring, 96–97 disadvantage, 95 estimates based on order statistics, 98–99 and confidence intervals, 99–102 Kaplan–Meier estimates, 102–107 Nelson cumulative hazard estimation method, 122–123 tolerance bounds construction, 107 cumulative probability of failure, 108 hazard function, 109 reliability value, 108–109 reliable life, 107 true cumulative failure probability, 107 TTT transforms cumulative failure probability, 109–111 definition, 110 properties, 110–111 scaled total time on test transform, 111–122 Nonrenewal and instantaneous repair equivalent age models device life distribution, 203 Kijima models, 203–210 quasi-renewal process, 210–214 imperfect repair models age-dependent imperfect repair, 202–203 age-independent perfect repair probabilities, 202 conditional intensity of occurrence of perfect repair, 200 cumulative intensity function, 202 371 distribution on time between perfect repairs, 201 equivalent age measures, 202 life and renewal time distributions, 201 survivor function, 200 Weibull distribution, 201–202 minimal repair models cumulative intensity function, 197–199 density function, 199 expectation of random variables, 195–196 failure intensity function, 194–195 gamma distribution, 197–200 hazard function, 197 inversion of cumulative rate function, 200 NHPP model, 194–195, 197–199 nonstationary counting process, 194 Weibull distribution, 199 nonstationary process, 193 Nonrenewal models, PM cost rate function, 275 imperfect PM models age-dependent imperfect repair model, 276 cost per unit time, 275–276 minimal repair cost, 276–277 nonhomogeneous Poisson process, 276 Kijima model cost per replacement cycle, 282 generalized renewal density, 282 generalized renewal function, 283 hazard function, 281 residual life distribution, 283 post maintenance failure distribution, 274 post PM equipment, 275 quasi-renewal process average availability, 278 cost rate model, 277 distribution on kth operating interval, 277 downtime-based availability function, 280–281 expected maintenance cost per unit time, 278 372 life distribution, 279–280 quasi-renewal function, 277, 280 repair time distribution, 279 service time distribution, 279–280 uptime-based availability function, 280 time-and calendar-based PM policies, 274 Normal distribution bivariate, 300, 338 confidence intervals, 106, 140 maximum likelihood, 155 reliability values, 158 sampling distribution, 141–142, 149 cumulative normal probabilities, 344 density function, 49, 343 distribution quantiles, 158 hazard function, 50–51 life length model, 51 method of moments, 137 parameters, 49 quasi-renewal process, 211–213 second partial derivatives, 157 standard normal distribution, 100, 308, 343 standard normal probabilities, 49 Numerical approximations gamma distribution, 344–345 normal distribution function, 343–344 psi (digamma) function, 345–346 student’s t distribution coordinates, 346 Index Opportunistic age replacement policies analysis availability functions, 273–274 life and repair time distributions, 272 limiting availability, 271–273 nested renewal, 272 renewal function, 272 replacement times, 273 Weibull life distributions, 273 density function, 259–260 failure model associated density function, 263 components failure probability, 262 limiting availability, 265 residual life distribution, 263 system availability, 264 total operating time distribution, 264 full opportunistic age replacement policy, 271 life and replacement time distributions, 259 nested renewal process major interval, 258, 262 minor interval, 259–261 operation, 258 opportunistic failure replacement policy, 265–268 partial opportunistic age replacement policy, 268–271 system-level availability, 262 time-dependent availability function, 257 Opportunistic failure replacement policy, 265–268 O Observable degradation processes density function, 294 disadvantage, 297 distribution function, 295 estimation equations, 295 failure threshold, 294, 296 likelihood function, 295 regression model, 297 residual life length, 296 residual reliability vs time, 297 P Parametric statistical methods graphical methods algebraic expression for slope, 130 cumulative hazard function and reliability function, 129 linear regression, 130 mean order statistic-based estimation equation, 131 numerical values, 133 Index ordered failure data, 130–131 regression solution, 133 Weibull distribution, 131, 134 Weibull failure data, 132, 134–135 maximum likelihood method binomial probability, 143–144 confidence interval for quantile, 155 with data censoring, 159–160 density function, 152, 156 derivative expressions, 150 exponential distribution, 145 extreme value distribution, 150–151 Fisher information matrix, 147–148, 150, 153 gamma distributions, 157 gamma function, 152 Hessian matrix, 147 joint distribution, 144 life test data, 156–157 marginal distributions, 145 maximum likelihood estimation equation, 144 normal distribution, 157–159 partial derivatives, 152 1–α probability confidence intervals, 149, 157 psi function, 154 variance and covariance values, 150 variance–covariance matrix, 147–148, 154 variance term, 149 Weibull distribution, 146–147, 157 method of moments with censored data, 161–163 confidence intervals, 139–143 distribution moments, 135 estimation expressions, 136–139 parametric reliability estimation methods, 129 step stress testing Arrhenius acceleration equation, 166 equivalent data set, 165 erroneous inference, 164 extrapolated failure time, 167 failure times, 165 overstress risk, 164 373 raw failure times, 166 three-level step stress regimen, 164 Weibull distribution, 165–166 Partial opportunistic age replacement policy, 268–271 Path vector, 12 PM, see Preventive maintenance Poissonian function, 347 Poisson process diffusion process models, 73–74 failure intensity, 180 HPP, 198–199, 309 NHPP construction, 194 cumulative intensity function, 197–198 failure intensity function, 309, 320 frequency probabilities, 197–198 gamma distribution, 197–200 hazard function, 194–195 imperfect PM models, 276 mapping, 199 minimal repair, 200 Weibull distribution, 199 point process, 177 shock and cumulative damage models, 67–69, 90, 324 Power law model, 72, 89 Predictive maintenance continuous process monitoring marker variable, 294 observable degradation processes, 294–297 unobservable degradation processes, 297–304 inspection scheduling, 289–290 instrumentation selection, 287 policy analysis average first cycle length, 293 cost function, 290 discrete time process, 292 failure threshold, 290 gamma distribution, 291–292 minimum inspection interval, 291 next inspection time, 291 replacement threshold, 290 stationary distribution, 292–293 system deterioration, 288–289 374 Preventive maintenance (PM), 287 advantages, 2–3 availability, 247 definition, 247 nonrenewal models cost rate function, 275 imperfect PM models, 275–277 Kijima model, 281–283 post maintenance failure distribution, 274 post PM equipment, 275 quasi-renewal process, 277–281 time-and calendar-based PM policies, 274 replacement policies age replacement, 248, 251–255, 289 block replacement, 248, 250, 255–257 convenient notation, 248–249 cost function, 252–253 cost models, 248, 250–251 cost per cycle, 251 expected cost per unit time, 252 field failure reduction, 250 IFR life distribution, 250 NBU life distributions, 249, 251 opportunistic age replacement, 257–274 optimal block replacement intervals, 251–252 random variable, 249 renewal density, 251 Weibull life distribution, 251 resource commitment, 247 safety and industrial productivity, Probability elements, 23 Prognostics, see Predictive maintenance Proportional hazards models, 326 advantages, 79 age acceleration, 91 base hazard function, 79 covariates/explanatory variables, 79 cumulative hazard function, 79 environment and failure process relationship, 78 hazard rate enhancement, 86 reliability function, 79 satellite-based electronic device, 79–80 Index Pro rata warranties automobile components, 314 characteristics, 314 discount factor, 319 expected profit, 319 vs full replacement warranties, 317–318 repurchase price, 318 repurchase probability, 318 residual value of failed unit, 318 Psi function definition, 345 maximum likelihood, 152–154 numerical approximation, 346 series expansion, 345 Q Quasi-renewal process availability analysis distributions on repair times, 235 downtime-based approximation, 238 downtime-based form, 237 exponential case values, 238, 240 Laplace transform, 236–237 operating intervals, 235 point availability, 238–239 uptime-based approximation, 238 uptime-based function, 237 average availability, 278 cost rate model, 277 distribution on kth operating interval, 277 downtime-based availability function, 280–281 expected maintenance cost per unit time, 278 geometric process, 210 intensity function, 212 life distribution, 279–280 normal density functions, 211 normal distribution functions, 211–212 normal life distribution, 213–214 postrepair device state, 203 quasi-renewal function, 212, 277, 280 repair time distribution, 279 sequence of operating intervals, 210 Index service time distribution, 279–280 time frequency relationship, 211 uptime-based availability function, 280 R Reliability engineering Boeing 787 aircraft, lithium-ion battery fire, 1986 Challenger accident, definition, 2002 Discovery accident, economic implications, failure features, human and software reliability, 3–4 in-service failure, life cycle costs, potential productivity gains, preventive maintenance, 2–4 reliability analysis, space shuttle accidents, Three Mile Island accident vs light bulb failure, Reliability growth AMSAA model, 321 Crow’s model, 320–321 cumulative mean time before failure, 320 embryonic models, 320 failure intensity function, 322–323 learning curve model, 320 TAAF process, 320, 322 test data, 322 Reliability over time bathtub curve, 42 cumulative hazard function, 41 density function, 40 distribution function, 40 failure rate, 40 functions of time, 39 hazard function, 40, 42 infant mortality period, 42 life distributions Birnbaum–Saunders distribution, 52 CFR distribution, 44 DFR distribution, 43–44 exponential distribution, 45–46 375 gamma distribution, 52 Gompertz distribution, 53 IDB distribution, 53 IFRA distribution, 44 IFR distribution, 43 lognormal distribution, 51–52 Makeham distribution, 52–53 normal distribution, 49–51 Weibull distribution, 46–49 life length, 39, 41 survivor (reliability) function, 40 system-level models density function, 54, 56 distribution, 57 hazard function, 55–56 IFRA, 54 reliability function, 54 standby redundant system, 57 Renewal and instantaneous repair life distributions classification bounds on renewal measures, 183–186 life lengths, 182 NBU, 182 NBUE, 182–184 nested ordering, 183 NWU, 182, 184 NWUE, 182–184 operating interval length, 181 preventive and corrective maintenance, 173 renewal process backward recurrence time, 176 Blackwell’s theorem, 181 definition, 174 density function, 177 exponential distribution, 177, 180 forward recurrence time, 176 gamma distribution, 177–178 graphical representation, 175 hazard function, 179 key renewal theorem, 178 k-fold convolution, 175 Laplace transform, 177, 180 modified/delayed renewal process, 175 nonnegative random variables, 174 ordinary renewal process, 175 376 point process, 174–175 Poisson process, 177, 180 probability distribution, 174, 176 rate of failure, 179 renewal density, 179–180 renewal function, 178, 180 Weibull life distribution, 181 residual life distribution backward recurrence time, 189 definition, 186 forward recurrence time, 189 key renewal theorem, 188 limiting forms, 188–189 random variable value, 187 renewal density, 189 Renewal density, 180, 189, 222, 336, 338 availability function, age replacement policy, 255–256 Kijima model, 282 major intervals, 262 minor intervals, 261, 264 renewal function, 179, 251 Renewal function, 219, 272, 283, 336–337, 353 age replacement policy, 254–256 bivariate, 335 definition, 178 failure time, 222 Laplace transform, 338 life distributions, 180 product life distribution function, 317 quasi-renewal function, 212, 236, 277, 280 renewal density, 179, 251 Weibull equivalent infinite series expansion, 186 gamma function–based quantities, 349–352 infinite sum representation, 347 k-fold convolution, 347 recursive definition, 349 renewal and convolution distribution functions, 348 renewal function and the renewal density coefficients, 349 values for coefficients, 352 Index Repairable systems definition, 173 nonrenewal and instantaneous repair equivalent age models, 203–214 imperfect repair models, 200–203 minimal repair models, 194–200 nonstationary process, 193 renewal and instantaneous repair life distributions classification, 181–186 preventive and corrective maintenance, 173 renewal processes, 174–181 residual life distribution, 186–189 repair duration expected number of failures, 219 expected number of repair completions, 219 failure times, 218 lengths of repair intervals, 219 operating times, 217 representative sample path, 217–218 total duration, 217 statistical analysis data from multiple identical systems, 310–313 data from single system, 307–310 S Scaled total time on test transform censored data sets, 118–122 complete data sets, 113–118 finite mean, 111 hazard functions, 112–113 scaling constant, 112 Weibull distribution, 112 Standard normal cumulative probabilities, 355 Statistical analysis, repairable system data from multiple identical systems confidence interval, 311 failure intensity function, 311–312 homogeneous failure times, 311–312 likelihood function, 311–312 Index nonhomogeneous failure times, 313 number of degrees of freedom, 313 number of failures, 310–311 from single system cumulative failure intensity, 309 distribution on order statistics, 309 failure intensity function, 307–309 homogeneous and nonhomogeneous failure process, 308–309 joint density, 310 NHPP, 309 (1–α) probability confidence bound, 309 (1–α) probability confidence interval, 310 standard normal distribution, 308 Status functions coherent systems, component status variables, component status vector, functional status of device, and system structures equivalent structures, 12–17 k-out-of-n system, 10–12 multistate components, 18–19 parallel system, 8–10 series systems, 7–8 Step stress testing, parametric statistical methods Arrhenius acceleration equation, 166 equivalent data set, 165 erroneous inference, 164 extrapolated failure time, 167 failure times, 165 overstress risk, 164 raw failure times, 166 three-level step stress regimen, 164 Weibull distribution, 165–166 Stress–strength models age acceleration, 91 interference model, 62–64, 67 377 Student’s t distribution coordinates, 346 probability, 356 System-level availability availability bounds, 229 availability function, 227 average availability, 230 average system uptime per cycle, 231 average time down per cycle, 231 component replacement rates, 231 individual failure and repair times, 230 limiting availabilities, 228, 230 limiting average system availability, 232 opportunistic age replacement policy, 262 point availabilities, 228 series system, 228 system status, 227 System structures equivalent structures algebraic effort, 27 computed values, 28 cut vector, 15 inequality, 29 lower bound, 28, 30 minimax bounds, 30–31 minimum cut set, 15–17 minimum paths and minimum cuts, 14–17, 26, 30 minimum path set, 13 minimum path vector, 13 path vector, 12 reliability values, 26–27 system reliability function, 27 upper bounds, 28–29 k-out-of-n system, 10–12, 25–26 modules, 31–32 multistate components, 18–19 parallel system, 8–10 parallel systems, 25 reliability allocation, 35–36 definition, importance, 32–35 series systems, 7–8, 24–25 378 T Test-analyze-and-fix (TAAF) process, 320, 322 Test unit status, Kaplan–Meier estimates continuous monitoring, 102–105 periodic monitoring, 105–107 Time series methods, degradation process, 298–299 Total time on test (TTT) transforms cumulative failure probability, 109–111 definition, 110 properties, 110–111 scaled TTT transform censored data sets, 118–122 complete data sets, 113–118 finite mean, 111 hazard functions, 112–113 scaling constant, 112 Weibull distribution, 112 TTT transforms, see Total time on test transforms U Unobservable degradation processes conditional probability methods, 300–304 marker variable, 297 time series methods, 298–299 V Variance–covariance matrix, 147–148, 154, 156, 160 W Warranties full replacement warranties expected profit function, 316–317 expected profit model, 315–316 fixed duration, 317 hazard function identity, 316 key feature, 314 optimal policy time, 317 product generation cost, 315 vs pro rata warranties, 317–318 Index Magnuson–Moss Act, 314 pro rata warranties automobile components, 314 characteristics, 314 discount factor, 319 expected profit, 319 vs full replacement warranties, 317–318 repurchase price, 318 repurchase probability, 318 residual value of failed unit, 318 Weibull distribution, 134 Cox’s result, 162 extreme distribution, 49 extreme value argument, 48 failure times, 48 function, 46 gamma distribution, 344 hazard functions, 47–49 life lengths, 184, 347 maximum likelihood function, 146–147, 152–153, 156, 159–160 mean and hazard function, 112 method of moments, 136, 138, 141–142, 165 Rayleigh type, 53 reliability function, interference model, 64–66 renewal function, 347 three-parameter Weibull life distribution, 46–47 two-parameter Weibull distribution function, 47 Weibull renewal functions equivalent infinite series expansion, 186 gamma function–based quantities, 349–352 infinite sum representation, 347 k-fold convolution, 347 recursive definition, 349 renewal and convolution distribution functions, 348 renewal function and the renewal density coefficients, 349 values for coefficients, 352 .. .RELIABILITY ENGINEERING Probabilistic Models and Maintenance Methods Second Edition RELIABILITY ENGINEERING Probabilistic Models and Maintenance Methods Second Edition... mathematical models of human and software reliability requires Reliability Engineering: Probabilistic Models and Maintenance Methods the acceptance of the view that probability models appropriately... development of new methods and models for reliability and maintenance has expanded our understanding significantly and is continuing The importance of preventive maintenance for safety and industrial