Handbook of reliability engineering

Contents PART I. System Reliability and Optimization 1 Multi-state k-out-of-n Systems Ming J. Zuo, Jinsheng Huang andWay Kuo . . . . . . . . . . . . . . . . . . . 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Relevant Concepts in Binary Reliability Theory . . . . . . . . . . . . . 3 1.3 Binary k-out-of-nModels . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 The k-out-of-n:G System with Independently and Identically DistributedComponents . . . . . . . . . . . . . . . . . . . . . 5 1.3.2 Reliability Evaluation Using Minimal Path or Cut Sets . . . . . 5 1.3.3 RecursiveAlgorithms . . . . . . . . . . . . . . . . . . . . . . 6 1.3.4 Equivalence Between a k-out-of-n:G System and an (n . k + 1)-out-of-n:Fsystem . . . . . . . . . . . . . . . . . . 6 1.3.5 The Dual Relationship Between the k-out-of-n G and F Systems 7 1.4 RelevantConceptsinMulti-stateReliabilityTheory . . . . . . . . . . 8 1.5 A Simple Multi-state k-out-of-n:GModel . . . . . . . . . . . . . . . . 10 1.6 A Generalized Multi-state k-out-of-n:GSystemModel . . . . . . . . . 11 1.7 Properties of Generalized Multi-state k-out-of-n:GSystems . . . . . . 13 1.8 Equivalence and Duality in Generalized Multi-state k-out-of-n Systems 15 2 Reliability of Systems with Multiple Failure Modes Hoang Pham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 TheSeriesSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 TheParallelSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 CostOptimization . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4 TheParallel.SeriesSystem . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 TheProfitMaximizationProblem . . . . . . . . . . . . . . . . 23 2.4.2 OptimizationProblem . . . . . . . . . . . . . . . . . . . . . . 24 2.5 TheSeries.ParallelSystem . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5.1 MaximizingtheAverageSystemProfit . . . . . . . . . . . . . 26 2.5.2 ConsiderationofTypeIDesignError . . . . . . . . . . . . . . 27 2.6 The k-out-of-n Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6.1 Minimizing the Average SystemCost . . . . . . . . . . . . . . 29 2.7 Fault-tolerantSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7.1 ReliabilityEvaluation . . . . . . . . . . . . . . . . . . . . . . 33 xii Contents 2.7.2 RedundancyOptimization . . . . . . . . . . . . . . . . . . . . 34 2.8 WeightedSystemswithThreeFailureModes . . . . . . . . . . . . . . 34 3 Reliabilities of Consecutive-k Systems Jen-Chun Chang and Frank K. Hwang . . . . . . . . . . . . . . . . . . . . . . 37 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.2 ComputationofReliability . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 TheRecursiveEquationApproach . . . . . . . . . . . . . . . 39 3.2.2 TheMarkovChainApproach . . . . . . . . . . . . . . . . . . 40 3.2.3 AsymptoticAnalysis . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 InvariantConsecutiveSystems . . . . . . . . . . . . . . . . . . . . . . 41 3.3.1 InvariantConsecutive-2Systems . . . . . . . . . . . . . . . . 41 3.3.2 Invariant Consecutive-k Systems . . . . . . . . . . . . . . . . 42 3.3.3 Invariant Consecutive-kGSystem. . . . . . . . . . . . . . . . 43 3.4 Component Importance and the Component Replacement Problem . 43 3.4.1 TheBirnbaumImportance . . . . . . . . . . . . . . . . . . . . 44 3.4.2 PartialBirnbaumImportance . . . . . . . . . . . . . . . . . . 45 3.4.3 TheOptimalComponentReplacement . . . . . . . . . . . . . 45 3.5 TheWeighted-consecutive-k-out-of-n System. . . . . . . . . . . . . . 47 3.5.1 The LinearWeighted-consecutive-k-out-of-n System . . . . . 47 3.5.2 The CircularWeighted-consecutive-k-out-of-n System . . . . 47 3.6 WindowSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.6.1 The f -within-consecutive-k-out-of-n System . . . . . . . . . 49 3.6.2 The 2-within-consecutive-k-out-of-n System . . . . . . . . . . 51 3.6.3 The b-fold-windowSystem . . . . . . . . . . . . . . . . . . . 52 3.7 NetworkSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.7.1 The Linear Consecutive-2 Network System . . . . . . . . . . . 53 3.7.2 The Linear Consecutive-kNetworkSystem . . . . . . . . . . . 54 3.7.3 The Linear Consecutive-k FlowNetworkSystem . . . . . . . . 55 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 Multi-state System Reliability Analysis and Optimization G. Levitin and A. Lisnianski . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Multi-state SystemReliabilityMeasures . . . . . . . . . . . . . . . . . 63 4.3 Multi-state System Reliability Indices Evaluation Based on the UniversalGeneratingFunction . . . . . . . . . . . . . . . . . . . . . . 64 4.4 Determination of u-function of ComplexMulti-state System Using CompositionOperators . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5 Importance and Sensitivity Analysis of Multi-state Systems . . . . . . 68 4.6 Multi-state SystemStructureOptimizationProblems . . . . . . . . . . 72 4.6.1 OptimizationTechnique . . . . . . . . . . . . . . . . . . . . . 73 4.6.1.1 GeneticAlgorithm . . . . . . . . . . . . . . . . . . 73 Contents xiii 4.6.1.2 Solution Representation and Decoding Procedure . 75 4.6.2 Structure Optimization of Series.Parallel System with Capacity-basedPerformanceMeasure . . . . . . . . . . . . . 75 4.6.2.1 ProblemFormulation . . . . . . . . . . . . . . . . . 75 4.6.2.2 Solution Quality Evaluation . . . . . . . . . . . . . 76 4.6.3 Structure Optimization of Multi-state System with Two Failure Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6.3.1 ProblemFormulation . . . . . . . . . . . . . . . . . 77 4.6.3.2 Solution Quality Evaluation . . . . . . . . . . . . . 80 4.6.4 Structure Optimization for Multi-state System with Fixed Resource Requirements and Unreliable Sources . . . . . . . . 83 4.6.4.1 ProblemFormulation . . . . . . . . . . . . . . . . . 83 4.6.4.2 Solution Quality Evaluation . . . . . . . . . . . . . 84 4.6.4.3 The Output Performance Distribution of a System Containing Identical Elements in the Main ProducingSubsystem . . . . . . . . . . . . . . . . . 85 4.6.4.4 The Output Performance Distribution of a System Containing Different Elements in the Main ProducingSubsystem . . . . . . . . . . . . . . . . . 85 4.6.5 Other Problems of Multi-state System Optimization . . . . . . 87 5 Combinatorial Reliability Optimization C. S. Sung, Y. K. Cho and S. H. Song . . . . . . . . . . . . . . . . . . . . . . . 91 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Combinatorial Reliability Optimization Problems of Series Structure . 95 5.2.1 OptimalSolutionApproaches . . . . . . . . . . . . . . . . . . 95 5.2.1.1 Partial Enumeration Method . . . . . . . . . . . . . 95 5.2.1.2 Branch-and-bound Method . . . . . . . . . . . . . . 96 5.2.1.3 DynamicProgramming . . . . . . . . . . . . . . . . 98 5.2.2 HeuristicSolutionApproach . . . . . . . . . . . . . . . . . . 99 5.3 Combinatorial Reliability Optimization Problems of a Non-series Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3.1 Mixed Series.Parallel System Optimization Problems . . . . . 102 5.3.2 General System Optimization Problems . . . . . . . . . . . . 106 5.4 Combinatorial Reliability Optimization Problems with Multiple-choiceConstraints . . . . . . . . . . . . . . . . . . . . . . . 107 5.4.1 One-dimensionalProblems . . . . . . . . . . . . . . . . . . . 108 5.4.2 Multi-dimensionalProblems . . . . . . . . . . . . . . . . . . 111 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 PART II. Statistical Reliability Theory 6 Modeling the Observed Failure Rate M. S. Finkelstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.2 Survival inthePlane . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 xiv Contents 6.2.1 One-dimensionalCase . . . . . . . . . . . . . . . . . . . . . . 118 6.2.2 FixedObstacles . . . . . . . . . . . . . . . . . . . . . . . . . . 119 6.2.3 FailureRateProcess . . . . . . . . . . . . . . . . . . . . . . . 121 6.2.4 MovingObstacles . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.3 MultipleAvailability . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.3.1 Statementof theProblem . . . . . . . . . . . . . . . . . . . . 124 6.3.2 OrdinaryMultipleAvailability . . . . . . . . . . . . . . . . . . 125 6.3.3 Accuracy of a Fast Repair Approximation . . . . . . . . . . . . 126 6.3.4 TwoNon-servicedDemandsinaRow . . . . . . . . . . . . . . 127 6.3.5 Not More than N Non-serviced Demands . . . . . . . . . . . 129 6.3.6 TimeRedundancy . . . . . . . . . . . . . . . . . . . . . . . . 130 6.4 ModelingtheMixtureFailureRate . . . . . . . . . . . . . . . . . . . . 132 6.4.1 Definitions and Conditional Characteristics . . . . . . . . . . 132 6.4.2 AdditiveModel . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.4.3 MultiplicativeModel . . . . . . . . . . . . . . . . . . . . . . . 133 6.4.4 SomeExamples . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.4.5 InverseProblem . . . . . . . . . . . . . . . . . . . . . . . . . 136 7 Concepts of Stochastic Dependence in Reliability Analysis C. D. Lai and M. Xie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 7.2 Important Conditions Describing Positive Dependence . . . . . . . . 142 7.2.1 SixBasicConditions . . . . . . . . . . . . . . . . . . . . . . . 143 7.2.2 The Relative Stringency of the Conditions . . . . . . . . . . . 143 7.2.3 Positive Quadrant Dependent in Expectation . . . . . . . . . . 144 7.2.4 AssociatedRandomVariables . . . . . . . . . . . . . . . . . . 144 7.2.5 Positively Correlated Distributions . . . . . . . . . . . . . . . 145 7.2.6 Summaryof Interrelationships . . . . . . . . . . . . . . . . . 145 7.3 PositiveQuadrantDependentConcept . . . . . . . . . . . . . . . . . 145 7.3.1 Constructions of Positive Quadrant Dependent Bivariate Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.3.2 Applications of Positive Quadrant Dependence Concept to Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.3.3 Effect of Positive Dependence on the Mean Lifetime of a ParallelSystem . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.3.4 Inequality Without Any Aging Assumption . . . . . . . . . . . 147 7.4 Families of Bivariate Distributions that are Positive Quadrant Dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.4.1 Positive Quadrant Dependent Bivariate Distributions with SimpleStructures . . . . . . . . . . . . . . . . . . . . . . . . 148 7.4.2 Positive Quadrant Dependent Bivariate Distributions with MoreComplicatedStructures . . . . . . . . . . . . . . . . . . 149 7.4.3 Positive Quadrant Dependent Bivariate Uniform Distributions 150 7.4.3.1 Generalized Farlie.Gumbel.Morgenstern Family of Copulas . . . . . . . . . . . . . . . . . . . . . . . . 151 7.5 Some Related Issues on Positive Dependence . . . . . . . . . . . . . . 152 Contents xv 7.5.1 Examples of Bivariate Positive Dependence Stronger than PositiveQuadrantDependentCondition . . . . . . . . . . . . 152 7.5.2 ExamplesofNegativeQuadrantDependence . . . . . . . . . . 153 7.6 PositiveDependenceOrderings . . . . . . . . . . . . . . . . . . . . . 153 7.7 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8 Statistical Reliability Change-point Estimation Models Ming Zhao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2 Assumptions in Reliability Change-point Models . . . . . . . . . . . . 158 8.3 SomeSpecificChange-pointModels . . . . . . . . . . . . . . . . . . . 159 8.3.1 Jelinski.Moranda De-eutrophication Model with a Change Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.3.1.1 ModelReview . . . . . . . . . . . . . . . . . . . . . 159 8.3.1.2 Model with One Change Point . . . . . . . . . . . . 159 8.3.2 Weibull Change-point Model . . . . . . . . . . . . . . . . . . 160 8.3.3 Littlewood Model with One Change Point . . . . . . . . . . . 160 8.4 MaximumLikelihoodEstimation . . . . . . . . . . . . . . . . . . . . 160 8.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 9 Concepts and Applications of Stochastic Aging in Reliability C. D. Lai and M. Xie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9.2 Basic Concepts for Univariate Reliability Classes . . . . . . . . . . . . 167 9.2.1 Some Acronyms and the Notions of Aging . . . . . . . . . . . 167 9.2.2 DefinitionsofReliabilityClasses . . . . . . . . . . . . . . . . 167 9.2.3 Interrelationships . . . . . . . . . . . . . . . . . . . . . . . . 169 9.3 Propertiesof theBasicConcepts . . . . . . . . . . . . . . . . . . . . . 169 9.3.1 Properties of Increasing and Decreasing Failure Rates . . . . . 169 9.3.2 Property of Increasing Failure Rate on Average . . . . . . . . . 169 9.3.3 Properties of NBU, NBUC, and NBUE . . . . . . . . . . . . . . 169 9.4 DistributionswithBathtub-shapedFailureRates . . . . . . . . . . . . 169 9.5 Life Classes Characterized by the Mean Residual Lifetime . . . . . . . 170 9.6 SomeFurtherClassesofAging . . . . . . . . . . . . . . . . . . . . . . 171 9.7 PartialOrderingofLifeDistributions . . . . . . . . . . . . . . . . . . 171 9.7.1 RelativeAging . . . . . . . . . . . . . . . . . . . . . . . . . . 172 9.7.2 ApplicationsofPartialOrderings . . . . . . . . . . . . . . . . 172 9.8 BivariateReliabilityClasses . . . . . . . . . . . . . . . . . . . . . . . 173 9.9 TestsofStochasticAging . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.9.1 AGeneralSketchofTests . . . . . . . . . . . . . . . . . . . . 174 9.9.2 Summary of Tests of Aging in Univariate Case . . . . . . . . . 177 9.9.3 Summary of Tests of Bivariate Aging . . . . . . . . . . . . . . 177 9.10 ConcludingRemarksonAging . . . . . . . . . . . . . . . . . . . . . . 177 xvi Contents 10 Class of NBU-t0 Life Distribution Dong Ho Park . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.2 Characterization of NBU-t0Class . . . . . . . . . . . . . . . . . . . . 182 10.2.1 Boundary Members of NBU-t0 and NWU-t0 . . . . . . . . . . 182 10.2.2 Preservation of NBU-t0 and NWU-t0 Properties under ReliabilityOperations . . . . . . . . . . . . . . . . . . . . . . 184 10.3 Estimation of NBU-t0 LifeDistribution . . . . . . . . . . . . . . . . . 186 10.3.1 Reneau.SamaniegoEstimator . . . . . . . . . . . . . . . . . . 186 10.3.2 Chang.RaoEstimator . . . . . . . . . . . . . . . . . . . . . . 188 10.3.2.1 Positively Biased Estimator . . . . . . . . . . . . . . 188 10.3.2.2 Geometric Mean Estimator . . . . . . . . . . . . . . 188 10.4 Tests for NBU-t0 LifeDistribution . . . . . . . . . . . . . . . . . . . . 189 10.4.1 Tests for NBU-t0 Alternatives Using Complete Data . . . . . . 189 10.4.1.1 Hollander.Park.Proschan Test . . . . . . . . . . . . 190 10.4.1.2 Ebrahimi.Habibullah Test . . . . . . . . . . . . . . 192 10.4.1.3 AhmadTest . . . . . . . . . . . . . . . . . . . . . . 193 10.4.2 Tests for NBU-t0 Alternatives Using Incomplete Data . . . . . 195 PART III. Software Reliability 11 Software Reliability Models: A Selective Survey and New Directions Siddhartha R. Dalal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 11.2 StaticModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 11.2.1 Phase-based Model: Gaffney and Davis . . . . . . . . . . . . . 203 11.2.2 Predictive Development Life Cycle Model: Dalal and Ho . . . . 203 11.3 Dynamic Models: Reliability Growth Models for Testing and OperationalUse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 11.3.1 AGeneralClassofModels . . . . . . . . . . . . . . . . . . . . 205 11.3.2 Assumptions Underlying the Reliability Growth Models . . . . 206 11.3.3 Caution in Using Reliability Growth Models . . . . . . . . . . 207 11.4 ReliabilityGrowthModelingwithCovariates . . . . . . . . . . . . . . 207 11.5 WhentoStopTestingSoftware . . . . . . . . . . . . . . . . . . . . . . 208 11.6 ChallengesandConclusions . . . . . . . . . . . . . . . . . . . . . . . 209 12 Software Reliability Modeling James Ledoux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.2 BasicConceptsofStochasticModeling . . . . . . . . . . . . . . . . . 214 12.2.1 Metrics with Regard to the First Failure . . . . . . . . . . . . . 214 12.2.2 StochasticProcessofTimesofFailure . . . . . . . . . . . . . . 215 12.3 Black-boxSoftwareReliabilityModels . . . . . . . . . . . . . . . . . . 215 12.3.1 Self-excitingPointProcesses . . . . . . . . . . . . . . . . . . . 216 12.3.1.1 Counting Statistics for a Self-exciting Point Process . 218 Contents xvii 12.3.1.2 Likelihood Function for a Self-exciting Point Process 218 12.3.1.3 Reliability and Mean Time to Failure Functions . . . 218 12.3.2 Classification of Software Reliability Models . . . . . . . . . . 219 12.3.2.1 0-Memory Self-exciting Point Process . . . . . . . . 219 12.3.2.2 Non-homogeneous Poisson Process Model: ƒÉ(t; Ht , F0) = f (t; F0) and is Deterministic . . . . 220 12.3.2.3 1-Memory Self-exciting Point Process with ƒÉ(t; Ht , F0) = f (N(t), t . TN(t), F0) . . . . . . . . 221 12.3.2.4 m . 2-Memory . . . . . . . . . . . . . . . . . . . . 221 12.4 White-boxModeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 12.5 CalibrationofModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 12.5.1 FrequentistProcedures . . . . . . . . . . . . . . . . . . . . . . 223 12.5.2 BayesianProcedure . . . . . . . . . . . . . . . . . . . . . . . 225 12.6 Current Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 12.6.1 Black-boxModeling . . . . . . . . . . . . . . . . . . . . . . . 225 12.6.1.1 Imperfect Debugging . . . . . . . . . . . . . . . . . 225 12.6.1.2 Early Prediction of Software Reliability . . . . . . . 226 12.6.1.3 Environmental Factors . . . . . . . . . . . . . . . . 227 12.6.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 228 12.6.2 White-boxModeling . . . . . . . . . . . . . . . . . . . . . . . 229 12.6.3 Statistical Issues . . . . . . . . . . . . . . . . . . . . . . . . . 230 13 Software Availability Theory and Its Applications Koichi Tokuno and Shigeru Yamada . . . . . . . . . . . . . . . . . . . . . . . 235 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 13.2 BasicModelandSoftwareAvailabilityMeasures . . . . . . . . . . . . 236 13.3 ModifiedModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 13.3.1 Model with Two Types of Failure . . . . . . . . . . . . . . . . 239 13.3.2 ModelwithTwoTypesofRestoration . . . . . . . . . . . . . . 240 13.4 AppliedModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 13.4.1 ModelwithComputationPerformance . . . . . . . . . . . . . 241 13.4.2 Model for Hardware.Software System . . . . . . . . . . . . . 242 13.5 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 14 Software Rejuvenation:Modeling and Applications Tadashi Dohi, Katerina Go.eva-Popstojanova, Kalyanaraman Vaidyanathan, Kishor S. Trivedi and Shunji Osaki . . . . . . . . . . . . . . . . . . . . . . . . 245 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 14.2 Modeling-basedEstimation . . . . . . . . . . . . . . . . . . . . . . . 246 14.2.1 Examples in Telecommunication Billing Applications . . . . . 247 14.2.2 Examples in a Transaction-based Software System . . . . . . . 251 14.2.3 ExamplesinaClusterSystem . . . . . . . . . . . . . . . . . . 255 14.3 Measurement-basedEstimation . . . . . . . . . . . . . . . . . . . . . 257 14.3.1 Time-basedEstimation . . . . . . . . . . . . . . . . . . . . . 258 14.3.2 Time andWorkload-based Estimation . . . . . . . . . . . . . 260 14.4 ConclusionandFutureWork . . . . . . . . . . . . . . . . . . . . . . . 262 xviii Contents 15 Software Reliability Management: Techniques and Applications Mitsuhiro Kimura and Shigeru Yamada . . . . . . . . . . . . . . . . . . . . . 265 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 15.2 Death Process Model for Software TestingManagement . . . . . . . . 266 15.2.1 ModelDescription . . . . . . . . . . . . . . . . . . . . . . . . 267 15.2.1.1 Mean Number of Remaining Software Faults/Testing Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 268 15.2.1.2 Mean Time to Extinction . . . . . . . . . . . . . . . 268 15.2.2 EstimationMethodofUnknownParameters . . . . . . . . . . 268 15.2.2.1 Case of 0<ƒ¿ . 1 . . . . . . . . . . . . . . . . . . . 268 15.2.2.2 Case of ƒ¿ = 0 . . . . . . . . . . . . . . . . . . . . . 269 15.2.3 SoftwareTestingProgressEvaluation . . . . . . . . . . . . . . 269 15.2.4 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . 270 15.2.5 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . 271 15.3 Estimation Method of Imperfect Debugging Probability . . . . . . . . 271 15.3.1 Hidden-Markov modeling for software reliability growth phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 15.3.2 EstimationMethodofUnknownParameters . . . . . . . . . . 272 15.3.3 Numerical Illustrations . . . . . . . . . . . . . . . . . . . . . 273 15.3.4 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . 274 15.4 Continuous State Space Model for Large-scale Software . . . . . . . . 274 15.4.1 ModelDescription . . . . . . . . . . . . . . . . . . . . . . . . 275 15.4.2 Nonlinear Characteristics of Software Debugging Speed . . . . 277 15.4.3 EstimationMethodofUnknownParameters . . . . . . . . . . 277 15.4.4 Software Reliability AssessmentMeasures . . . . . . . . . . . 279 15.4.4.1 Expected Number of Remaining Faults and Its Variance . . . . . . . . . . . . . . . . . . . . . . . . 279 15.4.4.2 Cumulative and Instantaneous Mean Time Between Failures . . . . . . . . . . . . . . . . . . . . . . . . 279 15.4.5 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . 280 15.5 DevelopmentofaSoftwareReliabilityManagementTool . . . . . . . . 280 15.5.1 Definitionof theSpecificationRequirement . . . . . . . . . . 280 15.5.2 Object-orientedDesign . . . . . . . . . . . . . . . . . . . . . 281 15.5.3 Examples of Reliability Estimation and Discussion . . . . . . 282 16 Recent Studies in Software Reliability Engineering Hoang Pham . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 16.1.1 SoftwareReliabilityConcepts . . . . . . . . . . . . . . . . . . 285 16.1.2 SoftwareLifeCycle . . . . . . . . . . . . . . . . . . . . . . . . 288 16.2 SoftwareReliabilityModeling . . . . . . . . . . . . . . . . . . . . . . 288 16.2.1 A Generalized Non-homogeneous Poisson Process Model . . . 289 16.2.2 Application 1: The Real-time Control System . . . . . . . . . . 289 16.3 Generalized Models with Environmental Factors . . . . . . . . . . . . 289 16.3.1 ParametersEstimation . . . . . . . . . . . . . . . . . . . . . . 292 16.3.2 Application 2: The Real-time Monitor Systems . . . . . . . . . 292 Contents xix 16.4 CostModeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 16.4.1 Generalized Risk.CostModels . . . . . . . . . . . . . . . . . 295 16.5 Recent Studies with Considerations of Random Field Environments . 296 16.5.1 AReliabilityModel . . . . . . . . . . . . . . . . . . . . . . . . 297 16.5.2 ACostModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 16.6 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 PART IV. Maintenance Theory and Testing 17 Warranty and Maintenance D. N. P. Murthy and N. Jack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 17.2 ProductWarranties:AnOverview . . . . . . . . . . . . . . . . . . . . 306 17.2.1 RoleandConcept . . . . . . . . . . . . . . . . . . . . . . . . . 306 17.2.2 ProductCategories . . . . . . . . . . . . . . . . . . . . . . . . 306 17.2.3 WarrantyPolicies . . . . . . . . . . . . . . . . . . . . . . . . . 306 17.2.3.1 Warranties Policies for Standard Products Sold Individually . . . . . . . . . . . . . . . . . . . . . . 306 17.2.3.2 Warranty Policies for Standard Products Sold in Lots 307 17.2.3.3 Warranty Policies for Specialized Products . . . . . 307 17.2.3.4 ExtendedWarranties . . . . . . . . . . . . . . . . . 307 17.2.3.5 Warranties for Used Products . . . . . . . . . . . . 308 17.2.4 IssuesinProductWarranty . . . . . . . . . . . . . . . . . . . 308 17.2.4.1 Warranty Cost Analysis . . . . . . . . . . . . . . . . 308 17.2.4.2 WarrantyServicing . . . . . . . . . . . . . . . . . . 309 17.2.5 ReviewofWarrantyLiterature . . . . . . . . . . . . . . . . . . 309 17.3 Maintenance:AnOverview . . . . . . . . . . . . . . . . . . . . . . . . 309 17.3.1 CorrectiveMaintenance . . . . . . . . . . . . . . . . . . . . . 309 17.3.2 PreventiveMaintenance . . . . . . . . . . . . . . . . . . . . . 310 17.3.3 ReviewofMaintenanceLiterature . . . . . . . . . . . . . . . . 310 17.4 WarrantyandCorrectiveMaintenance . . . . . . . . . . . . . . . . . 311 17.5 WarrantyandPreventiveMaintenance . . . . . . . . . . . . . . . . . 312 17.6 ExtendedWarrantiesandServiceContracts . . . . . . . . . . . . . . . 313 17.7 ConclusionsandTopicsforFutureResearch . . . . . . . . . . . . . . 314 18 Mechanical Reliability and MaintenanceModels Gianpaolo Pulcini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 18.2 StochasticPointProcesses . . . . . . . . . . . . . . . . . . . . . . . . 318 18.3 PerfectMaintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 18.4 MinimalRepair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 18.4.1 NoTrendwithOperatingTime . . . . . . . . . . . . . . . . . 323 18.4.2 Monotonic Trend with Operating Time . . . . . . . . . . . . . 323 18.4.2.1 ThePowerLawProcess . . . . . . . . . . . . . . . . 324 18.4.2.2 TheLog.LinearProcess . . . . . . . . . . . . . . . . 325 18.4.2.3 Bounded Intensity Processes . . . . . . . . . . . . . 326 xx Contents 18.4.3 Bathtub-type Intensity . . . . . . . . . . . . . . . . . . . . . . 327 18.4.3.1 NumericalExample . . . . . . . . . . . . . . . . . . 328 18.4.4 Non-homogeneous Poisson Process Incorporating Covariate Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 18.5 ImperfectorWorseRepair . . . . . . . . . . . . . . . . . . . . . . . . 330 18.5.1 Proportional Age Reduction Models . . . . . . . . . . . . . . 330 18.5.2 InhomogeneousGammaProcesses . . . . . . . . . . . . . . . 331 18.5.3 Lawless.ThiagarajahModels . . . . . . . . . . . . . . . . . . 333 18.5.4 Proportional IntensityVariationModel . . . . . . . . . . . . . 334 18.6 ComplexMaintenancePolicy . . . . . . . . . . . . . . . . . . . . . . . 335 18.6.1 Sequence of Perfect and Minimal RepairsWithout Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 18.6.2 Minimal Repairs Interspersed with Perfect Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 18.6.3 Imperfect Repairs Interspersed with Perfect Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 18.6.4 Minimal Repairs Interspersed with Imperfect Preventive Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 18.6.4.1 NumericalExample . . . . . . . . . . . . . . . . . . 341 18.6.5 Corrective Repairs Interspersed with PreventiveMaintenance WithoutRestrictiveAssumptions . . . . . . . . . . . . . . . . 342 18.7 ReliabilityGrowth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 18.7.1 ContinuousModels . . . . . . . . . . . . . . . . . . . . . . . . 344 18.7.2 DiscreteModels . . . . . . . . . . . . . . . . . . . . . . . . . 345 19 PreventiveMaintenanceModels: Replacement, Repair, Ordering, and Inspection Tadashi Dohi, Naoto Kaio and Shunji Osaki . . . . . . . . . . . . . . . . . . . 349 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 19.2 BlockReplacementModels . . . . . . . . . . . . . . . . . . . . . . . . 350 19.2.1 Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 19.2.2 Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 19.2.3 Model III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 19.3 AgeReplacementModels . . . . . . . . . . . . . . . . . . . . . . . . . 354 19.3.1 BasicAgeReplacementModel . . . . . . . . . . . . . . . . . . 354 19.4 OrderingModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 19.4.1 Continuous-timeModel . . . . . . . . . . . . . . . . . . . . . 357 19.4.2 Discrete-timeModel . . . . . . . . . . . . . . . . . . . . . . . 358 19.4.3 CombinedModelwithMinimalRepairs . . . . . . . . . . . . 359 19.5 InspectionModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 19.5.1 Nearly Optimal Inspection Policy by Kaio and Osaki (K&O Policy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 19.5.2 Nearly Optimal Inspection Policy by Munford and Shahani (M&SPolicy) . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 19.5.3 Nearly Optimal Inspection Policy by Nakagawa and Yasui (N&YPolicy) . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 19.6 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Contents xxi 20 Maintenance and Optimum Policy Toshio Nakagawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 20.2 ReplacementPolicies . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 20.2.1 AgeReplacement . . . . . . . . . . . . . . . . . . . . . . . . . 368 20.2.2 BlockReplacement . . . . . . . . . . . . . . . . . . . . . . . . 370 20.2.2.1 No Replacement at Failure . . . . . . . . . . . . . . 370 20.2.2.2 Replacement with Two Variables . . . . . . . . . . . 371 20.2.3 PeriodicReplacement . . . . . . . . . . . . . . . . . . . . . . 371 20.2.3.1 Modified Models with Two Variables . . . . . . . . . 372 20.2.3.2 Replacement at N Variables . . . . . . . . . . . . . 373 20.2.4 Other ReplacementModels . . . . . . . . . . . . . . . . . . . 373 20.2.4.1 ReplacementswithDiscounting . . . . . . . . . . . 373 20.2.4.2 Discrete Replacement Models . . . . . . . . . . . . 374 20.2.4.3 Replacements with Two Types of Unit . . . . . . . . 375 20.2.4.4 Replacement of a Shock Model . . . . . . . . . . . . 376 20.2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 20.3 PreventiveMaintenancePolicies . . . . . . . . . . . . . . . . . . . . . 378 20.3.1 One-unitSystem . . . . . . . . . . . . . . . . . . . . . . . . . 378 20.3.1.1 IntervalReliability . . . . . . . . . . . . . . . . . . 379 20.3.2 Two-unitSystem . . . . . . . . . . . . . . . . . . . . . . . . . 380