System Reliability Theory

MARVIN RAUSAND is Professor Emeritus in the department of Mechanical and Industrial Engineering at the Norwegian University of Science and Technology (NTNU), Norway, and author of Risk Assessment: Theory, Methods, and Applications and Reliability of Safety-Critical Systems: Theory and Applications, both published by Wiley. ANNE BARROS, PHD, is Professor in reliability and maintenance engineering at Ecole CentraleSupélec, University of Paris-Saclay, France. Her research focus is on degradation modeling, prognostics, condition based and predictive maintenance. She got a PHD then a professorship position at University of Technology of Troyes, France (2003 – 2014) and spent five years as a full-time professor at NTNU, Norway (2014 – 2019). She is currently heading a research group and holds an industrial chair at CentraleSupélec with the ambition to provide reliability assessment and maintenance modeling methods for systems of systems. The late ARNLJOT HØYLAND, PHD, was a Professor in the Department of Mathematical Sciences at the Norwegian University of Science and Technology.

• Preface xxiiiAbout the Companion Website xxix1 Introduction 11.1 What is Reliability? 11.1.1 Service Reliability 21.1.2 Past and Future Reliability 31.2 The Importance of Reliability 31.2.1 Related Applications 41.3 Basic Reliability Concepts 61.3.1 Reliability 61.3.2 Maintainability and Maintenance 81.3.3 Availability 81.3.4 Quality 91.3.5 Dependability 91.3.6 Safety and Security 101.3.7 RAM and RAMS 101.4 Reliability Metrics 111.4.1 Reliability Metrics for a Technical Item 111.4.2 Reliability Metrics for a Service 121.5 Approaches to Reliability Analysis 121.5.1 The Physical Approach to Reliability 131.5.2 Systems Approach to Reliability 131.6 Reliability Engineering 151.6.1 Roles of the Reliability Engineer 161.6.2 Timing of Reliability Studies 171.7 Objectives, Scope, and Delimitations of the Book 171.8 Trends and Challenges 191.9 Standards and Guidelines 201.10 History of System Reliability 201.11 Problems 26References 272 The Study Object and its Functions 312.1 Introduction 312.2 System and System Elements 312.2.1 Item 322.2.2 Embedded Item 332.3 Boundary Conditions 332.3.1 Closed and Open Systems 342.4 Operating Context 352.5 Functions and Performance Requirements 352.5.1 Functions 352.5.2 Performance Requirements 362.5.3 Classification of Functions 372.5.4 Functional Modeling and Analysis 382.5.5 Function Trees 382.5.6 SADT and IDEF 0 392.6 System Analysis 412.6.1 Synthesis 412.7 Simple, Complicated, and Complex Systems 422.8 System Structure Modeling 442.8.1 Reliability Block Diagram 442.8.2 Series Structure 462.8.3 Parallel Structure 462.8.4 Redundancy 472.8.5 Voted Structure 472.8.6 Standby Structure 482.8.7 More Complicated Structures 482.8.8 Two Different System Functions 492.8.9 Practical Construction of RBDs 502.9 Problems 51References 523 Failures and Faults 553.1 Introduction 553.1.1 States and Transitions 563.1.2 Operational Modes 563.2 Failures 573.2.1 Failures in a State 583.2.2 Failures During Transition 593.3 Faults 603.4 Failure Modes 603.5 Failure Causes and Effects 623.5.1 Failure Causes 623.5.2 Proximate Causes and Root Causes 633.5.3 Hierarchy of Causes 643.6 Classification of Failures and Failure Modes 643.6.1 Classification According to Local Consequence 653.6.2 Classification According to Cause 653.6.3 Failure Mechanisms 703.6.4 Software Faults 713.6.5 Failure Effects 713.7 Failure/Fault Analysis 723.7.1 Cause and Effect Analysis 733.7.2 Root Cause Analysis 743.8 Problems 76References 774 Qualitative System Reliability Analysis 794.1 Introduction 794.1.1 Deductive Versus Inductive Analysis 804.2 FMEA/FMECA 804.2.1 Types of FMECA 814.2.2 Objectives of FMECA 824.2.3 FMECA Procedure 834.2.4 Applications 874.3 Fault Tree Analysis 884.3.1 Fault Tree Symbols and Elements 884.3.2 Definition of the Problem and the Boundary Conditions 914.3.3 Constructing the Fault Tree 924.3.4 Identification of Minimal Cut and Path Sets 954.3.5 MOCUS 964.3.6 Qualitative Evaluation of the Fault Tree 984.3.7 Dynamic Fault Trees 1014.4 Event Tree Analysis 1034.4.1 Initiating Event 1044.4.2 Safety Functions 1054.4.3 Event Tree Construction 1064.4.4 Description of Resulting Event Sequences 1064.5 Fault Trees versus Reliability Block Diagrams 1094.5.1 Recommendation 1114.6 Structure Function 1114.6.1 Series Structure 1124.6.2 Parallel Structure 1124.6.3 koon:G Structure 1134.6.4 Truth Tables 1144.7 System Structure Analysis 1144.7.1 Single Points of Failure 1154.7.2 Coherent Structures 1154.7.3 General Properties of Coherent Structures 1174.7.4 Structures Represented by Paths and Cuts 1194.7.5 Pivotal Decomposition 1234.7.6 Modules of Coherent Structures 1244.8 Bayesian Networks 1274.8.1 Illustrative Examples 1284.9 Problems 131References 1385 Probability Distributions in Reliability Analysis 1415.1 Introduction 1415.1.1 State Variable 1425.1.2 Time-to-Failure 1425.2 A Dataset 1435.2.1 Relative Frequency Distribution 1435.2.2 Empirical Distribution and Survivor Function 1445.3 General Characteristics of Time-to-Failure Distributions 1455.3.1 Survivor Function 1475.3.2 Failure Rate Function 1485.3.3 Conditional Survivor Function 1535.3.4 Mean Time-to-Failure 1545.3.5 Additional Probability Metrics 1555.3.6 Mean Residual Lifetime 1575.3.7 Mixture of Time-to-Failure Distributions 1605.4 Some Time-to-Failure Distributions 1615.4.1 The Exponential Distribution 1615.4.2 The Gamma Distribution 1685.4.3 TheWeibull Distribution 1735.4.4 The Normal Distribution 1805.4.5 The Lognormal Distribution 1835.4.6 Additional Time-to-Failure Distributions 1885.5 Extreme Value Distributions 1885.5.1 The Gumbel Distribution of the Smallest Extreme 1905.5.2 The Gumbel Distribution of the Largest Extreme 1915.5.3 TheWeibull Distribution of the Smallest Extreme 1915.6 Time-to-Failure Models With Covariates 1935.6.1 Accelerated Failure Time Models 1945.6.2 The Arrhenius Model 1955.6.3 Proportional Hazards Models 1985.7 Additional Continuous Distributions 1985.7.1 The Uniform Distribution 1985.7.2 The Beta Distribution 1995.8 Discrete Distributions 2005.8.1 Binomial Situation 2005.8.2 The Binomial Distribution 2015.8.3 The Geometric Distribution 2015.8.4 The Negative Binomial Distribution 2025.8.5 The Homogeneous Poisson Process 2035.9 Classes of Time-to-Failure Distributions 2055.9.1 IFR and DFR Distributions 2065.9.2 IFRA and DFRA Distributions 2085.9.3 NBU and NWU Distributions 2085.9.4 NBUE and NWUE Distributions 2095.9.5 Some Implications 2095.10 Summary of Time-to-Failure Distributions 2105.11 Problems 210References 2186 System Reliability Analysis 2216.1 Introduction 2216.1.1 Assumptions 2226.2 System Reliability 2226.2.1 Reliability of Series Structures 2236.2.2 Reliability of Parallel Structures 2246.2.3 Reliability of koon Structures 2256.2.4 Pivotal Decomposition 2266.2.5 Critical Component 2276.3 Nonrepairable Systems 2286.3.1 Nonrepairable Series Structures 2286.3.2 Nonrepairable Parallel Structures 2306.3.3 Nonrepairable 2oo3 Structures 2346.3.4 A Brief Comparison 2356.3.5 Nonrepairable koon Structures 2366.4 Standby Redundancy 2376.4.1 Passive Redundancy, Perfect Switching, No Repairs 2386.4.2 Cold Standby, Imperfect Switch, No Repairs 2406.4.3 Partly Loaded Redundancy, Imperfect Switch, No Repairs 2416.5 Single Repairable Items 2426.5.1 Availability 2436.5.2 Average Availability with Perfect Repair 2446.5.3 Availability of a Single Item with Constant Failure and Repair Rates 2466.5.4 Operational Availability 2476.5.5 Production Availability 2486.5.6 Punctuality 2496.5.7 Failure Rate of Repairable Items 2496.6 Availability of Repairable Systems 2526.6.1 The MUT and MDT of Repairable Systems 2536.6.2 Computation Based on Minimal Cut Sets 2586.6.3 Uptimes and Downtimes for Reparable Systems 2606.7 Quantitative Fault Tree Analysis 2626.7.1 Terminology and Symbols 2636.7.2 Delimitations and Assumptions 2636.7.3 Fault Trees with a Single AND-Gate 2646.7.4 Fault Tree with a Single OR-Gate 2656.7.5 The Upper Bound Approximation Formula for Q0(t) 2656.7.6 The Inclusion–Exclusion Principle 2676.7.7 ROCOF of a Minimal Cut Parallel Structure 2716.7.8 Frequency of the TOP Event 2716.7.9 Binary Decision Diagrams 2736.8 Event Tree Analysis 2756.9 Bayesian Networks 2776.9.1 Influence and Cause 2786.9.2 Independence Assumptions 2786.9.3 Conditional Probability Table 2796.9.4 Conditional Independence 2806.9.5 Inference and Learning 2826.9.6 BN and Fault Tree Analysis 2826.10 Monte Carlo Simulation 2846.10.1 Random Number Generation 2856.10.2 Monte Carlo Next Event Simulation 2876.10.3 Simulation of Multicomponent Systems 2896.11 Problems 291References 2967 Reliability Importance Metrics 2997.1 Introduction 2997.1.1 Objectives of Reliability Importance Metrics 3007.1.2 Reliability Importance Metrics Considered 3007.1.3 Assumptions and Notation 3017.2 Critical Components 3027.3 Birnbaum’s Metric for Structural Importance 3047.4 Birnbaum’s Metric of Reliability Importance 3057.4.1 Birnbaum’s Metric in Fault Tree Analysis 3077.4.2 A Second Definition of Birnbaum’s Metric 3087.4.3 A Third Definition of Birnbaum’s Metric 3107.4.4 Computation of Birnbaum’s Metric for Structural Importance 3127.4.5 Variants of Birnbaum’s Metric 3127.5 Improvement Potential 3137.5.1 Relation to Birnbaum’s Metric 3147.5.2 A Variant of the Improvement Potential 3147.6 Criticality Importance 3157.7 Fussell–Vesely’s Metric 3177.7.1 Derivation of Formulas for Fussell–Vesely’s Metric 3177.7.2 Relationship to Other Metrics for Importance 3207.8 Differential Importance Metric 3237.8.1 Option 1 3237.8.2 Option 2 3247.9 Importance Metrics for Safety Features 3267.9.1 Risk AchievementWorth 3277.9.2 Risk ReductionWorth 3297.9.3 Relationship with the Improvement Potential 3307.10 Barlow–Proschan’s Metric 3317.11 Problems 333References 3358 Dependent Failures 3378.1 Introduction 3378.1.1 Dependent Events and Variables 3378.1.2 Correlated Variables 3388.2 Types of Dependence 3408.3 Cascading Failures 3408.3.1 Tight Coupling 3428.4 Common-Cause Failures 3428.4.1 Multiple Failures that Are Not a CCF 3448.4.2 Causes of CCF 3448.4.3 Defenses Against CCF 3458.5 CCF Models and Analysis 3468.5.1 Explicit Modeling 3478.5.2 Implicit Modeling 3488.5.3 Modeling Approach 3488.5.4 Model Assumptions 3498.6 Basic Parameter Model 3498.6.1 Probability of a Specific Multiplicity 3508.6.2 Conditional Probability of a Specific Multiplicity 3518.7 Beta-Factor Model 3528.7.1 Relation to the BPM 3548.7.2 Beta-Factor Model in System Analysis 3548.7.3 Beta-Factor Model for Nonidentical Components 3588.7.4 C-Factor Model 3608.8 Multi-parameter Models 3608.8.1 Binomial Failure Rate Model 3608.8.2 Multiple Greek Letter Model 3628.8.3 Alpha-Factor Model 3648.8.4 Multiple Beta-Factor Model 3658.9 Problems 366References 3689 Maintenance and Maintenance Strategies 3719.1 Introduction 3719.1.1 What is Maintenance? 3729.2 Maintainability 3729.3 Maintenance Categories 3749.3.1 Completeness of a Repair Task 3779.3.2 Condition Monitoring 3779.4 Maintenance Downtime 3789.4.1 Downtime Caused by Failures 3799.4.2 Downtime of a Series Structure 3819.4.3 Downtime of a Parallel Structure 3819.4.4 Downtime of a General Structure 3829.5 Reliability Centered Maintenance 3829.5.1 What is RCM? 3839.5.2 Main Steps of an RCM Analysis 3849.6 Total Productive Maintenance 3969.7 Problems 398References 39910 Counting Processes 40110.1 Introduction 40110.1.1 Counting Processes 40110.1.2 Basic Concepts 40610.1.3 Martingale Theory 40810.1.4 Four Types of Counting Processes 40910.2 Homogeneous Poisson Processes 41010.2.1 Main Features of the HPP 41110.2.2 Asymptotic Properties 41210.2.3 Estimate and Confidence Interval 41210.2.4 Sum and Decomposition of HPPs 41310.2.5 Conditional Distribution of Failure Time 41410.2.6 Compound HPPs 41510.3 Renewal Processes 41710.3.1 Basic Concepts 41710.3.2 The Distribution of Sn 41810.3.3 The Distribution of N(t) 42010.3.4 The Renewal Function 42110.3.5 The Renewal Density 42310.3.6 Age and Remaining Lifetime 42710.3.7 Bounds for the Renewal Function 43110.3.8 Superimposed Renewal Processes 43310.3.9 Renewal Reward Processes 43410.3.10 Delayed Renewal Processes 43610.3.11 Alternating Renewal Processes 43810.4 Nonhomogeneous Poisson Processes 44710.4.1 Introduction and Definitions 44710.4.2 Some Results 44910.4.3 Parametric NHPP Models 45210.4.4 Statistical Tests of Trend 45410.5 Imperfect Repair Processes 45510.5.1 Brown and Proschan’s model 45610.5.2 Failure Rate Reduction Models 45810.5.3 Age Reduction Models 46110.5.4 Trend Renewal Process 46210.6 Model Selection 46410.7 Problems 466References 47011 Markov Analysis 47311.1 Introduction 47311.1.1 Markov Property 47511.2 Markov Processes 47611.2.1 Procedure to Establish the Transition Rate Matrix 47911.2.2 Chapman–Kolmogorov Equations 48211.2.3 Kolmogorov Differential Equations 48311.2.4 State Equations 48411.3 Asymptotic Solution 48711.3.1 System Performance Metrics 49211.4 Parallel and Series Structures 49511.4.1 Parallel Structures of Independent Components 49511.4.2 Series Structures of Independent Components 49711.4.3 Series Structure of Components Where Failure of One Component Prevents Failure of the Other 49911.5 Mean Time to First System Failure 50111.5.1 Absorbing States 50111.5.2 Survivor Function 50411.5.3 Mean Time to the First System Failure 50511.6 Systems with Dependent Components 50711.6.1 Common Cause Failures 50811.6.2 Load-Sharing Systems 51011.7 Standby Systems 51211.7.1 Parallel System with Cold Standby and Perfect Switching 51311.7.2 Parallel System with Cold Standby and Perfect Switching (Item A is the Main Operating Item) 51511.7.3 Parallel System with Cold Standby and Imperfect Switching (Item A is the Main Operating Item) 51711.7.4 Parallel System with Partly Loaded Standby and Perfect Switching (Item A is the Main Operating Item) 51811.8 Markov Analysis in Fault Tree Analysis 51911.8.1 Cut Set Information 52011.8.2 System Information 52111.9 Time-Dependent Solution 52111.9.1 Laplace Transforms 52211.10 Semi-Markov Processes 52411.11 Multiphase Markov Processes 52611.11.1 Changing the Transition Rates 52611.11.2 Changing the Initial State 52711.12 Piecewise Deterministic Markov Processes 52811.12.1 Definition of PDMP 52911.12.2 State Probabilities 52911.12.3 A Specific Case 53011.13 Simulation of a Markov Process 53211.14 Problems 536References 54312 Preventive Maintenance 54512.1 Introduction 54512.2 Terminology and Cost Function 54612.3 Time-Based Preventive Maintenance 54812.3.1 Age Replacement 54912.3.2 Block Replacement 55312.3.3 P–F Intervals 55712.4 Degradation Models 56412.4.1 Remaining Useful Lifetime 56512.4.2 Trend Models; Regression-Based Models 56712.4.3 Models with Increments 56912.4.4 Shock Models 57112.4.5 Stochastic Processes with Discrete States 57312.4.6 Failure Rate Models 57412.5 Condition-Based Maintenance 57412.5.1 CBM Strategy 57512.5.2 Continuous Monitoring and Finite Discrete State Space 57612.5.3 Continuous Monitoring and Continuous State Space 58112.5.4 Inspection-Based Monitoring and Finite Discrete State Space 58312.5.5 Inspection-Based Monitoring and Continuous State Space 58612.6 Maintenance of Multi-Item Systems 58712.6.1 System Model 58712.6.2 Maintenance Models 58912.6.3 An Illustrative Example 59112.7 Problems 595References 60113 Reliability of Safety Systems 60513.1 Introduction 60513.2 Safety-Instrumented Systems 60613.2.1 Main SIS Functions 60713.2.2 Testing of SIS Functions 60813.2.3 Failure Classification 60913.3 Probability of Failure on Demand 61113.3.1 Probability of Failure on Demand 61213.3.2 Approximation Formulas 61713.3.3 Mean Downtime in a Test Interval 61813.3.4 Mean Number of Test Intervals Until First Failure 61913.3.5 Staggered Testing 62013.3.6 Nonnegligible Repair Time 62113.4 Safety Unavailability 62213.4.1 Probability of Critical Situation 62313.4.2 Spurious Trips 62313.4.3 Failures Detected by Diagnostic Self-Testing 62513.5 Common Cause Failures 62713.5.1 Diagnostic Self-Testing and CCFs 62913.6 CCFs Between Groups and Subsystems 63113.6.1 CCFs Between Voted Groups 63213.6.2 CCFs Between Subsystems 63213.7 IEC 61508 63213.7.1 Safety Lifecycle 63313.7.2 Safety Integrity Level 63413.7.3 Compliance with IEC 61508 63513.8 The PDS Method 63813.9 Markov Approach 63913.9.1 All Failures are Repaired After Each Test 64313.9.2 All Critical Failures Are Repaired after Each Test 64413.9.3 Imperfect Repair after Each Test 64413.10 Problems 644References 65214 Reliability Data Analysis 65514.1 Introduction 65514.1.1 Purpose of the Chapter 65614.2 Some Basic Concepts 65614.2.1 Datasets 65714.2.2 Survival Times 65814.2.3 Categories of Censored Datasets 66014.2.4 Field Data Collection Exercises 66214.2.5 At-Risk-Set 66314.3 Exploratory Data Analysis 66314.3.1 A Complete Dataset 66414.3.2 Sample Metrics 66514.3.3 Histogram 66914.3.4 Density Plot 67014.3.5 Empirical Survivor Function 67114.3.6 Q–Q Plot 67314.4 Parameter Estimation 67414.4.1 Estimators and Estimates 67514.4.2 Properties of Estimators 67514.4.3 Method of Moments Estimation 67714.4.4 Maximum Likelihood Estimation 68014.4.5 Exponentially Distributed Lifetimes 68614.4.6 Weibull Distributed Lifetimes 69214.5 The Kaplan–Meier Estimate 69614.5.1 Motivation for the Kaplan–Meier Estimate Based a Complete Dataset 69614.5.2 The Kaplan–Meier Estimator for a Censored Dataset 69714.6 Cumulative Failure Rate Plots 70114.6.1 The Nelson–Aalen Estimate of the Cumulative Failure Rate 70314.7 Total-Time-on-Test Plotting 70814.7.1 Total-Time-on-Test Plot for Complete Datasets 70814.7.2 Total-Time-on-Test Plot for Censored Datasets 72114.7.3 A Brief Comparison 72214.8 Survival Analysis with Covariates 72314.8.1 Proportional Hazards Model 72314.8.2 Cox Models 72614.8.3 Estimating the Parameters of the Cox Model 72714.9 Problems 730References 73615 Bayesian Reliability Analysis 73915.1 Introduction 73915.1.1 Three Interpretations of Probability 73915.1.2 Bayes’ Formula 74115.2 Bayesian Data Analysis 74215.2.1 Frequentist Data Analysis 74315.2.2 Bayesian Data Analysis 74315.2.3 Model for Observed Data 74515.2.4 Prior Distribution 74515.2.5 Observed Data 74615.2.6 Likelihood Function 74615.2.7 Posterior Distribution 74715.3 Selection of Prior Distribution 74915.3.1 Binomial Model 74915.3.2 Exponential Model – Single Observation 75215.3.3 Exponential Model – Multiple Observations 75315.3.4 Homogeneous Poisson Process 75515.3.5 Noninformative Prior Distributions 75715.4 Bayesian Estimation 75815.4.1 Bayesian Point Estimation 75815.4.2 Credible Intervals 76015.5 Predictive Distribution 76115.6 Models with Multiple Parameters 76215.7 Bayesian Analysis with R 76215.8 Problems 764References 76616 Reliability Data: Sources and Quality 76716.1 Introduction 76716.1.1 Categories of Input Data 76716.1.2 Parameters Estimates 76816.2 Generic Reliability Databases 76916.2.1 OREDA 77016.2.2 PDS Data Handbook 77216.2.3 PERD 77316.2.4 SERH 77316.2.5 NPRD, EPRD, and FMD 77316.2.6 GADS 77416.2.7 GIDEP 77416.2.8 FMEDA Approach 77516.2.9 Failure Event Databases 77516.3 Reliability Prediction 77516.3.1 MIL-HDBK-217 Approach 77616.3.2 Similar Methods 77816.4 Common Cause Failure Data 77816.4.1 ICDE 77916.4.2 IEC 61508 Method 77916.5 Data Analysis and Data Quality 78016.5.1 Outdated Technology 78016.5.2 Inventory Data 78116.5.3 Constant Failure Rates 78116.5.4 Multiple Samples 78316.5.5 Data From Manufacturers 78516.5.6 Questioning the Data Quality 78516.6 Data Dossier 78516.6.1 Final Remarks 785References 787Appendix A Acronyms 789Appendix B Laplace Transforms 793B.1 Important Properties of Laplace Transforms 794B.2 Laplace Transforms of Some Selected Functions 794Author Index 797Subject Index 803