System Reliability Theory: Models, Statistical Methods, and Applications

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Preface xxiii About the Companion Website xxix 1 Introduction 1 1.1 What is Reliability? 1 1.1.1 Service Reliability 2 1.1.2 Past and Future Reliability 3 1.2 The Importance of Reliability 3 1.2.1 Related Applications 4 1.3 Basic Reliability Concepts 6 1.3.1 Reliability 6 1.3.2 Maintainability and Maintenance 8 1.3.3 Availability 8 1.3.4 Quality 9 1.3.5 Dependability 9 1.3.6 Safety and Security 10 1.3.7 RAM and RAMS 10 1.4 Reliability Metrics 11 1.4.1 Reliability Metrics for a Technical Item 11 1.4.2 Reliability Metrics for a Service 12 1.5 Approaches to Reliability Analysis 12 1.5.1 The Physical Approach to Reliability 13 1.5.2 Systems Approach to Reliability 13 1.6 Reliability Engineering 15 1.6.1 Roles of the Reliability Engineer 16 1.6.2 Timing of Reliability Studies 17 1.7 Objectives, Scope, and Delimitations of the Book 17 1.8 Trends and Challenges 19 1.9 Standards and Guidelines 20 1.10 History of System Reliability 20 1.11 Problems 26 References 27 2 The Study Object and its Functions 31 2.1 Introduction 31 2.2 System and System Elements 31 2.2.1 Item 32 2.2.2 Embedded Item 33 2.3 Boundary Conditions 33 2.3.1 Closed and Open Systems 34 2.4 Operating Context 35 2.5 Functions and Performance Requirements 35 2.5.1 Functions 35 2.5.2 Performance Requirements 36 2.5.3 Classification of Functions 37 2.5.4 Functional Modeling and Analysis 38 2.5.5 Function Trees 38 2.5.6 SADT and IDEF 0 39 2.6 System Analysis 41 2.6.1 Synthesis 41 2.7 Simple, Complicated, and Complex Systems 42 2.8 System Structure Modeling 44 2.8.1 Reliability Block Diagram 44 2.8.2 Series Structure 46 2.8.3 Parallel Structure 46 2.8.4 Redundancy 47 2.8.5 Voted Structure 47 2.8.6 Standby Structure 48 2.8.7 More Complicated Structures 48 2.8.8 Two Different System Functions 49 2.8.9 Practical Construction of RBDs 50 2.9 Problems 51 References 52 3 Failures and Faults 55 3.1 Introduction 55 3.1.1 States and Transitions 56 3.1.2 Operational Modes 56 3.2 Failures 57 3.2.1 Failures in a State 58 3.2.2 Failures During Transition 59 3.3 Faults 60 3.4 Failure Modes 60 3.5 Failure Causes and Effects 62 3.5.1 Failure Causes 62 3.5.2 Proximate Causes and Root Causes 63 3.5.3 Hierarchy of Causes 64 3.6 Classification of Failures and Failure Modes 64 3.6.1 Classification According to Local Consequence 65 3.6.2 Classification According to Cause 65 3.6.3 Failure Mechanisms 70 3.6.4 Software Faults 71 3.6.5 Failure Effects 71 3.7 Failure/Fault Analysis 72 3.7.1 Cause and Effect Analysis 73 3.7.2 Root Cause Analysis 74 3.8 Problems 76 References 77 4 Qualitative System Reliability Analysis 79 4.1 Introduction 79 4.1.1 Deductive Versus Inductive Analysis 80 4.2 FMEA/FMECA 80 4.2.1 Types of FMECA 81 4.2.2 Objectives of FMECA 82 4.2.3 FMECA Procedure 83 4.2.4 Applications 87 4.3 Fault Tree Analysis 88 4.3.1 Fault Tree Symbols and Elements 88 4.3.2 Definition of the Problem and the Boundary Conditions 91 4.3.3 Constructing the Fault Tree 92 4.3.4 Identification of Minimal Cut and Path Sets 95 4.3.5 MOCUS 96 4.3.6 Qualitative Evaluation of the Fault Tree 98 4.3.7 Dynamic Fault Trees 101 4.4 Event Tree Analysis 103 4.4.1 Initiating Event 104 4.4.2 Safety Functions 105 4.4.3 Event Tree Construction 106 4.4.4 Description of Resulting Event Sequences 106 4.5 Fault Trees versus Reliability Block Diagrams 109 4.5.1 Recommendation 111 4.6 Structure Function 111 4.6.1 Series Structure 112 4.6.2 Parallel Structure 112 4.6.3 koon:G Structure 113 4.6.4 Truth Tables 114 4.7 System Structure Analysis 114 4.7.1 Single Points of Failure 115 4.7.2 Coherent Structures 115 4.7.3 General Properties of Coherent Structures 117 4.7.4 Structures Represented by Paths and Cuts 119 4.7.5 Pivotal Decomposition 123 4.7.6 Modules of Coherent Structures 124 4.8 Bayesian Networks 127 4.8.1 Illustrative Examples 128 4.9 Problems 131 References 138 5 Probability Distributions in Reliability Analysis 141 5.1 Introduction 141 5.1.1 State Variable 142 5.1.2 Time-to-Failure 142 5.2 A Dataset 143 5.2.1 Relative Frequency Distribution 143 5.2.2 Empirical Distribution and Survivor Function 144 5.3 General Characteristics of Time-to-Failure Distributions 145 5.3.1 Survivor Function 147 5.3.2 Failure Rate Function 148 5.3.3 Conditional Survivor Function 153 5.3.4 Mean Time-to-Failure 154 5.3.5 Additional Probability Metrics 155 5.3.6 Mean Residual Lifetime 157 5.3.7 Mixture of Time-to-Failure Distributions 160 5.4 Some Time-to-Failure Distributions 161 5.4.1 The Exponential Distribution 161 5.4.2 The Gamma Distribution 168 5.4.3 TheWeibull Distribution 173 5.4.4 The Normal Distribution 180 5.4.5 The Lognormal Distribution 183 5.4.6 Additional Time-to-Failure Distributions 188 5.5 Extreme Value Distributions 188 5.5.1 The Gumbel Distribution of the Smallest Extreme 190 5.5.2 The Gumbel Distribution of the Largest Extreme 191 5.5.3 TheWeibull Distribution of the Smallest Extreme 191 5.6 Time-to-Failure Models With Covariates 193 5.6.1 Accelerated Failure Time Models 194 5.6.2 The Arrhenius Model 195 5.6.3 Proportional Hazards Models 198 5.7 Additional Continuous Distributions 198 5.7.1 The Uniform Distribution 198 5.7.2 The Beta Distribution 199 5.8 Discrete Distributions 200 5.8.1 Binomial Situation 200 5.8.2 The Binomial Distribution 201 5.8.3 The Geometric Distribution 201 5.8.4 The Negative Binomial Distribution 202 5.8.5 The Homogeneous Poisson Process 203 5.9 Classes of Time-to-Failure Distributions 205 5.9.1 IFR and DFR Distributions 206 5.9.2 IFRA and DFRA Distributions 208 5.9.3 NBU and NWU Distributions 208 5.9.4 NBUE and NWUE Distributions 209 5.9.5 Some Implications 209 5.10 Summary of Time-to-Failure Distributions 210 5.11 Problems 210 References 218 6 System Reliability Analysis 221 6.1 Introduction 221 6.1.1 Assumptions 222 6.2 System Reliability 222 6.2.1 Reliability of Series Structures 223 6.2.2 Reliability of Parallel Structures 224 6.2.3 Reliability of koon Structures 225 6.2.4 Pivotal Decomposition 226 6.2.5 Critical Component 227 6.3 Nonrepairable Systems 228 6.3.1 Nonrepairable Series Structures 228 6.3.2 Nonrepairable Parallel Structures 230 6.3.3 Nonrepairable 2oo3 Structures 234 6.3.4 A Brief Comparison 235 6.3.5 Nonrepairable koon Structures 236 6.4 Standby Redundancy 237 6.4.1 Passive Redundancy, Perfect Switching, No Repairs 238 6.4.2 Cold Standby, Imperfect Switch, No Repairs 240 6.4.3 Partly Loaded Redundancy, Imperfect Switch, No Repairs 241 6.5 Single Repairable Items 242 6.5.1 Availability 243 6.5.2 Average Availability with Perfect Repair 244 6.5.3 Availability of a Single Item with Constant Failure and Repair Rates 246 6.5.4 Operational Availability 247 6.5.5 Production Availability 248 6.5.6 Punctuality 249 6.5.7 Failure Rate of Repairable Items 249 6.6 Availability of Repairable Systems 252 6.6.1 The MUT and MDT of Repairable Systems 253 6.6.2 Computation Based on Minimal Cut Sets 258 6.6.3 Uptimes and Downtimes for Reparable Systems 260 6.7 Quantitative Fault Tree Analysis 262 6.7.1 Terminology and Symbols 263 6.7.2 Delimitations and Assumptions 263 6.7.3 Fault Trees with a Single AND-Gate 264 6.7.4 Fault Tree with a Single OR-Gate 265 6.7.5 The Upper Bound Approximation Formula for Q0(t) 265 6.7.6 The Inclusion–Exclusion Principle 267 6.7.7 ROCOF of a Minimal Cut Parallel Structure 271 6.7.8 Frequency of the TOP Event 271 6.7.9 Binary Decision Diagrams 273 6.8 Event Tree Analysis 275 6.9 Bayesian Networks 277 6.9.1 Influence and Cause 278 6.9.2 Independence Assumptions 278 6.9.3 Conditional Probability Table 279 6.9.4 Conditional Independence 280 6.9.5 Inference and Learning 282 6.9.6 BN and Fault Tree Analysis 282 6.10 Monte Carlo Simulation 284 6.10.1 Random Number Generation 285 6.10.2 Monte Carlo Next Event Simulation 287 6.10.3 Simulation of Multicomponent Systems 289 6.11 Problems 291 References 296 7 Reliability Importance Metrics 299 7.1 Introduction 299 7.1.1 Objectives of Reliability Importance Metrics 300 7.1.2 Reliability Importance Metrics Considered 300 7.1.3 Assumptions and Notation 301 7.2 Critical Components 302 7.3 Birnbaum’s Metric for Structural Importance 304 7.4 Birnbaum’s Metric of Reliability Importance 305 7.4.1 Birnbaum’s Metric in Fault Tree Analysis 307 7.4.2 A Second Definition of Birnbaum’s Metric 308 7.4.3 A Third Definition of Birnbaum’s Metric 310 7.4.4 Computation of Birnbaum’s Metric for Structural Importance 312 7.4.5 Variants of Birnbaum’s Metric 312 7.5 Improvement Potential 313 7.5.1 Relation to Birnbaum’s Metric 314 7.5.2 A Variant of the Improvement Potential 314 7.6 Criticality Importance 315 7.7 Fussell–Vesely’s Metric 317 7.7.1 Derivation of Formulas for Fussell–Vesely’s Metric 317 7.7.2 Relationship to Other Metrics for Importance 320 7.8 Differential Importance Metric 323 7.8.1 Option 1 323 7.8.2 Option 2 324 7.9 Importance Metrics for Safety Features 326 7.9.1 Risk AchievementWorth 327 7.9.2 Risk ReductionWorth 329 7.9.3 Relationship with the Improvement Potential 330 7.10 Barlow–Proschan’s Metric 331 7.11 Problems 333 References 335 8 Dependent Failures 337 8.1 Introduction 337 8.1.1 Dependent Events and Variables 337 8.1.2 Correlated Variables 338 8.2 Types of Dependence 340 8.3 Cascading Failures 340 8.3.1 Tight Coupling 342 8.4 Common-Cause Failures 342 8.4.1 Multiple Failures that Are Not a CCF 344 8.4.2 Causes of CCF 344 8.4.3 Defenses Against CCF 345 8.5 CCF Models and Analysis 346 8.5.1 Explicit Modeling 347 8.5.2 Implicit Modeling 348 8.5.3 Modeling Approach 348 8.5.4 Model Assumptions 349 8.6 Basic Parameter Model 349 8.6.1 Probability of a Specific Multiplicity 350 8.6.2 Conditional Probability of a Specific Multiplicity 351 8.7 Beta-Factor Model 352 8.7.1 Relation to the BPM 354 8.7.2 Beta-Factor Model in System Analysis 354 8.7.3 Beta-Factor Model for Nonidentical Components 358 8.7.4 C-Factor Model 360 8.8 Multi-parameter Models 360 8.8.1 Binomial Failure Rate Model 360 8.8.2 Multiple Greek Letter Model 362 8.8.3 Alpha-Factor Model 364 8.8.4 Multiple Beta-Factor Model 365 8.9 Problems 366 References 368 9 Maintenance and Maintenance Strategies 371 9.1 Introduction 371 9.1.1 What is Maintenance? 372 9.2 Maintainability 372 9.3 Maintenance Categories 374 9.3.1 Completeness of a Repair Task 377 9.3.2 Condition Monitoring 377 9.4 Maintenance Downtime 378 9.4.1 Downtime Caused by Failures 379 9.4.2 Downtime of a Series Structure 381 9.4.3 Downtime of a Parallel Structure 381 9.4.4 Downtime of a General Structure 382 9.5 Reliability Centered Maintenance 382 9.5.1 What is RCM? 383 9.5.2 Main Steps of an RCM Analysis 384 9.6 Total Productive Maintenance 396 9.7 Problems 398 References 399 10 Counting Processes 401 10.1 Introduction 401 10.1.1 Counting Processes 401 10.1.2 Basic Concepts 406 10.1.3 Martingale Theory 408 10.1.4 Four Types of Counting Processes 409 10.2 Homogeneous Poisson Processes 410 10.2.1 Main Features of the HPP 411 10.2.2 Asymptotic Properties 412 10.2.3 Estimate and Confidence Interval 412 10.2.4 Sum and Decomposition of HPPs 413 10.2.5 Conditional Distribution of Failure Time 414 10.2.6 Compound HPPs 415 10.3 Renewal Processes 417 10.3.1 Basic Concepts 417 10.3.2 The Distribution of Sn 418 10.3.3 The Distribution of N(t) 420 10.3.4 The Renewal Function 421 10.3.5 The Renewal Density 423 10.3.6 Age and Remaining Lifetime 427 10.3.7 Bounds for the Renewal Function 431 10.3.8 Superimposed Renewal Processes 433 10.3.9 Renewal Reward Processes 434 10.3.10 Delayed Renewal Processes 436 10.3.11 Alternating Renewal Processes 438 10.4 Nonhomogeneous Poisson Processes 447 10.4.1 Introduction and Definitions 447 10.4.2 Some Results 449 10.4.3 Parametric NHPP Models 452 10.4.4 Statistical Tests of Trend 454 10.5 Imperfect Repair Processes 455 10.5.1 Brown and Proschan’s model 456 10.5.2 Failure Rate Reduction Models 458 10.5.3 Age Reduction Models 461 10.5.4 Trend Renewal Process 462 10.6 Model Selection 464 10.7 Problems 466 References 470 11 Markov Analysis 473 11.1 Introduction 473 11.1.1 Markov Property 475 11.2 Markov Processes 476 11.2.1 Procedure to Establish the Transition Rate Matrix 479 11.2.2 Chapman–Kolmogorov Equations 482 11.2.3 Kolmogorov Differential Equations 483 11.2.4 State Equations 484 11.3 Asymptotic Solution 487 11.3.1 System Performance Metrics 492 11.4 Parallel and Series Structures 495 11.4.1 Parallel Structures of Independent Components 495 11.4.2 Series Structures of Independent Components 497 11.4.3 Series Structure of Components Where Failure of One Component Prevents Failure of the Other 499 11.5 Mean Time to First System Failure 501 11.5.1 Absorbing States 501 11.5.2 Survivor Function 504 11.5.3 Mean Time to the First System Failure 505 11.6 Systems with Dependent Components 507 11.6.1 Common Cause Failures 508 11.6.2 Load-Sharing Systems 510 11.7 Standby Systems 512 11.7.1 Parallel System with Cold Standby and Perfect Switching 513 11.7.2 Parallel System with Cold Standby and Perfect Switching (Item A is the Main Operating Item) 515 11.7.3 Parallel System with Cold Standby and Imperfect Switching (Item A is the Main Operating Item) 517 11.7.4 Parallel System with Partly Loaded Standby and Perfect Switching (Item A is the Main Operating Item) 518 11.8 Markov Analysis in Fault Tree Analysis 519 11.8.1 Cut Set Information 520 11.8.2 System Information 521 11.9 Time-Dependent Solution 521 11.9.1 Laplace Transforms 522 11.10 Semi-Markov Processes 524 11.11 Multiphase Markov Processes 526 11.11.1 Changing the Transition Rates 526 11.11.2 Changing the Initial State 527 11.12 Piecewise Deterministic Markov Processes 528 11.12.1 Definition of PDMP 529 11.12.2 State Probabilities 529 11.12.3 A Specific Case 530 11.13 Simulation of a Markov Process 532 11.14 Problems 536 References 543 12 Preventive Maintenance 545 12.1 Introduction 545 12.2 Terminology and Cost Function 546 12.3 Time-Based Preventive Maintenance 548 12.3.1 Age Replacement 549 12.3.2 Block Replacement 553 12.3.3 P–F Intervals 557 12.4 Degradation Models 564 12.4.1 Remaining Useful Lifetime 565 12.4.2 Trend Models; Regression-Based Models 567 12.4.3 Models with Increments 569 12.4.4 Shock Models 571 12.4.5 Stochastic Processes with Discrete States 573 12.4.6 Failure Rate Models 574 12.5 Condition-Based Maintenance 574 12.5.1 CBM Strategy 575 12.5.2 Continuous Monitoring and Finite Discrete State Space 576 12.5.3 Continuous Monitoring and Continuous State Space 581 12.5.4 Inspection-Based Monitoring and Finite Discrete State Space 583 12.5.5 Inspection-Based Monitoring and Continuous State Space 586 12.6 Maintenance of Multi-Item Systems 587 12.6.1 System Model 587 12.6.2 Maintenance Models 589 12.6.3 An Illustrative Example 591 12.7 Problems 595 References 601 13 Reliability of Safety Systems 605 13.1 Introduction 605 13.2 Safety-Instrumented Systems 606 13.2.1 Main SIS Functions 607 13.2.2 Testing of SIS Functions 608 13.2.3 Failure Classification 609 13.3 Probability of Failure on Demand 611 13.3.1 Probability of Failure on Demand 612 13.3.2 Approximation Formulas 617 13.3.3 Mean Downtime in a Test Interval 618 13.3.4 Mean Number of Test Intervals Until First Failure 619 13.3.5 Staggered Testing 620 13.3.6 Nonnegligible Repair Time 621 13.4 Safety Unavailability 622 13.4.1 Probability of Critical Situation 623 13.4.2 Spurious Trips 623 13.4.3 Failures Detected by Diagnostic Self-Testing 625 13.5 Common Cause Failures 627 13.5.1 Diagnostic Self-Testing and CCFs 629 13.6 CCFs Between Groups and Subsystems 631 13.6.1 CCFs Between Voted Groups 632 13.6.2 CCFs Between Subsystems 632 13.7 IEC 61508 632 13.7.1 Safety Lifecycle 633 13.7.2 Safety Integrity Level 634 13.7.3 Compliance with IEC 61508 635 13.8 The PDS Method 638 13.9 Markov Approach 639 13.9.1 All Failures are Repaired After Each Test 643 13.9.2 All Critical Failures Are Repaired after Each Test 644 13.9.3 Imperfect Repair after Each Test 644 13.10 Problems 644 References 652 14 Reliability Data Analysis 655 14.1 Introduction 655 14.1.1 Purpose of the Chapter 656 14.2 Some Basic Concepts 656 14.2.1 Datasets 657 14.2.2 Survival Times 658 14.2.3 Categories of Censored Datasets 660 14.2.4 Field Data Collection Exercises 662 14.2.5 At-Risk-Set 663 14.3 Exploratory Data Analysis 663 14.3.1 A Complete Dataset 664 14.3.2 Sample Metrics 665 14.3.3 Histogram 669 14.3.4 Density Plot 670 14.3.5 Empirical Survivor Function 671 14.3.6 Q–Q Plot 673 14.4 Parameter Estimation 674 14.4.1 Estimators and Estimates 675 14.4.2 Properties of Estimators 675 14.4.3 Method of Moments Estimation 677 14.4.4 Maximum Likelihood Estimation 680 14.4.5 Exponentially Distributed Lifetimes 686 14.4.6 Weibull Distributed Lifetimes 692 14.5 The Kaplan–Meier Estimate 696 14.5.1 Motivation for the Kaplan–Meier Estimate Based a Complete Dataset 696 14.5.2 The Kaplan–Meier Estimator for a Censored Dataset 697 14.6 Cumulative Failure Rate Plots 701 14.6.1 The Nelson–Aalen Estimate of the Cumulative Failure Rate 703 14.7 Total-Time-on-Test Plotting 708 14.7.1 Total-Time-on-Test Plot for Complete Datasets 708 14.7.2 Total-Time-on-Test Plot for Censored Datasets 721 14.7.3 A Brief Comparison 722 14.8 Survival Analysis with Covariates 723 14.8.1 Proportional Hazards Model 723 14.8.2 Cox Models 726 14.8.3 Estimating the Parameters of the Cox Model 727 14.9 Problems 730 References 736 15 Bayesian Reliability Analysis 739 15.1 Introduction 739 15.1.1 Three Interpretations of Probability 739 15.1.2 Bayes’ Formula 741 15.2 Bayesian Data Analysis 742 15.2.1 Frequentist Data Analysis 743 15.2.2 Bayesian Data Analysis 743 15.2.3 Model for Observed Data 745 15.2.4 Prior Distribution 745 15.2.5 Observed Data 746 15.2.6 Likelihood Function 746 15.2.7 Posterior Distribution 747 15.3 Selection of Prior Distribution 749 15.3.1 Binomial Model 749 15.3.2 Exponential Model – Single Observation 752 15.3.3 Exponential Model – Multiple Observations 753 15.3.4 Homogeneous Poisson Process 755 15.3.5 Noninformative Prior Distributions 757 15.4 Bayesian Estimation 758 15.4.1 Bayesian Point Estimation 758 15.4.2 Credible Intervals 760 15.5 Predictive Distribution 761 15.6 Models with Multiple Parameters 762 15.7 Bayesian Analysis with R 762 15.8 Problems 764 References 766 16 Reliability Data: Sources and Quality 767 16.1 Introduction 767 16.1.1 Categories of Input Data 767 16.1.2 Parameters Estimates 768 16.2 Generic Reliability Databases 769 16.2.1 OREDA 770 16.2.2 PDS Data Handbook 772 16.2.3 PERD 773 16.2.4 SERH 773 16.2.5 NPRD, EPRD, and FMD 773 16.2.6 GADS 774 16.2.7 GIDEP 774 16.2.8 FMEDA Approach 775 16.2.9 Failure Event Databases 775 16.3 Reliability Prediction 775 16.3.1 MIL-HDBK-217 Approach 776 16.3.2 Similar Methods 778 16.4 Common Cause Failure Data 778 16.4.1 ICDE 779 16.4.2 IEC 61508 Method 779 16.5 Data Analysis and Data Quality 780 16.5.1 Outdated Technology 780 16.5.2 Inventory Data 781 16.5.3 Constant Failure Rates 781 16.5.4 Multiple Samples 783 16.5.5 Data From Manufacturers 785 16.5.6 Questioning the Data Quality 785 16.6 Data Dossier 785 16.6.1 Final Remarks 785 References 787 Appendix A Acronyms 789 Appendix B Laplace Transforms 793 B.1 Important Properties of Laplace Transforms 794 B.2 Laplace Transforms of Some Selected Functions 794 Author Index 797 Subject Index 803

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