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 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