Liudong Xing, PhD, is a Full Professor in the Department of Electrical and Computer Engineering at University of Massachusetts (UMass) Dartmouth, USA.Gregory Levitin, PhD, is a senior expert in the Reliability Department at The Israel Electric Corporation, Israel, and a distinguished visiting professor at University of Electronic Science and Technology of China. Chaonan Wang, PhD, is a Full Professor at the College of Information Science and Technology at Jinan University, Guangzhou, China.

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• Foreword ixPreface xiNomenclature xv1 Introduction 1References 42 Fundamental Reliability Theory 72.1 Basic Probability Concepts 72.1.1 Axioms of Probability 72.1.2 Conditional Probability 72.1.3 Total Probability Law 82.1.4 Bayes’ Theorem 92.1.5 Random Variables 92.2 Reliability Measures 102.2.1 Time to Failure 112.2.2 Failure Function 112.2.3 Reliability Function 112.2.4 Failure Rate 112.2.5 Mean Time to Failure 112.2.6 Mean Residual Life 122.3 Fault Tree Modeling 122.3.1 Static Fault Tree 132.3.2 Dynamic Fault Tree 132.3.3 Phased-Mission Fault Tree 142.3.4 Multi-State Fault Tree 152.4 Binary Decision Diagram 162.4.1 Basic Concept 172.4.2 ROBDD Generation 172.4.3 ROBDD Evaluation 182.4.4 Illustrative Example 192.5 Markov Process 202.6 Reliability Software 22References 223 Imperfect Fault Coverage 273.1 Different Types of IPC 273.2 ELC Modeling 283.3 Binary-State System 293.3.1 BDD Expansion Method 293.3.2 Simple and Efficient Algorithm 323.4 Multi-State System 343.4.1 MMDD-Based Method for MSS Analysis 353.4.2 Illustrative Example 363.5 Phased-Mission System 373.5.1 Mini-Component Concept 373.5.2 PMS SEA 383.5.3 PMS BDD Method 403.5.4 Summary of PMS SEA 423.5.5 Illustrative Example 423.6 Summary 43References 454 Modular Imperfect Coverage 494.1 Modular Imperfect Coverage Model 494.2 Non repairable Hierarchical System 514.3 Repairable Hierarchical System 554.4 Summary 58References 585 Functional Dependence 615.1 Logic OR Replacement Method 615.2 Combinatorial Algorithm 635.2.1 Task 1: Addressing UFs of Independent Trigger Components 635.2.2 Task 2: Generating Reduced Problems Without FDEP 635.2.3 Task 3: Solving Reduced Reliability Problems 645.2.3.1 Expansion Process 645.2.3.2 Reduced FT Generation Procedure 655.2.3.3 Dual Trigger-Basic Event Handling 655.2.3.4 Evaluation of P(system fails|ITEi) 655.2.4 Task 4: Integrating to Obtain Final System Unreliability 665.2.5 Algorithm Summary 665.2.6 Algorithm Complexity 665.3 Case Study 1: Combined Trigger Event 675.4 Case Study 2: Shared Dependent Event 705.5 Case Study 3: Cascading FDEP 735.5.1 Evaluation of P(system fails|ITE1) 745.5.2 Evaluation of P(system fails|ITE2) 755.5.3 Evaluation of URsystem 765.6 Case Study 4: Dual Event and Cascading FDEP 765.6.1 Evaluation of P(system fails|ITE1) 785.6.2 Evaluation of URsystem 795.7 Summary 79References 806 Deterministic Common-Cause Failure 836.1 Explicit Method 846.1.1 Two-Step Method 846.1.2 Illustrative Example 846.2 Efficient Decomposition and Aggregation Approach 856.2.1 Three-Step Method 866.2.2 Illustrative Example 876.3 Decision Diagram–Based Aggregation Method 896.3.1 Three-Step Method 896.3.2 Illustrative Example 916.4 Universal Generating Function–Based Method 946.4.1 System Model 946.4.2 u-Function Method for Series-Parallel Systems 956.4.3 u-Function Method for CCFs 976.4.4 Illustrative Example 996.5 Summary 104References 1047 Probabilistic Common-Cause Failure 1077.1 Single-Phase System 1077.1.1 Explicit Method 1087.1.2 Implicit Method 1107.1.3 Comparisons and Discussions 1157.2 Multi-Phase System 1157.2.1 Explicit Method 1157.2.2 Implicit Method 1197.2.3 Comparisons and Discussions 1237.3 Impact of PCCF 1247.4 Summary 125References 1258 Deterministic Competing Failure 1278.1 Overview 1278.2 PFGE Method 1288.2.1 s-Independent LF and PFGE 1288.2.2 s-Dependent LF and PFGE 1288.2.3 Disjoint LF and PFGE 1298.3 Single-Phase System with Single FDEP Group 1298.3.1 Combinatorial Method 1298.3.2 Case Study 1318.4 Single-Phase System with Multiple FDEP Groups 1358.4.1 Combinatorial Method 1358.4.2 Case Study 1378.5 Single-Phase System with PFs Having Global and Selective Effects 1418.5.1 Combinatorial Method 1418.5.2 Case Study 1448.6 Multi-Phase System with Single FDEP Group 1508.6.1 Combinatorial Method 1508.6.2 Case Study 1538.7 Multi-Phase System with Multiple FDEP Groups 1588.7.1 CTMC-Based Method 1588.7.2 Case Study 1598.8 Summary 166References 1679 Probabilistic Competing Failure 1699.1 Overview 1699.2 System with Single Type of Component Local Failures 1709.2.1 Combinatorial Method 1709.2.2 Case Study 1729.3 System with Multiple Types of Component Local Failures 1819.3.1 Combinatorial Method 1819.3.2 Case Study 1829.4 System with Random Failure Propagation Time 1909.4.1 Combinatorial Method 1909.4.2 Case Study: WSN System 1929.5 Summary 198References 19910 Dynamic Standby Sparing 20110.1 Types of Standby Systems 20110.2 CTMC-Based Method 20210.2.1 Cold Standby System 20310.2.2 Warm Standby System 20410.3 Decision Diagram−Based Method 20510.3.1 Cold Standby System 20510.3.2 Warm Standby System 20810.4 Approximation Method 21110.4.1 Homogeneous Cold Standby System 21210.4.2 Heterogeneous Cold Standby System 21410.5 Event Transition Method 21610.5.1 State-Space Representation of System Behavior 21710.5.2 Basic Steps 21810.5.3 Warm Standby System 21810.6 Overview of Optimization Problems 22010.7 Summary 222References 222Index 229

Liudong Xing, PhD, is a Full Professor in the Department of Electrical and Computer Engineering at University of Massachusetts (UMass) Dartmouth, USA.Gregory Levitin, PhD, is a senior expert in the Reliability Department at The Israel Electric Corporation, Israel, and a distinguished visiting professor at University of Electronic Science and Technology of China. Chaonan Wang, PhD, is a Full Professor at the College of Information Science and Technology at Jinan University, Guangzhou, China.

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• Foreword ixPreface xiNomenclature xv1 Introduction 1References 42 Fundamental Reliability Theory 72.1 Basic Probability Concepts 72.1.1 Axioms of Probability 72.1.2 Conditional Probability 72.1.3 Total Probability Law 82.1.4 Bayes’ Theorem 92.1.5 Random Variables 92.2 Reliability Measures 102.2.1 Time to Failure 112.2.2 Failure Function 112.2.3 Reliability Function 112.2.4 Failure Rate 112.2.5 Mean Time to Failure 112.2.6 Mean Residual Life 122.3 Fault Tree Modeling 122.3.1 Static Fault Tree 132.3.2 Dynamic Fault Tree 132.3.3 Phased-Mission Fault Tree 142.3.4 Multi-State Fault Tree 152.4 Binary Decision Diagram 162.4.1 Basic Concept 172.4.2 ROBDD Generation 172.4.3 ROBDD Evaluation 182.4.4 Illustrative Example 192.5 Markov Process 202.6 Reliability Software 22References 223 Imperfect Fault Coverage 273.1 Different Types of IPC 273.2 ELC Modeling 283.3 Binary-State System 293.3.1 BDD Expansion Method 293.3.2 Simple and Efficient Algorithm 323.4 Multi-State System 343.4.1 MMDD-Based Method for MSS Analysis 353.4.2 Illustrative Example 363.5 Phased-Mission System 373.5.1 Mini-Component Concept 373.5.2 PMS SEA 383.5.3 PMS BDD Method 403.5.4 Summary of PMS SEA 423.5.5 Illustrative Example 423.6 Summary 43References 454 Modular Imperfect Coverage 494.1 Modular Imperfect Coverage Model 494.2 Non repairable Hierarchical System 514.3 Repairable Hierarchical System 554.4 Summary 58References 585 Functional Dependence 615.1 Logic OR Replacement Method 615.2 Combinatorial Algorithm 635.2.1 Task 1: Addressing UFs of Independent Trigger Components 635.2.2 Task 2: Generating Reduced Problems Without FDEP 635.2.3 Task 3: Solving Reduced Reliability Problems 645.2.3.1 Expansion Process 645.2.3.2 Reduced FT Generation Procedure 655.2.3.3 Dual Trigger-Basic Event Handling 655.2.3.4 Evaluation of P(system fails|ITEi) 655.2.4 Task 4: Integrating to Obtain Final System Unreliability 665.2.5 Algorithm Summary 665.2.6 Algorithm Complexity 665.3 Case Study 1: Combined Trigger Event 675.4 Case Study 2: Shared Dependent Event 705.5 Case Study 3: Cascading FDEP 735.5.1 Evaluation of P(system fails|ITE1) 745.5.2 Evaluation of P(system fails|ITE2) 755.5.3 Evaluation of URsystem 765.6 Case Study 4: Dual Event and Cascading FDEP 765.6.1 Evaluation of P(system fails|ITE1) 785.6.2 Evaluation of URsystem 795.7 Summary 79References 806 Deterministic Common-Cause Failure 836.1 Explicit Method 846.1.1 Two-Step Method 846.1.2 Illustrative Example 846.2 Efficient Decomposition and Aggregation Approach 856.2.1 Three-Step Method 866.2.2 Illustrative Example 876.3 Decision Diagram–Based Aggregation Method 896.3.1 Three-Step Method 896.3.2 Illustrative Example 916.4 Universal Generating Function–Based Method 946.4.1 System Model 946.4.2 u-Function Method for Series-Parallel Systems 956.4.3 u-Function Method for CCFs 976.4.4 Illustrative Example 996.5 Summary 104References 1047 Probabilistic Common-Cause Failure 1077.1 Single-Phase System 1077.1.1 Explicit Method 1087.1.2 Implicit Method 1107.1.3 Comparisons and Discussions 1157.2 Multi-Phase System 1157.2.1 Explicit Method 1157.2.2 Implicit Method 1197.2.3 Comparisons and Discussions 1237.3 Impact of PCCF 1247.4 Summary 125References 1258 Deterministic Competing Failure 1278.1 Overview 1278.2 PFGE Method 1288.2.1 s-Independent LF and PFGE 1288.2.2 s-Dependent LF and PFGE 1288.2.3 Disjoint LF and PFGE 1298.3 Single-Phase System with Single FDEP Group 1298.3.1 Combinatorial Method 1298.3.2 Case Study 1318.4 Single-Phase System with Multiple FDEP Groups 1358.4.1 Combinatorial Method 1358.4.2 Case Study 1378.5 Single-Phase System with PFs Having Global and Selective Effects 1418.5.1 Combinatorial Method 1418.5.2 Case Study 1448.6 Multi-Phase System with Single FDEP Group 1508.6.1 Combinatorial Method 1508.6.2 Case Study 1538.7 Multi-Phase System with Multiple FDEP Groups 1588.7.1 CTMC-Based Method 1588.7.2 Case Study 1598.8 Summary 166References 1679 Probabilistic Competing Failure 1699.1 Overview 1699.2 System with Single Type of Component Local Failures 1709.2.1 Combinatorial Method 1709.2.2 Case Study 1729.3 System with Multiple Types of Component Local Failures 1819.3.1 Combinatorial Method 1819.3.2 Case Study 1829.4 System with Random Failure Propagation Time 1909.4.1 Combinatorial Method 1909.4.2 Case Study: WSN System 1929.5 Summary 198References 19910 Dynamic Standby Sparing 20110.1 Types of Standby Systems 20110.2 CTMC-Based Method 20210.2.1 Cold Standby System 20310.2.2 Warm Standby System 20410.3 Decision Diagram−Based Method 20510.3.1 Cold Standby System 20510.3.2 Warm Standby System 20810.4 Approximation Method 21110.4.1 Homogeneous Cold Standby System 21210.4.2 Heterogeneous Cold Standby System 21410.5 Event Transition Method 21610.5.1 State-Space Representation of System Behavior 21710.5.2 Basic Steps 21810.5.3 Warm Standby System 21810.6 Overview of Optimization Problems 22010.7 Summary 222References 222Index 229

Liudong Xing, PhD, is a Full Professor in the Department of Electrical and Computer Engineering at University of Massachusetts (UMass) Dartmouth, USA.Gregory Levitin, PhD, is a senior expert in the Reliability Department at The Israel Electric Corporation, Israel, and a distinguished visiting professor at University of Electronic Science and Technology of China. Chaonan Wang, PhD, is a Full Professor at the College of Information Science and Technology at Jinan University, Guangzhou, China.

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• Foreword ixPreface xiNomenclature xv1 Introduction 1References 42 Fundamental Reliability Theory 72.1 Basic Probability Concepts 72.1.1 Axioms of Probability 72.1.2 Conditional Probability 72.1.3 Total Probability Law 82.1.4 Bayes’ Theorem 92.1.5 Random Variables 92.2 Reliability Measures 102.2.1 Time to Failure 112.2.2 Failure Function 112.2.3 Reliability Function 112.2.4 Failure Rate 112.2.5 Mean Time to Failure 112.2.6 Mean Residual Life 122.3 Fault Tree Modeling 122.3.1 Static Fault Tree 132.3.2 Dynamic Fault Tree 132.3.3 Phased-Mission Fault Tree 142.3.4 Multi-State Fault Tree 152.4 Binary Decision Diagram 162.4.1 Basic Concept 172.4.2 ROBDD Generation 172.4.3 ROBDD Evaluation 182.4.4 Illustrative Example 192.5 Markov Process 202.6 Reliability Software 22References 223 Imperfect Fault Coverage 273.1 Different Types of IPC 273.2 ELC Modeling 283.3 Binary-State System 293.3.1 BDD Expansion Method 293.3.2 Simple and Efficient Algorithm 323.4 Multi-State System 343.4.1 MMDD-Based Method for MSS Analysis 353.4.2 Illustrative Example 363.5 Phased-Mission System 373.5.1 Mini-Component Concept 373.5.2 PMS SEA 383.5.3 PMS BDD Method 403.5.4 Summary of PMS SEA 423.5.5 Illustrative Example 423.6 Summary 43References 454 Modular Imperfect Coverage 494.1 Modular Imperfect Coverage Model 494.2 Non repairable Hierarchical System 514.3 Repairable Hierarchical System 554.4 Summary 58References 585 Functional Dependence 615.1 Logic OR Replacement Method 615.2 Combinatorial Algorithm 635.2.1 Task 1: Addressing UFs of Independent Trigger Components 635.2.2 Task 2: Generating Reduced Problems Without FDEP 635.2.3 Task 3: Solving Reduced Reliability Problems 645.2.3.1 Expansion Process 645.2.3.2 Reduced FT Generation Procedure 655.2.3.3 Dual Trigger-Basic Event Handling 655.2.3.4 Evaluation of P(system fails|ITEi) 655.2.4 Task 4: Integrating to Obtain Final System Unreliability 665.2.5 Algorithm Summary 665.2.6 Algorithm Complexity 665.3 Case Study 1: Combined Trigger Event 675.4 Case Study 2: Shared Dependent Event 705.5 Case Study 3: Cascading FDEP 735.5.1 Evaluation of P(system fails|ITE1) 745.5.2 Evaluation of P(system fails|ITE2) 755.5.3 Evaluation of URsystem 765.6 Case Study 4: Dual Event and Cascading FDEP 765.6.1 Evaluation of P(system fails|ITE1) 785.6.2 Evaluation of URsystem 795.7 Summary 79References 806 Deterministic Common-Cause Failure 836.1 Explicit Method 846.1.1 Two-Step Method 846.1.2 Illustrative Example 846.2 Efficient Decomposition and Aggregation Approach 856.2.1 Three-Step Method 866.2.2 Illustrative Example 876.3 Decision Diagram–Based Aggregation Method 896.3.1 Three-Step Method 896.3.2 Illustrative Example 916.4 Universal Generating Function–Based Method 946.4.1 System Model 946.4.2 u-Function Method for Series-Parallel Systems 956.4.3 u-Function Method for CCFs 976.4.4 Illustrative Example 996.5 Summary 104References 1047 Probabilistic Common-Cause Failure 1077.1 Single-Phase System 1077.1.1 Explicit Method 1087.1.2 Implicit Method 1107.1.3 Comparisons and Discussions 1157.2 Multi-Phase System 1157.2.1 Explicit Method 1157.2.2 Implicit Method 1197.2.3 Comparisons and Discussions 1237.3 Impact of PCCF 1247.4 Summary 125References 1258 Deterministic Competing Failure 1278.1 Overview 1278.2 PFGE Method 1288.2.1 s-Independent LF and PFGE 1288.2.2 s-Dependent LF and PFGE 1288.2.3 Disjoint LF and PFGE 1298.3 Single-Phase System with Single FDEP Group 1298.3.1 Combinatorial Method 1298.3.2 Case Study 1318.4 Single-Phase System with Multiple FDEP Groups 1358.4.1 Combinatorial Method 1358.4.2 Case Study 1378.5 Single-Phase System with PFs Having Global and Selective Effects 1418.5.1 Combinatorial Method 1418.5.2 Case Study 1448.6 Multi-Phase System with Single FDEP Group 1508.6.1 Combinatorial Method 1508.6.2 Case Study 1538.7 Multi-Phase System with Multiple FDEP Groups 1588.7.1 CTMC-Based Method 1588.7.2 Case Study 1598.8 Summary 166References 1679 Probabilistic Competing Failure 1699.1 Overview 1699.2 System with Single Type of Component Local Failures 1709.2.1 Combinatorial Method 1709.2.2 Case Study 1729.3 System with Multiple Types of Component Local Failures 1819.3.1 Combinatorial Method 1819.3.2 Case Study 1829.4 System with Random Failure Propagation Time 1909.4.1 Combinatorial Method 1909.4.2 Case Study: WSN System 1929.5 Summary 198References 19910 Dynamic Standby Sparing 20110.1 Types of Standby Systems 20110.2 CTMC-Based Method 20210.2.1 Cold Standby System 20310.2.2 Warm Standby System 20410.3 Decision Diagram−Based Method 20510.3.1 Cold Standby System 20510.3.2 Warm Standby System 20810.4 Approximation Method 21110.4.1 Homogeneous Cold Standby System 21210.4.2 Heterogeneous Cold Standby System 21410.5 Event Transition Method 21610.5.1 State-Space Representation of System Behavior 21710.5.2 Basic Steps 21810.5.3 Warm Standby System 21810.6 Overview of Optimization Problems 22010.7 Summary 222References 222Index 229