Preface xv References xvii Acknowledgements xix Part One INTRODUCTION and BACKGROUND 1 Introduction 2 1 Introduction to Importance Measures 5 References 11 2 Fundamentals of Systems Reliability 13 2.1 Block Diagrams 13 2.2 Structure Functions 14 2.3 Coherent Systems 17 2.4 Modules within a Coherent System 18 2.5 Cuts and Paths of a Coherent System 19 2.6 Critical Cuts and Critical Paths of a Coherent System 21 2.7 Measures of Performance 23 2.

7.1 Reliability for a mission time 24 2.7.2 Reliability function (of time t) 25 2.7.3 Availability function 27 2.8 Stochastic Orderings 28 2.9 Signature of Coherent Systems 28 2.

10 Multilinear Functions and Taylor (Maclaurin) Expansion 31 2.11 Redundancy 32 2.12 Reliability Optimization and Complexity 33 2.13 Consecutive-k-out-of-n Systems 34 2.14 Assumptions 35 References 36 Part Two PRINCIPLES of IMPORTANCE MEASURES 39 Introduction 40 3 The Essence of Importance Measures 43 3.1 ImportanceMeasures in Reliability 43 3.2 Classifications 44 3.3 c -type and p -type ImportanceMeasures 45 3.

4 ImportanceMeasures of a Minimal Cut and a Minimal Path 45 3.5 Terminology 45 References 46 4 Reliability Importance Measures 47 4.1 The B-reliability Importance 47 4.1.1 The B-reliability importance for system functioning and for system failure 52 4.1.2 The criticality reliability importance 52 4.1.

3 The Bayesian reliability importance 53 4.2 The FV Reliability Importance 53 4.2.1 The c-type FV (c-FV) reliability importance 54 4.2.2 The p-type FV (p-FV) reliability importance 54 4.2.3 Decomposition of state vectors 54 4.

2.4 Properties 56 References 57 5 Lifetime Importance Measures 59 5.1 The B-time-dependent-lifetime Importance 59 5.1.1 The criticality time-dependent lifetime importance 61 5.2 The FV Time-dependent Lifetime Importance 61 5.2.1 The c-FV time-dependent lifetime importance 61 5.

2.2 The p-FV time-dependent lifetime importance 63 5.2.3 Decomposition of state vectors 64 5.3 The BP Time-independent Lifetime Importance 64 5.4 The BP Time-dependent Lifetime Importance 69 5.5 Numerical Comparisons of Time-dependent Lifetime ImportanceMeasures 69 5.6 Summary 71 References 72 6 Structure Importance Measures 73 6.

1 The B-i.i.d. Importance and B-structure Importance 73 6.2 The FV Structure Importance 76 6.3 The BP Structure Importance 76 6.4 Structure ImportanceMeasures Based on the B-i.i.

d. importance 79 6.5 The Permutation Importance and Permutation Equivalence 80 6.5.1 Relations to minimal cuts and minimal paths 81 6.5.2 Relations to systems reliability 83 6.6 The Domination Importance 85 6.

7 The Cut Importance and Path Importance 86 6.7.1 Relations to the B-i.i.d. importance 87 6.7.2 Computation 89 6.

8 The Absoluteness Importance 91 6.9 The Cut-path Importance,Min-cut Importance, and Min-path Importance 92 6.10 The First-term Importance and Rare-event Importance 93 6.11 c-type and p-type of Structure ImportanceMeasures 93 6.12 Structure ImportanceMeasures for Dual Systems 94 6.13 Dominant Relations among ImportanceMeasures 96 6.13.1 The absoluteness importance with the domination importance 96 6.

13.2 The domination importance with the permutation importance 96 6.13.3 The domination importance with the min-cut importance and min-path importance 96 6.13.4 The permutation importance with the FV importance 96 6.13.5 The permutation importance with the cut-path importance, min-cut importance, and min-path importance 100 6.

13.6 The cut-path importance with the cut importance and path importance 101 6.13.7 The cut-path importance with the B-i.i.d. importance 101 6.13.

8 The B-i.i.d. importance with the BP importance 102 6.14 Summary 102 References 105 7 ImportanceMeasures of Pairs and Groups of Components 107 7.1 The Joint Reliability Importance and Joint Failure Importance 107 7.1.1 The joint reliability importance of dependent components 110 7.

1.2 The joint reliability importance of two gate events 110 7.1.3 The joint reliability importance for k-out-of-n systems 111 7.1.4 The joint reliability importance of order k 111 7.2 The Differential ImportanceMeasure 112 7.2.

1 The first-order differential importance measure 112 7.2.2 The second-order differential importance measure 113 7.2.3 The differential importance measure of order k 114 7.3 The Total Order Importance 114 7.4 The Reliability AchievementWorth and Reliability ReductionWorth 115 References 116 8 ImportanceMeasures for Consecutive- k -out-of- n Systems 119 8.1 Formulas for the B-importance 119 8.

1.1 The B-reliability importance and B-i.i.d. importance 119 8.1.2 The B-structure importance 122 8.2 Patterns of the B-importance for Lin/Con/k/n Systems 123 8.

2.1 The B-reliability importance 123 8.2.2 The uniform B-i.i.d. importance 124 8.2.

3 The half-line B-i.i.d. importance 126 8.2.4 The nature of the B-i.i.d.

importance patterns 126 8.2.5 Patterns with respect to p 128 8.2.6 Patterns with respect to n 129 8.2.7 Disproved patterns and conjectures 131 8.3 Structure ImportanceMeasures 135 8.

3.1 The permutation importance 135 8.3.2 The cut-path importance 135 8.3.3 The BP structure importance 135 8.3.4 The first-term importance and rare-event importance 136 References 137 Part Three IMPORTANCE MEASURES for RELIABILITY DESIGN 139 Introduction 140 References 141 9 Redundancy Allocation 143 9.

1 Redundancy ImportanceMeasures 144 9.2 A Common Spare 145 9.2.1 The redundancy importance measures 145 9.2.2 The permutation importance 147 9.2.3 The cut importance and path importance 147 9.

3 Spare Identical to the Respective Component 148 9.3.1 The redundancy importance measures 148 9.3.2 The permutation importance 149 9.4 Several Spares in a k-out-of-n System 150 9.5 Several Spares in an Arbitrary Coherent System 150 9.6 Cold Standby Redundancy 152 References 152 10 Upgrading System Performance 155 10.

1 Improving Systems Reliability 156 10.1.1 Same amount of improvement in component reliability 156 10.1.2 A fractional change in component reliability 157 10.1.3 Cold standby redundancy 158 10.1.

4 Parallel redundancy 158 10.1.5 Example and discussion 158 10.2 Improving Expected System Lifetime 159 10.2.1 A shift in component lifetime distributions 160 10.2.2 Exactly one minimal repair 160 10.

2.3 Reduction in the proportional hazards 167 10.2.4 Cold standby redundancy 168 10.2.5 A perfect component 170 10.2.6 An imperfect repair 170 10.

2.7 A scale change in component lifetime distributions 171 10.2.8 Parallel redundancy 171 10.2.9 Comparisons and numerical evaluation 172 10.3 Improving Expected System Yield 174 10.3.

1 A shift in component lifetime distributions 175 10.3.2 Exactly one minimal repair / cold standby redundancy / a perfect component / parallel redundancy 180 10.4 Discussion 182 References 182 11 Component Assignment in Coherent Systems 185 11.1 Description of Component Assignment Problems 186 11.2 Enumeration and Randomization Methods 187 11.3 Optimal Design based on the Permutation Importance and Pairwise Exchange 188 11.4 Invariant Optimal and InvariantWorst Arrangements 189 11.

5 Invariant Arrangements for Parallel-series and Series-parallel Systems 191 11.6 Consistent B-i.i.d. Importance Ordering and Invariant Arrangements 192 11.7 Optimal Design based on the B-reliability Importance 194 11.8 Optimal Assembly Problems 196 References 197 12 Component Assignment in Consecutive- k -out-of- n and Its Variant Systems 199 12.1 Invariant Arrangements for Con/k/n Systems 199 12.

1.1 Invariant optimal arrangements for Lin/Con/k/n systems 200 12.1.2 Invariant optimal arrangements for Cir/Con/k/n systems 200 12.1.3 Consistent B-i.i.d.

importance ordering and invariant arrangements 202 12.2 Necessary Conditions for Component Assignment in Con/k/n Systems 204 12.3 Sequential Component Assignment Problems in Con/2/n:F Systems 206 12.4 Consecutive-2 Failure Systems on Graphs 207 12.4.1 Consecutive-2 failure systems on trees 208 12.5 Series Con/k/n Systems 208 12.5.

1 Series Con/2/n:F systems 209 12.5.2 Series Lin/Con/k/n:G systems 209 12.6 Consecutive-k-out-of-r-from-n Systems 211 12.7 Two-dimensional and Redundant Con/k/n Systems 213 12.7.1 Con/(r, k)/(r, n) systems 214 12.8 Miscellaneous 216 References 217 13 B-importance based Heuristics for Component Assignment 219 13.

1 The Kontoleon Heuristic 219 13.2 The LK Type Heuristics 221 13.2.1 The LKA heuristic 221 13.2.2 Another three LK type heuristics 221 13.2.3 Relation to invariant optimal arrangements 221 13.

2.4 Numerical comparisons of the LK type heuristics 224 13.3 The ZK Type Heuristics 225 13.3.1 Four.