Optimal reliability modeling : principles and applications

Preface.Acknowledgments.1 Introduction.1.1 Needs for Reliability Modeling.1.2 Optimal Design.2 Reliability Mathematics.2.1 Probability and Distributions.2.1.1 Events and Boolean Algebra.2.1.2 Probabilities of Events.2.1.3 Random Variables and Their Characteristics.2.1.4 Multivariate Distributions.2.1.5 Special Discrete Distributions.2.1.6 Special Continuous Distributions.2.2 Reliability Concepts.2.3 Commonly Used Lifetime Distributions.2.4 Stochastic Processes.2.4.1 General Definitions.2.4.2 Homogeneous Poisson Process.2.4.3 Nonhomogeneous Poisson Process.2.4.4 Renewal Process.2.4.5 Discrete-Time Markov Chains.2.4.6 Continuous-Time Markov Chains.2.5 Complex System Reliability Assessment Using Fault Tree Analysis.3 Complexity Analysis.3.1 Orders of Magnitude and Growth.3.2 Evaluation of Summations.3.3 Bounding Summations.3.4 Recurrence Relations.3.4.1 Expansion Method.3.4.2 Guess-and-Prove Method.3.4.3 Master Method.3.5 Summary.4 Fundamental System Reliability Models.4.1 Reliability Block Diagram.4.2 Structure Functions.4.3 Coherent Systems.4.4 Minimal Paths and Minimal Cuts.4.5 Logic Functions.4.6 Modules within a Coherent System.4.7 Measures of Performance.4.8 One-Component System.4.9 Series System Model.4.9.1 System Reliability Function and MTTF.4.9.2 System Availability.4.10 Parallel System Model.4.10.1 System Reliability Function and MTTF.4.10.2 System Availability of Parallel System with Two i.i.d. Components.4.10.3 System Availability of Parallel System with Two Different Components.4.10.4 Parallel Systems with n i.i.d. Components.4.11 Parallel-Series System Model.4.12 Series-Parallel System Model.4.13 Standby System Model.4.13.1 Cold Standby Systems.4.13.2 Warm Standby Systems.5 General Methods for System Reliability Evaluation.5.1 Parallel and Series Reductions.5.2 Pivotal Decomposition.5.3 Generation of Minimal Paths and Minimal Cuts.5.3.1 Connection Matrix.5.3.2 Node Removal Method for Generation of Minimal Paths.5.3.3 Generation of Minimal Cuts from Minimal Paths.5.4 Inclusion-Exclusion Method.5.5 Sum-of-Disjoint-Products Method.5.6 Markov Chain Imbeddable Structures.5.6.1 MIS Technique in Terms of System Failures.5.6.2 MIS Technique in Terms of System Success.5.7 Delta-Star and Star-Delta Transformations.5.7.1 Star or Delta Structure with One Input Node and Two Output Nodes.5.7.2 Delta Structure in Which Each Node May Be either an Input Node or an Output Node.5.8 Bounds on System Reliability.5.8.1 IE Method.5.8.2 SDP Method.5.8.3 Esary-Proschan (EP) Method.5.8.4 Min-Max Bounds.5.8.5 Modular Decompositions.5.8.6 Notes.6 General Methodology for System Design.6.1 Redundancy in System Design.6.2 Measures of Component Importance.6.2.1 Structural Importance.6.2.2 Reliability Importance.6.2.3 Criticality Importance.6.2.4 Relative Criticality.6.3 Majorization and Its Application in Reliability.6.3.1 Definition of Majorization.6.3.2 Schur Functions.6.3.3 L-Additive Functions.6.4 Reliability Importance in Optimal Design.6.5 Pairwise Rearrangement in Optimal Design.6.6 Optimal Arrangement for Series and Parallel Systems.6.7 Optimal Arrangement for Series-Parallel Systems.6.8 Optimal Arrangement for Parallel-Series Systems.6.9 Two-Stage Systems.6.10 Summary.7 Thek-out-of-n System Model.7.1 System Reliability Evaluation.7.1.1 The k-out-of-n:G System with i.i.d. Components.7.1.2 The k-out-of-n:G System with Independent Components.7.1.3 Bounds on System Reliability.7.2 Relationship between k-out-of-n G and F Systems.7.2.1 Equivalence between k-out-of-n:G and (n - k + 1)-out-of-n:F Systems.7.2.2 Dual Relationship between k-out-of-n G and F Systems.7.3 Nonrepairable k-out-of-n Systems.7.3.1 Systems with i.i.d. Components.7.3.2 Systems with Nonidentical Components.7.3.3 Systems with Load-Sharing Components Following Exponential Lifetime Distributions.7.3.4 Systems with Load-Sharing Components Following Arbitrary Lifetime Distributions.7.3.5 Systems with Standby Components.7.4 Repairable k-out-of-n Systems.7.4.1 General Repairable System Model.7.4.2 Systems with Active Redundant Components.7.4.3 Systems with Load-Sharing Components.7.4.4 Systems with both Active Redundant and Cold Standby Components.7.5 Weighted k-out-of-n:G Systems.8 Design of k-out-of-n Systems.8.1 Properties of k-out-of-n Systems.8.1.1 Component Reliability Importance.8.1.2 Effects of Redundancy in k-out-of-n Systems.8.2 Optimal Design of k-out-of-n Systems.8.2.1 Optimal System Size n.8.2.2 Simultaneous Determination of n and k.8.2.3 Optimal Replacement Time.8.3 Fault Coverage.8.3.1 Deterministic Analysis.8.3.2 Stochastic Analysis.8.4 Common-Cause Failures.8.4.1 Repairable System with Lethal Common-Cause Failures.8.4.2 System Design Considering Lethal Common-Cause Failures.8.4.3 Optimal Replacement Policy with Lethal Common-Cause Failures.8.4.4 Nonlethal Common-Cause Failures.8.5 Dual Failure Modes.8.5.1 Optimal k or n Value to Maximize System Reliability.8.5.2 Optimal k or n Value to Maximize System Profit.8.5.3 Optimal k and n Values to Minimize System Cost.8.6 Other Issues.8.6.1 Selective Replacement Optimization.8.6.2 TMR and NMR Structures.8.6.3 Installation Time of Repaired Components.8.6.4 Combinations of Factors.8.6.5 Partial Ordering.9 Consecutive-k-out-of-n Systems.9.1 System Reliability Evaluation.9.1.1 Systems with i.i.d. Components.9.1.2 Systems with Independent Components.9.2 Optimal System Design.9.2.1 B-Importances of Components.9.2.2 Invariant Optimal Design.9.2.3 Variant Optimal Design.9.3 Consecutive-k-out-of-n:G Systems.9.3.1 System Reliability Evaluation.9.3.2 Component Reliability Importance.9.3.3 Invariant Optimal Design.9.3.4 Variant Optimal Design.9.4 System Lifetime Distribution.9.4.1 Systems with i.i.d. Components.9.4.2 System with Exchangeable Dependent Components.9.4.3 System with (k - 1)-Step Markov-Dependent Components.9.4.4 Repairable Consecutive-k-out-of-n Systems.9.5 Summary.10 Multidimensional Consecutive-k-out-of-n Systems.10.1 System Reliability Evaluation.10.1.1 Special Multidimensional Systems.10.1.2 General Two-Dimensional Systems.10.1.3 Bounds and Approximations.10.2 System Logic Functions.10.3 Optimal System Design.10.4 Summary.11 Other k-out-of-n and Consecutive-k-out-of-n Models.11.1 The s-Stage k-out-of-n Systems.11.2 Redundant Consecutive-k-out-of-n Systems.11.3 Linear and Circular m-Consecutive-k-out-of-n Model.11.4 The k-within-Consecutive-m-out-of-n Systems.11.4.1 Systems with i.i.d. Components.11.4.2 Systems with Independent Components.11.4.3 The k-within-(r, s)/(m, n):F Systems.11.5 Series Consecutive-k-out-of-n Systems.11.6 Combined k-out-of-n:F and Consecutive-kc-out-of-n:F System.11.7 Combined k-out-of-mn:F and Linear (r, s)/(m, n):F System.11.8 Combined k-out-of-mn:F, One-Dimensional Con/kc/n:F, and Two-Dimensional Linear (r, s)/(m, n):F Model.11.9 Application of Combined k-out-of-n and Consecutive-k-out-of-n Systems.11.10 Consecutively Connected Systems.11.11 Weighted Consecutive-k-out-of-n Systems.11.11.1 Weighted Linear Consecutive-k-out-of-n:F Systems.11.11.2 Weighted Circular Consecutive-k-out-of-n:F Systems.12 Multistate System Models.12.1 Consecutively Connected Systems with Binary System State and Multistate Components.12.1.1 Linear Multistate Consecutively Connected Systems.12.1.2 Circular Multistate Consecutively Connected Systems.12.1.3 Tree-Structured Consecutively Connected Systems.12.2 Two-Way Consecutively Connected Systems.12.3 Key Concepts in Multistate Reliability Theory.12.4 Special Multistate Systems and Their Performance Evaluation.12.4.1 Simple Multistate k-out-of-n:G Model.12.4.2 Generalized Multistate k-out-of-n:G Model.12.4.3 Generalized Multistate Consecutive-k-out-of-n:F System.12.5 General Multistate Systems and Their Performance Evaluation.12.6 Summary.Appendix: Laplace Transform.References.Bibliography.Index.