Statistical and Probabilistic Models in Reliability

Reliability Characteristics Estimators in Accelerated Life Testing.- 6.1 Introduction.- 6.2 Parametric Estimation.- 6.3 Nonparametric Estimation.- 6.4 Conclusion.- References.- 7 Chi-Squared Goodness of Fit Test for Doubly Censored Data With Applications in Survival Analysis and Reliability.- 7.1 Introduction.- 7.2 Weak Convergence of the Process Un(t).- 7.3 The Weak Convergence of the Process Un*(t).- 7.4 The Test Statistics.- References.- 8 Estimation of Kernel, Availability and Reliability of Semi-Markov Systems.- 8.1 Introduction.- 8.2 Estimator of the Semi-Markov Kernel.- 8.3 Estimation of the Markov Renewal Matrix and Its Asymptotic Properties.- 8.4 Estimation of the Semi-Markov Transition Matrix and Its Properties.- 8.5 Reliability and Availability Estimation.- 8.5.1 Availability.- 8.5.2 Reliability.- 8.5.3 Asymptotic properties of the estimators.- 8.6 Application.- References.- II: Probabilistic Methods.- 9 Stochastical Models of Systems in Reliability Problems.- 9.1 Introduction.- 9.2 Reliability Problem for a Redundant System.- 9.2.1 Repairable duplicated system.- 9.2.2 Sojourn time in a subset of states.- 9.3 Problems of Singular Perturbation.- 9.4 Analysis of Stochastic Systems.- 9.4.1 Phase merging scheme.- 9.4.2 Heuristic principles of phase merging.- 9.5 Diffusion Approximation Scheme.- References.- 10 Markovian Repairman Problems. Classification and Approximation.- 10.1 Introduction.- 10.2 Classification of Repairman Models.- 10.3 Asymptotical Analysis of Queueing Process.- References.- 11 On Limit Reliability Functions of Large Systems. Part I.- 11.1 Introduction.- 11.2 Limit Reliability Functions of Homogeneous Systems.- 11.3 Limit Reliability Functions of Nonhomogeneous Systems.- 11.4 Remarks on Limit Reliability Functions of Multi-State Systems.- 11.5 Summary.- References.- 12 On Limit Reliability Functions of Large Systems. Part II.- 12.1 Domains of Attraction of Limit Reliability Functions.- 12.2 Asymptotic Reliability Functions of a Regular Homogeneous Series-"k out of n" System.- 12.3 Limit Reliability Functions of Homogeneous Regular Series-Parallel Systems of Higher Order.- References.- 13 Error Bounds for a Stiff Markov Chain Approximation Technique and an Application.- 13.1 Introduction.- 13.2 Notations.- 13.3 Approximation Techniques.- 13.3.1 A path-based technique.- 13.3.2 Bobbio and Trivedi's algorithm.- 13.4 Main Results.- 13.4.1 Equivalence.- 13.4.2 A non-conservative case.- 13.4.3 Error bounds.- 13.5 Numerical Example.- 13.5.1 Model used.- 13.5.2 Results.- 13.6 Conclusion.- A.1 Proof of Proposition 13.3.1.- A.2 Proof of Proposition 13.4.1.- A.3 Proof of Theorem 13.4.1.- References.- 14 On the Failure Rate of Components Subjected to a Diffuse Stress Environment.- 14.1 Introduction.- 14.2 The Mathematical Model.- 14.3 General Results.- 14.3.1 The case of a stress starting from a fixed level.- 14.3.2 The case of a stationary stress process.- 14.4 Particular Case of Interest.- 14.4.1 Instantaneous action of the stress.- 14.4.2 Cumulative action of the stress.- 14.5 A Shot-Noise Model With Diffuse Stress.- 14.6 Conclusion.- Appendix (Proof of Lemma 14.3.1).- References.- 15 Modelling the Reliability of a Complex System Under Stress Environment.- 15.1 Introduction.- 15.2 Modelling the Stress.- 15.3 System of n Identical Components Subjected to an Homogeneous Poisson Stress Process.- 15.4 Some Particular Configurations of the n Identical Component System.- 15.5 Architecture and Stress Influence.- 15.6 Example - System of Two Identical Components Subjected to a Common, Homogeneous Poisson Stress Process.- 15.7 Conclusions.- References.- 16 On the Failure Rate.- 16.1 Introduction.- 16.2 Failure Process.- 16.3 Semi-Markov Process.- References.- 17 Asymptotic Results for the Failure Time of Consecutive k-out-of-n Systems.- 17.1 Introduction.- 17.2 Strong Laws for the Failure Time of the System.- References.- III: Special Techniques and Applications.- 18 Two-State Start-Up Demonstration Testing.- 18.1 Introduction.- 18.2 Probability Generating Function.- 18.3 Probabilities and Recurrence Relations.- References.- 19 Optimal Prophylaxis Policy for Systems With Partly Observable Parameters.- 19.1 Introduction.- 19.2 One-Server System.- 19.2.1 Mathematical model.- 19.2.2 Coefficient of readiness.- 19.3 Two-Server System.- 19.3.1 Mathematical model.- 19.3.2 Coefficient of readiness.- 19.4 Optimization.- 19.4.1 Functional equation.- 19.4.2 Continuous semi-Markov process.- 19.4.3 Evaluation of functionals.- 19.4.4 Process of maximal values.- 19.4.5 Inversed Gamma-process.- References.- 20 Exact Methods to Compute Network Reliability.- 20.1 Introduction.- 20.2 Definitions and Notation.- 20.3 Enumeration.- 20.3.1 State enumeration.- 20.3.2 Path enumeration-Cut enumeration.- 20.4 Reduction With Factoring.- 20.5 Decomposition.- 20.5.1 The principle.- 20.5.2 Algorithm implementation.- 20.5.3 Complexity.- 20.5.4 Adaptation for other relability problems.- 20.6 Conclusion.- References.- 21 On Matroid Base Families and the Reliability Computation of Totally Amenable Systems.- 21.1 Preliminaries.- 21.2 Algorithmic Complexity of Reliability Computation and Domination Theory.- 21.3 Matroid Base Families.- 21.4 On the Complexity of Computing the Reliability of Matroid Base Family Systems.- 21.5 Conclusions.- References.- 22 The Computer-Assisted Analysis of the Semi-Markovian Stochastic Petri Nets and an Application.- 22.1 Introduction.- 22.2 Background Material in the Stochastic Behavior of Petri Nets.- 22.3 Computer-Assisted Analysis of the Semi-Markovian Petri Nets.- 22.4 Application.- 22.5 Conclusions.- References.- 23 Incremental Approach for Building Stochastic Petri Nets for Dependability Modeling.- 23.1 Introduction.- 23.2 Presentaiton of the Incremental Approach.- 23.3 Guidelines for Modular Construction of GSPN Models.- 23.4 Example: Duplex System.- 23.5 Conclusions.- References.- 24 Lifetime of High Temperature Working Pipes.- 24.1 Introduction.- 24.2 Failure Risk.- 24.3 Defining Reliability.- 24.4 Mathematical Model for Lifetime Estimations.- 24.5 Simulating Reliability.- 24.6 Algorithm of Simulation.- 24.7 Simulating Reliability for Components.- 24.8 Simulating System Reliability.- References.