Statistical and Probabilistic Models in Reliability

This volume consists of twenty-four papers selected by the editors from the sixty-one papers presented at the 1st International Conference on Mathemati­ cal Methods in Reliability held at the Politehnica University of Bucharest from 16 to 19 September 1997. The papers have been divided into three sections: statistical methods, probabilistic methods, and special techniques and appli­ cations. Of course, as with any classification, some papers could be as well assigned to other sections. Problems in reliability are encountered in items in everyday usage. Relia­ bility is an important feature of household appliances, cars, telephones, power supplies, and so on, whether viewed from the vantage of the producer or the consumer. Important decisions are based on the reliability of the product. Obtaining systems that perform adequately for a specified period of time in a given environment is an important goal for both government and industry. Hence study and use of reliability theory, which can be applied in the research, development, and production phases of a system to enable the user to evaluate and improve performance, is a worthwhile venture. If reliability theory is to be useful, it must be quantitative in nature, because reliability must be demonstra­ ble. Subsequently probability and statistics, among others, play an important part in its development.

I: Statistical Methods.- 1 Statistical Modeling and Analysis of Repairable Systems.- 1.1 Introduction.- 1.2 “Major Events” in the History of Repairable Systems Reliability.- 1.3 Notation and Basic Definitions.- 1.4 Classification of Repair Actions.- 1.5 The Trend-Renewal Process.- 1.6 Statistical Inference in Trend-Renewal Processes.- 1.7 Trend Testing.- 1.8 Monte Carlo Trend Tests.- 1.9 Concluding Remarks and Topics for Further Study.- References.- 2 CPIT Goodness-of-Fit Tests for Reliability Growth Models.- 2.1 Introduction.- 2.2 The Conditional Probabilty Integral Transformation.- 2.3 CPIT GOF Tests for the Homogeneous Poisson Process.- 2.4 CPIT GOF Tests for the Jelinski-Moranda and Goel-Okumoto Models.- 2.5 CPIT GOF Tests for the Power-Law Process.- 2.6 Experimental Results.- 2.7 Conclusion.- References.- 3 On the Use of Minimally Informative Copulae in Competing Risk Problems.- 3.1 Competing Risk.- 3.2 Bounds Without Assumptions on a Dependence Structure.- 3.2.1 Peterson bounds.- 3.2.2 Crowder-Bedford-Meilijson bounds.- 3.3 Estimators Using Dependence Assumptions.- 3.3.1 The copula-graphic estimator.- 3.4 Minimallly Informative Copulae.- 3.5 Examples.- 3.5.1 Example 1.- 3.5.2 Example 2.- 3.6 Conclusions.- References.- 4 Model Building in Accelerated Experiments.- 4.1 Introduction.- 4.2 Additive Accumulation of Damages Model and Its Submodels.- 4.3 Generalized Multiplicative Models.- 4.4 Generalized Additive and Additive-Multiplicative Models.- 4.5 Models Describing the Influence of Stresses to the Shape and Scale of Distribution.- 4.6 The Model of Sedyakin and Its Generalizations.- 4.7 The Heredity Hypothesis.- References.- 5 On Semiparametric Estimation of Reliability From Accelerated Life Data.- 5.1 Introduction.- 5.2 Estimation in the AAD Model.- 5.3 Properties of Estimators.- 5.4 Estimation, When Stresses Change the Shape of Distribution.- 5.5 Estimation in AFT Model, When G is Completely Unknown and r is Parametrized.- References.- 6 Analysis of 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.