Dynamic reliability models with conditional proportional hazards

A dynamic approach to the stochastic modelling of reliability systems is further explored. This modelling approach is particularly appropriate for load-sharing, software reliability, and multivariate failure-time models, where component failure characteristics are affected by their degree of use, amount of load, or extent of stresses experienced. This approach incorporates the intuitive notion that when a set of components in a coherent system fail at a certain time, there is a ‘jump’ from one structure function to another which governs the residual lifetimes of the remaining functioning components, and since the component lifetimes are intrinsically affected by the structure function which they constitute, then at such a failure time there should also be a jump in the stochastic structure of the lifetimes of the remaining components. For such dynamically-modelled systems, the stochastic characteristics of their jump times are studied. These properties of the jump times allow us to obtain the properties of the lifetime of the system. In particular, for a Markov dynamic model, specific expressions for the exact distribution functions of the jump times are obtained for a general coherent system, a parallel system, and a series-parallel system. We derive a new family of distribution functions which describes the distributions of the jump times for a dynamically-modelled system.

[1]  I. Olkin,et al.  A Multivariate Exponential Distribution , 1967 .

[2]  Zvi Schechner.  A load‐sharing model: The linear breakdown rule , 1984 .

[3]  Z. Jelinski,et al.  Software reliability Research , 1972, Statistical Computer Performance Evaluation.

[4]  Bernard D. Coleman,et al.  Time Dependence of Mechanical Breakdown in Bundles of Fibers. I. Constant Total Load , 1957 .

[5]  Ilkka Norros,et al.  Life Lengths and Association: A Dynamic Approach , 1984, Math. Oper. Res..

[6]  N. Keiding,et al.  Survival models and martingale dynamics: original text, discussion and reply , 1989 .

[7]  S. Phoenix.  The Asymptotic Time to Failure of a Mechanical System of Parallel Members , 1978 .

[8]  Elja Arjas,et al.  The Failure and Hazard Processes in Multivariate Reliability Systems , 1981, Math. Oper. Res..

[9]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[10]  P. Brémaud.  Point Processes and Queues , 1981 .

[11]  Tony Lancaster,et al.  The econometric analysis of transition data , 1990 .

[12]  S. Ross.  A Model in Which Component Failure Rates Depend on the Working Set. , 1984 .

[13]  Douglas R. Miller,et al.  On the use and the performance of software reliability growth models , 1991 .

[14]  B. Bergman.  On reliability theory and its applications , 1985 .

[15]  A. V. Peterson,et al.  On the regression analysis of multivariate failure time data , 1981 .

[16]  Eric Slud,et al.  Multivariate Dependent Renewal Processes. , 1984 .

[17]  Moshe Shaked,et al.  On the First Failure Time of Dependent Multicomponent Reliability Systems , 1988, Math. Oper. Res..

[18]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[19]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[20]  Bev Littlewood,et al.  Theories of Software Reliability: How Good Are They and How Can They Be Improved? , 1980, IEEE Transactions on Software Engineering.

[21]  R. L. Smith,et al.  LOWER TAIL ANALYSIS OF THE DISTRIBUTION OF THE STRENGTH OF LOAD-SHARING SYSTEMS , 1983 .

[22]  J. E. Freund.  A Bivariate Extension of the Exponential Distribution , 1961 .

[23]  J. Jacod.  Un théorème de représentation pour les martingales discontinues , 1976 .

[24]  Douglas R. Miller.  Exponential order statistic models of software reliability growth , 1986, IEEE Transactions on Software Engineering.

[25]  S. C. Saunders,et al.  A Statistical Model for Life-Length of Materials , 1958 .

[26]  P. Brémaud.  Point processes and queues, martingale dynamics , 1983 .

[27]  J. Jacod.  Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales , 1975 .

[28]  G. Box.  Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, I. Effect of Inequality of Variance in the One-Way Classification , 1954 .

[29]  Richard E. Barlow,et al.  Statistical Theory of Reliability and Life Testing: Probability Models , 1976 .

[30]  Bernard D. Coleman,et al.  Time Dependence of Mechanical Breakdown in Bundles of Fibers. II. The Infinite Ideal Bundle under Linearly Increasing Loads , 1957 .