Stochastic Bounds and Dependence Properties of Survival Times in a Multicomponent Shock Model

Abstract

Consider a system that consists of several components. Shocks arrive according to a counting process (which may be non-homogeneous and with correlated interarrival times) and each shock may simultaneously destroy a subset of the components. Shock models of this type arise naturally in reliability modeling in dynamic environments. Due to correlated shock arrivals, individual component lifetimes are statistically dependent, which makes the explicit evaluation of the joint distribution intractable. To facilitate the development of easily computable tight bounds and good approximations, an analytic analysis of the dependence structure of the system is needed. The purpose of this paper is to provide a general framework for studying the correlation structure of shock models in the setup of a multivariate, correlated counting process and to systematically develop upper and lower bounds for its joint component lifetime distribution and survival functions. The thrust of the approach is the interplay between a newly developed notion, majorization with respect to weighted trees, and various stochastic dependence orders, especially orthant dependence orders of random vectors and orthant dependence orders of stochastic processes. It is shown that the dependence nature of the joint lifetime is inherited from spatial dependence and temporal dependence; that is, dependence among various components due to simultaneous arrivals and dependence over different time instants introduced by the shock arrival process. The two types of dependency are investigated separately and their joint impact on the performance of the system is analyzed. The results are used to develop computable bounds for the statistics of the joint component lifetimes, which are tighter than the productform bounds under certain conditions. The shock model with a non-homogeneous Poisson arrival process is studied as an illustrative example. The result is also applicable to the cumulative damage model with multivariate shock arrival processes.2000 Academic Press AMS 1991 subject classifications: 60E15; 60K10. Key words and phrases: shock models, correlated shock arrivals, Marshall Olkin distributions, majorization with respect to weighted trees, positive and negative orthant dependence order, orthant dependence in time, non-homogeneous Poisson arrival processes, bounds.

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