A threshold representation for the strength distribution of a complex load sharing system

Abstract

Abstract In many reliability contexts, the distribution, P(T⩽t), for a combination of k components with strengths (or lifetimes) X =(X 1 ,…,X k ) , is a first passage probability of the form P(T⩽t)=P( X ∈tC) for t⩾0, where C is a cone containing the origin. For example, in a load sharing system, a load sharing rule describes how load is transferred from a failed component to the collection of unfailed components as the pattern of breaks evolves. The rule quantifies the component interactions leading to failure and, with the component breaking patterns, characterize certain conical regions in the nonnegative orthant of Rk which define the system strength. In this paper we show that, for independent component strengths (or lifetimes) having exponential distributions, T has a threshold representation, S/θ, where S and θ are independent random variables with S having a gamma distribution with unit scale and shape parameter depending on the structure of the cone. The consequent mixture representation has two noteworthy implications: First, failure rate relationships are established between the distributions of θ and T which are related to some recent work on mixtures of increasing failure rates resulting in a decreasing failure rate. Second, it makes tools such as the EM-algorithm and the gradient function applicable for data analytic purposes. A methodology for analyzing this model and based on these tools is proposed and illustrated on some data sets.