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

• Published in: Journal of statistical planning and inference 2000, Vol.83 (1), p.25-46
• Main Authors: Durham, S.D., Lynch, J.D.
• Format: Article
• Language: English
• Subjects: Applications; Distribution theory; Exact sciences and technology; Load sharing; Load transfer; Mathematics; Mixed distributions; Multivariate analysis; Passage probabilities; Probability and statistics; Probability theory and stochastic processes; Reliability, life testing, quality control; Sciences and techniques of general use; Statistics; System reliability; System strength
• ISSN: 0378-3758; 1873-1171
• DOI: 10.1016/S0378-3758(99)00094-4

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 R k 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.