A Proposed Measure of Residual Life of Live Components of a Coherent System

The concept of the signature of a coherent system is useful to study the stochastic and aging properties of the system. Let X_1m, X_2:n, ⋯ X_n:n denote the ordered lifetimes of the components of a coherent system consisting of n i.i.d components. If T denotes the lifetime of the system, then the signature vector of the system is defined to be a probability vectors = (s₁, s₂, ⋯, s_n) such that s_i = P(T = X_i:n), i = 1, 2, ⋯,n. Here we consider a coherent system with sig- nature of the form s = (s₁, s₂, ⋯ S_i, 0 ... , 0), where s_k >; 0, k = 1,2, ⋯, i. Under the condition that the system is working at time t, we propose a time dependent measure to calculate the probability of residual life of live components of the system, i.e., X_k:n, k = i + 1 ⋯, n. Several stochastic and aging properties of the proposed measure are explored.