Preservation of multivariate stochastic orders under multivariate Poisson shock models

Abstract

Consider two systems, labeled system 1 and system 2, each with m components. Suppose component i in system k, k = 1, 2, is subjected to a sequence of shocks occurring randomly in time according to a non-explosive counting process {Γ i ( t ), t > 0}, i = 1, ···, m. Assume that Γ 1 , · ··, Γ m are independent of M k = ( M k, 1 , · ··, M k,m ), the number of shocks each component in system k can sustain without failure. Let Z k,i be the lifetime of component i in system k. We find conditions on processes Γ 1 , · ··, T m such that some stochastic orders between M 1 and M 2 are transformed into some stochastic orders between Z 1 and Z 2 . Most results are obtained under the assumption that Γ 1 , · ··, Γ m are independent Poisson processes, but some generalizations are possible and can be seen from the proofs of theorems.