Preservation of some partial orderings under Poisson shock models

Abstract

Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity λ. Let Pk denote the probability that the first device will survive the first k shocks and let denote such a probability for the second device. Let and denote the survival functions of the first and the second device respectively. In this note we show that some partial orderings, namely likelihood ratio ordering, failure rate ordering, stochastic ordering, variable ordering and mean residual-life ordering between the shock survival probabilities and are preserved by the corresponding survival functions and .

On preservation of some partial orderings under shock models

“…Let shocks occur according to a counting process such that P[N(t) = k] is TP 2 on R x NJ, where NJ = {O, 1, 2, ... }. Then the results (i), (ii) and (v) of Theorem 2.1 of Singh and Jain (1989) continue to hold.…”

“…Recently Singh and Jain (1989) have proved some interesting results on certain partial orderings of life distributions of two devices subjected to similar shocks occurring according to a homogeneous Poisson process. In this note it is shown that their results hold under more general shock models.…”

Failure rate properties of parallel systems

“…Partial ordering of lifetime distributions has been studied extensively by various authors (see for example Deshpande et al [18], Kochar and Wiens [23], Singh [41], Fagiuoli and Pellerey [21], and Shaked and Shanthikumar [40]) because of their applicability in a wide spectrum of different fields, such as econometrics (Whitmore [46]), reliability (Barlow and Proschan [8]), queues (Stoyan [43]), and other stochastic processes (Ross [37]). Singh and Jain [42] and Fagiuoli and Pellerey [21] have proposed an application to stochastic comparison between two devices that are subjected to Poisson shock models.…”

Partial orderings under cumulative damage shock models

“…Definitions of LR, FR and MR orders, for absolutely continuous and discrete non-negative random variables, can be found in Singh and Jain (1989). Definitions of random variables with PF z density and with the IFR or DMRL property can be found in Ross (1983) or Singh (1989).…”

“…Let also T 1 and T 2 be the lifetimes of the first and the second device, respectively. In Singh and Jain (1989) it is assumed that the underlying counting process is a homogeneous Poisson process, and it is shown that if U~V then T,~T 2 where (*) can be the likelihood ratio order (LR), the failure rate order (FR) or the mean residual life order (MR). In Kochar (1990) (see also the correction note on p. 1013 of this issue) it is shown…”