Generalized Poisson Shock Models

Abstract

Suppose that shocks hit a device in accordance with a nonhomogeneous Poisson process with intensity function $\lambda(t)$. The $i^{th}$ shock has a value $X_i$ attached to it. The $X_i$ are assumed to be independent and identically distributed positive random variables, and are also assumed independent of the counting process of shocks. Let $D(x_1, \ldots, x_n, \underline{0}) \equiv D(x_1, \ldots, x_n, 0, 0, 0, \ldots)$ denote the total damage when $n$ shocks having values $x_1, \ldots, x_n$ have occurred. It has previously been shown that the first time that $D$ exceeds a critical threshold value is an increasing failure rate average random variable whenever (i) $\int^t_0 \lambda(s) ds/t$ is nondecreasing in $t$ and (ii) $D(\underline{x}) = \sum x_i$. We extend this result to the case where $D(\underline{x})$ is a symmetric, nondecreasing function. The extension is obtained by making use of a recent closure result for increasing failure rate average stochastic processes.