Generalized Poisson Shock Models

Suppose that shocks hit a device in accordance with a nonhomogeneous Poisson process with intensity function λ(t). The iᵗʰ shock has a value Xᵢ attached to it. The Xᵢ are assumed to be independent and identically distributed positive random variables, and are also assumed independent of the counting process of shocks. Let D(x₁, …, xₙ, 0) ≡ D(x₁, …, xₙ, 0, 0, 0, …) denote the total damage when n shocks having values x₁, …, xₙ 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) ∫ᵗ₀ λ(s) ds/t is nondecreasing in t and (ii) D(x) = ∑ xᵢ. We extend this result to the case where D(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.