Suppose that shocks hit a device in accordance with a nonhomogeneous Poisson process with intensity function lambda(t). The ith shock causes a damage X sub i. The X sub 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 sub 1, ..., x sub n) denote the total damage when n shocks having damages x sub 1, ..., x sub n have occurred. It has previously been shown that the first time that D(X) exceeds a critical threshold value is an increasing, failure rate average random variable whenever lambda(t) = lambda and D(x) = sum over x sub i. This result is extended to the case where integral from 0 to t of (lambda(s)ds/t) is nondecreasing in t and 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.