## The Annals of Probability

### Shock Models and Wear Processes

#### Abstract

The life distribution $H(t)$ of a device subject to shocks governed by a Poisson process is considered as a function of the probabilities $P_k$ of not surviving the first $k$ shocks. Various properties of the discrete failure distribution $P_k$ are shown to be reflected in corresponding properties of the continuous life distribution $H(t)$. As an example, if $P_k$ has discrete increasing hazard rate, then $H(t)$ has continuous increasing hazard rate. Properties of $P_k$ are obtained from various physically motivated models, including that in which damage resulting from shocks accumulates until exceedance of a threshold results in failure. We extend our results to continuous wear processes. Applications of interest in renewal theory are obtained. Total positivity theory is used in deriving many of the results.

#### Article information

Source
Ann. Probab., Volume 1, Number 4 (1973), 627-649.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996891

Digital Object Identifier
doi:10.1214/aop/1176996891

Mathematical Reviews number (MathSciNet)
MR350893

Zentralblatt MATH identifier
0262.60067

JSTOR