Life Behavior of a System under Discrete Shock Model

There are various engineering systems which are subjected to shocks of random magnitudes at random times. The shock models can be classified in different ways. According to the cumulative shock model, the system breaks down because of a cumulative effect of shocks, while in an extreme shock model the system fails because of one single shock with large magnitude. See, for example, 1–9 for various problems on shock models. Most of the studies on shock models focus on the evaluation of system failure time in a continuous setup, that is, the shocks arrive according to a renewal process, and the times between successive shocks have a continuous probability distribution. Some results on discrete case are in 3, 7, 10 . Consider a system which is subjected to periodic random shocks. A shock occurs with probability p in each period n 1, 2, . . .. The period should be understood as hour, day, and so forth. The magnitude of the shock which occurs in period j is a random variable denoted by Bj . Assume that such a system fails if and only if the sum of the magnitudes of cumulative shocks exceed, the level k for k > 0. Let Ij be a binary random variable representing the shock occurrences that is, Ij 1 if a shock occurs in period j and Ij 0, otherwise. For j ≥ 1, define