Analysis of damage induced by two types of shocks

In the literature of reliability theory, many studies have concentrated on single types of shock (e.g., aging). Previous studies have investigated the following cases. (1) When a shock of a given magnitude is applied to a unit, it immediately fails. (2) When the shocks are additive and the total shock magnitude is greater than K, a unit fails. (3) When the shocks are not additive, and the magnitude of a given shock is greater than K, the unit fails. In this paper, we suppose case (2). In addition, suppose two types of shocks are applied to a single unit (i.e., not a single shock type). Each type of shock occurs by a different stochastic process, and the respective magnitude of each type of shock shows a different probabilistic distribution. We perform damage analysis for a single unit when both types of event occur by general stochastic processes and both shock magnitudes follow general probabilistic distributions. Thus, we perform a detailed analysis while the events occur by Poisson processes and both shock magnitudes show different exponential distributions.