A shock process with a non-cumulative damage

Two types of non-cumulative damage shock models are considered. Based on the distribution of damage, caused by a shock effecting a system, the intervals with small, intermediate and large damage are introduced. The initial homogeneous Poisson shock process is split into three homogeneous Poisson processes and studied independently. Several criteria of failure are considered, based on the assumption that shocks with a small level of damage are harmless for a system, shocks with a large level of damage results in the systemʹs failure and shocks with an intermediate level of damage can result in the systemʹs failure only with some probability. The second model is based on an assumption that shocks with a small level of damage are harmless to a system, if they are not too close to each other. The probability of the systemʹs failure-free performance in [0,t) is derived explicitly. Simple asymptotic exponential approximations are obtained The accuracy of this method is analyzed. Possible generalizations are discussed.