Simple bounds for terminating Poisson and renewal shock processes

A system subject to a point process of shocks is considered. The shocks occur in accordance with a renewal process or a nonhomogeneous Poisson process. Each shock independently of the previous history leads to a system failure with probability θ and is survived with a complimentary probability $\text{θ}\text{̄}$. A number of problems in reliability and safety analysis can be interpreted by means of this model. The exact solution for the probability of survival $\text{W}\text{̄}\text{(t,θ)}$ can be obtained only in the form of infinite series (renewal process of shocks). Approximate solutions and new simple bounds for the probability of survival are obtained. The introduced method is based on the notion of a stochastic hazard rate process. A supplementary characteristic in this analysis is the mean of the hazard rate process. This method makes it possible to consider a generalization important in practical applications when the probability of a system failure under the effect of a current shock depends on the time since the previous one.