The Branching Compound Nonhomogeneous Poisson Process and its Application to a Replacement Model

In studying and analysing the failure patterns of complex system, plausible stochastic models are needed to represent the sequence of events. A simple and frequently model is derived by the assumption that the times-between-failures of a system are exponentially distributed and independent. Experience has shown, however, that successive times-between-failures are not exponentially distributed and not independent. These deviations are due to imperfect search of failed components. We constitute a plausible stochastic process which describes the sequence of events, and obtain the interval reliability and the expected number of failures. As an application of these results, we deal with the replacement model where a system undergoes minimal repair before time T and is replaced at time T. We discuss the optimum policy minimizing the total expected cost per unit time