Bivariate preventive maintenance for repairable systems subject to random shocks

We consider a bivariate model for preventive maintenance for items operating in a random environment modeled by a Poisson process of shocks. An item is replaced on the predetermined replacement time or on a shock with the predetermined number, whichever comes first. Its failures are minimally repaired in-between. Each shock in our stochastic model has a double effect. First, it acts directly on the failure rate of an item, which results in the corresponding stochastic intensity process. Second, each shock causes additional damage, which can be attributed, for example, to a short drop in the output of a system or other adverse consequences. The corresponding bivariate optimization problem is considered and illustrated by detailed numerical examples.