Bivariate preventive maintenance of systems with lifetimes dependent on a random shock process

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 either on failure or on the predetermined replacement time or on a shock with the predetermined number, whichever comes first. 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, secondly, each shock causes additional 'damage', which can be attributed, e.g., to a short drop in the output of a system. The corresponding bivariate optimization problem is considered and illustrated by detailed numerical examples.