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

We consider preventive maintenance (age replacement) of 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, whichever comes first. Each shock in our stochastic model has a double effect. On one hand, it acts directly on the failure rate of an item, which results in the corresponding stochastic failure rate process. On the other hand, each shock causes additional ‘damage’, which can be attributed, e.g., to a short drop in the output of an item. The corresponding optimization problem is considered and illustrated by detailed numerical examples.