We consider a device that is subject to three types of failures: repairable, non-repairable and failures due to wear-out. This last type is also non-repairable. The times when the system is operative or being repaired follow phase type distributions. When a repairable failure occurs, the operating time of the device decreases, in that the lifetimes between failures are stochastically decreasing according to a geometric process. Following a non-repairable failure or after a previously fixed number of repairs occurs, the device is replaced by a new one. Under these conditions, the functioning of the device can be modelled by a Markov process. We consider two different models depending on whether or not the phase of the operational system at the instants of failure is remembered or not. For both models we derive the stationary distribution of the Markov process, the availability of the device, the rate of occurrence of the different types of failures, and certain quantities of interest.

In this article, we assume that the state of a system forms a continuous-time Markov chain or a higher-dimensional Markov process after introducing some supplementary variables. A formula for evaluating the rate of occurrence of failures for the system is derived. As an application of the theory, a maintenance model for a two-component system is also studied.