On the point process with finite memory and its application to optimal age replacement

There has been extensive study of various repair models in the literature, mostly under the assumption that these repairs are minimal or imperfect/better than minimal. Although this is often a realistic assumption, it may not be sufficient to model instances where the repair is worse than minimal. The generalized Polya process (GPP) that has been used to describe this type of repair takes into account all previous events/repairs, which is not often the case in practice. Therefore, in this paper, we define a new process with finite memory that starts as the GPP but, after a certain number of events or elapsed time, becomes the non-homogeneous Poisson process of repairs (minimal repairs). The corresponding age replacement policy is defined and the optimal solutions that minimize the long-run expected cost rate are analyzed. The detailed numerical examples illustrate our findings.