A generalized age and block replacement of a system subject to shocks

A system is subject to shocks that arrive according to a nonhomogeneous Poisson process. As shocks occur a system has two types of failures. Type I failure (minor failure) is removed by a minimal repair, whereas type II failure (catastrophic failure) is removed by an unplanned (or unscheduled) replacement. The choice of these two actions is based on some random mechanism which depends on the number of shocks suffered since the last replacement. For an age replacement maintenance policy, planned (or scheduled) replacements occur whenever an operating system reaches age T, whereas in the block replacement case, planned replacements occur every T units of time. The aim of this paper is to derive the expressions for the expected long-run cost per unit time and the total α-discounted cost for each policy. The optimal T* which would minimize the cost rate or the total α-discounted cost is discussed. Various special cases are detailed. A numerical example is given to illustrate the method.