Optimal Number of Repairs Before Replacement for a System Subject to Shocks of a Non-Homogeneous Pure Birth Process

We consider a system subject to shocks that arrive according to a nonhomogeneous pure birth process (NHPBP). As a shock occurs, the system has two types of failures. Type-I failure (minor failure) is rectified by a general repair, whereas type-II failure (catastrophic failure) is removed by an unplanned replacement. The probabilities of these two types of failures depend on the number of shocks since the last replacement. We consider a policy with which the system is replaced at the n th type-I failure, or at any type-II failure. The aim of this paper is to determine the optimal policy n*, the number of minor failures up to replacement that minimizes the expected cost rate of the system subject to NHPBP shocks. The model is a generalization of the existing models, and is more applicable in practice. We present some numerical examples, and show that several classical models are the special cases of our model.