Optimal number of repairs before replacement for a two-unit system subject to non-homogeneous pure birth process

We consider a discrete replacement model for a two-unit system subject to failure rate interaction and shocks. Two types of shocks occur according to a non-homogeneous pure birth process and can affect the two-unit system. Type I shock causes unit A to fail and can be rectified by a general repair, while type II shock results in a non-repairable failure and must be fixed by a replacement. Two-unit systems also exhibit failure rate interactions between the units: each failure of unit A causes some damage to unit B, while each failure of unit B causes unit A into an instantaneous failure. The occurrence of a particular type of shock is dependent on the number of shocks occurred since the last replacement. The objective of this paper is to determine the optimal number of minor failures before replacement that minimizes the expected cost rate. A numerical example is presented to illustrate application of the model.