Age replacement policy for a two-unit system subject to non-homogeneous pure birth shocks

When a two-unit system is subjected to shocks, it may fail due to interactions between the units. Two types of shocks stemming from a non-homogeneous pure birth process can affect a two-unit system. A Type I shock causes a minor failure of Unit A which affects Unit B and can be rectified with minimal repair. The Type II shock causes total (catastrophic) system failure, at which point an unplanned maintenance is required to replace the failed system. Two-unit systems also exhibit failure rate interactions between units: failure of Unit A causes an internal shock that increases the failure rate of Unit B, while failure of Unit B causes instantaneous failure of Unit A. The occurrence of a particular type of shock is based on a random mechanism dependent on the number of shocks that have occurred since the last replacement. The goal of this study is to derive the expected replacement cost rate by introducing relative costs as a factor, and determining the optimal replacement period T∗T∗ which minimizes cost. A numerical example is presented to illustrate application of the method.