A geometric process repair model for a two-component system with shock damage interaction

In this article, the repair-replacement problem for a two-component system with shock damage interaction and one repairman is studied. Assume that component 1 will be replaced as soon as it fails, and each failure of component 1 will induce a random shock to component 2. The shock damages may be accumulative, and whenever the total shock damage equals or exceeds a given threshold Delta, component 2 fails and the system breaks down. Component 2 is repairable, and it follows a geometric process repair. Under these assumptions, we consider a replacement policy N based on the number of failures of component 2. Our problem is to determine an optimal replacement policy N* such that the average cost rate (i.e. the long-run average cost per unit time) is minimised. The explicit expression of the average cost rate is derived by the renewal reward theorem, and the optimal replacement policy can be determined analytically or numerically. The existence and uniqueness of the optimal replacement policy N* is also proved under some mild conditions. Finally, two appropriate numerical examples are provided to show the effectiveness and applicability of the theoretic results in this article.