Optimal number of minimal repairs with cumulative repair cost limit for a two-unit system with failure rate interactions

A discrete replacement model is presented that includes a cumulative repair cost limit for a two-unit system with failure rate interactions between the units. We assume a failure in unit 1 causes the failure rate in unit 2 to increase, whereas a failure in unit 2 causes a failure in unit 1, resulting in a total system failure. If unit 1 fails and the cumulative repair cost till to this failure is less than a limit L, then unit 1 is repaired. If there is a failure in unit 1 and the cumulative repair cost exceeds L or the number of failures equals n, the entire system is preventively replaced. The system is also replaced at a total failure, and such replacement cost is higher than the preventive replacement cost. The long-term expected cost per unit time is derived using the expected costs as the optimality criterion. The minimum-cost policy is derived, and existence and uniqueness are proved.