Optimal order-replacement policy for a phase-type geometric process model with extreme shocks

A system is subject to random shocks that arrive according to a phase-type (PH) renewal process. As soon as an individual shock exceeds some given level the system will break down. The failed system can be repaired immediately. With the increasing number of repairs, the maximum shock level that the system can withstand will be decreasing, while the consecutive repair times after failure will become longer and longer. Undergoing a specified number of repairs, the existing system will be replaced by a new and identical one. The spare system for the replacement is available only by sending a purchase order to a supplier, and the duration of spare system procurement lead time also follows a PH distribution. Based on the number of system failures, a new order-replacement policy (also called $(K\text{,}N)$ policy) is proposed in this paper. Using the closure property of the PH distribution, the long-run average cost rate for the system is given by the renewal reward theorem. Finally, through numerical calculation, it is determined an optimal order-replacement policy such that the long-run expected cost rate is minimum.