A phase‐type geometric process repair model with spare device procurement and repairman’s multiple vacations

This paper analyzes a phase‐type geometric process repair model with spare device procurement lead time and repairman’s multiple vacations. The repairman may mean here the human beings who are used to repair the failed device. When the device functions smoothly, the repairman leaves the system for a vacation, the duration of which is an exponentially distributed random variable. In vacation period, the repairman can perform other secondary jobs to make some extra profits for the system. The lifetimes and the repair times of the device are governed by phase‐type distributions (PH distributions), and the condition of device following repair is not ‘as good as new’. After a prefixed number of repairs, the device is replaced by a new and identical one. The spare device for replacement is available only by an order and the procurement lead time for delivering the spare device also follows a PH distribution. Under these assumptions, the vector‐valued Markov process governing the system is constructed, and several important performance measures are studied in transient and stationary regimes. Furthermore, employing the standard results in renewal reward process, the explicit expression of the long‐run average profit rate for the system is derived. Meanwhile, the optimal maintenance policy is also numerically determined.