A Geometric Process $\delta$ -Shock Maintenance Model

A geometric process delta -shock maintenance model for a repairable system is introduced. If there exists no shock, the successive operating time of the system after repair will form a geometric process. Assume that the shocks will arrive according to a Poisson process. When the interarrival time of two successive shocks is smaller than a specified threshold, the system fails, and the latter shock is called a deadly shock. The successive threshold values are monotone geometric. The system will fail at the end of its operating time, or the arrival of a deadly shock, whichever occurs first. The consecutive repair time after failure will constitute a geometric process. A replacement policy N is adopted by which the system will be replaced by a new, identical one at the time following the N th failure. Then, for the deteriorating system, and the improving system, an optimal policy N* for minimizing the long-run average cost per unit time is determined analytically.