A bivariate replacement policy for an imperfect repair system based on geometric processes

A repair replacement model for a deteriorating system with delayed repair is studied, in which the successive working times after repair and the consecutive repair times of the system are described by geometric processes. The instantaneous availability is studied in the case of general distributions for the working time, repair time and delayed repair time. A bivariate replacement policy is considered, that is, the system is replaced whenever the working age of the system reaches T or at the first hitting time of the working time after repair with respect to the working time threshold tau, whichever occurs first. The explicit expression of the long-run average cost rate under the replacement policies is derived. The corresponding optimal replacement policy can be determined numerically, and numerical examples are presented to demonstrate the application of the developed model and approach. It is shown that the optimal solution and optimal value are sensitive to the tiny change in the ratios of the Geometric processes and the expectation of the delayed repair time.