A geometrical process repair model for a repairable system with delayed repair

In this paper, a repairable system consisting of one component and a single repairman (i.e. a simple repairable system) with delayed repair is studied. Assume that the working time distribution, the repair time distribution and the delayed repair time distribution of the system are all exponential. After repair, the system is not “as good as new”. Under these assumptions, by using the geometrical process and the supplementary variable technique, we derive some important reliability indices such as the system availability, rate of occurrence of failures (ROCOF), reliability and mean time to first failure (MTTFF). A repair replacement policy NN under which the system is replaced when the number of failures of the system reaches NN is also studied. The explicit expression for the average cost rate (i.e. the long-run average cost per unit time) of the system is derived, and the corresponding optimal replacement policy N∗N∗ can be found analytically or numerically. Finally, a numerical example for policy NN is given.