A bivariate optimal repair–replacement model using geometric processes for a cold standby repairable system

This article studies a cold standby repairable system consisting of two identical components and one repairman. It is assumed that each component after repair in the system is not ‘as good as new’ and the deterioration of the system is stochastic. Under these assumptions, by using a geometric process a bivariate replacement policy (T,N) based on the working age T and the failure number N of component 1 is considered. The problem is to choose an optimal replacement policy (T,N)* such that the long-run average loss per unit time of the system is minimized. The explicit expression for the long-run average loss per unit time of the system is evaluated, and the corresponding optimal replacement policy (T,N)* can be found analytically or numerically. Finally, under some mild conditions, it is proved that the optimal policy (T,N)* is better than the optimal policy N* for a cold standby repairable system.