Some indices of the steady-state and an optimization problem of a warm standby system with priority

In this paper we consider an optimization problem for a warm standby repairable system that consists of two dissimilar components and one repairman. Suppose that the life of each component satisfies the exponentially distribution and the repair time of the component satisfies the general distribution, the component 1 has priority in use and repair. Firstly, a mathematical model is built via the differential and partial differential equations, and some basic results such as the existence and uniqueness of the solution, the non-negative steady-state solution, the exponential stability of the system and some indices of the steady-state are derived. Based on the stability result, we discuss an optimization problem about the normal availability of the system.