The asymptotical stability of a warm standby system with priority in use and repair

In this paper we consider 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, then by using the C₀-semigroup theory and analyzing the spectra distribution of the system operator, the unique existence of the solution is derived. Then the non-negative steady-state solution and the asymptotic stability of the system are derived.