Asymptotic stability of a parallel repairable system with warm standby

This paper presents the asymptotic stability of a repairable system associated with two identical units parallel and one unit warm standby. It is shown that the system operator generates a positive C₀-semigroup in the state Banach space. The steady-state solution is shown to be the eigenvector of the system operator corresponding to the eigenvalue 0. By analysing the spectrum of the system, it is found that 0 is the unique spectral point of the system on the imaginary axis. As a result, the asymptotic stability of the system is obtained.