Exponential asymptotic stability of repairable system with randomly selected repairman

In this paper, the repairable system solution´s exponential asymptotic stability was discussed. First we prove that the positive contraction strongly continuous semigroup which is generated by the operator corresponding to these equations describing a system with randomly selected repairman is a quasi-compact operator. Following the result that 0 is an eigenvalue of the operator with algebraic index one and the strongly continuous semigroup is contraction, we deduce that the spectral bound of the operator is zero. By the above results we obtain easily the exponential asymptotic stability of the solution of the repairable system.