Exponential stability of a repairable system with N failure modes and K standby units

This paper is devoted to studying the existence, uniqueness and exponential stability of repairable system´s time-dependent solution. By analysis the spectrum of the system operator generated by the repairable system, we derive that zero is the unique spectral point on the imaginary axis and the other spectral lie on the left half plane. The positive contraction strongly continuous semigroup which is generated by the operator corresponding to these equation is a quasi-compact operator. As a result, we get the desired result that the dynamic solution of the system exponentially tends to the steady solution.