Exponential Stability of a Repairable System with N Failure Modes and One Standby Unit

The existence of a unique nonnegative time-dependent solution to the repairable system with N failure modes and one standby unit is obtained by using C
₀
-semigroup theory. The spectral properties and the essential spectrum bound of the stand-by system are discussed by using perturbation method. In the end, we get the desired result that the dynamic solution of the system exponentially tends to the steady solution.