Dynamic Analysis of a System with Warm Standby and Common-Cause Failure

In this paper we analyze the dynamic behavior of a two unit parallel system with warm standby and common-cause failure. By the semigroup theory of linear operators on the Banach space, we give the wellposedness of the system and then prove the existence of the nonnegative dynamic solution and the steady solution of system. By spectral analysis of the system operator, we show that all the spectrum points of system operator besides 0 are in the left half-plane, hence we obtain the asymptotic stability of the system. Further we prove that 0 is a dominant eigenvalue of the system. Especially we discuss the essential spectral bound of the system operator and the radius of the essential spectrum of the semigroup associated with the system. Those results show that the dynamic solution of the system converges exponentially to the steady solution. Finally, we analyze the some reliability indic of the system. Key–Words: a two unit parallel system; steady solution; Asymptotic stability; Exponential stability; Reliability