The Asymptotic Stability And Reliability Of Repairable Systems With Some Character

The asymptotic behavior of cold standby repairable system with multiple non-critical and critical errors, While the switching mechanism of the system subjected to failure ,is presented. A failed system will be repair times arbitrarily distributed. Firstly, the system is transformed into Volterra operator equation in a Banach space, and the proof of existence and uniqueness of nonnegative solution of the system is gotten. Secondly,by the positive Co-semigroup which is generated by the A + E determined by the system, We show that there exists a nonnegative solution of the standby system which is just the normalized eigenvector of operator A + E corresponding to eigenvalue 0. By spectral analysis of A + E, We prove that all spectrum of A + E lie in the left half plane and there is no spectrum on the imaginary axis except 0. As a result of the stability of semigroup theory of linear operators,we get the stability of this system.