Exponential Stability Of A Parallel Repairable System With Common-Cause Failure And Critical Human Error

This paper presents the exponential stability of a system with n identical units parallel repairable system subject to common-cause failure and critical human error. Some of the common-cause failures may occur due to:equipment design deficiency, operations and main-tenance error, external normal environment, functional errors, etc. A critical human error could also cause the failure of the entire system.We first obtain the existence and uniqueness of the nonnegative strong solution of the system by use of the Volterra operator equation in a space and elementary techniques. Then by the theory of nonnegative Co-semigroup which is generated by A+E determined by the system, we analysis the spectrum distraction of the system operator. As a result of the stability of Co-semigroup theory of linear operators, we obtain the asymptotic stability of the solution of the repairable system. Finally, by using the method of functional analysis, especially, the linear operator theory and Co-semigroup theory on Banach space, we obtain that 0 is the strictly dominant eigenvalue of the system. By studying the essential spectral spectrum of the system operator before and after perturbation, we show that the dynamic solution of the system converges exponentially to its steady-state solution.