Exponential Stability of a Parallel Repairable System with Common-cause Failure and Critical Human Error

摘要

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. A common-cause failure is defined as any instance where multiple units fail due to a sin gle cause. In this paper, we first obtain the existence and uniqueness of the nonnegative time-dependent weak solution of the system by use of the C_0-semigroup theory of linear operator in Banach space. Then by the theory of nonnegative C_0-semigroup which is gen erated by A + E determined by the system, we discuss the spectrum distribution of the system operator. As a result of the stability of C_0-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 C_0-semigroup theory on Banach space, we obtain that 0 is the strictly dominant eigenvalue of the sys tem. 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.