The spectral analysis of k out of N: G redundant system

Abstract

In this paper, k out of N: G redundant system with repair facilities and multiple critical and non-critical errors is investigated. Firstly, we formulate the problem into a suitable Banach space. Then we carry out a detailed spectral analysis of the system operator. Based on the spectral analysis and semigroup theory, we prove the linear stability and the exponential stability of the system. As a consequence, we get that its dynamic solutions converge exponentially to the steady-state solution. Finally, we derive some reliability indices of the system.