Well-posedness and stability of the repairable system with N failure modes and one standby unit

Abstract

The well-posedness and stability of the repairable system with N failure modes and one standby unit were discussed by applying the c 0 semigroups theory of function analysis. Firstly, the integro-differential equations described the system were transformed into some abstract Cauchy problem of Banach space. Secondly, the system operator generates positive contractive c 0 semigroups T (t) and so the well-posedness of the system was obtained.Finally, the spectral distribution of the system operator was analyzed. It was proven that 0 is strictly dominant eigenvalue of the system operator and the dynamic solution of the system converges to the steady-state solution. The steady-state solution was shown to be the eigenvector of the system operator corresponding to the eigenvalue 0. At the same time the dynamic solution exponentially converges to the steady-state solution.

Convergence of positive operator semigroups

“…For several of these models one can prove convergence of the semigroup with respect to the operator norm as e.g. in [56,51,55]. On the other hand, there also exist models for which one can only show strong stability; in [27,26,28] such results are proved by employing spectral analysis of the generator and a version of the ABLV-theorem.…”

Reliability of a repairable system

“…In most of these articles, methods used in the existing literature dealing with non-Markov systems involving many general random variables include the regenerative point technique (RPT) [6] and the supplementary variables method (SVM) [8]- [11]. In order to use the RPT, one has to correctly formulate and solve a system of Markov renewal equations, usually using an analytical method which is difficult for a non-Markov repairable system with only a few renewal points.…”