Stationary distribution for repairable systems (Q2744949)

This paper is a development of the previous results by \textit{V. S. Korolyuk} [see, for instance, ``Maintenance problem for repairable Markov systems'', in: Statistical and probabilistic models in reliability, ed. by C. Ionescu and N. Limnios (Boston, 1999)]. The Markov repairable system with limited service and spare devices is considered. The main goal of this paper is to describe the stationary distribution of this system. For this purpose the stationary distribution is transformed to the Gibbs measure and properties of the Gibbs potential are studied. Then the authors prove that the limit distribution is concentrated on the equilibrium set (set of the global minima for the limit Gibbs potential). As an application the reliability of \(k\)-out-of-\(n\) system is studied.