A path-based algorithm to evaluate asymptotic unavailability for large Markov models

Abstract

Markov chains are commonly used to study the dependability of complex systems. Nevertheless, the explosion of the number of states when the modeled system becomes too large is still a major problem. In such cases, reliability and availability cannot be calculated using conventional methods based on the construction of the state graph. One of the possible solutions to avoid this problem is to use only a local description of the system: the Markov chain is not actually constructed, but the knowledge of the rules which govern its evolution enable exploration of the state graph step by step. This idea already led to efficient algorithms for the computation of reliability. In this paper, we propose a method exploiting this path-based approach to evaluate the asymptotic unavailability of a system which is completely and quickly repairable. Then we show on a simple example that the more reliable the system, the better the approximation given by our method. Finally, we apply the presented algorithm to an electrical power system, much too large to enable the use of conventional methods.