Approximations for steady-state measures in highly dependable systems with generally distributed component failure and repair times

In this paper we present an approach for approximating steady-state measures, i.e., steady-state mean time between failures and the steady-state unavailability, in models of repairable, highly dependable systems where the components have generally distributed failure times and repair times. Exact solutions for these measures, even for problems with a small number of components, is very difficult as it involves solving a set of integro-differential equations. The approach we follow is to use a ratio representation of these measures and the fact that there are relatively few paths to system failure that are probabilistically dominant. We apply the approach to models of computer systems that were previously evaluated using simulation, and we find that most of our approximations are within 5% of those estimates. As a consequence of this study one also finds that for a large class of such models the steady-state measures depend mainly on the first moment of the component failure time random variables. Intuitive justifications for the approximations are provided.