Accelerating mean time to failure computations

In this paper we consider the problem of numerical computation of the mean time to failure (MTTF) in Markovian dependability and/or performance models. The problem can be cast as a system of linear equations which is solved using an iterative method preserving sparsity of the Markov chain matrix. For highly dependable systems, system failure is a rare event and the above system solution can take an extremely large number of iterations. We propose to solve the problem by dividing the computation in two parts. First, by making some of the high probability states absorbing, we compute the MTTF of the modified Markov chain. In a subsequent step, by solving another system of linear equations, we are able to compute the MTTF of the original model. We prove that for a class of highly dependable systems, the resulting method can speed up computation of the MTTF by orders of magnitude. Experimental results supporting this claim are presented. We also obtain bounds on the convergence rate for computing the mean entrance time of a rare set of states in a class of queueing models.