Analysis of sparse numerical methods for dependability evaluation

Abstract

Homogeneous, continuous-time Markov chains are an usual mathematical tool for dependability evaluation of computer systems. Complex systems rise the large state space problem, thus making mandatory the use of efficient numerical methods. This paper analyzes state of the art sparse methods in the context of the development of an enhanced version of an existing tool. We first consider point SOR for the evaluation of the stationary probability and mean time before absorption vectors. By dynamic adjustment of the relaxation parameter a very high convergence rate is attained in many cases. The convergence rate is however rather poor when the Markov chain contains highly recurrent subsets. In order to improve the method we propose the integration of a novel, exact, one-step aggregation technique with block SOR. For the evaluation of the transient regime, two methods are compared: randomization and implicit integration. Scenarios in which one of them is clearly preferable are described.

Postprint (published version)