Parallel Implementation of Uniformization to Compute the Transient Solution of Stochastic Automata Networks

Abstract

Analysis of Stochastic Automata Networks (SAN) is a well established approach for modeling the behaviour of computing networks and systems, particularly parallel systems. The transient study of performance measures leads us to time and space complexity problems as well as error control of the numerical results. The SAN theory presents some advantages such as avoiding to build the entire infinitesimal generator and facing the time complexity problem thanks to the tensor algebra properties. The aim of this study is the computation of the transient state probability vector of SAN models. We first select and modify the (stable) uniformization method in order to compute that vector in a sequential way. We also propose a new efficient algorithm to compute a product of a vector by a tensor sum of matrices. Then, we study the contribution of parallelism in front of the increasing execution time for stiff models by developing a parallel algorithm of the uniformization. The latter algorithm is efficient and allows to process, within a fair computing time, systems with more than one million states and large mission time values.