Product form approximations for communicating Markov processes

Product form solutions have been found for several classes of stochastic models including some networks of stochastic automata or communicating Markov chains. In this paper a theory of approximate and higher order product forms is presented. The idea is to define an approximate product form solution as a Kronecker product of vectors that minimizes the Euclidean norm of the residual vector for arbitrary networks of communicating Markov processes. If the residual becomes zero, the product form becomes exact. By adopting ideas from numerical analysis to approximate a matrix by a sum of Kronecker products of small matrices, higher order product forms that result in better approximations are defined.This paper presents the general theory of product form approximations for communicating Markov processes and it introduces first algorithms to compute product form solutions. By means of some examples it is shown that the approach allows one to compute approximations with increasing accuracy by increasing the order of the product form.