An algebraic condition for product form in stochastic automata networks without synchronizations

We consider Stochastic Automata Networks (SANs) in continuous time and we prove a sufficient condition for the steady-state distribution to have product form. We consider synchronization-free SANs in which the transitions of one automaton may depend upon the states of the other automata. This model can represent efficiently multidimensional Markov chains whose transitions are limited to one component but whose rates may depend on the state of the chain.The sufficient condition we obtain is quite simple and our theorem generalizes former results on SANs as well as results on modulated Markovian queues, such as Boucherie’s theory on competing Markov chain, on reversible queues considered by Kelly and on modulated Jackson queueing networks studied by Zhu. The sufficient condition and the proof are purely algebraic and are based on the intersection of kernels for a certain set of matrices.