Product form for stochastic automata networks

We consider Stochastic Automata Networks (SAN) in continuous time and we prove a sufficient condition for the steady-state distribution to have product form. We consider SAN without synchronizations where the transitions of one automaton may depend of the states of the other automata. Even with this restriction, this sufficient condition is quite simple and this theorem generalizes former results on SAN but also on modulated Markovian queues, such as the Boucherie's theory on competing Markov chain, or on reversible queues considered by Kelly. The sufficient condition and the proof are purely algebraic.