Markovian queueing network with complex synchronizations: Product form and tensor

We consider complex synchronizations in a generalized network of queues with signals (Gnetwork) or Stochastic Automata Networks without functions. Both models allow to describe their continuous time Markov chain as a summation of tensor (or Kronecker) products and sums of local description of queues (or automata). We give a purely algebraic proof of the product form results based on properties of the tensor products. These results generalize many well-known results in queueing theory but also on all the models which allow a tensor based representation such as Stochastic Petri Nets or Stochastic Process Algebra.