Unbounded product-form Petri nets

Computing steady-state distributions in infinite-state stochastic systems is in general a very dificult task. Product-form Petri nets are those Petri nets for which the steady-state distribution can be described as a natural product corresponding, up to a normalising constant, to an exponentiation of the markings. However, even though some classes of nets are known to have a product-form distribution, computing the normalising constant can be hard. The class of (closed) {\Pi}3-nets has been proposed in an earlier work, for which it is shown that one can compute the steady-state distribution efficiently. However these nets are bounded. In this paper, we generalise queuing Markovian networks and closed {\Pi}3-nets to obtain the class of open {\Pi}3-nets, that generate infinite-state systems. We show interesting properties of these nets: (1) we prove that liveness can be decided in polynomial time, and that reachability in live {\Pi}3-nets can be decided in polynomial time; (2) we show that we can decide ergodicity of such nets in polynomial time as well; (3) we provide a pseudo-polynomial time algorithm to compute the normalising constant.