On the tra c equations for batch routing queueing networks and stochastic Petri nets

The tra c equations are a set of linear equations, which are the basis for the exact analysis of product form queueing networks, and the approximate analysis of non-product form queueing networks. Conditions characterising the structure of the network that guarantees the existence of a solution for the tra c equations are therefore of great importance. This note provides a necessary and su cient condition on the structure of the network for a solution of the tra c equations to exist. The basis of this structural characterisation is the equivalence between batch routing queueing networks and stochastic Petri nets at the level of the underlying stochastic process. Based on new and known results for stochastic Petri nets, this note shows that the new condition stating that each transition is covered by a minimal closed support T -invariant is necessary and su cient for the existence of a solution for the tra c equations for batch routing queueing networks and stochastic Petri nets. 1991 AMS Subject Classi cation: Primary: 60K25, Secondary: 90B22