continuous models can be found in [@vazquez2013introduction], while a more extensive reference is [@david2010discrete].

In order to evaluate the long-term performance of Petri nets, one has to characterize the stationary or steady states of the Petri nets dynamics. Cohen, Gaubert and Quadrat [@cohen1995asymptotic] introduced an approximation of a discrete Petri net by a fluid, piecewise affine dynamics with finite delays, and showed that the limit throughput does exist for a class of consistent and free choice Petri nets. In the more recent work of Gaujal and Giua [@gaujal2004optimal], the result is extended to larger classes of Petri nets, and the stationary throughputs are computed as the solutions of a linear program. The results obtained using this fluid approximation hardly apply to the discrete model, up to a remarkable exception identified by Bouillard, Gaujal and Mairesse [@bouillard2006extremal] (bounded Petri nets under total allocation). This reference illustrates the many difficulties that arise from the discrete setting (*e.g.*, some firing sequences may lead to a deadlock).

In the continuous dynamics setting, with time attached to transitions, Recalde and Silva [@recalde2000pn] showed that the steady states of free choice Petri nets as well as upper bounds of the throughputs in larger classes of Petri 