Petri nets and integrality relaxations: A view of continuous petri net models

Petri nets are formalisms for the modeling of discrete event dynamic systems (DEDS). The integrality of the marking and of the transitions firing counters is a clear reflection of this. To reduce the computational complexity of the analysis or synthesis of Petri nets, two relaxations have been introduced at two different levels: (1) at net level, leading to continuous [1]-[3] net systems; (2) at state equation (or fundamental equation) level, which has allowed to obtain systems of linear inequalities, or linear programming problems [4]-[6]. These relaxations are mainly related to the fractional firing of transitions, which implies the existence of non-integer markings. For instance, a first-order continuization transformsa discrete and stochastic model, into a continuous and deterministic approximated model. The purpose of our work is to give an overview of this emerging field. It is focused on the relationship between the properties of (discrete) PNs and the corresponding properties of their continuous approximation. Through the interleaving of qualitative and quantitative techniques, surprising results can be obtained from the analysis of these continuous systems. For these approximations to be acceptable, it is necessary that large markings (populations) exist. It will also be seen, however, that not every populated net system can be continuized. In fact, there exist systems with large populations for which continuization does not make sense. The possibility of expressing non-linear behaviors may lead to deterministic continuous differential systems with complex behaviors (orbits, limit cycles, different attractors, and chaos).