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 net systems; (2) at state equation level, which has allowed to obtain systems of linear inequalities, or linear programming problems. These relaxations are mainly related to the fractional firing of transitions, which implies the existence of non-integer markings. We 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 can also be seen, however, that not every populated net system can be continuized. In fact, there exist systems with "large" populations for which continuation does not make sense. The possibility of expressing nonlinear behaviors may lead to deterministic continuous differential systems with complex behaviors.

A class of synchronized queueing networks with deterministic routing is identified to be equivalent to a subclass of timed Petri nets called marked graphs. Some structural and behavioral properties of marked graphs are used to show interesting properties of this class of performance models. In particular, ergodicity is derived from the boundedness and liveness of the underlying Petri net representation. In the case of unbounded (i.e., nonstrongly connected) marked graphs, ergodicity is computed as a function of the average transition firing delays. For steady-state performance, linear programming problems defined on the incidence matrix of the underlying Petri nets are used to compute tight (i.e., attainable) bounds for the throughput of transitions for marked graphs with deterministic or stochastic time associated with transitions. These bounds depend on the initial marking and the mean values of the delays but not on the probability distortion functions. The benefits of interleaving qualitative and quantitative analysis of marked graph models are shown.<>