nets can be determined by linear programming. However, in general, the asymptotic throughputs are non-monotone with respect to the initial marking or the firing rates of the transitions [@mahulea2006performance]. An example of oscillations in infinite time around a steady state is also given in [@mahulea2008steady].

Contributions {#contributions .unnumbered}
-------------

We propose a continuous dynamics of Petri nets where time is attached to places and not to transitions. The main novelty is that it handles a class of Petri nets in which tokens can be routed according to priority rules (Section \[sec:semantics\]). We initially studied this class in [@allamigeon2015performance] in the discrete setting, motivated by an application to the performance analysis of an emergency call center.

We show that the continuous dynamics can equivalently be expressed in terms of [*policies*]{}. A policy is a map associating with every transition one of its upstream places. In this way, the dynamics of the Petri net can be written as an infimum of the dynamics of subnets induced by the different policies. The policies reaching the infimum indicate the places which are bottleneck in the Petri net. On any time interval in which a fixed policy reaches the infimum, the dynamics reduces to a 