of Petri nets with priorities in the discrete setting, and applied it to the performance analysis of an emergency call center. The discrete dynamics is shown there to be given by piecewise affine equations (tropical analogues of rational equations). The idea of modeling priority rules by piecewise affine dynamics originated from Farhi, Goursat and Quadrat [@farhi], who applied it to a special class of road traffic models. In the discrete setting, limit time-periodic behaviors can occur. They may lead to asymptotic throughputs different from the affine stationary solutions of the dynamics, a pathology which motivates our study of a continuous version of the dynamics.

The “continuization” of our dynamics draws inspiration from the original continuous model where time is attached to transitions. In particular, the situation in which the routing of a token at a given place is influenced by the firing times of the output transitions through a race policy has received much attention, see [@vazquez2013introduction]. Here, we address the situation in which the routing is specified by priority or preselection rules which are independent of the processing rates. To do so, it is convenient to attach times to places, instead of attaching firing rates to transitions. We point 