  We study timed Petri nets, with preselection and priority routing. We
represent the behavior of these systems by piecewise affine dynamical systems.
We use tools from the theory of nonexpansive mappings to analyze these systems.
We establishan equivalence theorem between priority-free fluid timed Petri nets
and semi-Markov decision processes, from which we derive the convergence to a
periodic regime and the polynomial-time computability of the throughput. More
generally, we develop an approach inspired by tropical geometry, characterizing
the congestion phases as the cells of a polyhedral complex. We illustrate these
results by a current application to the performance evaluation of emergency
call centers in the Paris area. We show that priorities can lead to a
paradoxical behavior: in certain regimes, the throughput of the most prioritary
task may not be an increasing function of the resources.
