Piecewise Affine Dynamical Models of Timed Petri Nets -- Application to Emergency Call Centers

We study timed Petri nets, with preselection and priority routing. Werepresent 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 netsand semi-Markov decision processes, from which we derive the convergence to aperiodic regime and the polynomial-time computability of the throughput. Moregenerally, we develop an approach inspired by tropical geometry, characterizingthe congestion phases as the cells of a polyhedral complex. We illustrate theseresults by a current application to the performance evaluation of emergencycall centers in the Paris area. We show that priorities can lead to aparadoxical behavior: in certain regimes, the throughput of the most prioritarytask may not be an increasing function of the resources.Comment: To appear in a special issue of Fundamenta Informatica