Correspondence:
[†]
Address for correspondence: INRIA and CMAP, CNRS, École polytechnique, IP Paris, France.
Note: [*] The second author is supported by a joint PhD grant of DGA and INRIA. The authors have been partially supported by the “Investissement d’avenir”, référence ANR-11-LABX-0056-LMH, LabEx LMH and by a project of “Centre des Hautes Études du Ministère de l’Intérieur” (CHEMI).
Abstract: 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 establish an 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.
Keywords: Timed Petri net, Performance evaluation, Markov decision process, Tropical geometry, Emergency call center
