Abstract
A fundamental issue in the analysis of emergency call centers is to estimate the time needed to return to a congestion-free regime after an unusual event with a massive arrival of calls. Call centers can generally be represented by timed Petri nets with a hierarchical structure, in which several layers describe the successive steps of treatments of calls. We study a continuous approximation of the Petri net dynamics (with infinitesimal tokens). Then, we show that a counter function, measuring the deviation to the stationary regime, coincides with the value function of a semi-Markov decision problem. We establish a finite time convergence result, exploiting the hierarchical structure of the Petri net. We obtain an explicit bound for the transience time, as a function of the initial marking and sojourn times. This is based on methods from the theory of stochastic shortest paths and non-linear Perron–Frobenius theory. We illustrate the bound on a case study of a medical emergency call center.
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Acknowledgements
The second author was partially supported by the PhD fellowship program of DGA. We also acknowledge the support of IHEMI, and of the IROE-PRMO program of Fondation Mathématique Jacques Hadamard, sponsored by EDF. The modelling of the medical emergency call center described in Sect. 2 was carried out as part of a collaboration with the SAMU of AP-HP. We thank especially, Pr. P. Carli, Dr. E. Chanzy, Dr. E. Lecarpentier, Dr. Ch. Leroy, Dr. Th. Loeb, Dr. J.-S. Marx, Dr. N. Poirot and Dr. C. Telion for making this work possible, for their support and for insightful comments. We also thank all the other personals of the SAMU, in particular Dr. J. Boutet, J.-M. Gourgues, I. Lhomme, F. Linval and Th. Pérennou. This work also strongly benefited from the experience acquired, since 2014, on the analysis of the new platform “PFAU” (answering to the emergency numbers 17, 18 and 112), developed by PP. We thank especially Lcl S. Raclot and R. Reboul.
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Allamigeon, X., Boyet, M., Gaubert, S. (2022). Computing Transience Bounds of Emergency Call Centers: A Hierarchical Timed Petri Net Approach. In: Bernardinello, L., Petrucci, L. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2022. Lecture Notes in Computer Science, vol 13288. Springer, Cham. https://doi.org/10.1007/978-3-031-06653-5_5
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