Computing Transience Bounds of Emergency Call Centers: a Hierarchical Timed Petri Net Approach

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. Then, 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.