Performance evaluation of an emergency call center: tropical polynomial systems applied to timed Petri nets

TL;DR

This paper models an emergency call center using timed Petri nets and tropical polynomial systems, providing explicit formulas for throughput and congestion phases, and validating the fluid approximation with numerical experiments.

Contribution

It introduces a novel approach combining timed Petri nets with tropical polynomial systems to analyze throughput and congestion in emergency call centers.

Findings

Explicit formulas for throughput as a piecewise linear function.

Identification of different congestion phases.

Fluid model approximates real throughput effectively.

Abstract

We analyze a timed Petri net model of an emergency call center which processes calls with different levels of priority. The counter variables of the Petri net represent the cumulated number of events as a function of time. We show that these variables are determined by a piecewise linear dynamical system. We also prove that computing the stationary regimes of the associated fluid dynamics reduces to solving a polynomial system over a tropical (min-plus) semifield of germs. This leads to explicit formul{\ae} expressing the throughput of the fluid system as a piecewise linear function of the resources, revealing the existence of different congestion phases. Numerical experiments show that the analysis of the fluid dynamics yields a good approximation of the real throughput.