Computing the Congestion Phases of Dynamical Systems with Priorities and Application to Emergency Departments

TL;DR

This paper presents a novel method to compute congestion phases in complex priority-based dynamical systems, with applications to optimizing resource allocation in emergency departments.

Contribution

It introduces a polynomial time algorithm for analyzing achievable policies and congestion phases in nonmonotone piecewise linear systems modeled by Petri nets with priorities.

Findings

Developed a polynomial time feasibility testing algorithm.

Applied the method to a real-world emergency department case study.

Mapped congestion phases to resource bottlenecks.

Abstract

Medical emergency departments are complex systems in which patients must be treated according to priority rules based on the severity of their condition. We develop a model of emergency departments using Petri nets with priorities, described by nonmonotone piecewise linear dynamical systems. The collection of stationary solutions of such systems forms a "phase diagram", in which each phase corresponds to a subset of bottleneck resources (like senior doctors, interns, nurses, consultation rooms, etc.). Since the number of phases is generally exponential in the number of resources, developing automated methods is essential to tackle realistic models. We develop a general method to compute congestion diagrams. A key ingredient is a polynomial time algorithm to test whether a given "policy" (configuration of bottleneck tasks) is achievable by a choice of resources. This is done by reduction to a feasibility problem for an unusual class of lexicographic polyhedra. Furthermore, we show that each policy uniquely determines the system's throughput. We apply our approach to a case study, analyzing a simplified model of an emergency department from Assistance Publique - H\^opitaux de Paris.