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

Xavier Allamigeon, Pascal Capetillo, Stéphane Gaubert

TL;DR

The paper tackles computing congestion phase diagrams for dynamics of timed Petri nets with priorities, modeling emergency departments under priority routing. It leverages a max-plus piecewise linear framework, reducing stationary regimes to affine forms $z_i(t)=u_i+\rho_i t$ and recasting feasibility as lexicographic polyhedra problems, solved in polynomial time via a max-front algorithm. A key theoretical advance is proving throughput is uniquely determined by a strictly feasible policy under generic delays when $1$ is semisimple, enabling explicit throughput expressions. The approach is demonstrated on an ED model from Assistance Publique -- Hôpitaux de Paris, yielding interpretable phase diagrams that identify bottlenecks and inform resource dimensioning with potential for broader DES applications.

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 - Hôpitaux de Paris.

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

TL;DR

The paper tackles computing congestion phase diagrams for dynamics of timed Petri nets with priorities, modeling emergency departments under priority routing. It leverages a max-plus piecewise linear framework, reducing stationary regimes to affine forms and recasting feasibility as lexicographic polyhedra problems, solved in polynomial time via a max-front algorithm. A key theoretical advance is proving throughput is uniquely determined by a strictly feasible policy under generic delays when is semisimple, enabling explicit throughput expressions. The approach is demonstrated on an ED model from Assistance Publique -- Hôpitaux de Paris, yielding interpretable phase diagrams that identify bottlenecks and inform resource dimensioning with potential for broader DES applications.

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 - Hôpitaux de Paris.
Paper Structure (9 sections, 3 theorems, 10 equations, 5 figures, 1 table)

This paper contains 9 sections, 3 theorems, 10 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Context: Piecewise Linear Models of Timed Discrete Event Systems.
  3. Contributions.
  4. Related Works.
  5. Petri Net Model of an Emergency Department
  6. Principle of the Algorithm
  7. Piecewise Linear Dynamics of Petri Nets
  8. Stationary Regimes and Lexicographic Constraints
  9. Solving Lexicographic Inequalities in Polynomial Time

Key Result

lemma thmcounterlemma

The front $f^\star$ defined by is the greatest front $f$ such that $\mathcal{P}^\mathrm{lex} \subset \mathcal{P}^f$.

Figures (5)

  • Figure 1: Petri net model of an emergency department. Parameters prefixed by $\tau$ stand for holding times associated with places, those prefixed by $\pi$ for routing proportions, and those prefixed by $N$ for resources (initial marking).
  • Figure 2: Petri net of the administrative registration, triage, nursing, and exam procedures.
  • Figure 3: The dynamical system over the counter functions of the ED Petri net.
  • Figure 4: (a) The lex-polyhedron defined by $(x_1, x_2) \mathrel{\geq_\mathrm{lex}} (0, 1)$, $(x_1, x_1) \mathrel{\geq_\mathrm{lex}} (-x_2, 1)$. In the boundary, solid lines and points are included, while dash lines are excluded. (b) The max-front polyhedron $\mathcal{P}^{f^\star}$ and (c) its relative interior.
  • Figure 5: Computing the max-front

Theorems & Definitions (4)

  • lemma thmcounterlemma
  • proposition thmcounterproposition
  • theorem thmcountertheorem
  • remark thmcounterremark