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A dynamic model of congestion

Hector Andres Chang-Lara, Sergio David Zapeta-Tzul

TL;DR

A dynamic version of the Beckmann problem is introduced, for which the corresponding discrete partial differential equations governing the evolution of the system are derived and these equations enable the size of the support of the edge flow to be estimated.

Abstract

We revisit the classic problem of determining optimal routes in a graph for transporting two given distributions defined on its nodes, originally studied by Wardrop and Beckmann in the 1950s. The global congestion profile at any given time defines a dynamic metric on the graph, for which the routes must be geodesics. Our first contribution is the introduction of a dynamic version of the Beckmann problem, for which we derive the corresponding discrete partial differential equations governing the evolution of the system. These equations enable us to estimate the size of the support of the edge flow. Finally, we present some numerical simulations to illustrate the behavior of efficient equilibria in a dynamic setting with non-local interactions.

A dynamic model of congestion

TL;DR

A dynamic version of the Beckmann problem is introduced, for which the corresponding discrete partial differential equations governing the evolution of the system are derived and these equations enable the size of the support of the edge flow to be estimated.

Abstract

We revisit the classic problem of determining optimal routes in a graph for transporting two given distributions defined on its nodes, originally studied by Wardrop and Beckmann in the 1950s. The global congestion profile at any given time defines a dynamic metric on the graph, for which the routes must be geodesics. Our first contribution is the introduction of a dynamic version of the Beckmann problem, for which we derive the corresponding discrete partial differential equations governing the evolution of the system. These equations enable us to estimate the size of the support of the edge flow. Finally, we present some numerical simulations to illustrate the behavior of efficient equilibria in a dynamic setting with non-local interactions.

Paper Structure

This paper contains 36 sections, 46 theorems, 296 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Stochastic geodesic problem
  5. Congestion
  6. The potential
  7. The Beckmann problem
  8. The Smirnov decomposition
  9. The constitutive relations
  10. The long-term problem
  11. The stationary minimal-time mean field game
  12. The general capacity problem
  13. From the constitutive relation to Wardrop equilibria
  14. A bound on the support of the edge flow
  15. Existence of the multiplier
  16. ...and 21 more sections

Key Result

Lemma 2.1

Given $\xi \colon E\to [0,\infty)$, a path profile $q\in \mathcal{P}(\operatorname{Path}(G))$ satisfies $\{q>0\}\subseteq \operatorname{Geod}(G,\xi)$ if and only if $\mathbb E_q(L_\xi)={\mathbb E}_{\gamma[q]}(d_\xi)$, or equivalently

Figures (5)

  • Figure 1: A graph modelling the intersection of two roads. In order to describe the congestion effects of this intersection, one should consider that the cost of crossing the edge $(a_1,b_1)$ should depend on the flows over the edges $(a_1,b_1)$ and $(a_2,b_2)$.
  • Figure 2: For the construction of the extended graph $G^T$ from the graph $G$ we consider $T+1$ copies of the nodes and join two nodes $(x,t-1)$ and $(y,t)$ if and only if $e=(x,y)$ is an edge of $G$.
  • Figure 3: A graph modelling a simple road.
  • Figure 4: Illustration of the minimization of the quadratic function $E$ over the rectangle $[j_1^+,j_1^-]\times [j_2^+,j_2^-]$.
  • Figure 5: Solutions of the dynamic problem computed by an iterative method for the graph in Figure \ref{['fig:intersection']}, $T=10$, $m_1=2$, $m_2=3$, $\varepsilon=0.1$, and $\gamma$ decreasing uniformly from 2 to 0.

Theorems & Definitions (98)

  • Definition 2.1: Transport plan
  • Lemma 2.1: Characterization of path profiles supported on geodesics
  • proof
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5: Existence of geodesic profiles
  • proof
  • Corollary 2.6
  • ...and 88 more