A dynamic model of congestion

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.