On The Microscopic Modeling of Vehicular Traffic on General Networks

Abstract

We introduce a formalism to deal with the microscopic modeling of vehicular traffic on a road network. Traffic on each road is uni-directional, and the dynamics of each vehicle is described by a Follow-the-Leader model. From a mathematical point of view, this amounts to define a system of ordinary differential equations on an arbitrary network. A general existence and uniqueness result is provided, while priorities at junctions are shown to hinder the stability of solutions. We investigate the occurrence of the Braess paradox in a time-dependent setting within this model. The emergence of Nash equilibria in a non-stationary situation results in the appearance of Braess type paradoxes, and this is supported by numerical simulations.

Microscopic derivation of a traffic flow model with a bifurcation

“…Our starting point is a microscopic model. Before describing it, let us recall that few discrete traffic flow models with a junction or a local perturbation exist in the literature: [25] discusses an interesting leader follower model with a junction including several incoming and outgoing roads: the model we present in the present paper shares similar flavors, but in the much simpler setting of a single incoming road; [5] presents a microscopic model of traffic with a flow limitation at a point and formally justifies the derivation of a conservation law with a discontinuous flux (but leaves the rigorous proof as an open problem); [30] describes a traffic flow model with (deterministic) traffic lights and derives rigorously the continuous model (in terms of a flux limited Hamilton-Jacobi equation on the line). The only model proving micro-macro derivation in the case of a bifurcation is [28]: in [28] there are two outgoing roads and it is assumed (no too realistically) that every second vehicle takes a given road.…”