On the micro-to-macro limit for first-order traffic flow models on networks

TL;DR

This paper establishes the connection between microscopic follow-the-leader models and macroscopic traffic flow models on networks, addressing the challenges posed by junctions and ill-posedness in existing models.

Contribution

It demonstrates that a natural extension of the microscopic model converges to a known macroscopic multi-path model on networks as the number of vehicles increases.

Findings

Microscopic model converges to macroscopic multi-path model

Addresses junction ambiguity in traffic models

Extends single-road results to networks

Abstract

Connections between microscopic follow-the-leader and macroscopic fluid-dynamics traffic flow models are already well understood in the case of vehicles moving on a single road. Analogous connections in the case of road networks are instead lacking. This is probably due to the fact that macroscopic traffic models on networks are in general ill-posed, since the conservation of the mass is not sufficient alone to characterize a unique solution at junctions. This ambiguity makes more difficult to find the right limit of the microscopic model, which, in turn, can be defined in different ways near the junctions. In this paper we show that a natural extension of the first-order follow-the-leader model on networks corresponds, as the number of vehicles tends to infinity, to the LWR-based multi-path model introduced in [Bretti et al., Discrete Contin. Dyn. Syst. Ser. S, 7 (2014)] and [Briani and Cristiani, Netw. Heterog. Media, 9 (2014)].