Solving the additive eigenvalue problem associated to a dynamics of a 2D-traffic system

This is a technical note where we solve the additive eigenvalue problem associated to a dynamics of a 2D-traffic system. The traffic modeling is not explained here. It is available in \cite{Far08}. It consists of a microscopic road traffic model of two circular roads crossing on one junction managed with the priority-to-the-right rule. It is based on Petri nets and minplus algebra. One of our objectives in \cite{Far08} was to derive the fundamental diagram of 2D-traffic, which is the relation between the density and the flow of vehicles. The dynamics of this system, derived from a Petri net design, is non monotone and additively homogeneous of degree 1. In this note, we solve the additive eigenvalue problem associated to this dynamics.