Modeling and control of elementary 2D-Traffic systems using Petri nets and minplus algebra

We present in this paper the modeling of an elementary 2D-Traffic system, basing on Petri nets and minplus algebra. Our aim is to derive fundamental diagrams of 2D-traffic on a system of one circular road with one junction without turn, taking shape of the number eight (8). The fundamental diagram is the relation between the flow and the density of vehicles on a road. Two possibilities of managing the junction are considered: using the priority-to-the-right rule, or using a signal traffic light control. We show that in both two cases, modeling a traffic junction needs the use of negative weights on Petri nets. Although our models are simplistic, they permit to obtain significant qualitative results. In the first case where the junction is managed by the priority-to-the-right rule, some analytical results are given, which are completed and confirmed by numerical results. The second case, where the junction is controlled by using a signal light, is solved analytically. The resulting fundamental diagrams are given and compared.