Derivation of the fundamental traffic diagram for two circular roads and a crossing using minplus algebra and Petri net modeling.

The fundamental diagram gives the relation between the flow and the density of vehicles for the car traffic on a road. We extend this diagram to the case of two circular roads with an intersection. From a Petri net model we derive the dynamics in terms of the composition of a standard linear operator and a minplus operator which determines uniquely the system trajectory. We show, experimentally, that there exists an average flow which becomes asymptotically independent of the initial position of the vehicles when the size of the system goes to infinity. We study the corresponding fundamental diagram and analyze the different phases that can appear.