A min-plus derivation of the fundamental car-traffic law

We give deterministic and stochastic models of the traffic on a circular road without overtaking. From this model the mean speed is derived as an eigenvalue of the min-plus matrix describing the dynamics of the system in the deterministic case and as the Lyapunov exponent of a min-plus stochastic matrix in the stochastic case. The eigenvalue and the Lyapunov exponent are computed explicitly. From these formulas, we derive the fundamental law that links the flow to the density of vehicles on the road. Numerical experiments using the MAXPLUS toolbox of SCILAB confirm the theoretical results obtained.