Traffic Flow Models Considering an Internal Degree of Freedom

We solve numerically the integrodifferential equation for the equilibrium case of Paveri–Fontana's Boltzmann-like traffic equation. Beside space and actual velocity, Paveri–Fontana used an additional phase space variable, the desired velocity, to distinguish between the various driver characters. We refine his kinetic equation by introducing a modified cross section in order to incorporate finite vehicle length. We then calculate from the equilibrium solution the mean-velocity–density relation and investigate its dependence on the imposed desired velocity distribution. A further modification is made by modeling the interaction as an imperfect showing-down process. We find that the velocity cumulants of the stationary homogeneous solution essentially only depend on the first two cumulants, but not on the exact shape of the imposed desired velocity distribution. The equilibrium solution can therefore be approximated by a bivariate Gaussian distribution which is in agreement with empirical velocity distributions. From the improved kinetic equation we then derive a macroscopic model by neglecting third and higher order cumulants. The equilibrium solution of the macroscopic model is compared with the cumulants of the kinetic equilibrium solution and shows good agreement, thus justifying the closure assumption.