A macro traffic flow model with probability distribution function

In this paper, we introduce three probability distribution functions into the dynamic equation and propose a macro traffic flow model to investigate the impacts of the probability distribution functions on the evolutions of traffic flow under three typical states (i.e., uniform flow, shock, rarefaction wave, and small perturbation). The numerical results indicate that the probability distribution functions do not change the density and speed distributions of uniform flow, produce a two-layer shock but have no prominent effects on rarefaction wave, and have little effect on small perturbation.