Stochastic modeling of the equilibrium speed-density relationship

As the graphical and mathematical representation of relationships among traffic flow, speed, and density, the fundamental diagram has been the foundation of traffic flow theory and transportation engineering. Underlying the fundamental diagram is the speed-density relationship which was originally documented in Greenshields' seminal work and followed by a variety of equilibrium models over the past 75 years. Observed in these efforts was their deterministic nature striving to pursue two seemingly competing goals: mathematical elegance and empirical accuracy, the former of which is attractive to mathematical modeling of traffic dynamics, and the latter is required if such modeling is meant to be realistic. As a continued effort of such a pursuit, this paper presents a stochastic speed-density model. The motivation is twofold: first, it is desirable to have a model which achieves both goals reasonably and second, the stochastic model can potentially lead to probabilistic traffic flow modeling and prediction which is typically not offered by a macroscopic approach.