Analytical formulation for explaining the variations in traffic states: A fundamental diagram modeling perspective with stochastic parameters

Despite the simplicity and practicality of (deterministic) fundamental diagram models in highway traffic flow theory, the wide scattering effect observed in empirical data remains highly controversial, particularly for explaining traffic state variations. Owing to the analytical properties of the fundamental diagram modeling approach, in this study, we proposed an analytical and quantitative method for analyzing traffic state variations. We investigated the scattering effect in the fundamental diagram and proposed two stochastic fundamental diagram (SFD) models with lognormal and skew-normal distributions to explain the variations in traffic states. The first SFD model assumes that the scattering effect results from stochasticity in both the free-flow speed and the speed at critical density. Both random variables were assumed to follow the lognormal distribution. In the second SFD model, an integrated error term that was assumed to follow the skew-normal distribution over different density ranges was appended to the deterministic fundamental diagram. The properties of these two SFD models were analyzed and compared, and the parameters in these SFD models were calibrated using real-world loop detector data. The observed scatters from the empirical data were reproduced well by the simulated fundamental diagram model, indicating the validity of the proposed SFD models for explaining traffic state variations. Using these two analytical SFD models, we can analyze the stochastic capacity of freeways with closed forms. More importantly, the sources of stochasticity in freeway capacity can be traced in terms of randomly distributed parameters in fundamental diagram models.