On the stochastic fundamental diagram: A general micro-macroscopic traffic flow modeling framework

The stochastic fundamental diagram (SFD), which describes the stochasticity of the macroscopic relations of traffic flow, plays a crucial role in understanding the uncertainty of traffic flow evolution and developing robust traffic control strategies. Although many efforts have been made to reproduce the SFD via various methods, few studies have focused on the analytical modeling of the SFD, particularly linking the macroscopic relations with microscopic behaviors. This study fills this gap by proposing a general micro-macroscopic modeling approach, which uses probabilistic leader–follower behavior to derive the macroscopic relations of a platoon and is referred to as the leader–follower conditional distribution-based stochastic traffic modeling (LFCD-STM) framework. Specifically, we first define a conditional probability distribution of speed for the leader‒follower pair according to Brownian dynamics, which is proven to be a general representation of the longitudinal interaction and compatible with classical car-following models. As a result, we can describe the joint distribution of vehicle speeds of the platoon through Markov chain modeling and further derive the macroscopic relations (e.g., the mean flow‒density relation and its variance) under equilibrium conditions. On the basis of this general micro-macroscopic framework, we utilize the maximum entropy approach to theoretically derive the SFD model, in which we provide a specific conditional distribution for longitudinal interaction and thus solve the analytical functions of the mean and variance of FD. The performance of the maximum entropy-based SFD model is thoroughly validated with the NGSIM I-80, US-101 and HighD datasets. The high consistency between the theoretical results and empirical results demonstrates the soundness of the LFCD-STM framework and the maximum entropy-based SFD model. Finally, the proposed SFD model has practical implications for promoting smoother driving behaviors to suppress stochasticity and improve traffic flow.