Stochastic Fundamental Diagram for Probabilistic Traffic Flow Modeling
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Stochastic Fundamental Diagram for Probabilistic Traffic Flow Modeling

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      Flowing water in river, transported gas or oil in pipe, electric current in wire, moving goods on conveyor, molecular motors in living cell, and driving vehicles on a highway are various kinds of flow from physical or non-physical systems, yet each exhibits distinct characteristics. One of the attributes that distinguishes vehicular traffic flow from other flows as a special kind of 'fluid' is the so-called fundamental diagram - the relationships among traffic flow characteristics (e.g. flow, speed, and density) which are typically represented graphically, see an illustration in Figure 1. The fundamental diagram plays an essential role in transportation analysis and operations. For example, the study on traffic flow dynamics relies on input from the speed-density relationship to understand how a perturbation propagates among vehicles; a highway capacity analysis utilizes the speed-flow relationship to determine the level of service that the highway provides. Hence, sound mathematical models that better represent these relationships build a solid foundation for traffic flow analysis and efficient traffic control. Such an observation has motivated a variety of speed-density models since the path-breaking attempt by Greenshields in 1935. Departing from the Greenshields model, a number of models are proposed with varying degrees of success. Note that these models are in a single-equation form (single-regime model). Further improvements are made by decomposing the speed-density relationship into multiple pieces for better fitting. However, no matter single- or multi-regime, these models are deterministic in nature which essentially describes average system behaviors from a statistical perspective.
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