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Analytical stochastic macroscopic fundamental diagram driven by Wiener process

The macroscopic fundamental diagram (MFD) is a powerful and popular tool that describes a network scale traffic operational state and serve as the plant model of perimeter control. As both the supply and the demand suffer from random disturbances, the traffic flow dynamics cannot be said to be deterministic. A stochastic MFD model can generate a stochastic evolution of the system state with desired distribution of aggregated variables is still lacking. A stochastic formulation of MFD, that considers the accumulation-dependent variations, is proposed to fill this gap. The model is based on the stochastic differential equation (SDE) theory. First, the exit flow variation is formulated as a Wiener-driven process, which admits the accumulation-of dependent variations. The stochastic MFD model is then constructed by combining the exit flow variations model. The solution of the system state is derived by the forward Fokker-Planck equation. The stability of the model is analyzed, and the parameters of a calibration method are provided. Several cases of the model are then tested. The results show that the model can be applied to different functional MFD forms, and the hysteresis and gridlock phenomenon is reproduced. The proposed MFD model can be used in the network analysis and control that considers the system's stochastic evolution.