Analysis of LWR model with fundamental diagram subject to uncertainties

The LWR model is of interest since it is simple and can successfully reproduce some essential features of traffic flow, such as the formation and propagation of traffic disturbances. In this paper, we investigate the LWR model from an uncertainty perspective. We attempt to analyze how reliable the LWR model prediction will be if the fundamental diagram (FD) in use is not accurately specified. To fulfill this end, we postulate a flux function (equivalently a fundamental diagram) driven by a random free flow speed, which accommodates the scattering feature observed in the speed-density data. We provide essential mathematical properties and solution schemes of the LWR model with the probabilistic fundamental diagram. In case studies, the approach to evaluate the uncertainty of traffic disturbance propagation with this model is presented. We find that if fundamental diagram in a LWR model cannot be perfectly specified, the uncertainty associated with the location of a traffic disturbance would increase over time. In contrast, the magnitude of the traffic disturbance can still be accurately predicted.