Research on Bathtub Models of Urban Traffic with Different Velocity-Density Relation

In recent years, the problem of urban traffic congestion has increasingly become the focus of scholars, but there are few studies on the phenomenon of hypercongestion when the traffic density in the city center is negatively correlated with the traffic flow during rush hours. Bathtub model is a suitable and effective model to simulate traffic congestion under this background, but it develops slowly because of the difficulty in solving partial differential equations and other mathematical problems in bathtub model. In this paper, a continuous scheduling preference bathtub model is constructed under Greenberg’ Relation. Then, the velocity, density, departure rate, arrival rate and cumulative departure are calculated in different periods. Finally, a numerical example is used to analyze the characteristics of each index under the Greenberg’ Relation under the equilibrium state, and a comparative analysis is made with the Greenshields‘ Relation. It is found that the Greenberg’ Relation can better fit the phenomenon of hypercongestion, but it can not describe the process of congestion dissipation well. This study extends the theory of bathtub model.