A bathtub model with nonlinear velocity–density relation

The bathtub model is an applicable and effective tool for simulating the characteristics of traffic dynamics in downtown areas during morning rush hour, especially the characteristics of hypercongested traffic. In order to reduce the complexity of the research, previous studies have adopted a linear velocity–density relationship. This has limitations in analyzing the cost and duration of hypercongestion in traffic dynamics. In this paper, we develop a bathtub model with one kind of nonlinear velocity–density Relation, i.e., Pipes–Munjal Relation, based on continuous scheduling preferences and analyze its solutions in both no-toll user equilibrium and social optimum states. Due to the intractability in solving this bathtub model, there exists local analytical solutions, but for a complete solution of the model, only numerical solutions are available. The results indicate that as the value of n increases, the model can simulate traffic dynamics with higher cost and longer duration of hypercongestion under equilibrium, and can simulate traffic states with lower system cost under social optimum. We also explore the results of implementing optimal time-varying toll to realize social optimum. This study expands the relevant theories of bathtub models and provides a new perspective for traffic congestion.