In this paper, a modified macroscopic traffic flow model is presented. The term of the density-dependent relaxation time is introduced here. The relation between the relaxation time and the density in traffic flow is presented quantitatively. Besides, a factor R depicting varied properties of traffic flow in different traffic states is also introduced in the formulation of the model. Furthermore, the evolvement law of traffic flow with distinctly initial density distribution and boundary perturbations is emphasized.

Motivated by the previous traffic flow model considering the real-time traffic state, a modified macroscopic traffic flow model is established. The periodic boundary condition is applied to the car-following model. Besides, the traffic state factor R is defined in order to correct the real traffic conditions in a more reasonable way. It is a key step that we introduce the relaxation time as a density-dependent function and provide corresponding evolvement of traffic flow. Three different typical initial densities, namely the high density, the medium one and the low one, are intensively investigated. It can be found that the hysteresis loop exists in the proposed periodic-boundary system. Furthermore, the linear and nonlinear stability analyses are performed in order to test the robustness of the system.

In this paper, a modified microscopic traffic flow model accounting for the optimal velocity has been proposed. Different with previous models, drivers’ response ability and the maximum of accelerations are considered in the term of the optimal velocity. The effect of parameters in the term of the optimal velocity on bifurcations in the rotary traffic is studied here. Besides, the evolvement of bifurcations in the system is calculated by performing numerical simulation experiments. Moreover, the linear stability analysis of the proposed model is presented.