The optimal velocity traffic flow models with open boundaries

The effects of the open boundaries on the dynamical behavior of the optimal velocity traffic flow models with a delay time $\tau$ allowing the car to reach its optimal velocity is studied using numerical simulations. The particles could enter the chain with a given injecting rate probability $\alpha$ , and could leave the system with a given extracting rate probability $\beta $ . In the absence of the variation of the delay time $\Delta\tau$ , it is found that the transition from unstable to metastable and from metastable to stable state occur under the effect of the probabilities rates $\alpha$ and $\beta $ . However, for a fixed value of $\alpha$ , there exist a critical value of the extraction rate $\beta_{c_1}$ above which the wave density disappears and the metastable state appears and a critical value $\beta_{c_2}$ above which the metastable state disappears while the stable state appears. $\beta_{c_1}$ and $\beta_{c_2}$ depend on the values of $\alpha$ and the variation of the delay time $\Delta\tau$ . Indeed $\beta_{c_1}$ and $\beta_{c_2}$ increase when increasing $\alpha$ and/or decreasing $\Delta\tau$ . The flow of vehicles is calculated as a function of $\alpha$ , $\beta $ and $\Delta\tau$ for a fixed value of $\tau$ . Phase diagrams in the ( $\alpha,\beta$ ) plane exhibits four different phases namely, unstable, metastable, stable. The transition line between stable phase and the unstable one is curved and it is of first order type. While the transition between stable (unstable) phase and the metastable phase are of second order type. The region of the metastable phase shrinks with increasing the variation of the delay time $\Delta\tau$ and disappears completely above a critical value $\Delta\tau_{c}$ . Copyright Springer-Verlag Berlin/Heidelberg 2003