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 Deltatau, 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(c1) above which the wave density disappears and the metastable state appears and a critical value beta(c2) above which the metastable state disappears while the stable state appears. beta(c1) and beta(c2) depend on the values of alpha and the variation of the delay time Deltatau. Indeed beta(c1) and beta(c2) increase when increasing alpha and/or decreasing Deltatau. The flow of vehicles is calculated as a function of alpha, beta and Deltatau 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 Deltatau and disappears completely above a critical value Deltatau(c).