Upper Bounds for the Critical Car Densities in Traffic Flow Problems

In most models of traffic flow, the car density p is the only free parameter in determining the average car velocity 〈v〉. The critical car density pc, which is defined to be the car density separating the jamming phase (with 〈v〉 = 0) and the moving phase (with 〈v〉 > 0), is an important physical quantity to investigate. By means of simple statistical argument, we show that pc < 1 for the Biham-Middleton-Levine model of traffic flow in two or higher spatial dimensions. In particular, we show that pc ≤ 11/12 in 2 dimension and pc ≤ 1− ( D−1 2D D in D (D > 2) dimensions.