Upper Bounds for the Critical Car Densities in Traffic Flow Problems

Abstract

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 $p_c$, 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 $p_c<1$ for the Biham-Middleton-Levine model of traffic flow in two or higher spatial dimensions. In particular, we show that $p_c\le 11/12$ in 2 dimension and $p_c\le 1-{\left(\frac{D-1}{2D}\right)}^D$ in $D$ ($D>2$) dimensions.

REVTEX 3.0, 5 pages with 1 figure appended at the back, Minor revision, to be published in the Sept issue of J.Phys.Soc.Japan