Stationary velocity distributions in traffic flows

We introduce a traffic flow model that incorporates clustering and passing. We obtain analytically the steady state characteristics of the flow from a Boltzmann-like equation. A single dimensionless parameter, R = c(0) upsilon(0)t(0) with c(0) the concentration, upsilon(0) the velocity range, and t(0)(-1) the passing rate, determines the nature of the steady state. When R much less than 1, uninterrupted flow with single cars occurs. When R much greater than 1, large clusters with average mass [m] similar to R-alpha form, and the flux is J similar to R-gamma. The initial distribution of slow cars governs the statistics. When P-0(upsilon) similar to upsilon(mu) as upsilon --> 0, the scaling exponents are gamma = 1/(mu + 2), alpha = 1/2 when mu > 0, and alpha = (mu + 1)/(mu + 2) when mu < 0.