Statistical properties of low-density traffic

This paper considers an infinitely long line of traffic moving on a highway without traffic lights or other inhomogeneities. It is assumed that each car travels at a constant speed which is a random variable. A further assumption is that when one car overtakes another, passing is always possible and occurs without change of speed. it is shown that any initial headway distribution must relax to a negative exponential distribution in the limit of $t$ becoming infinite. The statistics of passing events are examined, and it is shown that the probability of passing (or being passed by) $n$ cars In time $t$ is described by a Poisson distribution.