Abstract
Firstly, we consider a unidirectional flux 
of vehicles, each of which is characterized by its 'natural' velocity
v drawn from a
distribution P(v). The traffic flow is modeled as a collection of straight 'world lines' in the
time–space plane, with overtaking events represented by a fixed queuing time
τ
imposed on the overtaking vehicle. This geometrical model exhibits platoon formation and
allows, among many other things, for the calculation of the effective average velocity
of a vehicle of natural velocity
v. Secondly, we extend the model to two opposite lanes, A and B. We argue that the queuing time
τ
in one lane is determined by the traffic density in the opposite lane. On the basis of
reasonable additional assumptions we establish a set of equations that couple the two lanes
and can be solved numerically. It appears that above a critical value 
of the control parameter the symmetry between the lanes is spontaneously broken: there is a slow lane where long
platoons form behind the slowest vehicles, and a fast lane where overtaking is
easy due to the wide spacing between the platoons in the opposite direction. A
variant of the model is studied in which the spatial vehicle density 
rather than the flux is the control parameter. Unequal fluxes 
and 
in the two lanes are also considered. The symmetry breaking phenomenon exhibited by
this model, even though no doubt hard to observe in pure form in real-life traffic,
nevertheless indicates a tendency of such traffic.
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