  We present an exact solution of a probabilistic cellular automaton for
traffic with open boundary conditions, e.g. cars can enter and leave a part of
a highway with certain probabilities. The model studied is the asymmetric
exclusion process (ASEP) with {\it simultaneous} updating of all sites. It is
equivalent to a special case ($v_{\rm max}=1$) of the Nagel-Schreckenberg model
for highway traffic, which has found many applications in real-time traffic
simulations. The simultaneous updating induces additional strong short range
correlations compared to other updating schemes. The stationary state is
written in terms of a matrix product solution. The corresponding algebra, which
expresses a system-size recursion relation for the weights of the
configurations, is quartic, in contrast to previous cases, in which the algebra
is quadratic. We derive the phase diagram and compute various properties such
as density profiles, two point functions and the fluctuations in the number of
particles (cars) in the system. The current and the density profiles can be
mapped onto the ASEP with other time discrete updating procedures. Through use
of this mapping, our results also give new results for these models.
