Exact solution of a cellular automaton for traffic

We present an exact solution of a probabilistic cellular automaton for traf fic 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 simultaneous updating of all sites. It is equ ivalent to a special case (upsilon(max) = 1) of the Nagel-Schreckenberg mod el for highway traffic. which has found many applications in real-time traf fic 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 algebr a, which expresses a system-size recursion relation for the weights of the configurations. is quartic, in contrast to previous cases, in which the alg ebra is quadratic, Wt derive the phase diagram and compute various properti es such as density profiles. two-point functions, and the fluctuations in t he number of particles (cars) in the system. Thr current and the density pr ofiles can be mapped onto the ASEP with other time-discrete updating proced ures. Through use of this mapping. our results also give new results for th ese models.