A combinatorial approach to jumping particles: The parallel TASEP

In this paper we continue the combinatorial study of models of particles jumping on a row of cells which we initiated with the standard totally asymmetric exclusion process or TASEP (Journal of Combinatorial Theory, Series A, to appear). We consider here the parallel TASEP, in which particles can jump simultaneously. On the one hand, the interest in this process comes from highway traffic modeling: it is the only solvable special case of the Nagel-Schreckenberg automaton, the most popular model in that context. On the other hand, the parallel TASEP is of some theoretical interest because the derivation of its stationary distribution, as appearing in the physics literature, is harder than that of the standard TASEP. We offer here an elementary derivation that extends the combinatorial approach we developed for the standard TASEP. In particular we show that this stationary distribution can be expressed in terms of refinements of Catalan numbers.