  The asymmetric exclusion process (ASEP) has attracted a lot of interest not
only because its many applications, e.g. in the context of the kinetics of
biopolymerization and traffic flow theory, but also because it is a
paradigmatic model for nonequilibrium systems. Here we study the ASEP for
different types of updates, namely random-sequential, sequential,
sublattice-parallel and parallel. In order to compare the effects of the
different update procedures on the properties of the stationary state, we use
large-scale Monte Carlo simulations and analytical methods, especially the
so-called matrix-product Ansatz (MPA). We present in detail the exact solution
for the model with sublattice-parallel and sequential updates using the MPA.
For the case of parallel update, which is important for applications like
traffic flow theory, we determine the phase diagram, the current, and density
profiles based on Monte Carlo simulations. We furthermore suggest a MPA for
that case and derive the corresponding matrix algebra.
