  We study mixing times of the symmetric and asymmetric simple exclusion
process on the segment where particles are allowed to enter and exit at the
endpoints. We consider different regimes depending on the entering and exiting
rates as well as on the rates in the bulk, and show that the process exhibits
pre-cutoff and in some cases cutoff. Our main contribution is to study mixing
times for the asymmetric simple exclusion process with open boundaries. We show
that the order of the mixing time can be linear or exponential in the size of
the segment depending on the choice of the boundary parameters, proving a
strikingly different (and richer) behavior for the simple exclusion process
with open boundaries than for the process on the closed segment. Our arguments
combine coupling, second class particle and censoring techniques with current
estimates. A novel idea is the use of multi-species particle arguments, where
the particles only obey a partial ordering.
