  We consider the discrete Hammersley-Aldous-Diaconis process (HAD) and the
totally asymmetric simple exclusion process (TASEP) in Z. The basic coupling
induces a multiclass process which is useful in discussing shock measures and
other important properties of the processes. The invariant measures of the
multiclass systems are the same for both processes, and can be constructed as
the law of the output process of a system of multiclass queues in tandem; the
arrival and service processes of the queueing system are a collection of
independent Bernoulli product measures. The proof of invariance involves a new
coupling between stationary versions of the processes called a multi-line
process; this process has a collection of independent Bernoulli product
measures as an invariant measure. Some of these results have appeared elsewhere
and this paper is partly a review, with some proofs given only in outline.
However we emphasize a new approach via dual points: when the graphical
construction is used to construct a trajectory of the TASEP or HAD process as a
function of a Poisson process in ZxR, the dual points are those which govern
the time-reversal of the trajectory. Each line of the multi-line process is
governed by the dual points of the line below. We also mention some other
processes whose multiclass versions have the same invariant measures, and we
note an extension of Burke's theorem to multiclass queues which follows from
the results.
