"Multiclass processes, dual points and multitype
customer queues"
Pablo Ferrari, Universidade de Sao Paulo
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.
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.