  We consider totally asymmetric simple exclusion processes with n types of
particle and holes ($n$-TASEPs) on $\mathbb {Z}$ and on the cycle $\mathbb
{Z}_N$. Angel recently gave an elegant construction of the stationary measures
for the 2-TASEP, based on a pair of independent product measures. We show that
Angel's construction can be interpreted in terms of the operation of a
discrete-time $M/M/1$ queueing server; the two product measures correspond to
the arrival and service processes of the queue. We extend this construction to
represent the stationary measures of an n-TASEP in terms of a system of queues
in tandem. The proof of stationarity involves a system of n 1-TASEPs, whose
evolutions are coupled but whose distributions at any fixed time are
independent. Using the queueing representation, we give quantitative results
for stationary probabilities of states of the n-TASEP on $\mathbb {Z}_N$, and
simple proofs of various independence and regeneration properties for systems
on $\mathbb {Z}$.
