  We investigate a Markov chain defined by Thomas Lam, which generalizes the
multi-type TASEP on a ring to any Weyl group. For groups of type C we define an
analogue of the multiline queues of Ferrari and Martin (which compute the
stationary distribution for the classical TASEP). While our construction does
not suffice for finding the stationary distribution, the construction gives the
stationary distribution of a certain projection of Lam's chain. Also, our
approach is incremental, in the sense that the construction appears to fit into
a pattern of 'conjugation matrices', which remains to be fully worked out.
Finally, we prove a theorem for the classical TASEP which fits into the picture
of viewing TASEP in a permutation-free way.
