The Totally Asymmetric Simple Exclusion Process (TASEP) is a simple stochastic model for particles moving in a lattice with very rich behavior. We will examine the case when the lattice is on a ring. Here the steady-state distribution can be given combinatorially using multiline queues from the work of Ferrari and Martin. In this talk, we will describe some additional weighting parameters on multiline queues, prove the commutativity conjecture of Arita et al., and as time permits, show a relation with crystal bases. This is joint work with Erik Aas and Darij Grinberg.