Stationary distributions of multi-type totally asymmetric exclusion processes

Abstract

We consider totally asymmetric simple exclusion processes with n types of particle and holes (n-TASEPs) on ℤ and on the cycle 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 ℤ_N, and simple proofs of various independence and regeneration properties for systems on ℤ.