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 $\mathbb{Z}$ 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 $\mathbb{Z}$.

Published at http://dx.doi.org/10.1214/009117906000000944 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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