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 ZN . 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 ZN , and simple proofs of various independence and regeneration properties for systems on Z.= {1, ∞} Z N . We set u(i) = 1 if

Intracellular transport driven by cytoskeletal motors: General mechanisms and defects

“…Some exact results have been obtained for similar systems in continuous space and time (kinetic models), first for a one lane model in which passing is allowed with a certain rate [35], or for a two lane model with bidirectional trafic, under a mean field coupling assumption [17]. Back in the frame of TASEP-based models, some recent analytical results were obtained for a one-lane model where particles have the same speed, but have different priorities for passing each other [124,117,18,19] -a feature that results into different effective velocities. By contrast with the particle-wise disorder TASEP, the TASEP with site-disorder is not exactly solvable even in the stationary state for periodic boundary conditions, except in some quite special cases (for example for a single site disorder with alternated parallel update [333]).…”

Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

“…Fix α , β ∈ (Z ≥0 ) n . Set r = |β| and consider Corollary 5.4 with respect to this r: The first term here is calculated by applying Proposition 5.2 as 6) where the sum is over γ, δ ∈ (Z ≥0 ) n and ⊗ n≥i≥1 is arranged from i = n in the left to i = 1 in the right. As this term exemplifies, the rightmost n − 1 + N components only yield a common overall factor free from x, y for all the terms appearing in (5.5).…”

An inhomogeneous multispecies TASEP on a ring

“…By the inverse of the ringing path construction mentioned in Theorem 2.5(1), see [12,Prop 3.3] there is exactly one such transition. Thus, there are k − 1 incoming transitions with effective rate 1 and exactly one with rate x 1 .…”

“…The general solution of the stationary distribution for any number of different classes of particles came from Ferrari and Martin [12], who built on their own work on multiline queues [11] and used ideas from Ferrari, Fontes and Kohayakawa [10] and Angel [2]. Building on this work, the matrix ansatz solution for the general TASEP was constructed in [9], and for the general PASEP in [20].…”