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Multi-floor generalization of TASEP

Yuliy Baryshnikov, Alexander Stolyar

Abstract

We consider an interacting particle system, which generalizes the classical totally asymmetric simple exclusion process (TASEP), in that each site can contain up to a fixed finite number of particles, and the particle movement is governed by a {\em back-pressure} (BP) algorithm (also often called {\em MaxWeight}). There are $N$ sites (with $N$ finite or infinite), each may contain at most $c$ particles, $1 \le c < \infty$. New particles enter the system at the left-most site $1$ as a Poisson process of rate $α\le 1$, unless site $1$ has $c$ particles. Particles (if any) are removed from the right-most site $N$ as a Poisson process of rate $β\le 1$. The left-to-right movement of particles between neighboring sites is governed by the BP rule: one particle moves from site $n$ to $n+1$ at epochs of a rate $1$ Poisson process, as long as the former site has strictly more particles than the latter. When $c=1$, this is the standard TASEP. Our main results address the asymptotics of the stationary distribution of a finite system, and especially the limit of the flux (current) as $N\to\infty$. In particular, we prove that interesting non-trivial phase transitions take place in a system with $c>1$. For example, if $c>1$ and $1/2 \le β\le 1$, the maximum limiting flux $1/4$ is achieved as long as $α\ge α_c^*$, where $α_c^* < 1/2$ is some non-trivial threshold. (For the standard TASEP the threshold is $1/2$.) We also put forward a general conjecture about the stationary distribution asymptotics under an arbitrary parameter setting. We illustrate our formal results and the conjecture by simulations, and identify interesting directions for further research.

Multi-floor generalization of TASEP

Abstract

We consider an interacting particle system, which generalizes the classical totally asymmetric simple exclusion process (TASEP), in that each site can contain up to a fixed finite number of particles, and the particle movement is governed by a {\em back-pressure} (BP) algorithm (also often called {\em MaxWeight}). There are sites (with finite or infinite), each may contain at most particles, . New particles enter the system at the left-most site as a Poisson process of rate , unless site has particles. Particles (if any) are removed from the right-most site as a Poisson process of rate . The left-to-right movement of particles between neighboring sites is governed by the BP rule: one particle moves from site to at epochs of a rate Poisson process, as long as the former site has strictly more particles than the latter. When , this is the standard TASEP. Our main results address the asymptotics of the stationary distribution of a finite system, and especially the limit of the flux (current) as . In particular, we prove that interesting non-trivial phase transitions take place in a system with . For example, if and , the maximum limiting flux is achieved as long as , where is some non-trivial threshold. (For the standard TASEP the threshold is .) We also put forward a general conjecture about the stationary distribution asymptotics under an arbitrary parameter setting. We illustrate our formal results and the conjecture by simulations, and identify interesting directions for further research.
Paper Structure (20 sections, 8 theorems, 16 equations, 15 figures)

This paper contains 20 sections, 8 theorems, 16 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Model and motivations
  3. Setup
  4. Overview of results and paper structure
  5. One-sided model
  6. Definitions, basic properties and main result
  7. Coupling constructions
  8. Coupling of processes under back-pressure algorithm with particles of different priority classes
  9. Coupling of processes with different rates $\alpha$
  10. Coupling of processes with different number $c$ of floors
  11. Coupling of processes with one being "impeded" with respect to another
  12. Proof of Theorem \ref{['th-main-res']}(i)
  13. Proof of Theorem \ref{['th-main-res']}(ii)
  14. Proof of Lemma \ref{['lem-phi-props']}
  15. Left- and right-side asymptotic limits (special case)
  16. ...and 5 more sections

Key Result

Theorem 1

(i) The sequence $\alpha_1^*=1/2, \alpha_2^*, \ldots$ is strictly decreasing and $\alpha_c^* \downarrow 1/4$. (ii) For any $c\ge 2$, under LIM ${\cal L}(\alpha,c)$, we have the following. For any $\alpha < \alpha_{c-1}^*$ the effective floor is $1$; and for any $\alpha \ge \alpha_{c-1}^*$, ${\cal L}

Figures (15)

  • Figure 1: Top: 2-floor TASEP, with all particles moving to the right. Bottom: Interpretation in terms of positive "particles" moving to the right and negative "particles" moving to the left. The move shown on the top picture corresponds to the annihilation of the two colliding "particles" on the bottom picture.
  • Figure 2: The projections $A_m, B_m$ are shown for $m=3$.
  • Figure 3: $\alpha=1, \beta=1, c=2$
  • Figure 4: $\alpha=1, \beta=1, c=3$
  • Figure 5: $\alpha=0.45, \beta=1, c=2$
  • ...and 10 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Proposition 6: Corollary from corollary 3.17 in Liggett-1975
  • Theorem 7
  • Lemma 8
  • Conjecture 9