Multi-floor generalization of TASEP

TL;DR

This paper studies a generalized TASEP model with multi-particle sites and back-pressure dynamics, revealing phase transitions and flux behavior as system size grows, extending classical TASEP results.

Contribution

It introduces a multi-particle site TASEP with back-pressure rules, analyzes its stationary distribution, and uncovers new phase transition phenomena for systems with site capacity greater than one.

Findings

Non-trivial phase transitions in flux for c>1

Maximum flux of 1/4 achieved under certain conditions

Thresholds for parameters differ from classical 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 $N$ 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 $\alpha\le 1$, unless site $1$ has $c$ particles. Particles (if any) are removed from the right-most site $N$ as a Poisson process of rate $\beta \le 1$. 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 $1/2 \le \beta \le 1$, the maximum limiting flux $1/4$ is achieved as long as $\alpha \ge \alpha_c^*$, where $\alpha_c^* < 1/2$ 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.