Exclusive Queueing Process with Discrete Time

TL;DR

This paper extends the exclusive queueing process to discrete time with parallel updates, deriving its stationary state, phase diagram, and key metrics, thus advancing understanding of exclusion effects in queueing models.

Contribution

It introduces a discrete-time version of the exclusive queueing process with parallel updates and derives its stationary state and phase diagram.

Findings

Stationary state obtained in matrix product form.

Phase diagram identified with a critical line based on stationary current.

Average system length and particle number calculated.

Abstract

In a recent study [C Arita, Phys. Rev. E 80, 051119 (2009)], an extension of the M/M/1 queueing process with the excluded-volume effect as in the totally asymmetric simple exclusion process (TASEP) was introduced. In this paper, we consider its discrete-time version. The update scheme we take is the parallel one. A stationary-state solution is obtained in a slightly arranged matrix product form of the discrete-time open TASEP with the parallel update. We find the phase diagram for the existence of the stationary state. The critical line which separates the parameter space into the regions with and without the stationary state can be written in terms of the stationary current of the open TASEP. We calculate the average length of the system and the average number of particles.