Queueing process with excluded-volume effect

TL;DR

This paper extends the M/M/1 queueing model by incorporating spatial structure and excluded-volume effects, using TASEP rules, and analyzes its stationary states, phase transitions, and generalizations.

Contribution

It introduces a novel queueing process with spatial and exclusion effects, providing a matrix product solution and analyzing phase behavior and system properties.

Findings

Derived the stationary-state solution using matrix product form.

Identified the critical line separating different phases.

Calculated average length and particle number, showing monotonicity.

Abstract

We introduce an extension of the M/M/1 queueing process with a spatial structure and excluded- volume effect. The rule of particle hopping is the same as for the totally asymmetric simple exclusion process (TASEP). A stationary-state solution is constructed in a slightly arranged matrix product form of the open TASEP. We obtain the critical line that separates the parameter space depending on whether the model has the stationary state. We calculate the average length of the model and the number of particles and show the monotonicity of the probability of the length in the stationary state. We also consider a generalization of the model with backward hopping of particles allowed and an alternate joined system of the M/M/1 queueing process and the open TASEP.