Zero-automatic queues and product form

TL;DR

This paper introduces 0-automatic queues, a new model with a unique buffering mechanism, demonstrating that all stable instances have a product form stationary distribution and Poisson output, unifying several queue types.

Contribution

The paper defines 0-automatic queues, proves their stability and product form stationary distribution, and connects them to classical queue models like M/M/1 and G-queues.

Findings

All stable 0-automatic queues have a product form stationary distribution.

Stable 0-automatic queues produce a Poisson output process.

Special cases recover M/M/1 and Gelenbe's G-queue models.

Abstract

We introduce and study a new model: 0-automatic queues. Roughly, 0-automatic queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The salient result is that all stable 0-automatic queues have a product form stationary distribution and a Poisson output process. When considering the two simplest and extremal cases of 0-automatic queues, we recover the simple M/M/1 queue, and Gelenbe's G-queue with positive and negative customers.