Zero-automatic queues and product form

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.

💡 Research Summary

The paper introduces a novel class of queueing systems called zero‑automatic queues (0‑automatic queues). In these systems the buffering mechanism evolves as a random walk on an infinite group or monoid, and each arrival or service event corresponds to a group (or monoid) operation on the current buffer content. The state of the system is therefore represented by a word (or product) of group elements, with the empty word (the identity element) denoting an empty buffer. Arrivals are modeled as a Poisson process whose marks belong to a finite label set Σ; the mark distribution ν determines the probabilities of the different group generators. Service times are exponential with rate μ, and a service removes the “last” element according to the inverse group operation. This construction makes the transition kernel of the underlying Markov chain identical to the transition matrix of a random walk on the Cayley graph of the chosen group/monoid.

The authors first formalize the notion of 0‑automaticity: the initial state is the identity, and the transition probabilities are completely determined by ν and the algebraic structure of the group. They then derive a stability condition that generalizes the classic traffic intensity ρ = λ/μ < 1. In the group‑theoretic setting the drift of the random walk, δ = E