Stability Results on Synchronized Queues in Discrete-Time for Arbitrary Dimension

TL;DR

This paper analyzes the stability of synchronized queues in discrete-time, showing they are quasi-stable for two or three queues but unstable for four or more, using Markov chain and random walk theory.

Contribution

It establishes the stability thresholds for synchronized queues of different sizes and connects queue stability to properties of associated Markov chains and random walks.

Findings

Quasi-stability for 2 and 3 queues

Instability for 4 or more queues

Connection to Pólya's theorem on random walks

Abstract

In a batch of synchronized queues, customers can only be serviced all at once or not at all, implying that service remains idle if at least one queue is empty. We propose that a batch of $n$ synchronized queues in a discrete-time setting is quasi-stable for $n \in \{2,3\}$ and unstable for $n \geq 4$. A correspondence between such systems and a random-walk-like discrete-time Markov chain (DTMC), which operates on a quotient space of the original state-space, is derived. Using this relation, we prove the proposition by showing that the DTMC is transient for $n \geq 4$ and null-recurrent (hence quasi-stability) for $n \in \{2,3\}$ via evaluating infinite power sums over skewed binomial coefficients. Ignoring the special structure of the quotient space, the proposition can be interpreted as a result of P\'olya's theorem on random walks, since the dimension of said space is $d-1$.