Exponential Tail Bounds on Queues: A Confluence of Non-Asymptotic Heavy Traffic and Large Deviations

TL;DR

This paper develops non-asymptotic exponential tail bounds for queue lengths under heavy traffic, applicable to various queueing models, bridging classical heavy traffic and large deviations regimes with sharper results.

Contribution

It introduces a novel method for deriving sharper, non-asymptotic tail bounds in queueing systems under heavy traffic, applicable to multiple models and regimes, including large system limits.

Findings

Provides exponential tail bounds for queue lengths in different models.

Bridges the gap between heavy traffic and large deviations regimes.

Offers refined analysis of state space collapse in JSQ systems.

Abstract

In general, obtaining the exact steady-state distribution of queue lengths is not feasible. Therefore, we establish bounds for the tail probabilities of queue lengths. Specifically, we examine queueing systems under Heavy-Traffic (HT) conditions and provide exponentially decaying bounds for the probability $\mathbb P(\epsilon q > x)$, where $\epsilon$ is the HT parameter denoting how far the load is from the maximum allowed load. Our bounds are not limited to asymptotic cases and are applicable even for finite values of $\epsilon$, and they get sharper as $\epsilon \to 0$. Consequently, we derive non-asymptotic convergence rates for the tail probabilities. Unlike other approaches such as moment bounds based on drift arguments and bounds on Wasserstein distance using Stein's method, our method yields sharper tail bounds. Furthermore, our results offer bounds on the exponential rate of decay of the tail, given by $-\frac{1}{x} \log \mathbb P(\epsilon q > x)$ for any finite value of $x$. These can be interpreted as non-asymptotic versions of Large Deviation (LD) results. We demonstrate our approach by presenting tail bounds for: (i) a continuous time Join-the-shortest queue (JSQ) load balancing system, (ii) a discrete time single-server queue and (iii) an $M/M/n$ queue. We not only bridge the gap between classical-HT and LD regimes but also explore the large system HT regimes for JSQ and $M/M/n$ systems. In these regimes, both the system size and the system load increase simultaneously. Our results also close a gap in the existing literature on the limiting distribution of JSQ in the super-NDS (a.k.a. super slowdown) regime. This contribution is of an independent interest. Here, a key ingredient is a more refined characterization of state space collapse for JSQ system, achieved by using an exponential Lyapunov function designed to approximate the $\ell_{\infty}$ norm.