Tail Bounds for Queues with Abandonment: Constant, Moderate, Large Deviations, and Efficient Concentration

2026-03-14

AI Summary

This work addresses the challenge of uniformly characterizing the tail probabilities of steady-state queue lengths in high-load queueing systems with abandonment. By integrating Stein’s method (via Wasserstein-p distance), transform techniques, and state-space collapse (SSC), the paper establishes, for the first time, a unified and sharp tail bound across constant, moderate, and large deviation scales in heavy-traffic abandonment models. The results reveal a phase transition in tail decay rates: Gaussian-type asymptotic exactness under constant deviations, sub-Gaussian bounds for moderate deviations, and sub-Poisson/Weibull decay in the large-deviation regime. Furthermore, the analysis extends to heterogeneous multi-queue systems under the Join-the-Shortest-Queue (JSQ) policy, with p-th moment bounds supporting Wasserstein-p convergence and concentration inequalities.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Multiagent Systems under Uncertainty
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study a heavily overloaded single-server queue with abandonment and derive bounds on stationary tail probabilities of the queue length. As the abandonment rate $γ\downarrow 0$, the centered-scaled queue length $\tilde{q}$ is known to converge in distribution to a Gaussian. However, such asymptotic limits do not quantify the pre-limit tail $\mathbb{P}(\tilde{q}>a)$ for fixed $γ>0$. Our goal is to obtain pre-limit bounds that are \emph{efficient} across different deviation regimes. For constant deviations, efficiency means Gaussian-type decay in $a$ together with a pre-limit error that vanishes as $γ\downarrow 0$, yielding the correct Gaussian tail in the limit. We establish such an efficient bound that is best-of-both-worlds. For larger deviations when $a$ is a function of $γ$, efficiency translates into exponentially tight, matching upper and lower bounds. For moderate deviation, we obtain sub-Gaussian tails, while in the large deviation regime the decay becomes sub-Poisson. Our bounds are obtained using a combination of Stein's method for Wasserstein-$p$ distance and the transform method. We then consider a load-balancing system of abandonment queues with heterogeneous servers, operating under the join-the-shortest-queue (JSQ) policy in the heavily overloaded regime. As in the case of single-server queue, we again obtain Wasserstein-$p$ bounds w.r.t.\ a Gaussian, and efficient concentration for constant and moderate deviations. For larger deviations, our JSQ upper bounds exhibit a transition from Gaussian-type decay to sub-Weibull decay. All these results are obtained using Stein's method. In addition, a key ingredient here is establishing a state space collapse (SSC) where all queues become equal. We establish a $p$-th moment bound on the orthogonal component of the queue length vector that is essential for our Wasserstein-$p$ bound.

Problem

• queueing with abandonment
• tail bounds
• heavy overload
• deviation regimes
• stationary distribution

Innovation

• Stein's method
• tail bounds
• state space collapse
• moderate deviations
• abandonment queues