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

Zedong Wang, Siva Theja Maguluri

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.

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

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 , the centered-scaled queue length is known to converge in distribution to a Gaussian. However, such asymptotic limits do not quantify the pre-limit tail for fixed . 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 together with a pre-limit error that vanishes as , yielding the correct Gaussian tail in the limit. We establish such an efficient bound that is best-of-both-worlds. For larger deviations when is a function of , 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- 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- 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 -th moment bound on the orthogonal component of the queue length vector that is essential for our Wasserstein- bound.
Paper Structure (95 sections, 23 theorems, 312 equations, 1 figure, 3 tables)

This paper contains 95 sections, 23 theorems, 312 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Organization of the Paper
  4. Model and Assumptions
  5. Modeling and Notation
  6. Notation
  7. Assumptions
  8. Remarks
  9. Summary and Analogy of Results
  10. Ratio Convergence
  11. Larger Deviation
  12. Main Result
  13. Tail Bound for Single Server Queue with Abandonment
  14. Constant Deviation
  15. Large Deviation
  16. ...and 80 more sections

Key Result

Proposition 3.1

We assume the heavily overloaded regime ass:heavy_overload with $\gamma\leq \gamma_0$ (see eq: gamma assumption for SSQ W-p). We denote $\tilde{q}:=\frac{\sqrt{\gamma}}{\sqrt{\lambda}}(q-\frac{\lambda-\mu}{\gamma})$. Then , with $a_\gamma = D_{\delta}/\gamma^\delta$ for appropriate choice of constan

Figures (1)

  • Figure 1: Phase Transition of Tail Bounds for SSQ, with $x$-axis representing $\delta$ in $a_\gamma = \Theta(1/\gamma^\delta)$.

Theorems & Definitions (30)

  • Proposition 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3: Tail Bounds for Load Balancing Policy
  • Proposition 4.4: Refinement for Large Deviation
  • Theorem 4.5: State Space Collapse for JSQ
  • Theorem 4.6
  • Lemma 5.1
  • Definition 5.3: Test functions
  • Lemma 5.5: Identity for Kramers-Moyal coefficients
  • ...and 20 more