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.

翻译：我们研究了一个严重过载且带有放弃机制的单服务器队列，并推导了队列长度的平稳尾部概率界限。当放弃率 $γ\downarrow 0$ 时，中心化-缩放后的队列长度 $\tilde{q}$ 已知依分布收敛于高斯分布。然而，此类渐近极限无法量化固定 $γ>0$ 下的预极限尾部概率 $\mathbb{P}(\tilde{q}>a)$。我们的目标是获得在不同偏差体制下\textit{高效}的预极限界限。对于恒定偏差，高效性意味着 $a$ 呈高斯型衰减，且预极限误差随 $γ\downarrow 0$ 消失，从而在极限中得到正确的高斯尾部。我们建立了这样一个兼取两者之长的高效界限。对于更大的偏差（即 $a$ 为 $γ$ 的函数），高效性转化为指数紧的、匹配的上界和下界。在中等偏差下，我们得到亚高斯尾部；而在大偏差体制下，衰减变为亚泊松型。这些界限是通过结合 Wasserstein-$p$ 距离的 Stein 方法和变换方法得到的。随后，我们考虑一个由异质服务器组成的放弃队列负载均衡系统，该系统在严重过载体制下采用最短队列（JSQ）策略。与单服务器队列类似，我们再次得到相对于高斯分布的 Wasserstein-$p$ 界限，以及恒定和中等偏差下的高效集中性。对于更大的偏差，我们的 JSQ 上界显示出从高斯型衰减到亚威布尔型衰减的转变。所有这些结果都是通过 Stein 方法获得的。此外，这里一个关键要素是建立状态空间坍缩（SSC），使得所有队列变得相等。我们建立了队列长度向量正交分量的 $p$ 阶矩界限，这对于我们的 Wasserstein-$p$ 界限至关重要。