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：我们研究一个严重过载的、具有弃置行为的单服务器队列，并推导了队列长度稳态尾概率的界。当弃置率 $γ\downarrow 0$ 时，已知中心化缩放后的队列长度 $\tilde{q}$ 依分布收敛于一个高斯分布。然而，此类渐近极限无法量化固定 $γ>0$ 时的极限前尾部概率 $\mathbb{P}(\tilde{q}>a)$。我们的目标是获得在不同偏差机制下均具有 \emph{高效性} 的极限前界。对于常数偏差，高效性意味着关于 $a$ 的高斯型衰减，以及一个随 $γ\downarrow 0$ 而消失的极限前误差，从而在极限下得到正确的高斯尾部。我们建立了这样一个"两全其美"的高效界。对于当 $a$ 是 $γ$ 的函数时的更大偏差，高效性转化为指数级紧致的、匹配的上界和下界。对于中度偏差，我们得到了亚高斯尾部，而在大幅偏差机制下，衰减变为亚泊松型。我们的界是通过结合用于 Wasserstein-$p$ 距离的 Stein 方法和变换方法得到的。随后，我们考虑一个具有异质服务器的弃置队列负载均衡系统，在严重过载机制下运行于加入最短队列（JSQ）策略。与单服务器队列的情况类似，我们再次获得了相对于高斯分布的 Wasserstein-$p$ 界，以及常数和中度偏差下的高效集中性。对于更大偏差，我们的 JSQ 上界展现出从高斯型衰减到亚威布尔衰减的转变。所有这些结果都是使用 Stein 方法获得的。此外，这里的一个关键要素是建立状态空间坍缩（SSC），即所有队列变得相等。我们为队列长度向量的正交分量建立了一个 $p$ 阶矩界，这对于我们的 Wasserstein-$p$ 界至关重要。