In many different settings, requests for service can arrive in near or true simultaneity with one another. This creates batches of arrivals to the underlying queueing system. In this paper, we study the staffing problem for the batch arrival queue. We show that batches place a dangerous and deceptive stress on services, requiring a high amount of resources and exhibiting a fundamentally larger tail in those demands. This uncovers a service regime in which a system with large batch arrivals may have low utilization but will still have non-trivial waiting. Methodologically, these staffing results follow from novel large batch and large batch-and-rate limits of the multi-server queueing model. In the large batch limit, we establish the first formal connection between general multi-server queues and storage processes, another family of stochastic models. By consequence, we show that the batch scaled queue length process is not asymptotically normal, and that, in fact, the fluid and diffusion-type limits coincide. Hence, the (safety) staffing of this system must be directly proportional to the batch size just to achieve a non-degenerate probability of wait. In exhibition of the existence and insights of this large batch regime, we apply our results to data on Covid-19 contact tracing in New York City. In doing so, we identify significant benefits produced by the tracing agency's decision to staff above national recommendations, and we also demonstrate that there may have been an opportunity to further improve the operation by optimizing the arrival pattern in the public health data pipeline.

翻译：在许多不同场景中，服务请求可能几乎同时或完全同时到达，这会对底层排队系统形成批量到达。本文研究了批量到达排队系统的员工配置问题。研究表明，批量到达会给服务带来危险且具有欺骗性的压力，不仅需要大量资源，还会导致需求尾部显著增大。这一现象揭示了一种服务模式：在批量到达较大的系统中，即便利用率较低，仍可能出现不可忽视的等待时间。在方法论层面，这些员工配置结论源于对多服务台排队模型的新颖大规模批量极限与大规模批量-到达率极限分析。在大规模批量极限下，我们首次建立了通用多服务台排队系统与存储过程（另一类随机模型）之间的形式化联系。由此证明，经批量规模标度的队长过程不具有渐近正态性，事实上其流体极限与扩散极限是重合的。因此，该系统的（安全）员工配置必须与批量规模成正比，才能实现非退化的等待概率。为展示大规模批量极限的存在性与洞见，我们将研究成果应用于纽约市新冠接触追踪数据。通过分析，我们识别出追踪机构按高于国家建议标准配置员工所产生的重要效益，同时证明公共卫生数据管道中可能存在通过优化到达模式来进一步改善运营的机会。