  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.
