Economies-of-scale in many-server queueing systems: tutorial and partial review of the Qed Halfin-Whitt heavy-traffic regime

Multiserver queueing systems describe situations in which users require service from multiple parallel servers. Examples include check-in lines at airports, waiting rooms in hospitals, queues in contact centers, data buffers in wireless networks, and delayed service in cloud data centers. These are all situations with jobs (clients, patients, tasks) and servers (agents, beds, processors) that have large capacity levels, ranging from the order of tens (checkouts) to thousands (processors). This survey investigates how to design such systems to exploit resource pooling and economies-of-scale. In particular, we review the mathematics behind the quality- and efficiency-driven (QED) regime, which lets the system operate close to full utilization, while the number of servers grows simultaneously large and delays remain manageable. Aimed at a broad audience, we describe in detail the mathematical concepts for the basic Markovian many-server system, and we provide only sketches or references for more advanced settings related to, e.g., load balancing, overdispersion, parameter uncertainty, general service requirements, and queueing networks. While serving as a partial survey of a massive body of work, the tutorial is not meant to be exhaustive.