Diffusion-Based Staffing for Multitasking Service Systems with Many Servers
研究了每个服务器可并行服务多个顾客的多任务排队系统，在重流量极限下推导出不同顾客分配策略的扩散极限，并指出将顾客分配给最不忙或最忙的服务器分别在服务率凹或凸时最优。
We consider a many-server queue in which each server can serve multiple customers in parallel. Such multitasking phenomena occur in various applications areas (e.g., in hospitals and contact centers), although the impact of the number of customers who are simultaneously served on system efficiency may vary. We establish diffusion limits of the queueing process under the quality-and-efficiency-driven scaling and for different policies of assigning customers to servers depending on the number of customers they serve. We show that for a broad class of routing policies, including routing to the least busy server, the same one-dimensional diffusion process is obtained in the heavy-traffic limit. In case of assignment to the most busy server, there is no state-space collapse, and the diffusion limit involves a custom regulator mapping. Moreover, we also show that assigning customers to the least (most) busy server is optimal when the cumulative service rate per server is concave (convex), motivating the routing policies considered. Finally, we also derive diffusion limits in the nonheavy-traffic scaling regime and in the heavy-traffic scaling regime where customers can be reassigned during service. Funding: The research of J. Storm is partly funded by the Netherlands Organization for Scientific Research (NWO) Gravitation project Networks [Grant 024.002.003].