Dynamic Server Assignment in Multiclass Queues with Shifts, with Applications to Nurse Staffing in Emergency Departments

Many service systems are staffed by workers who work in shifts. In this article, we study the dynamic assignment of servers to different areas of a service system at the beginning of discrete time intervals, namely, shifts. The ability to reassign servers at discrete intervals, rather than continuously, introduces a partial flexibility that provides an opportunity for reducing the expected waiting time of customers. The problem is primarily motivated by an application to nurse staffing in emergency departments (EDs), where nurses can work in different areas of the ED, but their assignment can only be changed at the beginning of their shifts (typically 8-12 hours). To investigate the reassignment decision and its potential benefits, we consider a multiclass queueing system, where customers of each class differ in terms of their average service requirements and the holding cost incurred as they wait in the queues. We study a discrete-time fluid control problem to minimize transient holding costs over a finite horizon and show that an appropriate "translation" of the solution to the fluid control problem is asymptotically optimal for the original stochastic system. Through analysis of the fluid control problem we further obtain insights on the structure of "good" policies in the presence of the shift constraint. Leveraging these insights, we develop heuristic policies and use simulation to demonstrate their effectiveness in systems with dynamics often observed in EDs. We find that, in a parameter regime relevant to our motivating application, the partial flexibility introduced by reassigning servers at the beginning of shifts can substantially reduce the expected cost of the system-by 10%-50% in some parameter regimes-compared with the status quo, dedicated staffing.