A Queueing Theoretic Approach to Set Staffing Levels in TimeDependent DualClass Service Systems<link href=#deci12236-note-0001/>

This article addresses the optimal staffing problem for a nonpreemptive priority queue with two customer classes and a time-dependent arrival rate. The problem is related to several important service settings such as call centers and emergency departments where the customers are grouped into two classes of “high priority” and “low priority,” and the services are typically evaluated according to the proportion of customers who are responded to within targeted response times. To date, only approximation methods have been explored to generate staffing requirements for time-dependent dual-class services, but we propose a tractable numerical approach to evaluate system behavior and generate safe minimum staffing levels using mixed discrete-continuous time Markov chains (MDCTMCs). Our approach is delicate in that it accounts for the behavior of the system under a number of different rules that may be imposed on staff if they are busy when due to leave and involves explicitly calculating delay distributions for two customer classes. Ultimately, we embed our methodology in a proposed extension of the Euler method, coined Euler Pri, that can cope with two customer classes, and use it to recommend staffing levels for the Welsh Ambulance Service Trust (WAST). [Submitted: January 29, 2015. Revised: May 10, 2016. Accepted: May 16, 2016.] Subject Areas: Exhaustive Service Discipline, Markov Chain, Numerical Methods, Priority Queues, Staffing, Time-Dependent Queues, and Waiting Time Distribution. ∗This research was supported by GS501100000266 EPSRC http://dx.doi.org/10.13039/501100000266 grant GS501100000266 EP/F033338/1 as part of the LANCS initiative. This research was also funded by EPSRC grant EP/F033338/1 (part of the LANCS initiative) and the data underpinning the project was provided by the Welsh Ambulance Services NHS Trust (WAST). The authors would particularly like to thank WAST Clinical R&D Manager Richard Whitfield for his useful feedback at various stages of this research and the anonymous referees for their helpful comments and insights on this paper †Corresponding author. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.