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We consider the control of patient flow through physicians in emergency departments (EDs). The physicians must choose between catering to patients right after triage, who are yet to be checked, and those that are in-process (IP), who are occasionally returning to be checked. Physician capacity is thus modeled as a queueing system with multiclass customers, where some of the classes face deadline constraints on their time-till-first-service, while the other classes feedback through service while incurring congestion costs. We consider two types of such costs: first, costs that are incurred at queue-dependent rates, and second, costs that are functions of IP sojourn time. The former is our base-model, which paves the way for the latter (perhaps more ED-realistic). In both cases, we propose and analyze scheduling policies that, asymptotically in conventional heavy-traffic, minimize congestion costs while adhering to all deadline constraints. Our policies have two parts: the first chooses between triage and IP patients; assuming triage patients are chosen, the physicians serve the one who is closest to violating the deadline; alternatively, IP patients are served according to a Gcµ rule, in which µ is simply modified to account for feedbacks. For our proposed policies, we establish asymptotic optimality, and develop some congestion laws (snapshot principles) that support forecasting of waiting and sojourn times. Simulation then shows that these policies outperform some commonly-used ones. It also validates our laws and demonstrates that some ED features, the complexity of which reaches beyond our model (e.g., time-varying arrival rates, Leave-Without-Being-Seen (LWBS) or Leave-Against-Medical-Advice (LAMA)), do not lead to significant performance degradation.

We revisit many-server approximations for the well-studied Erlang-A queue. This is a system with a single pool of i.i.d. servers that serve one class of impatient i.i.d. customers. Arrivals follow a Poisson process and service times are exponentially distributed as are the customers' patience times. We propose a diffusion approximation that applies simultaneously to all existing many-server heavy-traffic regimes: quality and efficiency driven, efficiency driven, quality driven, and nondegenerate slowdown. We prove that the approximation provides accurate estimates for a broad family of steady-state metrics. Our approach is "metric-free" in that we do not use the specific formulas for the steady-state distribution of the Erlang-A queue. Rather, we study excursions of the underlying birth-and-death process and couple these to properly defined excursions of the corresponding diffusion process. Regenerative process and martingale arguments, together with derivative bounds for solutions to certain ordinary differential equations, allow us to control the accuracy of the approximation. We demonstrate the appeal of universal approximation by studying two staffing optimization problems of practical interest.

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