Control of Patient Flow in Emergency Departments, or Multiclass Queues with Deadlines and Feedback

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 who are in process (IP) and 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, whereas 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 G cμ 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.