Emergency Department Patient Flow Optimization with an Alternative Care Threshold Policy

Emergency department (ED) overcrowding and patient boarding represent critical systemic challenges that compromise care quality. We propose a threshold-based admission policy that redirects non-urgent patients to alternative care pathways, such as telemedicine, during peak congestion. The ED is modeled as a two-class $M/M/c$ preemptive-priority queuing system, where high-acuity patients are prioritized and low-acuity patients are subject to state-dependent redirection. Analyzed via a level-dependent Quasi-Birth-Death (QBD) process, the model determines the optimal threshold by maximizing a long-run time-averaged objective function comprising redirection-affected revenue and costs associated with patient balking and system occupancy. Numerical analysis using national healthcare data reveals that optimal policies are highly context-dependent. While rural EDs generally optimize at lower redirection thresholds, urban EDs exhibit performance peaks at moderate thresholds. Results indicate that our optimal policy yields significant performance gains of up to $4.84\%$ in rural settings and $5.90\%$ in urban environments. This research provides a mathematically rigorous framework for balancing clinical priority with operational efficiency across diverse ED settings.