Physician Staffing in Emergency Rooms (ERs) : Opening the Black-box of ER Care via a Multi-Class Multi-Stage Network

Problem Definition and Relevance: Characterized by time-varying arrivals, multi-stage service, and multi-class patient population, emergency rooms (ERs) are complex healthcare delivery systems, where optimizing the staffing levels of physicians is a challenge. In collaboration with Mayo Clinic, we study a staffing and an associated routing problem for ER physicians and propose a new staffing rule to meet tail probability of delay (TPoD) type service targets. Methodology: We capture the time-varying patient flow in the ER with a multi-class multi-stage queuing network, describing the ER care as sequences of treatment queues, where physicians serve, and groups of diagnostic medical processes. Treatment stations in ER are busy-server queues, where waiting before service is common, but experience negligible abandonments. Motivated by these queues, we propose a new staffing algorithm, translating the offered load into staffing decisions for efficiency-driven queues with TPoD targets and perfectly patient customers. Results: We analytically show the asymptotic effectiveness of our staffing rule on stabilizing TPoD for M/M/s queues, operating in efficiency-driven mode, and numerically demonstrate its robustness and optimality in various time-varying ER settings via realistic and data-driven simulation experiments. We further show that as the service complexity of an ER increases, hybrid routing rules, using pre-determined (static) priorities and (dynamic) current system state jointly, become necessary to meet TPoD targets. Managerial Implications: Instead of treating the entire patient length of stay as a black-box service model, our multi-stage network model provides more managerial control over the internal components of ER service and is easily implementable in practice