On Fair Routing from Emergency Departments to Hospital Wards: QED Queues with Heterogeneous Servers

The process of patients’ routing from an Emergency Department (ED) to Internal Wards (IWs) is often a hospital’s bottleneck. Motivated by this process in Anonymous Hospital, we analyze queueing systems with heterogeneous server pools, where the pools represent the wards and servers are beds. Our queueing system, with a single centralized queue and several server pools, forms an inverted-V model. We introduce the Randomized Most-Idle (RMI) routing policy and analyze it in the QED (Quality and Efficiency Driven) regime, which is natural in our setting. The RMI policy results in the same server fairness (measured by idleness ratios) as the Longest-Idle Server First (LISF) policy, which is commonly used in call centers and considered fair. However, RMI utilizes only the information on the number of idle servers in different pools while LISF requires information that is unavailable in hospitals on a real-time basis.