A Two-Time-Scale Approach to Time-Varying Queues in Hospital Inpatient Flow Management

We analyze a time-varying M-peri/Geo(2timeScale)/N queueing system. The arrival process is periodic Poisson. The service time of a customer has components in different time scales: length of stay (LOS) in days and departure time (h(dis)) in hours. This queueing system has been used to study patient flows from the emergency department (ED) to hospital inpatient wards. In that setting, the LOS of a patient is simply the number of days she spends in a ward, and her departure time h(dis) is the discharge hour on the day of her discharge. We develop a new analytical framework that can perform exact analysis on this novel queueing system. This framework has two steps: first analyze the midnight customer count process and obtain its stationary distribution, then analyze the time-dependent customer count process to compute various performance measures. We also develop approximation tools that can significantly reduce the computational time. In particular, via Stein's method, we derive explicit expressions to approximate the stationary distribution of the midnight count. We provide error bounds for these approximations and numerically demonstrate that they are remarkably accurate for systems with various sizes and load conditions. Our theoretical and numerical analysis have produced a number of insights that can be used to improve hospital inpatient flow management. We find that the LOS term affects the overnight wait caused by the mismatch between daily arrivals and discharges, whereas the h(dis) term affects the intraday wait caused by the nonsynchronization between the arrival and discharge time patterns. Thus, reducing LOS or increasing capacity can impact the daily average performance significantly; shifting the discharge timing to earlier times of a day can alleviate the peak congestion in the morning and mainly affects the time-dependent performance.