Steady-state Diffusion Approximations for Discrete-time Queue in Hospital Inpatient Flow Management

In this paper, we analyze a discrete-time queue that is motivated from studying hospital inpatient flow management, where the customer count process captures the midnight inpatient census. The stationary distribution of the customer count has no explicit form and is difficult to compute in certain parameter regimes. Using the Stein's method framework, we identify a continuous random variable to approximate the steady-state customer count. The continuous random variable corresponds to the stationary distribution of a diffusion process with state-dependent diffusion coefficients. We characterize the error bounds of this approximation under a variety of system load conditions -- from lightly loaded to heavily loaded. We also identify the critical role that the service rate plays in the convergence rate of the error bounds. We perform extensive numerical experiments to support the theoretical findings and to demonstrate the approximation quality. In particular, we show that our approximation performs better than those based on constant diffusion coefficients when the number of servers is small, which is relevant to decision making in a single hospital ward.