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In this paper, we analyze a discrete-time queue that is motivated fromstudying hospital inpatient flow management, where the customer count processcaptures the midnight inpatient census. The stationary distribution of thecustomer count has no explicit form and is difficult to compute in certainparameter regimes. Using the Stein's method framework, we identify a continuousrandom variable to approximate the steady-state customer count. The continuousrandom variable corresponds to the stationary distribution of a diffusionprocess with state-dependent diffusion coefficients. We characterize the errorbounds of this approximation under a variety of system load conditions -- fromlightly loaded to heavily loaded. We also identify the critical role that theservice rate plays in the convergence rate of the error bounds. We performextensive numerical experiments to support the theoretical findings and todemonstrate the approximation quality. In particular, we show that ourapproximation performs better than those based on constant diffusioncoefficients when the number of servers is small, which is relevant to decisionmaking in a single hospital ward.

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