Abstract We study a discrete‐time queueing network with blocking that is primarily motivated by outpatient network management. To tackle the curse of dimensionality in performance analysis, we develop a refined mean‐field approximation that deals with changing population size, a nonconventional feature that makes the analysis challenging within the existing literature. We explicitly quantify the convergence rate for this approximation as with being the system size. Not only is this convergence better than the convergence proven in prior work, but our approximation shows a significant improvement in performance prediction accuracy when the system size is small, compared to the conventional (unrefined) mean‐field approximation. This accuracy makes our approximation appealing to support decision‐making in practice.