We consider the work flow in a medical teaching facility, examining the process that involves an initial patient exam by a resident physician, a subsequent conference between the resident and the attending physician, and the attending physician's visit with the patient. We create an analytical model of a tandem queue with finite buffer space to analyze the impact of different work prioritization policies on the throughput and the flow time of patients in the facility—measures that influence both the facility's finances and patients' satisfaction. We derive throughput-optimal policies and show that these policies involve dynamic batching. This finding is interesting because our model does not include any setup times, and setup times normally imply batching; rather it is the uncertain service times and the requirement for simultaneous service in the conference step that make batching optimal. The optimal dynamic batching policy is complex, so we consider a simpler static batching policy. We show that, in systems with limited buffer space, large batches can sometimes degrade efficiency by simultaneously increasing flow time and decreasing throughput. However, in general, both flow time and throughput increase with batch size. Flow time increases at a faster rate than throughput, so hospital management may want to consider what batch size is optimal given the value it places on the two measures.