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Telemedical Physician Triage (TPT) is an example of a Hierarchical Knowledge-Based Service System (HKBSS), in which a second level of decision agent (telemedical physician) renders a decision on cases referred to him/her by the primary level agents (triage nurses). Managing the speed-versus-quality tradeoff in such systems presents a unique challenge because of the interplay between agent knowledge and flow of work between the two levels. We develop a novel model of agent knowledge, based on the beta distribution, and deploy it in a Partially Observable Markov Decision Process (POMDP) model to describe the optimal policy for deciding which cases (patients) to refer to the second level for further evaluation. We show that this policy has a monotone control-limit structure that reduces the fraction of decisions made at the upper level as workload increases. Because the optimal policy is complex, we use structural insights from it to design two practical heuristics. These heuristics enable an HKBSS to adapt efficiently to workload shifts by adjusting the criteria for referring decisions to the upper level based on partial real-time queue length information. Finally, we conduct analytic and numerical analyses to derive insights into the management of a TPT system. We ?nd that (1) the telemedical physician should evaluate more patients as congestion in the emergency room waiting area increases,(2) training that improves accuracy of the physician and/or nurses can be effective even if it only does so for a single patient type, but training that improves consistency must do so for all patient types to be effective, and (3) patient classification in triage should consider environmental and operational conditions in addition to the patient’s medical condition.