The median problem has been generalized to include queueing-like congestion of facilities, which are assumed to have finite numbers of servers. In one statement of the problem, a closest available server is assumed to handle each service request. More general server assignment policies are allowed, however. The analysis requires keeping track of the states (available or unavailable) of all servers. Paralleling the standard deterministic median problem, the objective is to minimize the expected travel time associated with a random service request, weighted appropriately by the equilibrium state probabilities of the system. Under suitable conditions, it is known that at least one set of optimal locations exists solely on the nodes of the network. This analysis ties together previously disparate efforts in network analysis and spatial queueing analysis. Twelve footnotes are provided. (Author abstract modified)