An efficient buffer-management algorithm is developed for queues that handle distortion-tolerant data under finite memory limitations. We avoid overflows and realize significant performance gains through the use of multiresolution source codes. These codes enable us to reduce the fidelity of signal descriptions in a controlled progressive manner. The proposed approach is universal, i.e., it works without knowledge of queue arrival and departure statistics. More strongly, we show that its performance is sample-path optimal, i.e., it achieves an average distortion equal to the best achievable by any algorithm, including those designed with full noncausal knowledge of queue arrival and service times.