Packet loss due to network congestion causes degradation in the quality of networked video transmitted over IP networks. Previous buffer management methods have been designed to prevent network congestion in order to reduce packet loss. However, a low packet loss ratio by itself does not necessarily translate to high video quality, so these methods do not ensure user-expected video quality; and hence, there is a need for alternative approaches to achieve user-expected video quality. This paper proposes a new buffer management scheme that focuses on achieving user-expected video quality, rather than just aiming at reducing packet loss. The proposed scheme allows the videos that share an output buffer to be guaranteed an appropriate share of the buffer when the buffer faces overflow, yet allows a complete sharing of the buffer space when it is not overflowing. The scheme also provides a mechanism that gives parts of a video lower loss than other parts. Simulation experiments show that the Quality of Service requirements of multiple videos and effective utilization of network resources can both be achieved.