A model of a switching component in a packet switching network is considered. Packets from several incoming channels arrive and must be routed to the appropriate outgoing port according to a service policy. A task confronting the designer of such a system is the selection of policy and the determination of the corresponding input buffer requirements which will prevent packet loss. One natural choice is the Longest Queue First discipline, and a tight bound on the size of the largest buffer required under this policy is obtained. The bound depends on the channel speeds and is logarithmic in the number of channels. As a consequence, Longest Queue First is shown to require less storage than Exhaustive Round Robin and First Come First Served in preventing packet overflow.