We study a basic problem in Multi-Queue switches. A switch connects m input ports to a single output port. Each input port is equipped with an incoming FIFO queue with bounded capacity B. A switch serves its input queues by transmitting packets arriving at these queues, one packet per time unit. Since the arrival rate can be higher than the transmission rate and each queue has limited capacity, packet loss may occur as a result of insufficient queue space. The goal is to maximize the number of transmitted packets. This general scenario models most current networks (e.g., IP networks) which only support a "best effort" service in which all packet streams are treated equally. A 2-competitive algorithm for this problem was designed in  for arbitrary B. Recently, a 17/9 ≈ 1.89-competitive algorithm was presented for B > 1 in . Our main result in this paper shows that for B which is not too small our algorithm can do better than 1.89, and approach a competitive ratio of e/e-1 ≈ 1.58.