We study an optimal choice of the buffer size in the Internet routers. The objective is to determine the minimum value of the buffer size required in order to fully utilize the link capacity. The reare some empirical rules for the choice of the buffer size. The most known rule of thumb states that the buffer length should be set to the bandwidth delay product of the network, i.e., the product between the average round trip time in the network and the capacity of the bottleneck link. Several recent works have suggested that as a consequence of the traffic aggregation, the buffer size should be set to smaller values. In this paper we propose an analytical framework for the optimal choice of the router buffer size. We formulate this problem as a multi-criteria optimization problem, in which the Lagrange function corresponds to a linear combination of the average sending rate and average delay in the queue. The solution to this optimization problem provides further evidence that indeed the buffer size should be reduced in the presence of traffic aggregation. Furthermore, our result states that the minimum required buffer is smaller than what previous studies suggested. Our analytical results are confirmed by simulations performed with the NS simulator.