We investigate the validity of reducing router buffer size in a large-scale network that includes both core networks and edge networks. We first devise a novel mathematical analysis method of estimating the average behavior of TCP connections in a network with 100/1,000/10,000 routers/endhosts/links and 100,000 concurrent TCP connections. By applying our analysis method to the Abilene-inspired network, we demonstrate the influence of small buffer on link utilization, packet loss ratio and the performance of TCP connections passing through the router. One important result is that, especially when the edge network becomes faster, decreasing buffer size at core routers causes unfairness between TCP connections that traverse the core network and those do not traverse the core network.