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This paper investigates the effect of changes in router buffer size on throughput and fairness characteristics of high-speed variants of TCP. Whereas for fairness, round-triptime (R)-bias has been focussed. A generalized model for R-bias has been developed, while using Droptail and Random Early Detection (RED) routers,, at a bottleneck link. An existing optimal buffer sizing scheme has been also generalized to include the case of high-speed variants of TCP. In addition, through NS-2 based simulations, we have found that R-bias for most of the high-speed TCPs, under Droptail, increases as buffer size grows; whereas Hamilton TCP and Compound TCP have shown the best performance under smaller buffer size. On the other hand, while using a RED router, R-bias for all the high-speed TCPs has been shown to be much less than Droptail, though with a slight drop in total throughput of all TCP connections. An interesting observation is that R-bias for HighSpeed TCP and Scalable TCP, initially increases as buffer size grows, which later on start decreasing and reach almost to negligible values for the larger size of buffers for RED router; thus providing a quite good intra-protocol fairness. Another contribution of this paper is that R-bias in high-speed TCPs can be effectively controlled by employing delay-loss based congestion avoidance,, as well as by active queue management schemes, such as RED.