In this work we address the problem of fast and fair transmission of ows in a router, which is a fundamental issue in networks like the Internet. We model the interaction between a TCP source and a bottleneck queue with the objective of designing optimal packet admission controls in the bottleneck queue. We focus on the relaxed version of the problem obtained by relaxing the fixed buffer capacity constraint that must be satisfied at all time epoch. The relaxation allows us to reduce the multi-ow problem into a family of single-ow problems, for which we can analyze both theoretically and numerically the existence of optimal control policies of special structure. In particular, we show that for a variety of parameters, TCP ows can be optimally controlled in routers by so-called index policies. We have implemented index policies in Network Simulator-3 (NS-3) and compared its performance with DropTail and RED buffers. The simulation results show that the index policy has several desirable properties with respect to fairness and efficiency.