In this paper we address the problem of fast and fair transmission of flows 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-flow problem into a family of single-flow 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 flows can be optimally controlled in routers by so-called index policies, but not always by threshold policies. We have also implemented index policies in Network Simulator-3 and tested in a simple topology their applicability in real networks. The simulation results show that the index policy covers a big range of desirable properties with respect to fairness between different versions of TCP models, across users with different round-trip-time and minimum buffer required to achieve full utility of the queue.