We focus on fairness and stability of the congestion control mechanisms adopted in several versions of TCP by investigating their time-transient behavior through an analytic approach. In addition to TCP Tahoe, TCP Reno, and TCP Vegas, we consider enhanced TCP Vegas which is proposed in this paper for fairness enhancements. We consider the homogeneous case, where two connections have the equivalent propagation delays, and the heterogeneous case, where each connection has different propagation delay. We show that TCP Tahoe and TCP Reno can achieve fairness among connections in the homogeneous case, but cannot in heterogeneous case. We also show that TCP Vegas can provide almost fair service among connection, but there is some unfairness caused by the essential nature of TCP Vegas. Finally, we explain the effectiveness of our enhanced TCP Vegas in terms of fairness and throughput.