The development of personalized medical systems should be supported by a fast and stable network system. The FAST TCP network system is the appropriate support system for this purpose. However, when the FAST TCP is deployed, the static mapping selection method for protocol parameters is unable to guarantee the small queuing delay and fast convergence of the network simultaneously. By conducting theoretical analysis and simulation experiments, the relationships among FAST TCP protocol slow start condition, control law gain parameters, and FAST TCP system convergence rate were examined. To ensure the stability of the FAST TCP system and to select the smallest protocol parameters, an improved method to effectively accelerate the convergence velocity of the FAST TCP system is proposed in this study. In this method, the number of packets for staying in the buffer for FAST TCP connections was taken as the criterion of the slow start, and the gain parameter of the control law was dynamically adjusted according to the local information of each FAST TCP connection. Using this improved method, the FAST TCP system can achieve a stable and small queuing delay, whilst the FAST TCP system could converge quickly to the equilibrium point simultaneously.