TCP flow control based on an instantaneous model

With TCP the network congestion is controlled by adjusting the transmission rate through congestion window (cwnd). The cwnd either increases or decreases depending on ACKs (acknowledgements), which reflect the status of link´s buffer queues. This TCP dynamics has been modelled mathematically with various queue management principle of the link. However, the existing results are mostly average dynamic models. But, we raise a question about the validity of existing average models. The dynamics of TCP can be modelled by switching between the two sub-dynamic models: dynamics of congestion avoidance and dynamics of fast recovery. The existing average model would be valid only if one parameter groups of the vector fields for two sub-dynamic models commute. The commutativity is checked by Lie bracket operation, but the result shows that they are not commutative. We propose an instantaneous model for TCP congestion control algorithm with the use of a switching function. This proposed model describes the dynamic behavior of cwnd and queue length versus time. We check the validity of the model by comparing simulation result with that of ns simulator.