A simple fluid model for the joint throughput evolution of a set of TCP sessions sharing a common bottleneck router has been introduced in \citeaimd based on products of random matrices. This paper studies the asymptotics of the aimd model when the number of TCP sessions $N$ goes to infinity. We show that this limit process can be defined by a recurrence relation in $\R$ and we show how to characterize its stationary behavior very simply based on the recurrence relation. We study the accuracy of this limit model and the impact of the synchronization effect not only on the mean throughput but also on its QoS.LOC