We study the performance of several TCP connections subjected to the bottleneck of a slow network, accessed via a single queue with high but finite capacity. Using the mean-field methodology and fluid approximation, we establish asymptotic results on the queue length distribution and the window size distribution. We prove that the difference between the buffer capacity and the actual queue length tends to be exponentially distributed. Based on these results, we analyze the autocorrelation function of a single TCP connection, and we show that several TCP connections with heavy tailed round trip delays generate traffic with long term dependencies.