TCP is max-plus linear and what it tells us on its throughput

We give a representation of the packet-level dynamical behavior of the Reno and Tahoe variants of TCP over a single end-to-end connection. This repres entation allows one to consider the case when the connection involves a net work made of several, possibly heterogeneous, deterministic or random route rs in series. It is shown that the key features of the protocol and of the network can be expressed via a linear dynamical system in the so called max -plus algebra. This opens new ways of both analytical evaluation and fast s imulation based on products of matrices in this algebra. This also leads to closed form formulas for the throughput allowed by TCP under natural assum ptions on the behavior of the routers and on the detection of losses and ti meouts; these new formulas are shown to refine those obtained from earlier models which either assume that the network could be reduced to a single bo ttleneck router and/or approximate the packets by a fluid.