TCP is max-plus linear and what it tells us on its throughput. - Baccelli, François (Computer communication review.)
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 representation allows one to consider the case when the connection involves a network made of several, possibly heterogeneous, deterministic or random routers 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 simulation based on products of matrices in this algebra. This also leads to closed form formulas for the throughput allowed by TCP under natural assumptions on the behavior of the routers and on the detection of losses and timeouts; these new formulas are shown to refine those obtained from earlier models which either assume that the network could be reduced to a single bottleneck router and/or approximate the packets by a fluid.