The Transmission Control Protocol (TCP) is one of the main protocols of
the Internet protocol suite and major internet applications rely on it. The TCP
protocol provides reliability, flow control and congestion control. Alongside the
TCP, the User Datagram Protocol (UDP) is another transport protocol which, in
contrast to TCP, is a simplified request-response protocol that does not have any
connection setup time and does not provide any flow, congestion or error controls.
We consider in this presentation a fluid model for multiple TCP and UDP
connections interacting through a network of queues. We suppose that the
connections are randomly routed according to a dynamical routing table protocol
which takes into account the topology of the network and adapts the routing
dynamically. Our model extends the multi-class model studied in Graham et al
(2009). The dynamic of the TCP flows follows the additive increase/multiplicative-
decrease (AIMD) protocol and is represented by a stochastic differential equation
w.r.t. a Poisson random measure and the UDP flows are represented by simple point
processes. Using an adequate scaling, a mean-field result is proved where, as the
number of connections goes to infinity, the behaviour of the different connections
can be represented by the solution of an original nonlinear stochastic differential
equation. The existence and uniqueness of the solution of this equation are derived.
Moreover, we discuss some open problems and possible extensions. This talk is
based on a current ongoing joint work with Donald A. Dawson and Yiqiang Q. Zhao. |