Convergence of the fixed point algorithm of analytical models of reliable internet protocols (TCP)

Abstract

Analytical models are important tools for the performance investigation of the Internet. The literature shows that the fixed point algorithm (FPA) is one of the most useful ways of solving analytical models of Internet performance. Apart from what is observed in experimental literature, no comprehensive proof of the convergence and uniqueness of the FPA is given. In this paper we show how analytical models of reliable Internet protocols (TCP) converge to a unique fixed point. Unlike previous work in the literature the basic principles of our proof apply to both single and multiple bottleneck networks, to short and long-lived TCP connections and to both Drop Tail and Active Queue Management (AQM) routers. Our proof of convergence is based on a well known fixed point theorem and our uniqueness proof exploits the feedback nature of the reliable protocol. The paper specifies conditions under which the FPA of analytical models of TCP converges to a unique point. The concepts used in the proof can also be extended to analyze the equilibrium, stability and global uniqueness issues of TCP, other reliable protocols and the Internet as a whole. © Springer-Verlag Berlin Heidelberg 2006.