Optimization and Stability of TCP/IP with Delay-Sensitive Utility Functions

Abstract

TCP/IP can be interpreted as a distributed primal-dual algorithm to maximize aggregate utility over source rates. It has recently been shown that an equilibrium of TCP/IP, if exists, maximize the same aggregate utility function over both source rates and routes, provided pure congestion prices are used as link costs in the shortest-path calculation of IP. Moreover, the utility functions are delay-insensitive, i.e., they are functions of rates only. We extend this result in several ways. First, we show that if utility functions are delay-insensitive, then there are networks for which TCP/IP optimizes aggregate utility only if routing is based on pure congestion prices. Routing based on the weighted sum of congestion prices and propagation delays optimizes aggregate utility for general networks only if the utility functions are delay-sensitive. Moreover, we identify such a class of delay-sensitive utility functions that is implicitly optimized by TCP/IP. As for the delay-insensitive case, we show for this class of utility functions, equilibrium of TCP/IP exists if and only if the optimization problem has zero duality gap. In that case, there is no penalty for not splitting the traffic. We exhibit some counter-intuitive properties of this class of utility functions. We also prove that any class of delay-sensitive utility functions that are optimized by TCP/IP necessarily possess some strange properties. We prove that, for general networks, if the weight on congestion prices is small enough, only minimum-propagation-delay paths are selected. Hence if all source-destination pairs have unique minimum-propagation-delay paths, then equilibrium of TCP/IP exists and is asymptotically stable. For general networks, their equilibrium properties are the same as a modified network where paths with non-minimum propagation delays are deleted and routing is based on pure congestion prices. It is commonly believed that there is generally an inevitable tradeoff between utility maximization and stability in TCP/IP networks. In particular, as the weight on congestion prices increases, the routing will change from stable to unstable. We exhibit a counterexample where routing changes from stable to unstable and then to stable again, as the weight on congestion prices increases. Moreover, one can construct a network with any given utility profile as a function of the weight on congestion prices.