A duality model of TCP and queue management algorithms

We propose a duality model of end-to-end congestion control and apply it to understanding the equilibrium properties of TCP and active queue management schemes. The basic idea is to regard source rates as primal variables and congestion measures as dual variables, and congestion control as a distributed primal-dual algorithm over the Internet to maximize aggregate utility subject to capacity constraints. The primal iteration is carried out by TCP algorithms such as Reno or Vegas, and the dual iteration is carried out by queue management algorithms such as DropTail, RED or REM. We present these algorithms and their generalizations, derive their utility functions, and study their interaction

Equilibrium bandwidth and buffer allocations for elastic traffics

Consider a set of users sharing a network node under an allocation scheme that provides each user with a fixed minimum and a random extra amount of bandwidth and buffer. Allocations and prices are adjusted to adapt to resource availability and user demands. Equilibrium is achieved when all users optimize their utility and demand equals supply for nonfree resources. We analyze two models of user behavior. We show that at equilibrium expected return on purchasing variable resources can be higher than that on fixed resources. Thus users must balance the marginal increase in utility due to higher return on variable resources and the marginal decrease in utility due to their variability. For the first user model we further show that at equilibrium where such tradeoff is optimized all users hold strictly positive amounts of variable bandwidth and buffer. For the second model we show that if both variable bandwidth and buffer are scarce then at equilibrium every user either holds both variable resources or none

Convex Relaxation of Optimal Power Flow, Part II: Exactness

This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.Comment: Citation: IEEE Transactions on Control of Network Systems, June 2014. This is an extended version with Appendex VI that proves the main results in this tutoria

Convex Relaxation of Optimal Power Flow, Part I: Formulations and Equivalence

This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.Comment: Citation: IEEE Transactions on Control of Network Systems, 15(1):15-27, March 2014. This is an extended version with Appendices VIII and IX that provide some mathematical preliminaries and proofs of the main result

Optimization flow control with Newton-like algorithm

We proposed earlier an optimization approach to reactive flow control where the objective of the control is to maximize the aggregate utility of all sources over their transmission rates. The control mechanism is derived as a gradient projection algorithm to solve the dual problem. In this paper we extend the algorithm to a scaled gradient projection. The diagonal scaling matrix approximates the diagonal terms of the Hessian and can be computed at individual links using the same information required by the unscaled algorithm. We prove the convergence of the scaled algorithm and present simulation results that illustrate its superiority to the unscaled algorithm

Reverse Engineering TCP/IP-like Networks using Delay-Sensitive Utility Functions

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 it exists, maximizes the same delay-insensitive utility over both source rates and routes, provided pure congestion prices are used as link costs in the shortest-path calculation of IP. In practice, however, pure dynamic routing is never used and link costs are weighted sums of both static as well as dynamic components. In this paper, we introduce delay-sensitive utility functions and identify a class of utility functions that such a TCP/IP equilibrium optimizes. We exhibit some counter-intuitive properties that any class of delay-sensitive utility functions optimized by TCP/IP necessarily possess. We prove a sufficient condition for global stability of routing updates for general networks. We construct example networks that defy conventional wisdom on the effect of link cost parameters on network stability and utility