We propose an optimization approach to flow control where the objective is to maximize the aggregate source utility over their transmission rates. We view network links and sources as processors of a distributed computation system to solve the dual problem using gradient projection algorithm. In this system sources select transmission rates that maximize their own benefits, utility minus bandwidth cost, and network links adjust bandwidth prices to coordinate the sources' decisions. We allow feedback delays to be different, substantial and time-varying, and links and sources to update at different times and with different frequencies. We provide asynchronous distributed algorithms and prove their convergence in a static environment. We present measurements obtained from a preliminary prototype to illustrate the convergence of the algorithm in a slowly time-varying environment.

We propose a duality model of congestion control and apply it to understand the equilibrium properties of TCP and active queue management schemes. Congestion control is the interaction of source rates with certain congestion measures at network links. 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 carried out 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 such as DropTail, RED or REM. We present these algorithms and their generalizations, derive their utility functions, and study their interaction.

We formulate and solve a problem of allocating resources among competing services differentiated by user traffic characteristics and maximum end-to-end delay. The solution leads to an alternative approach to service provisioning in an ATM network, in which the network offers directly for rent its bandwidth and buffers and users purchase freely resources to meet their desired quality. Users make their decisions based on their own traffic parameters and delay requirements and the network sets prices for those resources. The procedure is iterative in that the network periodically adjusts prices based on monitored user demand, and is decentralized in that only local information is needed for individual users to determine resource requests. We derive network's adjustment scheme and users' decision rule and establish their optimality. Since our approach does not require the network to know user traffic and delay parameters, it does not require traffic policing on the part of the network. 1 I...

Abstract — TCP–AQM can be interpreted as distributed primal-dual algorithms to maximize aggregate utility over source rates. We show that an equilibrium of TCP/IP, if exists, maximizes aggregate utility over both source rates and routes, provided congestion prices are used as link costs. An equilibrium exists if and only if this utility maximization problem and its Lagrangian dual have no duality gap. In this case, TCP/IP incurs no penalty in not splitting traffic across multiple paths. Such an equilibrium, however, can be unstable. It can be stabilized by adding a static component to link cost, but at the expense of a reduced utility in equilibrium. If link capacities are optimally provisioned, however, pure static routing, which is necessarily stable, is sufficient to maximize utility. Moreover single-path routing again achieves the same utility as multipath routing at optimality. Index Terms — Utility optimization, congestion control, TCP-

We propose earlier an optimization based flow control for the Internet called Random Early Marking (REM). In this paper we propose and evaluate an enhancement that attempts to speed up the convergence of REM in the face of large feedback delays. REM can be regarded as an implementation of an optimization algorithm in a distributed network. The basic idea is to treat the optimization algorithm as a discrete time system and apply linear control techniques to stabilize its transient. We show that the modified algorithm is stable globally and converges exponentially locally. This algorithm translates into an enhanced REM scheme and we illustrate the performance improvement through simulation. Keywords---Internet flow control, pricing, optimization flow control, marking, REM, feedback delay I. INTRODUCTION We proposed earlier a flow control scheme for the Internet called Random Early Marking (REM) [1]. It is derived from an optimization model where each source is characterized by a utili...

Network protocols in layered architectures have historically been obtained on an ad hoc basis, and many of the recent cross-layer designs are conducted through piecemeal approaches. They may instead be holistically analyzed and systematically designed as distributed solutions to some global optimization problems. This paper presents a survey of the recent efforts towards a systematic understanding of “layering ” as “optimization decomposition”, where the overall communication network is modeled by a generalized Network Utility Maximization (NUM) problem, each layer corresponds to a decomposed subproblem, and the interfaces among layers are quantified as functions of the optimization variables coordinating the subproblems. There can be many alternative decompositions, each leading to a different layering architecture. This paper summarizes the current status of horizontal decomposition into distributed computation and vertical decomposition into functional modules such as congestion control, routing, scheduling, random access, power control, and channel coding. Key messages and methods arising from many recent work are listed, and open issues discussed. Through case studies, it is illustrated how “Layering as Optimization Decomposition” provides a common language to think

We proposed earlier an optimization approach to reactive flow control where the objective of the control is to maximize the total 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 consider two extensions to the basic algorithm. First, the basic algorithm requires communication from sources of their rates to links in their paths in order to carry out the gradient projection algorithm. We prove that it is possible for the links to estimate the gradient using only local information, thus eliminating the need for explicit communication. Second, the basic algorithm assumes that each source is served by a single path. We generalize the model to the case where there are multiple paths between a source--destination pair. This allows flow control and routing to be jointly optimized. 1 Introduction We have proposed previously an optimization approach to flow control where the cont...

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. Keywords: Flow control, optimization flow control, Newton algorithm 1. Introduction We have proposed previously an optimization approach to flow control where the control mechanism is derived as a gradient projection algorithm to solve the dual of a global optimization problem [18,22]. An important feature is that the problem is decomposed into simple algorithms that are executed at individual links and sources using `local' information. It is well known that Newton method, where the gradient is scaled by the inverse of the second derivative matrix, typically enjoys a m

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 non-free 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 variab...

A duality model of flow control is proposed in Part I of this paper and leads to a basic flow control algorithm. In this sequel we develop a practical implementation of the basic algorithm, Random Exponential Marking (REM). It consists of a link algorithm, that probabilistically marks packets inside the network, and a source algorithm, that adapts source rate to observed marking. REM has three advantages. First the marking probability is exponential in a link congestion measure, so that the end--to--end marking probability observed at a source is exponential in its path congestion measure. Marking allows the source to estimate its path congestion measure and adjusts its rate in a way that aligns individual optimality with social optimality. Second REM achieves high link utilization with very low backlog, and hence negligible loss and queueing delay. Third sources stabilize around a globally optimal equilibrium, thus avoiding the perpetual cycle of sinking into and recovering from congestion. Moreover the equilibrium can be chosen to achieve different fairness criteria. We present extensive simulation results to demonstrate that REM is not only stable and fair, but more importantly, scalable and robust. Finally, the link algorithm itself can also be used for active queue management that interact with existing source algorithms. We compare the performance of Reno, Reno/RED and Reno/REM. I.