Optimal flow control and routing in multi-path networks

doi:10.1016/S0166-5316(02)00176-1

Performance Evaluation

Volume 52, Issues 2–3, April 2003, Pages 119-132

https://doi.org/10.1016/S0166-5316(02)00176-1 Get rights and content

Abstract

We propose two flow control algorithms for networks with multiple paths between each source–destination pair. Both are distributed algorithms over the network to maximize aggregate source utility. Algorithm 1 is a first order Lagrangian method applied to a modified objective function that has the same optimal solution as the original objective function but has a better convergence property. Algorithm 2 is based on the idea that, at optimality, only paths with the minimum price carry positive flows, and naturally decomposes the overall decision into flow control (determines total transmission rate based on minimum path price) and routing (determines how to split the flow among available paths). Both algorithms can be implemented as simply a source-based mechanism in which no link algorithm nor feedback is needed. We present numerical examples to illustrate their behavior.

Introduction

Flow control can be regarded as a distributed computation over a network to solve an optimization problem [8], [9], [12], [13], [14], [15], [19], [21]. In this formulation, each source is characterized by a utility function of its transmission rate and the goal is to maximize aggregate utility. Indeed, one can interpret major TCP congestion control protocols, such as Reno [10], Vegas [4], RED [7], and REM [2], within this framework where different protocols are merely different algorithms to solve the same prototype problem with different utility functions [18].

Most of these papers assume that there is a unique path between source and destination, and the issue is to determine the source rate based on network congestion. On the other hand, multi-path routing problem has received significant attention in recent literature, e.g. Refs. [5], [6], [23], where the issue is to determine efficient loop-free multi-paths. In this paper, we propose algorithms that attempt to jointly optimize flow control and routing when multiple paths are available between source and destination.

This problem has been studied in Ref. [13] using a penalty function approach, in Ref. [16] using sliding mode control [22], and in Ref. [11] using a subgradient method. The main obstacle in the multi-path case is that, even if the objective function is strictly concave in the total source rates, it is not strictly concave in path rates. Then the optimal path rates are non-unique and the dual problem becomes non-differentiable. Lagrangian multiplier method generally converges only when the objective function is strictly concave.

In the next section, we describe the optimization model. We present two algorithms in Section 3. The first algorithm is derived as the first order Lagrangian method on a modified objective function that has the same optimal solution as the original objective function but apparently better convergence property. The second method is a subgradient-like method based on the idea that only paths that are least congested carry positive flows at optimality. We discuss an implementation method in Section 4 that uses queuing delay to implicit feeds back congestion information. Finally, we present numerical results to illustrate performance of these algorithms in Section 5, and conclude with some limitations of this paper.

Section snippets

The optimization problem

Consider a network whose links are denoted by $L ={1,2,…,L}$ . Let c_l be the capacity of link $l∈ L$ and c=[c₁,c₂,…,c_L]^T. Let $S ={1,2,…,S}$ be the set of sources. Each source s has n_s available paths or routes from the source to the destination. Let the L×1 vector $R_{s,i}$ denote the set of links used by source $s∈ S$ on its path i∈{1,2,…,n_s}, whose lth element equals 1 if the path contains link l and 0 otherwise. The set of all the available paths of user s is defined by $R_{s} =[R_{s,1}, R_{s,2},…, R_{s,n_{s}}]$ and the total

Algorithms

In this section, we present two distributed algorithms for the solution of problem , , . When the objective function is not strictly concave, such as ours, it is well known that a first order Lagrangian algorithm may oscillate.

End-to-end implementation

Both Algorithm 1, Algorithm 2 require communication between sources and links: source s must obtain the sum p^r_s,i(t) of link prices in its paths $R_{s,i}$ for i=1,2,…,n_s, and link l must obtain the aggregate source rate x^l(t). Note that x^l(t) is the sum of transmission rates at the source and is not the aggregate input flow rate observable at link l, because queuing and multiplexing at upstream links distorts the flow of s from x_s(t) at the source into some other fluid flow. To eliminate the need

Numerical examples

In this section we present numerical results on two network topologies. In the first example both Algorithm 1, Algorithm 2 converge; we also show the behavior of Lagrangian method applied to the original objective function as opposed to the modified objective function used in Algorithm 1 and the behavior of the recently proposed algorithm by Kar et al. [11].

Example 1

Consider the following simple network which consists three unidirectional links labeled L1, L2 and L3 with capacities c=(1,2,3) as shown in

Conclusion

In this paper, we propose two flow control algorithms for networks with multiple paths between source–destination pairs. Algorithm 1 is a first order Lagrangian method on a modified objective function that has the same optimal solution as the original objective function but has a better convergence property. Algorithm 2 is based on the idea that, at optimality, only paths with the minimum price carry positive flows, and naturally decomposes the overall decision into flow control (determines

Acknowledgements

The authors are very grateful to the anonymous reviewers for their valuable suggestions on and corrections of the manuscript.

Wei-Hua Wang received his B.Eng. from Xi’an Jiaotong University, P.R. China in 1992, M.Eng. from Northeastern University, P.R. China in 1995 and from Nanyang Technological University, Singapore in 1997, all in Electrical Engineering. From 1997 to 1999, he was a lecturer at the Department of Electronic Engineering, Fudan University, P.R. China. Since 1999, he has been a Ph.D. candidate at the Department of Electrical and Electronic Engineering, the University of Melbourne, Australia. His

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Marimuthu Palaniswami obtained his B.E. (Hons) from the University of Madras, M.Eng. Sc. from the University of Melbourne, and Ph.D. from the University of Newcastle, Australia. He is an Associate Professor at the University of Melbourne, Australia.

His research interests are in the fields of computational intelligence, computer networks, nonlinear dynamics, intelligent control and Internet computing. He has published more than 180 conference and journal papers in these topics. He was an Associate Editor of the IEEE Tran. on Neural Networks and is on the editorial board of a few computing and electrical engineering journals. He served as a Technical Program Co-chair for the IEEE International Conference on Neural Networks, 1995 and has served on the programme committees of a number of international conferences. His invited presentations include several key note lectures and invited tutorials, in the areas of Machine Learning, Bio Medical engineering, and Control. He has completed several industry sponsored projects for National Australia Bank, Broken Hill Propriety Limited, Defence Science and Technology Organisation, Integrated Control Systems Pty Ltd, and Signal Processing Associates Pty Ltd. He has also been supported with several Australian Research Council Grants, Industry Research and Development grants and industry research contracts. He was also a recipient of foreign specialist award from the Ministry of Education, Japan.

Steven H. Low received his B.S. degree from Cornell University and Ph.D. from the University of California-Berkeley both in electrical engineering. He was with AT&T Bell Laboratories, Murray Hill, from 1992 to 1996, and was with the University of Melbourne, Australia, from 1996 to 2000, and is now an Associate Professor at the California Institute of Technology, Pasadena. He was a co-recipient of the IEEE William R. Bennett Prize Paper Award in 1997 and the 1996 R&D 100 Award. He is on the editorial board of IEEE/ACM Transactions on Networking. He has been a guest editor of the IEEE Journal on Selected Area in Communications and on the program committee of many conferences. His research interests are in the control and optimization of communications networks and protocols. His home is netlab.caltech.edu and email is [email protected].

^☆: An earlier version of this paper was presented at the Internet Performance and Control of Network Systems Conference, R.D. van der Mei, F. Huebner-Szaba de Bucs (Eds.), Proceedings of SPIE, vol. 4523, Denver, CO, 21–22 August 2001, The International Society for Optical Engineering. This work was supported by the Australian Research Council through grant A49930405.

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Optimal flow control and routing in multi-path networks☆

Abstract

Introduction

Section snippets

The optimization problem

Algorithms

End-to-end implementation

Numerical examples

Conclusion

Acknowledgements

Automatica

Microprocess. Microsyst.

TCP Vegas: end to end congestion avoidance on a global Internet

IEEE J. Select. Areas Commun.

Random early detection gateways for congestion avoidance

IEEE/ACM Trans. Networking