Abstract
We consider in this paper the problem of combined flow control and routing in a noncooperative setting, where each user is faced with a multi-criteria optimization problem, formulated as the minimization of one criterion subject to constraints on others. We address here the basic questions of existence and uniqueness of equilibrium. We show that an equilibrium indeed exists, but it may not be unique due to the multi-criteria nature of the problem. We are able, however, to obtain uniqueness in some weaker sense under appropriate conditions; we show in particular that the link utilizations are uniquely determined at equilibrium and the normalized Nash equilibrium is unique.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Altman, E. and Kameda, H. (2001). Equilibria for Multiclass Routing Problems in Multi-Agent Networks. Submitted in 2001; http://www-sop.inria.fr/mistral/personnel/Eitan.Altman/ntkgame.html.
Altman, E., Başar, T., and Srikant, R. (2002). Nash equilibria for combined flow control and routing in networks: Asymptotic behavior for a large number of users. IEEE Transactions on Automatic Control, 47(6):917–930; Proceedings of the 38th IEEE Conference on Decision and Control, pp. 4002–4007, Phoenix, Arizona, USA, 1999.
Altman, E., El Azouzi, R., and Vyacheslav, V. (2002). Non-cooperative routing in loss networks. Performance Evaluation, 49(1–4):257–272.
Arrow, K., Hurwicz, L., and Uzawa, H. (1961). Constraint qualifications in maximization problems. Naval Research Logistics Quarterly, 8:175–191.
Aubin, J.P. (1980). Mathematical Methods of Game and Economic Theory. Elsevier, Amsterdam.
Ching, W.K. (1999). A note on the convergence of asynchronous greedy algorithm with relaxation in a multiclass queueing environment. IEEE Communications Letters, 3(2):34–36.
El Azouzi, R. and Altman, E. (2003). Constrained traffic equilibrium in routing. IEEE Transaction on Automatic Control,48(9):1656–1660.
Gibbons, R. (1992). Game Theory of Applied Economists. Princeton University Press.
Haurie, A. and Marcotte, P. (1985). On the relationship between Nash-Cournot and wardrop equilibria. Networks, 15:295–308.
Lazar, A., Orda, A., and Pendarakis, D.E. (1997). Virtual path bandwidth allocation in multiuser networks. IEEE/ACM Transaction on Networking, 5(6):861–871.
Lutton, J.L. (2001). On Link Costs in Networks. Private communication, France Telecom.
Nikaido, H. and Isoda, K. (1955). Note on cooperative convex games. Pacific Journal of Mathematics, 5(1):807–815.
Orda, A., Rom, R., and Shimkin, N. (1993). Competitive routing in multi-user communication networks. IEEE/ACM Transactions on Networks, 1:510–520.
Patriksson, M. (1994). The Traffic Assignment Problem: Models and Methods. VSP BV, Utrecht, The Netherlands.
Rhee, S.H. and Konstantopoulos, T. (1998). Optimal flow control and capacity allocation in multi-service networks. 37th IEEE Conf. Decision and Control, Tampa, Florida.
Rhee, S.H. and Konstantopoulos, T. (1999). Decentralized optimal flow control with constrained source rates. IEEE Communications Letters, 3(6):188–200.
Rosen, J. (1965). Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33:520–534.
The ATM Forum Technical Committee. (1999). Traffic Management Specification, Version 4.1, AF-TM-0121.000.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
El Azouzi, R., El Kamili, M., Altman, E., Abbad, M., Başar, T. (2005). Combined Competitive Flow Control and Routing in Networks with Hard Side Constraints. In: Boukas, E.K., Malhamé, R.P. (eds) Analysis, Control and Optimization of Complex Dynamic Systems. Springer, Boston, MA. https://doi.org/10.1007/0-387-25477-3_7
Download citation
DOI: https://doi.org/10.1007/0-387-25477-3_7
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-25475-3
Online ISBN: 978-0-387-25477-7
eBook Packages: Business and EconomicsBusiness and Management (R0)