This chapter provides approaches on how to deal with games in more complicated network topologies, starting from the basic games in single point-to-point WDM fiber links studied in Chap.  7. The multi-link topologies studied are representative for selected paths extracted from a mesh configuration in which no closed loops are being formed by channel optical paths. In network configurations, coupled constraints are propagated along fiber links and constraint functions become complicated from the end-to-end point of view. The non-convexity introduces additional complexities for analysis. In this chapter, we present a partition approach. More precisely, we partition the general multi-link structure into stages each stage being a single link. Then we formulate a partitioned Nash game for general multi-link topologies composed of ladder-nested stage Nash games. We also show that convexity is ensured in single-sink topologies, so that a partition approach could be based on single-sink stages.


Mesh Topology Stage Game Channel Power Fiber Link Hierarchical Decomposition 
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  1. 2.
    Agrawal, G.P.: Fiber-Optic Communication Systems, 3rd edn. Wiley, New York (2002) CrossRefGoogle Scholar
  2. 21.
    Basar, T., Srikant, R.: Revenue-maximizing pricing and capacity expansion in a many-users regime. In: Proceedings of the 21st IEEE Conference on Computer Communications (INFOCOM), June 2002, pp. 294–301 (2002) Google Scholar
  3. 24.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Nashua (1999) MATHGoogle Scholar
  4. 68.
    Kelly, F.P., Maulloo, A.K., Tan, D.: Rate control for communication networks: shadow prices, proportional fairness and stability. J. Oper. Res. Soc. 49(3), 237–252 (1998) MATHGoogle Scholar
  5. 70.
    Kinderlehrer, D.: An Introduction to Variational Inequalities and Their Applications. Academic Press, San Diego (1980) MATHGoogle Scholar
  6. 71.
    Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Èkon. Mat. Metody 12, 747–756 (1976) [translated as Matecon.] MATHGoogle Scholar
  7. 77.
    Laplante, P.A.: Comprehensive Dictionary of Electrical Engineering. Springer, Berlin (1999) Google Scholar
  8. 81.
    Low, S.H., Lapsley, D.E.: Optimization flow control-I: basic algorithm and convergence. IEEE/ACM Trans. Netw. 7(6), 861–874 (1999) CrossRefGoogle Scholar
  9. 101.
    Pan, Y.: A game theoretical approach to constrained OSNR optimization problems in optical networks. Ph.D. Thesis, University of Toronto, Toronto, Canada (2009) Google Scholar
  10. 106.
    Pan, Y., Pavel, L.: Games with coupled propagated constraints in optical networks: the multi-link case. In: Proceedings of the 46th IEEE Conference on Decision and Control, December, pp. 3443–3449 (2007) CrossRefGoogle Scholar
  11. 111.
    Pan, Y., Pavel, L.: Games with coupled propagated constraints in general topology optical networks. In: Proceedings of the 1st International Conference on Game Theory in Networks (GameNets), May, pp. 549–558 (2009) CrossRefGoogle Scholar
  12. 112.
    Pan, Y., Pavel, L.: Games with coupled propagated constraints in optical networks with multi-link topologies. Automatica 45(4), 871–880 (2009) MathSciNetMATHCrossRefGoogle Scholar
  13. 117.
    Pavel, L.: A nested noncooperative game formulation for OSNR optimization in distributed optical links. In: Proceedings of the 44th IEEE Conference on Decision and Control, December, pp. 6958–6965 (2005) CrossRefGoogle Scholar
  14. 120.
    Pavel, L.: An extension of duality to a game-theoretic framework. Automatica 43(2), 226–237 (2007) MathSciNetMATHCrossRefGoogle Scholar
  15. 121.
    Pavel, L.: A nested noncooperative OSNR game in distributed WDM optical links. IEEE Trans. Commun. 55(6), 1220–1230 (2007) CrossRefGoogle Scholar
  16. 126.
    Ramaswami, R., Sivarajan, K.N.: Optical Networks: A Practical Perspective, 2nd edn. Academic Press, San Diego (2002) Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of TorontoTorontoCanada

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