Continuum equilibria and global optimization for routing in dense static ad hoc networks

  • Authors:
  • Alonso Silva;Eitan Altman;Pierre Bernhard;Mérouane Debbah

  • Affiliations:
  • INRIA, B.P. 93, 2004 Route des Lucioles, 06902 Sophia-Antipolis Cedex, France and Alcatel-Lucent Chair in Flexible Radio - SUPELEC, 91192 Gif sur Yvette, France;INRIA, B.P. 93, 2004 Route des Lucioles, 06902 Sophia-Antipolis Cedex, France;I3S, Université de Nice-Sophia Antipolis and CNRS, 940 Route des Colles, B.P. 145, 06903 Sophia-Antipolis Cedex, France;Alcatel-Lucent Chair in Flexible Radio - SUPELEC, 91192 Gif sur Yvette, France

  • Venue:
  • Computer Networks: The International Journal of Computer and Telecommunications Networking
  • Year:
  • 2010

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Abstract

We consider massively dense ad hoc networks and study their continuum limits as the node density increases and as the graph providing the available routes becomes a continuous area with location and congestion dependent costs. We study both the global optimal solution as well as the non-cooperative routing problem among a large population of users where each user seeks a path from its origin to its destination so as to minimize its individual cost. Finally, we seek for a (continuum version of the) Wardrop equilibrium. We first show how to derive meaningful cost models as a function of the scaling properties of the capacity of the network and of the density of nodes. We present various solution methodologies for the problem: (1) the viscosity solution of the Hamilton-Jacobi-Bellman equation, for the global optimization problem, (2) a method based on Green's Theorem for the least cost problem of an individual, and (3) a solution of the Wardrop equilibrium problem using a transformation into an equivalent global optimization problem.