Continuum Equilibria for Routing in Dense Static Ad-hoc Networks Eitan Altman ∗ , Pierre Bernhard † , Merouane Debbah ‡ , Alonso Silva ∗ 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 source to its destination so as to minimize its individual cost. 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-Bellman-Jacobi equation, for the global optimization problem, (2) a method based on Green 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. Keywords: Routing, Ad-hoc networks, equilibrium. 1 Introduction In the design and analysis of wireless networks, researchers frequently stumble on the scala- bility problem that can be summarized in the following sentence: “As the number of nodes in the network increases, problems become harder to solve” [26]. The sentence takes its meaning from several issues. Some examples are the following: • In Routing: As the network size increases, routes consists of an increasing number of nodes, and so they are increasingly susceptible to node mobility and channel fading [22]. • In Transmission Scheduling: The determination of the maximum number of non- conflicting transmissions in a graph is a NP-complete problem [29]. • In Capacity of Wireless Networks: As the number of nodes increases, the determination of the precise capacity becomes an intractable problem. * INRIA, B.P.93, 2004 Route des Lucioles, 06902 Sophia-Antipolis Cedex, France. Email: {eitan.altman,alonso.silva}@sophia.inria.fr † I3S, Universit´e de Nice-Sophia Antipolis and CNRS, 940 Route des Colles, B.P. 145, 06903 Sophia- Antipolis Cedex, France. Email: Pierre.Bernhard@polytech.unice.fr ‡ Chaire Alcatel Lucent, Sup´elec, 91192 Gif sur Yvette, France. Email: merouane.debbah@supelec.fr 1