Continuum Equilibria and Global Optimization for Routing in Dense Static Ad Hoc Networks

TL;DR

This paper models and analyzes routing in extremely dense static ad hoc networks using continuum limits, deriving solutions for global optimization, individual routing, and equilibrium states based on advanced mathematical methods.

Contribution

It introduces a continuum framework for dense ad hoc networks and develops novel solution methods for global, individual, and equilibrium routing problems.

Findings

Derived cost models based on network capacity and node density.

Presented solution methodologies including Hamilton-Jacobi-Bellman equations.

Established a connection between Wardrop equilibrium and global optimization.

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.