Abstract
This article presents a multicommodity, discrete-time, distributed, and noncooperative routing algorithm, which is proved to converge to an equilibrium in the presence of heterogeneous, unknown, time-varying but bounded delays. Under mild assumptions on the latency functions, which describe the cost associated with the network paths, two algorithms are proposed: The former assumes that each commodity relies only on measurements of the latencies associated with its own paths; the latter assumes that each commodity has (at least indirectly) access to the measures of the latencies of all the network paths. Both algorithms are proven to drive the system state to an invariant set that approximates and contains the Wardrop equilibrium, defined as a network state in which no traffic flow over the network paths can improve its routing unilaterally, with the latter achieving a better reconstruction of the Wardrop equilibrium. Numerical simulations show the effectiveness of the proposed approach.
Highlights
WARDROP equilibria are a game-theoretical concept, originally introduced for network games when modelling transportation networks with congestion [1]
The convergence of the controlled network state to a set that approximates the Wardrop equilibrium is proven by means of standard control theory arguments, derived from LaSalle’s invariance principle
The network is modelled as a time-invariant communication graph and the total flow demand is constituted by various constant traffic flows, or commodities, each one characterised by a source and a destination node
Summary
WARDROP equilibria are a game-theoretical concept, originally introduced for network games when modelling transportation networks with congestion [1]. The dynamic, discrete-time, algorithm presented in this paper deals with a multi-commodity flow problem that consists of distributing a flow demand, split between various source and destination facilities, over a communication network. It is assumed that the latency measures are subject to time-varying, unknown but bounded delays Under such assumptions, the proposed routing algorithm is proved to converge to an approximation of the Wardrop equilibrium. The rest of the paper is organised as follows: Section II presents the state of the art on selfish routing solutions and their relation with Wardrop equilibria, and highlights the contributions of this work; Section III contains the selfish routing problem formulation; Section IV presents the proposed discrete-time control law and discusses some useful lemmata; Section V reports the convergence analysis of the proposed. Control solution and proves the convergence of the network to a set of approximated Wardrop equilibria; Section VI proposes an algorithm improvement in presence of (limited) information exchanges; Section VII validates the approach by discussing the results of numerical simulations, while Section VIII draws the conclusions and discusses possible future research directions
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