On Unique Ergodicity Of Coupled AIMD Flows

TL;DR

This paper proves that coupled AIMD networks inherit ergodic properties of individual networks, which is crucial for their convergence in large-scale optimization and control applications, correcting previous errors in the literature.

Contribution

It establishes the ergodic behavior of coupled AIMD systems, extending their theoretical understanding and ensuring convergence in distributed optimization.

Findings

Coupled AIMD networks inherit ergodic properties of individual networks.

Results have implications for convergence in large-scale optimization algorithms.

The paper corrects previous conceptual and technical errors.

Abstract

The AIMD algorithm, which underpins the Transmission Control Protocol (TCP) for transporting data packets in communication networks, is perhaps the most successful control algorithm ever deployed. Recently, its use has been extended beyond communication networks, and successful applications of the AIMD algorithm have been reported in transportation, energy, and mathematical biology. A very recent development in the use of AIMD is its application in solving large-scale optimization and distributed control problems without the need for inter-agent communication. In this context, an interesting problem arises when multiple AIMD networks that are coupled in some sense (usually through a nonlinearity). The purpose of this note is to prove that such systems in certain settings inherit the ergodic properties of individual AIMD networks. This result has important consequences for the convergence of the aforementioned optimization algorithms. The arguments in the paper also correct conceptual and technical errors in [1].