A stochastic analysis of resource sharing with logarithmic weights

TL;DR

This paper analyzes a resource sharing algorithm where nodes receive capacity proportional to the logarithm of their load, revealing complex multi-scale dynamics and establishing a heavy traffic limit theorem.

Contribution

It provides a detailed fluid scaling analysis of a two-node network with logarithmic weights and generalizes the model to other increasing functions.

Findings

Interaction of multiple time scales affects system evolution

Unusual scaling behaviors of stochastic processes are observed

Heavy traffic limit theorem for invariant distribution is proved

Abstract

The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has $x$ requests to transmit, then it receives a fraction of the capacity proportional to $\log(1+x)$, the logarithm of its current load. A detailed fluid scaling analysis of such a network with two nodes is presented. It is shown that the interaction of several time scales plays an important role in the evolution of such a system, in particular its coordinates may live on very different time and space scales. As a consequence, the associated stochastic processes turn out to have unusual scaling behaviors. A heavy traffic limit theorem for the invariant distribution is also proved. Finally, we present a generalization to the resource sharing algorithm for which the $\log$ function is replaced by an increasing function. Possible generalizations of these results with $J>2$ nodes or with the function $\log$ replaced by another slowly increasing function are discussed.