Qualitative properties of $\alpha$-fair policies in bandwidth-sharing networks

TL;DR

This paper analyzes the long-term behavior of $ ext{alpha}$-fair bandwidth sharing policies in networks, providing bounds on flow numbers, state space collapse, and validating diffusion approximations for steady-state distributions.

Contribution

It offers new bounds and properties for $ ext{alpha}$-fair policies, including state space collapse and exponential tail bounds, extending understanding of network flow dynamics.

Findings

Bounds on maximum flows for $ ext{alpha} \\geq 1$

Full state space collapse for all $ ext{alpha} \\geq 1$

Validation of diffusion approximation for $ ext{alpha}=1$

Abstract

We consider a flow-level model of a network operating under an $\alpha$-fair bandwidth sharing policy (with $\alpha>0$) proposed by Roberts and Massouli\'{e} [Telecomunication Systems 15 (2000) 185-201]. This is a probabilistic model that captures the long-term aspects of bandwidth sharing between users or flows in a communication network. We study the transient properties as well as the steady-state distribution of the model. In particular, for $\alpha\geq1$, we obtain bounds on the maximum number of flows in the network over a given time horizon, by means of a maximal inequality derived from the standard Lyapunov drift condition. As a corollary, we establish the full state space collapse property for all $\alpha\geq1$. For the steady-state distribution, we obtain explicit exponential tail bounds on the number of flows, for any $\alpha>0$, by relying on a norm-like Lyapunov function. As a corollary, we establish the validity of the diffusion approximation developed by Kang et al. [Ann. Appl. Probab. 19 (2009) 1719-1780], in steady state, for the case where $\alpha=1$ and under a local traffic condition.