State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy

TL;DR

This paper analyzes a network congestion model under fair bandwidth sharing, demonstrating that in heavy traffic, the flow process simplifies to a diffusion approximation related to workload, revealing a state space collapse phenomenon.

Contribution

It establishes a multiplicative state space collapse for a fair bandwidth sharing network under heavy traffic, linking flow counts to workload via diffusion approximation.

Findings

Proves state space collapse in the diffusion scale.

Shows flow process can be approximated by workload in heavy traffic.

Extends fluid model analysis to diffusion scale.

Abstract

We consider a connection-level model of Internet congestion control, introduced by Massouli\'{e} and Roberts [Telecommunication Systems 15 (2000) 185--201], that represents the randomly varying number of flows present in a network. Here, bandwidth is shared fairly among elastic document transfers according to a weighted $\alpha$-fair bandwidth sharing policy introduced by Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556--567] [$\alpha\in (0,\infty)$]. Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [Ann. Appl. Probab. 14 (2004) 1055--1083] by two of the authors. Here, we use the long-time behavior of the solutions of the fluid model established in that paper to derive a property called multiplicative state space collapse, which, loosely speaking, shows that in diffusion scale, the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process.