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

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

2009-10-20

arxiv.org/abs/0910.3821v2

Annals of Applied Probability 2009, Vol. 19, No. 5, 1719-1780

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/08-AAP591 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/08-AAP591

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.

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