## The Annals of Applied Probability

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

#### 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 Massoulié [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.

#### Article information

Source
Ann. Appl. Probab., Volume 24, Number 1 (2014), 76-113.

Dates
First available in Project Euclid: 9 January 2014

https://projecteuclid.org/euclid.aoap/1389278720

Digital Object Identifier
doi:10.1214/12-AAP915

Mathematical Reviews number (MathSciNet)
MR3161642

Zentralblatt MATH identifier
1304.60102

#### Citation

Shah, D.; Tsitsiklis, J. N.; Zhong, Y. Qualitative properties of $\alpha$-fair policies in bandwidth-sharing networks. Ann. Appl. Probab. 24 (2014), no. 1, 76--113. doi:10.1214/12-AAP915. https://projecteuclid.org/euclid.aoap/1389278720

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