## The Annals of Applied Probability

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

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

D. Shah, J. N. Tsitsiklis, and Y. Zhong

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/12-AAP915

**Mathematical Reviews number (MathSciNet)**

MR3161642

**Zentralblatt MATH identifier**

1304.60102

**Subjects**

Primary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx] 68M12: Network protocols

Secondary: 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

**Keywords**

Bandwidth-sharing policy state space collapse $\alpha$-fair heavy traffic

#### 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