## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 19, Number 1 (2009), 243-280.

### Fluid limits for networks with bandwidth sharing and general document size distributions

H. Christian Gromoll and Ruth J. Williams

#### Abstract

We consider a stochastic model of Internet congestion control, introduced by Massoulié and Roberts [*Telecommunication Systems* **15** (2000) 185–201], that represents the randomly varying number of flows in a network where bandwidth is shared among document transfers. In contrast to an earlier work by Kelly and Williams [*Ann. Appl. Probab.* **14** (2004) 1055–1083], the present paper allows interarrival times and document sizes to be generally distributed, rather than exponentially distributed. Furthermore, we allow a fairly general class of bandwidth sharing policies that includes the weighted *α*-fair policies of Mo and Walrand [*IEEE/ACM Transactions on Networking* **8** (2000) 556–567], as well as certain other utility based scheduling policies. To describe the evolution of the system, measure valued processes are used to keep track of the residual document sizes of all flows through the network. We propose a fluid model (or formal functional law of large numbers approximation) associated with the stochastic flow level model. Under mild conditions, we show that the appropriately rescaled measure valued processes corresponding to a sequence of such models (with fixed network structure) are tight, and that any weak limit point of the sequence is almost surely a fluid model solution. For the special case of weighted *α*-fair policies, we also characterize the invariant states of the fluid model.

#### Article information

**Source**

Ann. Appl. Probab., Volume 19, Number 1 (2009), 243-280.

**Dates**

First available in Project Euclid: 20 February 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1235140339

**Digital Object Identifier**

doi:10.1214/08-AAP541

**Mathematical Reviews number (MathSciNet)**

MR2498678

**Zentralblatt MATH identifier**

1169.60025

**Subjects**

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]

Secondary: 60F17: Functional limit theorems; invariance principles 90B15: Network models, stochastic

**Keywords**

Bandwidth sharing α-fair flow level Internet model congestion control simultaneous resource possession fluid model workload measure valued process invariant manifold

#### Citation

Gromoll, H. Christian; Williams, Ruth J. Fluid limits for networks with bandwidth sharing and general document size distributions. Ann. Appl. Probab. 19 (2009), no. 1, 243--280. doi:10.1214/08-AAP541. https://projecteuclid.org/euclid.aoap/1235140339