## Institute of Mathematical Statistics Collections

- Collections
- Volume 4, 2008, 253-265

### Fluid Model for a Data Network with *α*-Fair Bandwidth Sharing and General Document Size Distributions: Two Examples of Stability

H. C. Gromoll and R. J. Williams

#### Abstract

The design and analysis of congestion control mechanisms for modern data networks such as the Internet is a challenging problem. Mathematical models at various levels have been introduced in an effort to provide insight to some aspects of this problem. A model introduced and studied by Roberts and Massoulié [13] aims to capture the dynamics of document arrivals and departures in a network where bandwidth is shared fairly amongst flows that correspond to continuous transfers of individual elastic documents. Here we consider this model under a family of bandwidth sharing policies introduced by Mo and Walrand [14]. With generally distributed interarrival times and document sizes, except for a few special cases, it is an open problem to establish stability of this stochastic flow level model under the nominal condition that the average load on each resource is less than its capacity. As a step towards the study of this model, in a separate work [8], we introduced a measure valued process to describe the dynamic evolution of the residual document sizes and proved a fluid limit result: under mild assumptions, rescaled measure valued processes corresponding to a sequence of flow level models (with fixed network structure) are tight, and any weak limit point of the sequence is almost surely a solution of a certain fluid model. The invariant states for the fluid model were also characterized in [8]. In this paper, we review the structure of the stochastic flow level model, describe our fluid model approximation and then give two interesting examples of network topologies for which stability of the fluid model can be established under a nominal condition. The two types of networks are linear networks and tree networks.

#### Chapter information

**Source***Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008)

**Dates**

First available in Project Euclid: 28 January 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.imsc/1233152946

**Digital Object Identifier**

doi:10.1214/074921708000000417

**Mathematical Reviews number (MathSciNet)**

MR2574235

**Zentralblatt MATH identifier**

1166.90312

**Rights**

Copyright © 2008, Institute of Mathematical Statistics

#### Citation

Gromoll, H. C.; Williams, R. J. Fluid Model for a Data Network with α -Fair Bandwidth Sharing and General Document Size Distributions: Two Examples of Stability. Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz, 253--265, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000417. https://projecteuclid.org/euclid.imsc/1233152946

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