### Abstract

In this paper tools are developed to analyse a recently proposed random matrix model of communication networks that employ additive-increase multiplicative-decrease (AIMD) congestion control algorithms. We investigate properties of the Markov process describing the evolution of the window sizes of network users. Using paracontractivity properties of the matrices involved in the model, it is shown that the process has a unique invariant probability, and the support of this probability is characterized. Based on these results we obtain a weak law of large numbers for the average distribution of resources between the users of a network. This shows that under reasonable assumptions such networks have a well-defined stochastic equilibrium. ns2 simulation results are discussed to validate the obtained formulae. (The simulation program ns2, or network simulator, is an industry standard for the simulation of Internet dynamics.)

Original language | English |
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Pages (from-to) | 703-723 |

Number of pages | 21 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 28 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2006 |

Externally published | Yes |

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### Keywords

- AIMD congestion control
- Communication networks
- Infinite products of positive matrices
- Law of large numbers
- Markov e-chain
- Positive matrices

### ASJC Scopus subject areas

- Analysis

### Cite this

*SIAM Journal on Matrix Analysis and Applications*,

*28*(3), 703-723. https://doi.org/10.1137/040620953