### Abstract

In a general model (AIMD) of transmission control protocol (TCP) used in internet traffic congestion management, the time dependent data flow vector x(t) > 0 undergoes a biased random walk on two distinct scales. The amount of data of each component x_{i}(t) goes up to x_{i}(t)+a with probability 1-ζ_{i}(x) on a unit scale or down to γx _{i}(t), 0 < γ < 1 with probability ζ_{i}(x) on a logarithmic scale, where ζ_{i} depends on the joint state of the system x. We investigate the long time behavior, mean field limit, and the one particle case. According to c = lim inf _{|X|→∞} ζl(x)_{1} the process drifts to ∞ in the subcritical c < c_{+}(n, γ) case and has an invariant probability measure in the supercritical case c > c_{+}(n, γ). Additionally, a scaling limit is proved when ζ_{i}(x) and a are of order N^{-1} and t → Nt, in the form of a continuum model with jump rate α(x).

Original language | English (US) |
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Pages (from-to) | 271-285 |

Number of pages | 15 |

Journal | ESAIM - Probability and Statistics |

Volume | 14 |

Issue number | 4 |

DOIs | |

State | Published - Oct 29 2010 |

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### Keywords

- AIMD
- Fluid limit
- Mean field interaction
- TCP

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*ESAIM - Probability and Statistics*,

*14*(4), 271-285. https://doi.org/10.1051/ps:2008029