### Abstract

This chapter gives an overview of some properties of the storage occupancy process in a buffer fed with “fractional Brownian traffic,” a Gaussian self‐similar process. This model, called here “fractional Brownian storage,” is the logically simplest long‐range‐dependent (LRD) storage system having strictly self‐similar input variation. The impact of the self‐similarity parameter H can be very clearly illustrated with this model. Even in this case, all the known explicitly calculable formulas for quantities like the storage occupancy distribution are only limit results, for example, large deviation asymptotics. Scaling formulas, on the other hand, hold exactly for this model.

The simplicity is won at the price that the input model is not meaningful at the smallest time scales, where half of the “traffic” is negative. The model can be justified by rigorous limit theorems, but it should be emphasized that this involves not only a central limit theorem (CLT) argument for Gaussianity but also a heavy traffic limit. From a less rigorous, practical viewpoint one can say that fractional Brownian storage gives usable results when, at time scales relevant for queueing phenomena, the traffic consists of independent streams such that a large number of them are simultaneously active, and second‐order self‐similarity holds.

This chapter is structured as follows. Definitions are given and then some basic scaling formulas are derived. Results based on large deviations in path space are presented. Finally, some other approaches are outlined.

The simplicity is won at the price that the input model is not meaningful at the smallest time scales, where half of the “traffic” is negative. The model can be justified by rigorous limit theorems, but it should be emphasized that this involves not only a central limit theorem (CLT) argument for Gaussianity but also a heavy traffic limit. From a less rigorous, practical viewpoint one can say that fractional Brownian storage gives usable results when, at time scales relevant for queueing phenomena, the traffic consists of independent streams such that a large number of them are simultaneously active, and second‐order self‐similarity holds.

This chapter is structured as follows. Definitions are given and then some basic scaling formulas are derived. Results based on large deviations in path space are presented. Finally, some other approaches are outlined.

Original language | English |
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Title of host publication | Self-Similar Network Traffic and Performance Evaluation |

Editors | Kihong Park, Walter Willinger |

Place of Publication | Canada |

Publisher | Wiley |

Chapter | 4 |

Pages | 101-114 |

ISBN (Electronic) | 978-0-471-20644-6 |

ISBN (Print) | 0-471-31974-0, 978-0-471-31974-0 |

DOIs | |

Publication status | Published - 2000 |

MoE publication type | A3 Part of a book or another research book |

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## Cite this

Norros, I. (2000). Queueing behavior under fractional brownian traffic. In K. Park, & W. Willinger (Eds.),

*Self-Similar Network Traffic and Performance Evaluation*(pp. 101-114). Wiley. https://doi.org/10.1002/047120644X.ch4