This chapter gives an overview of some properties of the storage occupancy process in a buffer fed with ``fractional Brownian traf®c,'' 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. . | Self-Similar Network Traffic and Performance Evaluation Edited by Kihong Park and Walter Willinger Copyright 2000 by John Wiley Sons Inc. Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X 4 QUEUEING BEHAVIOR UNDER FRACTIONAL BROWNIAN TRAFFIC ILKKA NORROS VTT Information Technology Espoo Finland INTRODUCTION 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 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 see Chapter 5. 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 see Chapter 1 holds. Chapter 7 describes many features of storage processes with finitely aggregated on off input traffic which differ qualitatively from those of fractional Brownian 101 102 QUEUEING BEHAVIOR UNDER FRACTIONAL BROWNIAN TRAFFIC storage. For example the correlation function of the input process remains unchanged if one changes the distributions of on and
