Limits of on/off hierarchical product models for data transmission
type: Item
instance of: scholarly article
title: Limits of on/off hierarchical product models for data transmission
zbMATH Open document ID: 1042.90004
author: Gennady Samorodnitsky
author: Sidney I. Resnick
published in: The Annals of Applied Probability
publication date: 2004-03-30T00:00:00Z
review text: A hierarchial product model seeks to model network traffic as a product of independent on/off processes. Previous studies have assumed a Markovian structure for component processes amounting to assuming that exponential distributions govern on and off periods. However, if the number of factor processes grows and input rates are stabilized by allowing the period distribution to change suitably, a limiting on/off process can be obtained which has exponentially distributed on periods and whose off periods are equal in distribution to the busy period of an M/G/\(\infty\) queue. The authors give a complete study of the possible limits of the product process as the number of the factors grows. A process \(\{ Z^{(n)}(t) = \prod_{j=1}^n I_j^{(n)} (t),\, t \geq 0 \}\) where \(I_j^{(n)} (\cdot)\), \(j=1, \ldots, n\) are i.i.d. on/off processes is considered. The authors propose the general theory of processes \(Z^{(n)} (t).\) Under some assertions they give the nessesary and sufficient conditions for \(Z^{(n)} (\cdot)\) to converge to a limiting approximation \(Z^{(\infty)} (\cdot)\) in the sense of convergence of finite dimensional distribution and they provide various interpretations for the limiting process \(Z^{(\infty)} (\cdot).\) Information about dependence structure of the limits process \(Z^{(\infty)} (\cdot)\) is presented as well. In particular various facts about the decay of the correlation functions are given.
Mathematics Subject Classification ID: 90B18
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60K05
zbMATH DE Number: 2063742
zbMATH Keywords: fluid queue
zbMATH Keywords: M/G/\(\infty\) queue
zbMATH Keywords: heavy tails
zbMATH Keywords: long-range dependence
zbMATH Keywords: product models
zbMATH Keywords: infinite divisibility
zbMATH Keywords: renewal theorems
reviewed by: Yu. V. Kozachenko
describes a project that uses: longmemo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1069786502
OpenAlex ID: W2166480152
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DOI: 10.1214/AOAP/1069786502
description: scientific article; zbMATH DE number 2063742