## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 13, Number 4 (2003), 1355-1398.

### Limits of on/off hierarchical product models for data transmission

Sidney Resnick and Gennady Samorodnitsky

#### Abstract

A hierarchical 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, but this is not in good agreement with traffic
measurements. However,
if the number of factor processes grows and input rates are stabilized
by allowing the *on* 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. We give a
fairly complete study of the possible limits of the product process
as the number of factors
grows and offer various characterizations of the approximating
processes. We also study the dependence structure of the approximations.

#### Article information

**Source**

Ann. Appl. Probab., Volume 13, Number 4 (2003), 1355-1398.

**Dates**

First available in Project Euclid: 25 November 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1069786502

**Digital Object Identifier**

doi:10.1214/aoap/1069786502

**Mathematical Reviews number (MathSciNet)**

MR2023880

**Zentralblatt MATH identifier**

1042.90004

**Subjects**

Primary: 90B15: Network models, stochastic

Secondary: 60K25: Queueing theory [See also 68M20, 90B22]

**Keywords**

Fluid queue $M/G/\infty$ queue heavy tales long-range dependence steady state distribution product models infinite divisibility renewal theorems

#### Citation

Resnick, Sidney; Samorodnitsky, Gennady. Limits of on/off hierarchical product models for data transmission. Ann. Appl. Probab. 13 (2003), no. 4, 1355--1398. doi:10.1214/aoap/1069786502. https://projecteuclid.org/euclid.aoap/1069786502