The influence of dependence on data network models

Abstract

Consider an infinite-source marked Poisson process to model end user inputs to a data network. At Poisson times, connections are initated. The connection is characterized by a triple (F, L, R) denoting the total quantity of transmitted data in a connection, the length or duration of the connection, and the transmission rate; the three quantities are related by F = LR. How critical is the dependence structure of the mark for network characteristics such as burstiness, distribution tails of cumulative input, and longrange dependence properties of traffic measured in consecutive time slots? In a previous publication (D'Auria and Resnick (2006)) we assumed that F and R were independent. Here we assume that L and R are independent. The change in dependence assumptions means that the model properties change dramatically: tails of cumulative input per time slot are dramatically heavier, traffic cannot be approximated by a Gaussian distribution, and the decay of dependence cannot be measured in the traditional way using correlation functions. Different network applications are likely to have different mark dependence structure. We argue that the present independence assumption on L and R is likely to be appropriate for network applications such as streaming media or peer-to-peer networks. Our conclusion is that it is desirable to separate network traffic by application and to model each application with its own appropriate dependence structure.

On beta-product convolutions

“…In a financial or insurance framework, the random scaling model (1.1) appears naturally with W the deflated risk arising from some loss or investment R which is independent from the random scaling/deflating factor S. Other prominent applications in the literature concern modeling of network data (see e.g., D'Auria and Resnick (2006Resnick ( , 2008); random difference equations (see e.g., Mikosch and Konstantinides (2004), Denisov and Zwart (2007)); insurance and finance applications (see e.g., Tsitsiashvili (2003, 2004), Tang (2006Tang ( , 2006, Piterbarg et al (2009), Liu and Tang (2010), Tang and Vernic (2010), Zhang (2010)); approximation of multivariate distributions (see e.g., Hashorva (2007), Charpentier and Segers (2009), McNeil and Nešlehová (2009), Balakrishnan and Hashorva (2010)). A monograph treatment rich in applications and references is Galambos and Simonelli (2004).…”