Modeling of Dependence in a Peer-to-Peer Video Application Natalia M. Markovich Institute of Control Sciences of Russian Academy of Sciences Profsoyuznaya 65, Moscow 117997, Russia +7(495)3348820 markovic@ipu.rssi.ru ABSTRACT We consider the underlying network characteristics of an IPTV trace. The pairwise dependence in a triple (X,Y,R) denoting the inter-arrival time between packets, the packet length and the rate of transmission in a peer-to-peer IPTV session is detected and modeled. The three quantities are re- lated by R = Y/X. We argue that the inter-arrival time and the packet length as well as the rate and the packet length are dependent, and the rate and the inter-arrival time are almost independent random variables. The Pickand’s func- tion and an empirical copula are used as measures of depen- dence. A Clayton copula provides an appropriate model of all underlying pair-wise dependencies. Categories and Subject Descriptors C.4 [Performance of Systems]: Modeling techniques; G.3 [Probability and Statistics]: Multivariate statistics General Terms Measurement, Performance, Theory Keywords Peer-to-peer, IPTV video packet flow, statistical traffic mod- eling and characterization 1. INTRODUCTION Peer-to-peer (P2P) multimedia applications like Skype and IPTV are used now as prominent tools of voice and video transport via the Internet. The random structure of the corresponding overlay networks creates the issue to opti- mize their operation. Typical features of P2P applications are the burstiness, long-range dependence and heaviness of tails, [12]. Here dependencies of pairs in the triple (X,Y,R) of an IPTV video packet flow are investigated. {Xi } and {Yi } denote the inter-arrival times (IATs) between pack- ets at a measurement point and the packet lengths (PLs), Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. IWCMC ’10, June 28- July 2, 2010, Caen, France Copyright 2010 ACM 978-1-4503-0062-9/10/06/ ...$10.00. respectively. The rate of the packet transmission can be approximated by the ratio Ri = Yi /Xi . We are studying technologies which generate random IATs and PLs. The dependence structure of network characteristics causes a clustering in the data or conglomerates of observations with homogeneous properties, [11]. As a consequence tradi- tional models accepted for input processes (e.g., Gaussian) and decay of dependence have to be revised. In [2], [3] the independence of the transmission rate R and the transmitted file size S as well as the rate and the duration D of a connec- tion have been assumed and the impact of such possible in- dependence on network traffic models have been considered. It was concluded that the dependence can vary for different applications. Hence, it is required to separate traffic by the application and to model each application separately with its own dependence structure. The independence of D and R is shown to be realistic for streaming media and P2P networks since the transmission durations are given by the lifetime of a user on the P2P networks, while the rates are given by the maximum upload bandwidth, [3]. By studying aggregated flows generated by the same user and by checking the condi- tion ED · ER/ES = 1, Mandjes et al. [13] discovered that R and D were dependent in 80% of the investigated traces. In [11] the video traffic with a strong burstiness was classified by the extremal index, which is a dependence measure of the random sequence in consecutive time slots. In [10] the weak dependence between R and S and almost independence be- tween R and D arising from the TCP-flow of a Web session is proved using the Pickand’s function which is another mea- sure of dependence. The objective of the paper is to detect and to model the pairwise dependence in the triple (X,Y,R) of the underly- ing IPTV data set. The dependence might be required to evaluate the joint distribution of the underlying r.v.s and their quantiles and to classify the traffic. In [16] the joint DF of PLs and IATs of the four P2P IPTV applications PPlive, PPStream, SopCast, and TVants allows to separate two classes of packets, namely, signaling and video pack- ets. It is known that the bivariate distribution function (DF) can be represented as a product of marginal DFs, i.e., P{X ≤ x, Y ≤ y} = P{X ≤ x}P{Y ≤ y} only holds if the r.v.s X and Y are independent. In case of dependent r.v.s the joint DF may be expressed by copulas [14] and by the Pickand’s function related to copulas [1], [10]. Copu- las and the Pickand’s function indicate the dependence of r.v.s. Note that well-known rank correlations like Kendall’s tau and Spearman’s rho do not indicate dependence since a rank correlation of zero does not necessarily imply the inde-