978-1-4244-7173-7/10/$26.00 ©2010 IEEE Packet Variation Delay Distribution Discrimination based on Kullback-Leibler Divergence Luis Rizo-Dominguez Department of Basic Science and Engineering University of Caribe Cancun, Mexico lrizo@ucaribe.edu.mx Deni Torres-Roman CINVESTAV Unidad Guadalajara Guadalajara, Mexico dtorres@gdl.cinvestav.mx David Munoz-Rodriguez Department of Electrical and Computer Engineering Tecnologico de Monterrey Monterrey, Mexico dmunoz@itesm.mx Cesar Vargas-Rosales Department of Electrical and Computer Engineering Tecnologico de Monterrey Monterrey, Mexico cvargas@itesm.mx Abstract— In many real-time applications the Quality of Service (QoS) is dominated by jitter. Currently jitter generators are based on Laplace distribution. Nevertheless, the observed jitter measurements depart from that distribution. As a matter of fact data resembles those of a t-Student distribution, and it is also known that Internet traffic presents heavy tailed behavior that can be modeled with alpha-stable distributions. The intention of this work is to find which of these distributions has the best fitting. For this purpose, we collect extensive (round trip time) RTT measurements, and show that the alpha-stable distribution models the jitter adequately. Fitness test included Kullback- Leibler divergence and P-P plot criteria. Alpha-stable parameters are dependent on the transmission hop numbers. Keywords- jitter; alpha-stable model; QoS. I. INTRODUCTION Some approaches to packet variation delay have been reported in the literature such as the jitter analytical approximation for periodic, Poisson and Markovian traffic [1]; the jitter control through buffer sizing [2]; the jitter modeling and prediction based on Deviation-Lag Function (DLF) [3]; the relation between jitter and the characteristics of background traffic such as burstiness, traffic load and traffic burst length [4]. Currently the jitter generator is based on the Laplace distribution [5]; nevertheless, our observations in this work do not agree with this one. The motivation of this work is to find which distribution has the best fitting for the packet variation delay. For this purpose, we collected extensive Round Trip Time (RTT) measurements to other countries and carried out tests with the Kullback-Leibler divergence and P-P plot criteria. The remainder of this work is organized as follows: section two gives definitions of jitter and models to fit. Section three presents measurement methodologies and environment. Section four describes the comparisons of jitter measurements and proposed models. Finally, section five presents conclusions. II. STATISTICAL MODEL A. Jitter Definition 1: jitter is a random variable (r.v.) defined as the time difference between consecutive packets and is given by ) 1 ( ) ( ) (    k D k D k J (1) where D(k) is the delay of k-th packet. In an ideal scenario J(k) is equal to zero meaning that there is no difference between interdeparture and interarrival times. Nevertheless, when jitter is greater than zero a packet clustering occurs; and if jitter is less than zero there is packet spreading. The packet spreading degrades the Internet performance more than packet clustering. Figure 1 shows the jitter phenomenon, on an end-to-end connection, where k t is associated to the departure time of the k -th packet and () Dk is de delay produced by queuing along the path; note that the packet delay variation is related to the time difference between the delay of the k -th and ( 1 k  )-st packets. Each real-time application has its requirements of maximum jitter for an adequately performance, i.e., the jitter- QoS requirement for Video over IP (VIP) is (() 30 ) 0.99 PJk ms   and for Voice over IP (VoIP) { ( )} 30 EJk ms  [6]. B. Distributions Definition 2: An r.v. () Jk is said to have a t-Student distribution, if the equation (2) holds