Effects of Spatial Aggregation on the Characteristics of Origin-Destination Pair Traffic in Funet

Abstract

In this paper we analyze measurements from the Finnish University Network (Funet) and study the effect of spatial aggregation on the origin-destination flows. The traffic is divided into OD pairs based on IP addresses, using different prefix lengths to obtain data sets with various aggregation levels. We find that typically the diurnal pattern of the total traffic is followed more closely by the OD pairs as their volume increases, but there are many exceptions. Gaussian assumption holds well for all OD pairs when the aggregation level is high enough, and we find an approximate threshold for OD pair traffic volume after which they tend to be Gaussian. Also the functional mean-variance relation holds better when the aggregation level is higher.

Blind maximum likelihood estimation of traffic matrices under long-range dependent traffic

“…The main conclusion is that stationarity can be reasonably assumed to hold within periods of 30-90 min. The empirical studies performed in [15,11,16] for Funet data essentially confirm the stationarity assumption in [7] (again, a time aggregation from 1 s to 300 s is used).…”

“…To justify this statement, we first remark that the expectations l X i s and the variances r 2 X i s are related by a monotone relationship: the larger the mean l X i , the larger the variance r 2 X i . More precisely, a power law r 2 X i ¼ cl 1=q X i (with c a suitable constant) is usually considered, with q = 1 as in [6], or q = 2/ 3, as in [16]. As a consequence, denoting by Nð0; 1Þ a standard normal distribution, we have…”

“…Set an initial estimate for the expected value and the variance of traffic X repeat Compute for each OD pair the (identity) autocorrelation matrix R (expressions (4), with H = 0.5) Compute the mean value of the OD traffic conditioned on the traffic on links through expression (16) Compute the covariance matrix of the OD traffic conditioned on the traffic on links through expression (17) Compute the expected value of traffic by expression (27) Find the values of the variances of the OD pairs maximizing the likelihood function (by Nelder and Mead's method [47]) ? new estimate of the variances of the OD pairs Update estimate of traffic on links by expression (6) until Convergence…”

“…Such an assumption has been considered in [7,14,15,11,16], and validated via QQ-plots and related correlation tests comparing the empirical distribution with a fitted Gaussian distribution. The data at hand are obtained from the traffic observed on one link, which is partitioned into OD traffic based on source and destination IP addresses (time aggregation from 1 s to 300 s), except in [7] where the data are the traffic observed on the links and OD pairs of a onerouter network with 5 min time aggregation, as provided by SNMP.…”

Estimation of traffic matrices in the presence of long memory traffic

“…The assumption of Gaussianity of the S-D bits (packets) arrival process is justified by (space/time) aggregation obtained by superimposing independent traffic processes (independent sources) satisfying the usual conditions of the functional central limit theorem (see Norros and Pruthi, 1996;Taqqu et al, 1997). The assumption has been considered in Cao et al (2000), Norros and Kilpi (2002), Juva et al (2005), Susitaival et al (2006) and Juva et al (2007), and validated via QQ-plots and related correlation tests that compare the empirical distribution with a fitted Gaussian distribution. The data at hand is traffic observed on one link partioned into S-D traffic based on source and destination IP address (time aggregation from 1 second to 300 seconds), except in Cao et al (2000) where data is traffic observed on the links and S-D pairs of a one-router network with 5 minutes time aggregation, as provided by Simple Network Management Protocol (SNMP).…”