doi:10.1016/j.peva.2007.12.003
Copyright © 2007 Elsevier Ltd All rights reserved.
Rate-interval curves — A tool for the analysis and monitoring of network traffic
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G. Giorgi
, a,
and C. Narduzzia, 
aDepartment of Information Engineering, University of Padova, via Gradenigo 6/b, 35100 Padova, Italy
Received 26 October 2007;
accepted 7 December 2007.
Available online 15 December 2007.
Abstract
In this paper we propose a method, related to the theory of Network Calculus, for the analysis of aggregate network traffic by means of empirical rate-interval curves obtained from experimental data. The algorithm on which it is based differs from the commonly employed wavelet-based approach, although it retains some of its multiresolution features. We briefly introduce the theoretical aspects, analyze measurement accuracy and present results, obtained both by simulation and by the analysis of real traffic traces, which provide an assessment of the strengths and weaknesses of the proposed method.
Rate-interval curve analysis provides very robust and acceptably accurate estimates of the Hurst parameter value and, even in the presence of flow irregularities, results can be proved to be correct as far as scaling properties are concerned.
Further analyzes concerning peaks, bursts and similar localized phenomena that may have a significant impact on the performances of a network are allowed by considering maximal rate envelopes, showing the potential of this approach for monitoring applications.
Keywords: Traffic measurement; Flow rate; Uncertainty evaluation
Fig. 1. Theoretical and empirical curves for a fractional Brownian motion traffic model.
Fig. 2. Rate-interval curves for a fractional Brownian motion traffic model plotted in a RIC log-scale diagram. Theoretical upper and lower bounds, Yup and Ylow, are drawn by a solid line.
Fig. 3. RIC log-scale diagram.
(a) DWT diagram.
(b) RIC diagram.
Fig. 4. Log-scale diagrams.
(a) Scale j=6.
(b) Scale j=8.
(c) Scale j=10.
Fig. 5. Cumulative density functions at different scales.
(a) Scale j=6.
(b) Scale j=8.
(c) Scale j=10.
Fig. 6. Normal probability papers at different scales.
Fig. 7. RIC log-scale diagram.
(a) Scale j=10.
(b) Scale j=12.
(c) Scale j=14.
Fig. 8. Cumulative density functions at different scales.
Fig. 9. Aggregate time series at T=1 ms. The presence of both burst and peaks can be noted in this trace.
Fig. 10. Rate envelope calculated for a violation probability γ=0.10 against the maximal rate envelope.
Fig. 11. Rate-interval curves plotted in a RIC log-scale diagram. Theoretical upper and lower bounds, Yup and Ylow, are drawn by a solid line.
(a) Aggregate time series at
. The presence of several bursts can be noted.
(b) Estimated Hurst values.
Fig. 12. Comparison between the aggregate time series and the series of the estimated values of the Hurst parameter.
Fig. 13. Aggregate time series of the “well-behaved” packets at
. No burst is present.
Fig. 14. Analysis of the time series of “well-behaved”packets only. Rate-interval curves plotted in a RIC log-scale diagram. Theoretical upper and lower bounds, Yup and Ylow, are drawn by a solid line.
Table 1.
Estimation of the Hurst parameter H by rate-interval curves

Table 2.
Estimation of the Hurst parameter by DWT-based tool

Expanded version of the paper “Analysis of Traffic Flow Measurements by Rate-Interval Curves” was presented at the Valuetools 2006 Conference (Pisa, Italy, 12–14 October 2007).

Corresponding author.