[Paper Review] Fractal Analysis On Internet Traffic Time Series

K. B. Chong, K. Y. Choo|ArXiv.org|Jun 5, 2002

Complex Systems and Time Series Analysis1 references3 citations

TL;DR

This study applies fractal analysis—using power-spectral analysis (PSA), detrended fluctuation analysis (DFA), and time-scale analysis (TSA)—to internet traffic time series from the Library of Congress (LOC) server. It demonstrates that both LOC(request) and LOC(send) traffic exhibit long-range dependence and self-similarity, with spectral exponent β ∈ (1,2), Holder exponent H ∈ (0,0.5), fractal dimension D ∈ (1,2), and negative correlation ρ ∈ (−0.5,0), confirming fractal characteristics across multiple scales.

ABSTRACT

Fractal behavior and long-range dependence have been observed in tele-traffic measurement and characterization. In this paper we show results of application of the fractal analysis to internet traffic via various methods. Our result demonstrate that the internet traffic exhibits self-similarity. Time-scale analysis show to be an effective way to characterize the local irregularity. Based on the result of this study, these two Internet time series exhibit fractal characteristic with long-range dependence.

Motivation & Objective

• To investigate fractal and long-range dependent characteristics in internet traffic time series.
• To compare the dynamics of incoming (LOC(request)) and outgoing (LOC(send)) traffic at the LOC server.
• To evaluate the effectiveness of multiple fractal analysis techniques in capturing complex temporal behavior.
• To quantify local and global scaling properties using time-scale analysis (TSA), revealing multifractal and non-stationary features.
• To provide a foundation for improved network modeling and optimization through advanced fractal characterization.

Proposed method

• Power-spectral analysis (PSA) is used to estimate the spectral exponent β from the power-law decay of the power spectrum S(ω) ∝ ω^−β.
• Detrended fluctuation analysis (DFA) computes the α-exponent to identify long-range dependence, with crossovers observed at segment lengths of 60 and 400.
• Time-scale analysis (TSA) computes local Holder exponent H(t) across time windows, revealing non-stationary, multifractal behavior.
• The relationship between Holder exponent H and fractal dimension D is applied via D = 2 − H, with H ∈ (0, 0.5) implying D ∈ (1, 2).
• Correlation coefficient ρ is derived from ρ = 2^(2H−1) − 1, confirming anti-persistent behavior (ρ < 0) for both traffic types.
• Wavelet-based TSA enables local singularity analysis, capturing time-varying scaling behavior beyond global averages.

Experimental results

Research questions

• RQ1 Do internet traffic time series (LOC(request) and LOC(send)) exhibit long-range dependence and self-similarity?
• RQ2 What are the scaling exponents (β, H, D, ρ) characterizing the fractal nature of these traffic streams?
• RQ3 How do the local scaling properties (H(t)) differ between incoming and outgoing traffic, and what explains the complexity?
• RQ4 What is the significance of crossover phenomena observed in DFA, and how do they reflect underlying network dynamics?
• RQ5 Can time-scale analysis (TSA) reveal multifractal and non-stationary features not captured by global methods like PSA and DFA?

Key findings

• The spectral exponent β lies in the range 1 < β < 2, confirming long-range dependence in both LOC(request) and LOC(send) traffic.
• The Holder exponent H is in the range 0 < H < 0.5, indicating anti-persistent behavior and fractal dynamics with D = 2 − H ∈ (1, 2).
• Correlation coefficient ρ is negative, with −0.5 < ρ < 0, consistent with anti-persistent fractional Brownian motion.
• DFA reveals crossover behavior at segment lengths of 60 and 400, with α ≈ 1.0 (white noise), α ≈ 1.0 (1/f process), and α ≈ 2.0 (smooth process).
• TSA shows local Holder exponents H(t) span from −0.49 to 1.48 for LOC(request) and −0.26 to 1.15 for LOC(send), indicating complex, non-stationary, multifractal dynamics.
• LOC(request) exhibits higher local complexity than LOC(send), likely due to network congestion at the server gateway from massive incoming data streams.