Fractional Brownian motion (fBm) has been used as a theoretical framework to study real-time series appearing in diverse scientific fields. Because of its intrinsic nonstationarity and long-range dependence, its characterization via the Hurst parameter, H, requires sophisticated techniques that often yield ambiguous results. In this work we show that fBm series map into a scale-free visibility graph whose degree distribution is a function of H. Concretely, it is shown that the exponent of the power law degree distribution depends linearly on H. This also applies to fractional Gaussian noises (fGn) and generic f−β noises. Taking advantage of these facts, we propose a brand new methodology to quantify long-range dependence in these series. Its reliability is confirmed with extensive numerical simulations and analytical developments. Finally, we illustrate this method quantifying the persistent behavior of human gait dynamics.

https://iopscience.iop.org/article/10.1209/0295-5075/86/30001/pdf

The visibility graph: A new method for estimating the Hurst exponent of fractional Brownian motion

Reliable performance evaluations require the use of representative workloads. This is no easy task since modern computer systems and their workloads are complex, with many interrelated attributes and complicated structures. Experts often use sophisticated mathematics to analyze and describe workload models, making these models difficult for practitioners to grasp. This book aims to close this gap by emphasizing the intuition and the reasoning behind the definitions and derivations related to the workload models. It provides numerous examples from real production systems, with hundreds of graphs. Using this book, readers will be able to analyze collected workload data and clean it if necessary, derive statistical models that include skewed marginal distributions and correlations, and consider the need for generative models and feedback from the system. The descriptive statistics techniques covered are also useful for other domains.

Workload Modeling for Computer Systems Performance Evaluation

http://staff.iium.edu.my/sakib/ndclab/papers/SI-MCM_FINAL-ALL--Camera.pdf

Accurate modeling of VoIP traffic QoS parameters in current and future networks with multifractal and Markov models

This paper revisits three important concepts in fractal type network traffic, namely, self-similarity (SS), long-range dependence (LRD), and local self-similarity (LSS). Based on those concepts, we address the reason why the local properties of fractional Gaussian noise (fGn) are contained in the global properties of fGn and vice versa, which may be a limitation of fGn in data traffic modeling. Key-Words: Fractal time series; Network traffic; Fractal dimension; The Hurst parameter.

http://www.wseas.us/e-library/conferences/2009/hangzhou/MUSP/MUSP02.pdf

Self-similarity and long-range dependence in teletraffic

This paper proposes a multifractal model, with the aim of providing a possible explanation for the locality phenomenon that appears in the estimation of the Hurst exponent in stationary second order temporal series representing self-similar traffic flows in current high-speed computer networks. It is shown analytically that this phenomenon occurs if the network flow consists of several components with different Hurst exponents.

/pdf/a-simplified-multifractal-model-for-self-similar-traffic-39tlblbydz.pdf

G. Horn

Papers

An empirical comparison of generators for self similar simulated traffic