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Irregularities and scaling in signal and image processing: Multifractal analysis
 M. FRAME ED., BENOIT MANDELBROT: A LIFE IN MANY DIMENSIONS, WORLD SCIENTIFIC
, 2012
"... B. Mandelbrot gave a new birth to the notions of scale invariance, selfsimilarity and noninteger dimensions, gathering them as the founding cornerstones used to build up fractal geometry. The first purpose of the present contribution is to review and relate together these key notions, explore thei ..."
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B. Mandelbrot gave a new birth to the notions of scale invariance, selfsimilarity and noninteger dimensions, gathering them as the founding cornerstones used to build up fractal geometry. The first purpose of the present contribution is to review and relate together these key notions, explore their interplay and show that they are different facets of a same intuition. Second, it will explain how these notions lead to the derivation of the mathematical tools underlying multifractal analysis. Third, it will reformulate these theoretical tools into a wavelet framework, hence enabling their better theoretical understanding as well as their efficient practical implementation. B. Mandelbrot used his concept of fractal geometry to analyze realworld applications of very different natures. As a tribute to his work, applications of various origins, and where multifractal analysis proved fruitful, are revisited to illustrate the theoretical developments proposed here.
Modeling TCP Throughput: an Elaborated LargeDeviationsBased Model and its Empirical Validation *
"... Abstract In today's Internet, a large part of the traffic is carried using the TCP transport protocol. Characterization of the variations of TCP traffic is thus a major challenge, both for resource provisioning and Quality of Service purposes. However, most existing models are limited to the p ..."
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Abstract In today's Internet, a large part of the traffic is carried using the TCP transport protocol. Characterization of the variations of TCP traffic is thus a major challenge, both for resource provisioning and Quality of Service purposes. However, most existing models are limited to the prediction of the (almostsure) mean TCP throughput and are unable to characterize deviations from this value. In this paper, we propose a method to describe the deviations of a long TCP flow's throughput from its almostsure mean value. This method relies on an ergodic largedeviations result, which was recently proved to hold on almost every single realization for a large class of stochastic processes. Applying this result to a Markov chain modeling the congestion window's evolution of a longlived TCP flow, we show that it is practically possible to quantify and to statistically bound the throughput's variations at different scales of interest for applications. Our Markovchain model can take into account various network conditions and we demonstrate the accuracy of our method's prediction in different situations using simulations, experiments and realworld Internet traffic. In particular, in the classical case of Bernoulli losses, we demonstrate: i) the consistency of our method with the widelyused squareroot formula predicting the almostsure mean throughput, and ii) its ability to additionally predict finer properties reflecting the traffic variability at different scales.
WAVELET SPECTRUM FOR INVESTIGATING STATISTICAL CHARACTERISTICS OF UDPBASED INTERNET TRAFFIC
"... ABSTRACT In this paper, we consider statistical characteristics of real User Datagram Protocol (UDP) traffic. Four main issues in the study include(i) the presence of long rangedependence (LRD) in the UDP traffic,(ii) the marginal distribution of the UDP traces,(iii) dependence structure of wavelet ..."
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ABSTRACT In this paper, we consider statistical characteristics of real User Datagram Protocol (UDP) traffic. Four main issues in the study include(i) the presence of long rangedependence (LRD) in the UDP traffic,(ii) the marginal distribution of the UDP traces,(iii) dependence structure of wavelet coefficients,(iv) and performance evaluation of the Hurst parameter estimation based on different numbers of vanishing moments of the mother wavelet. By analyzing a large set of real traffic data, it is evident that the UDP Internet traffic reveals the LRD properties with considerably high nonstationary processes.Furthermore, it exhibits nonGaussian marginal distributions. However, by increasing the number of vanishing moments,it is impossible to achieve reduction fromLRD to
Tail Dependence for Regularly Varying Time Series
"... We use tail dependence functions to study tail dependence for regularly varying RV time series. First, tail dependence functions about RV time series are deduced through the intensity measure. Then, the relation between the tail dependence function and the intensity measure is established: they are ..."
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We use tail dependence functions to study tail dependence for regularly varying RV time series. First, tail dependence functions about RV time series are deduced through the intensity measure. Then, the relation between the tail dependence function and the intensity measure is established: they are biuniquely determined. Finally, we obtain the expressions of the tail dependence parameters based on the expectation of the RV components of the time series. These expressions are coincided with those obtained by the conditional probability. Some simulation examples are demonstrated to verify the results we established in this paper.
Nine Years of Observing Traffic Anomalies: Trending Analysis in Backbone Networks
"... Abstract—We present the longitudinal trending analysis of traffic anomalies on a transPacific backbone network over nine years. Throughout our analysis, we try to answer several questions: how frequent do such anomalies appear and how long do they last? Does a set of anomalous hosts occur correspon ..."
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Abstract—We present the longitudinal trending analysis of traffic anomalies on a transPacific backbone network over nine years. Throughout our analysis, we try to answer several questions: how frequent do such anomalies appear and how long do they last? Does a set of anomalous hosts occur correspondingly? We answer these by applying the stateoftheart anomaly detectors to (un)anonymized packet traces and look into interesting insights from the longterm analysis. The key observations are as follow. The sources of anomalies are decreasing over the recent years, but take a significant portion of traffic volume during the measurement period (i.e., 0.03 % of all IP addresses take upto 30 % of traffic volume). The frequency analysis reveals that there is a clear periodicity of anomalies and anomalous host occurrences in various durations. Finally, we find the influences of anomaly detectors to the overall trending and how they differ from each other. I.
ITHEA 55 MODELING TELECOMMUNICATIONS TRAFFIC USING THE STOCHASTIC MULTIFRACTAL CASCADE PROCESS
"... Abstract: In this work the simulation of realizations of telecommunications traffic, which has multifractal properties. The mathematical model of traffic is based on a stochastic binomial multiplicative cascade process with betadistributed weighting coefficients. There was carry out computer simula ..."
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Abstract: In this work the simulation of realizations of telecommunications traffic, which has multifractal properties. The mathematical model of traffic is based on a stochastic binomial multiplicative cascade process with betadistributed weighting coefficients. There was carry out computer simulation of model multifractal traffic advancing over the communication channel. The emergence of queuing in the infinite buffer size and number of losses with limited buffer size has been studied.
Grenoble RhôneAlpes
"... Optimized protocols and software for high performance networks ..."
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Grenoble RhôneAlpes
"... Optimized protocols and software for high performance networks ..."
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HProbe: Estimating Traffic Correlations from Sampling and Active Network Probing
"... Abstract—An extensive body of research deals with estimating the correlation and the Hurst parameter of Internet traffic traces. The significance of these statistics is due to their fundamental impact on network performance. The coverage of Internet traffic traces is, however, limited since acquirin ..."
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Abstract—An extensive body of research deals with estimating the correlation and the Hurst parameter of Internet traffic traces. The significance of these statistics is due to their fundamental impact on network performance. The coverage of Internet traffic traces is, however, limited since acquiring such traces is challenging with respect to, e.g., confidentiality, logging speed, and storage capacity. In this work, we investigate how the correlation of Internet traffic can be reliably estimated from random traffic samples. These samples are observed either by passive monitoring within the network, or otherwise by active packet probes at end systems. We analyze random sampling processes with different intersample distributions and show how to obtain asymptotically unbiased estimates from these samples. We quantify the inherent limitations that are due to limited observations and explore the influence of various parameters, such as sampling intensity, network utilization, or Hurst parameter on the estimation accuracy. We design an active probing method which enables simple and lightweight traffic sampling without support from the network. We verify our approach in a controlled network environment and present comprehensive Internet measurements. We find that the correlation exhibits properties such as long range dependence as well as periodicities and that it differs significantly across Internet paths and observation times. I.