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64
A multifractal wavelet model with application to TCP network traffic
 IEEE TRANS. INFORM. THEORY
, 1999
"... In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the mo ..."
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Cited by 171 (30 self)
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In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing Npoint data sets. We study both the secondorder and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variancetime plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.
Multifractional Brownian motion: definition and preliminary results

, 1995
"... We generalize the definition of the fractional Brownian motion of exponent H to the case where H is no longer a constant, but a function of the time index of the process. This allows us to model non stationary continuous processes, and we show that H(t) and 2 \Gamma H(t) are indeed respectively t ..."
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Cited by 50 (3 self)
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We generalize the definition of the fractional Brownian motion of exponent H to the case where H is no longer a constant, but a function of the time index of the process. This allows us to model non stationary continuous processes, and we show that H(t) and 2 \Gamma H(t) are indeed respectively the local Holder exponent and the local box and Hausdorff dimension at point t. Finally, we propose a simulation method and an estimation procedure for H(t) for our model.
Fractional Brownian motion and data traffic modeling: The other end of the spectrum
 Fractals in Engineering
, 1997
"... Introduction Fractal analysis of computer traffic has received considerable attention since the seminal work of Leland and al. [11] who provided experimental evidence that some traces of data traffic exhibit long range dependence (LRD). This is a typical fractal feature which is not found with the ..."
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Cited by 39 (13 self)
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Introduction Fractal analysis of computer traffic has received considerable attention since the seminal work of Leland and al. [11] who provided experimental evidence that some traces of data traffic exhibit long range dependence (LRD). This is a typical fractal feature which is not found with the classical Poisson models. An important issue since then has been to propose "physical" models that lead to such fractal behavior. A popular model [27] is based on the superposition of simple i.i.d ON/OFF sources which ON and/or OFF periods follow a heavy tailed law (P r(X ? ) ¸ c \Gammaff ; 1 ! ff ! 2). When properly normalized, the resulting traffic is a fractional Brownian motion (fBm) of LRD exponent H = (3 \Gamma ff)=2. Several practical implications of LRD traffic have consequently been investigated, e.g. the queuing behavior [15] (see
Multifractal Processes
, 1999
"... This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and sel ..."
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Cited by 28 (6 self)
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This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and selfsimilar processes with a special eye on the use of wavelets. Particular attention is given to a novel class of multifractal processes which combine the attractive features of cascades and selfsimilar processes. Statistical properties of estimators as well as modelling issues are addressed.
Multifractal Measures and a Weak Separation Condition
, 1999
"... We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the wellknown class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of ..."
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Cited by 28 (11 self)
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We define a new separation property on the family of contractive similitudes that allows certain overlappings. This property is weaker than the open set condition of Hutchinson. It includes the wellknown class of infinite Bernoulli convolutions associated with the P.V. numbers and the solutions of the twoscale dilation equations. Our main purpose in this paper is to prove the multifractal formalism under such condition.
LongRange Dependence and Data Network Traffic
, 2001
"... This is an overview of a relatively recent application of longrange dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in highspeed data networks such as the Internet. We demonstrate that this new application area off ..."
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Cited by 23 (1 self)
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This is an overview of a relatively recent application of longrange dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in highspeed data networks such as the Internet. We demonstrate that this new application area offers unique opportunities for significantly advancing our understanding of LRD and related phenomena. These advances are made possible by moving beyond the conventional approaches associated with the widespread "blackbox" perspective of traditional time series analysis and exploiting instead the physical mechanisms that exist in the networking context and that are intimately tied to the observed characteristics of measured network traffic. In order to describe this complexity we provide a basic understanding of the design, architecture and operations of data networks, including a description of the TCP/IP protocols used in today's Internet. LRD is observed in the large scale behavior of the data traffic and we provide a physical explanation for its presence. LRD tends to be caused by user and application characteristics and has little to do with the network itself. The network affects mostly small time scales, and this is why a rudimentary understanding of the main protocols is important. We illustrate why multifractals may be relevant for describing some aspects of the highly irregular traffic behavior over small time scales. We distinguish between a timedomain and waveletdomain approach to analyzing the small time scale dynamics and discuss why the waveletdomain approach appears to be better suited than the timedomain approach for identifying features in measured traffic (e.g., relatively regular traffic patterns over certain time scales) that have a direct networking interpretation (e....
Combining multifractal additive and multiplicative chaos
 COMMUN. MATH. PHYS
, 2005
"... In this work, we study the new class of multifractal measures, which combines additive and multiplicative chaos, defined by νγ,σ = X b j≥1 −jγ j2 X 0 ≤ k ≤bj µ([kb −1 −j, (k + 1)b −j)) σ δkb−j (γ ≥ 0, σ ≥ 1), where µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2. The singularities ..."
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Cited by 20 (12 self)
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In this work, we study the new class of multifractal measures, which combines additive and multiplicative chaos, defined by νγ,σ = X b j≥1 −jγ j2 X 0 ≤ k ≤bj µ([kb −1 −j, (k + 1)b −j)) σ δkb−j (γ ≥ 0, σ ≥ 1), where µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2. The singularities analysis of the measures νγ,σ involves new results on the mass distribution of µ when µ describes large classes of multifractal measures. These results generalize ubiquity theorems associated with the Lebesgue measure. Under suitable assumptions on µ, the multifractal spectrum of νγ,σ is linear on [0, hγ,σ] for some critical value hγ,σ, and then it is strictly concave on the right of hγ,σ, and deduced from the one of µ by an affine transformation. This untypical shape is the result of the combination between Dirac masses and atomless multifractal measures. These measures satisfy multifractal formalisms. These measures open interesting perspectives in modeling discontinuous phenomena.
The singularity spectrum of Lévy processes in multifractal time
, 2007
"... The interest for multifractal stochastic processes is mainly motivated by the need for accurate models in the study of the variability of wild signals. These locally irregular signals come from physical phenomena such as fully developed turbulence, TCP Internet traffic, variations of financial price ..."
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Cited by 18 (12 self)
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The interest for multifractal stochastic processes is mainly motivated by the need for accurate models in the study of the variability of wild signals. These locally irregular signals come from physical phenomena such as fully developed turbulence, TCP Internet traffic, variations of financial prices, or heart beats.
Multifractal Analysis of Choquet Capacities: Preliminary Results
, 1995
"... A multifractal analysis is defined for sequences of Choquet capacities with respect to a general class of measures, and some preliminary results are presented concerning the usual spectra. In particular, we show how to construct a sequence of capacities whose Hölder spectrum is, under mild conditio ..."
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Cited by 15 (2 self)
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A multifractal analysis is defined for sequences of Choquet capacities with respect to a general class of measures, and some preliminary results are presented concerning the usual spectra. In particular, we show how to construct a sequence of capacities whose Hölder spectrum is, under mild conditions, prescribed.
The multifractal nature of heterogeneous sums of Dirac masses
 MATH. PROC. CAMBRIDGE PH. SOC
, 2008
"... This article investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masses ν = ∑ n≥0 wn δxn, where (xn)n≥0 is a sequence of points in [0, 1] d and (wn)n≥0 is a positive sequence of weights such that ∑ n≥0 wn < ∞. We consider the case where the point ..."
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Cited by 14 (8 self)
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This article investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masses ν = ∑ n≥0 wn δxn, where (xn)n≥0 is a sequence of points in [0, 1] d and (wn)n≥0 is a positive sequence of weights such that ∑ n≥0 wn < ∞. We consider the case where the points xn are roughly uniformly distributed in [0, 1] d, and the weights wn depend on a random selfsimilar measure µ, a parameter ρ ∈ (0, 1], and a sequence of positive radii (λn)n≥1 converging to 0 in the following way wn = λ d(1−ρ) n µ ( B(xn, λ ρ n) )  log λn  −2. The measure ν has a rich multiscale structure. The computation of its multifractal spectrum is related to heterogeneous ubiquity properties of the system {(xn, λn)}n with respect to µ.