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52
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 173 (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.
Scaling Analysis of Conservative Cascades, With Applications to Network Traffic
, 1999
"... Recent studies have demonstrated that measured widearea network traffic such as Internet traffic exhibits locally complex irregularities, consistent with multifractal behavior. It has also been shown that the observed multifractal structure becomes most apparent when analyzing measured network tr ..."
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Cited by 45 (7 self)
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Recent studies have demonstrated that measured widearea network traffic such as Internet traffic exhibits locally complex irregularities, consistent with multifractal behavior. It has also been shown that the observed multifractal structure becomes most apparent when analyzing measured network traffic at a particular layer in the welldefined protocol hierarchy that characterizes modern data networks, namely the transport or TCP layer. To investigate this new scaling phenomenon associated with the dynamics of measured network traffic over small time scales, we consider a class of multiplicative processes, the socalled conservative cascades, that serves as a cascade paradigm for and is motivated by the networking application. We present a waveletbased time/scale analysis of these cascades to determine rigorously their global and local scaling behavior. In particular, we prove that for the class of multifractals generated by these conservative cascades the multifractal formal...
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 30 (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.
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 24 (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....
Directed polymers in random environment: path localization and strong disorder
 Bernoulli
, 2003
"... We consider directed polymers in random environment. Under mild assumptions on the environment, we prove here: (i) equivalence of decay rate of the partition function with some natural localization properties of the path, (ii) quantitative estimates of the decay of the partition function in dimensio ..."
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Cited by 19 (4 self)
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We consider directed polymers in random environment. Under mild assumptions on the environment, we prove here: (i) equivalence of decay rate of the partition function with some natural localization properties of the path, (ii) quantitative estimates of the decay of the partition function in dimensions one or two, or at sufficiently low temperature, (iii) existence of quenched free energy. In particular, we generalize to general environments, some of the results recently obtained by P. Carmona and Y. Hu for a Gaussian environment. We do not discuss here superdiffusivity or critical exponents.
Multifractal products of stochastic processes: Part II, 2003. Under construction
"... some basic properties ..."
MAJORIZING MULTIPLICATIVE CASCADES FOR DIRECTED POLYMERS IN RANDOM MEDIA
, 2005
"... In this note we give upper bounds for the free energy of discrete directed polymers in random media. The bounds are given by the socalled generalized multiplicative cascades from the statistical theory of turbulence. For the polymer model, we derive that the quenched free energy is different from ..."
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Cited by 17 (2 self)
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In this note we give upper bounds for the free energy of discrete directed polymers in random media. The bounds are given by the socalled generalized multiplicative cascades from the statistical theory of turbulence. For the polymer model, we derive that the quenched free energy is different from the annealed one in dimension 1, for any finite temperature and general environment. This implies localization of the polymer.
A probabilistic analysis of some tree algorithms, in "Annals of Applied Probability
, 2005
"... In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily without resorting to the usual complex analysis techniques. Thi ..."
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Cited by 16 (5 self)
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In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily without resorting to the usual complex analysis techniques. This approach gives a unified probabilistic treatment of these questions. It simplifies and extends some of the results known in this domain. 1. Introduction. A
A Stochastic Fixed Point Equation Related to Weighted Branching
"... For real numbers C, T1, T2,... we find all solutions µ to the stochastic fixed point equation W � j≥1 TjWj + C, where W, W1, W2,... are independent realvalued random variables with distribution µ and means equality in distribution. All solutions are infinitely divisible. The set of solutions depend ..."
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Cited by 13 (3 self)
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For real numbers C, T1, T2,... we find all solutions µ to the stochastic fixed point equation W � j≥1 TjWj + C, where W, W1, W2,... are independent realvalued random variables with distribution µ and means equality in distribution. All solutions are infinitely divisible. The set of solutions depends on the closed multiplicative subgroup of R ∗ = R\{0} generated by the Tj. If this group is continuous, i.e. R ∗ itself or the positive halfline R>, then all nontrivial fixed points are stable laws. In the remaining (discrete) cases further periodic solutions arise. A key observation is that the Lévy measure of any fixed point is harmonic with respect to Λ = � j≥1 δT, i.e. Γ = Γ ⋆ Λ, where ⋆ means multiplicative j convolution. This will enable us to apply the powerful ChoquetDeny theorem.