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35
SelfSimilarity Through HighVariability: Statistical Analysis of Ethernet LAN Traffic at the Source Level
 IEEE/ACM TRANSACTIONS ON NETWORKING
, 1997
"... A number of recent empirical studies of traffic measurements from a variety of working packet networks have convincingly demonstrated that actual network traffic is selfsimilar or longrange dependent in nature (i.e., bursty over a wide range of time scales)  in sharp contrast to commonly made tr ..."
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Cited by 616 (24 self)
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A number of recent empirical studies of traffic measurements from a variety of working packet networks have convincingly demonstrated that actual network traffic is selfsimilar or longrange dependent in nature (i.e., bursty over a wide range of time scales)  in sharp contrast to commonly made traffic modeling assumptions. In this paper, we provide a plausible physical explanation for the occurrence of selfsimilarity in LAN traffic. Our explanation is based on new convergence results for processes that exhibit high variability (i.e., infinite variance) and is supported by detailed statistical analyses of realtime traffic measurements from Ethernet LAN's at the level of individual sources. This paper is an extended version of [53] and differs from it in significant ways. In particular, we develop here the mathematical results concerning the superposition of strictly alternating ON/OFF sources. Our key mathematical result states that the superposition of many ON/OFF sources (also k...
Wavelet Analysis of Long Range Dependent Traffic
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1998
"... A Wavelet based tool for the analysis of long range dependence is introduced and a related semiparametric estimator of the Hurst parameter. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing t ..."
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Cited by 228 (19 self)
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A Wavelet based tool for the analysis of long range dependence is introduced and a related semiparametric estimator of the Hurst parameter. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the long range dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono vs multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.
Multiresolution markov models for signal and image processing
 Proceedings of the IEEE
, 2002
"... This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coheren ..."
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Cited by 121 (17 self)
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This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and conceptsâ€“in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for selfsimilar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden
Wavelet Analysis of LongRangeDependent Traffic
, 1998
"... A waveletbased tool for the analysis of longrange dependence and a related semiparametric estimator of the Hurst parameter is introduced. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing ..."
Abstract

Cited by 106 (1 self)
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A waveletbased tool for the analysis of longrange dependence and a related semiparametric estimator of the Hurst parameter is introduced. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing the direct analysis of very large data sets, and is highly robust against the presence of deterministic trends, as well as allowing their detection and identification. Statistical, computational, and numerical comparisons are made against traditional estimators including that of Whittle. The estimator is used to perform a thorough analysis of the longrange dependence in Ethernet traffic traces. New features are found with important implications for the choice of valid models for performance evaluation. A study of mono versus multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.
SemiSelfsimilar Processes
, 1999
"... A notion of semiselfsimilarity of R d valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semiselfsimilar processes are studied: 1. The existence of the exponent of semiselfsimilar processes. 2. Characterization for semiselfsimilar processes ..."
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Cited by 59 (3 self)
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A notion of semiselfsimilarity of R d valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semiselfsimilar processes are studied: 1. The existence of the exponent of semiselfsimilar processes. 2. Characterization for semiselfsimilar processes as scaling limits. 3. Relationship between semiselfsimilar processes with independent increments and semiselfdecomposable distributions. 4. Construction of semiselfsimilar processes with stationary increments. Semistable processes where all joint distributions are multivariate semistable are also discussed in connection with semiselfsimilar processes.
On the Relative Lengths of Excursions Derived From a Stable Subordinator
, 1996
"... Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times T the distribution of relative excursi ..."
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Cited by 14 (6 self)
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Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times T the distribution of relative excursion lengths prior to T is the same as if T were a fixed time. It follows that the generalized arcsine laws of Lamperti extend to such random times T . For some other random times T , absolute continuity relations are obtained which relate the law of the relative lengths at time T to the law at a fixed time. 1 Introduction Following Lamperti [10], Wendel [24], Kingman [7], Knight [8], PermanPitman Yor [12, 13, 15], consider the sequence V 1 (T ) V 2 (T ) \Delta \Delta \Delta (1) of ranked lengths of component intervals of the set [0; T ]nZ, where T is a strictly positive random time, and Z is the zero set of a Markov process X started at zero, such as a Brownian motion or Bessel process,...
Random Discrete Distributions Derived From SelfSimilar Random Sets
 Electronic J. Probability
, 1996
"... : A model is proposed for a decreasing sequence of random variables (V 1 ; V 2 ; \Delta \Delta \Delta) with P n V n = 1, which generalizes the PoissonDirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let V n be the length ..."
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Cited by 14 (10 self)
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: A model is proposed for a decreasing sequence of random variables (V 1 ; V 2 ; \Delta \Delta \Delta) with P n V n = 1, which generalizes the PoissonDirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let V n be the length of the nth longest component interval of [0; 1]nZ, where Z is an a.s. nonempty random closed of (0; 1) of Lebesgue measure 0, and Z is selfsimilar, i.e. cZ has the same distribution as Z for every c ? 0. Then for 0 a ! b 1 the expected number of n's such that V n 2 (a; b) equals R b a v \Gamma1 F (dv) where the structural distribution F is identical to the distribution of 1 \Gamma sup(Z " [0; 1]). Then F (dv) = f(v)dv where (1 \Gamma v)f(v) is a decreasing function of v, and every such probability distribution F on [0; 1] can arise from this construction. Keywords: interval partition, zero set, excursion lengths, regenerative set, structural distribution. AMS subject classificat...
On the distribution of ranked heights of excursions of a Brownian bridge
 In preparation
, 1999
"... The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where th ..."
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Cited by 11 (6 self)
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The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where the distribution of M br+ 1 = sup 0t1 B br t is given by L'evy's formula P (M br+ 1 ? x) = e \Gamma2x 2 . The probability density of the height M br j of the jth highest maximum of excursions of the reflecting Brownian bridge (jB br t j; 0 t 1) is given by a modification of the known `function series for the density of M br 1 = sup 0t1 jB br t j. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a selfsimilar recurrent Markov process. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, selfsimilar recurrent Markov process, Bessel p...
Lower Tails of SelfSimilar Stable Processes
, 1996
"... this paper we address the question of the socalled lower tails of Hsssi SffS processes. That is, we are interested in the behavior of the "small ball" probability ..."
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Cited by 10 (0 self)
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this paper we address the question of the socalled lower tails of Hsssi SffS processes. That is, we are interested in the behavior of the "small ball" probability