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33
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 597 (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...
Experimental Queueing Analysis with LongRange Dependent Packet Traffic
 IEEE/ACM Transactions on Networking
, 1996
"... Recent traffic measurement studies from a wide range of working packet networks have convincingly established the presence of significant statistical features that are characteristic of fractal traffic processes, in the sense that these features span many time scales. Of particular interest in packe ..."
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Cited by 295 (13 self)
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Recent traffic measurement studies from a wide range of working packet networks have convincingly established the presence of significant statistical features that are characteristic of fractal traffic processes, in the sense that these features span many time scales. Of particular interest in packet traffic modeling is a property called longrange dependence, which is marked by the presence of correlations that can extend over many time scales. In this paper, we demonstrate empirically that, beyond its statistical significance in traffic measurements, longrange dependence has considerable impact on queueing performance, and is a dominant characteristic for a number of packet traffic engineering problems. In addition, we give conditions under which the use of compact and simple traffic models that incorporate longrange dependence in a parsimonious manner (e.g., fractional Brownian motion) is justified and can lead to new insights into the traffic management of highspeed networks. 1...
Fitting Mixtures Of Exponentials To LongTail Distributions To Analyze Network Performance Models
, 1997
"... Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and interva ..."
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Cited by 144 (13 self)
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Traffic measurements from communication networks have shown that many quantities characterizing network performance have longtail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and intervals between connection requests in Internet traffic all have been found to have longtail distributions, being well described by distributions such as the Pareto and Weibull. It is known that longtail distributions can have a dramatic effect upon performance, e.g., longtail servicetime distributions cause longtail waitingtime distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component longtail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a longtail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove tha...
A Wavelet Based Joint Estimator of the Parameters of LongRange Dependence.
, 1998
"... A joint estimator is presented for the two parameters that define the longrange dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a recently proposed waveletbased estimator of the scaling parameter [4], as well as ..."
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Cited by 60 (10 self)
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A joint estimator is presented for the two parameters that define the longrange dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a recently proposed waveletbased estimator of the scaling parameter [4], as well as extending it to include the associated power parameter. An important feature is its conceptual and practical simplicity, consisting essentially in measuring the slope and the intercept of a linear fit after a discrete wavelet transform is performed, a very fast (O(n)) operation. Under well justified technical idealisations the estimator is shown to be unbiased and of minimum or close to minimum variance for the scale parameter, and asymptotically unbiased and efficient for the second parameter. Through theoretical arguments and numerical simulations it is shown that in practice, even for small data sets, the bias is very small and the variance close to optimal for both parameters. Closed for...
Waitingtime tail probabilities in queues with longtail servicetime distributions
 QUEUEING SYSTEMS
, 1994
"... We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails to ..."
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Cited by 55 (21 self)
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We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails to have a finite moment generating function. We have developed algorithms for computing the waitingtime distribution by Laplace transform inversion when the Laplace transforms of the interarrivaltime and servicetime distributions are known. One algorithm, exploiting Pollaczek’s classical contourintegral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient twoparameter family of longtail distributions on the positive half line with explicit Laplace transforms. This family is a Pareto mixture of exponential (PME) distributions. These PME distributions have monotone densities and Paretolike tails, i.e., are of order x − r for r> 1. We use this family of longtail distributions to investigate the quality of approximations based on asymptotics for P(W> x) as x → ∞. We show that the asymptotic approximations with these longtail servicetime distributions can be remarkably inaccurate for typical x values of interest. We also derive multiterm asymptotic expansions for the waitingtime tail probabilities in the M/G/1 queue. Even three terms of this expansion can be remarkably inaccurate for typical x values of interest. Thus, we evidently must rely on numerical algorithms for determining the waitingtime tail probabilities in this case. When working with servicetime data, we suggest using empirical Laplace transforms.
Heavy Tail Modeling And Teletraffic Data
 Annals of Statistics
, 1997
"... . Huge data sets from the teletraffic industry exhibit many nonstandard characteristics such as heavy tails and long range dependence. Various estimation methods for heavy tailed time series with positive innovations are reviewed. These include parameter estimation and model identification methods ..."
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Cited by 54 (4 self)
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. Huge data sets from the teletraffic industry exhibit many nonstandard characteristics such as heavy tails and long range dependence. Various estimation methods for heavy tailed time series with positive innovations are reviewed. These include parameter estimation and model identification methods for autoregressions and moving averages. Parameter estimation methods include those of YuleWalker and the linear programming estimators of Feigin and Resnick as well estimators for tail heaviness such as the Hill estimator and the qqestimator. Examples are given using call holding data and interarrivals between packet transmissions on a computer network. The limit theory makes heavy use of point process techniques and random set theory. 1. Introduction Classical queuing and network stochastic models contain simplifying assumptions guaranteeing the Markov property and insuring analytical tractability. Frequently interarrivals and service times are assumed to be iid and typically underlyi...
A waveletbased joint estimator of the parameters of longrange dependence
 IEEE Trans. Inform. Theory
, 1999
"... Abstract—A joint estimator is presented for the two parameters that define the longrange dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a recently proposed waveletbased estimator of the scaling parameter [4], as ..."
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Cited by 37 (8 self)
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Abstract—A joint estimator is presented for the two parameters that define the longrange dependence phenomenon in the simplest case. The estimator is based on the coefficients of a discrete wavelet decomposition, improving a recently proposed waveletbased estimator of the scaling parameter [4], as well as extending it to include the associated power parameter. An important feature is its conceptual and practical simplicity, consisting essentially in measuring the slope and the intercept of a linear fit after a discrete wavelet transform is performed, a very fast (O(n)) operation. Under welljustified technical idealizations the estimator is shown to be unbiased and of minimum or close to minimum variance for the scale parameter, and asymptotically unbiased and efficient for the second parameter. Through theoretical arguments and numerical simulations it is shown that in practice, even for small data sets, the bias is very small and the variance close to optimal for both parameters. Closedform expressions are given for the covariance matrix of the estimator as a function of data length, and are shown by simulation to be very accurate even when the technical idealizations are not satisfied. Comparisons are made against two maximumlikelihood estimators. In terms of robustness and computational cost the wavelet estimator is found to be clearly superior and statistically its performance is comparable. We apply the tool to the analysis of Ethernet teletraffic data, completing an earlier study on the scaling parameter alone. Index Terms—Hurst parameter, longrange dependence, packet traffic, parameter estimation, telecommunications networks, timescale analysis, wavelet decomposition. I.
Limit Theory For Bilinear Processes With Heavy Tailed Noise
 Ann. Appl. Probab
, 1995
"... . We consider a simple stationary bilinear model X t = cX t\Gamma1 Z t\Gamma1 + Z t ; t = 0; \Sigma1; \Sigma2; : : : generated by heavy tailed noise variables fZ t g. A complete analysis of weak limit behavior is given by means of a point process analysis. A striking feature of this analysis is th ..."
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Cited by 35 (15 self)
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. We consider a simple stationary bilinear model X t = cX t\Gamma1 Z t\Gamma1 + Z t ; t = 0; \Sigma1; \Sigma2; : : : generated by heavy tailed noise variables fZ t g. A complete analysis of weak limit behavior is given by means of a point process analysis. A striking feature of this analysis is that the sample correlation converges in distribution to a nondegenerate limit. A warning is sounded about trying to detect nonlinearities in heavy tailed models by means of the sample correlation function. 1. Introduction. Current efforts in time series analysis attempt to deal with data which exhibit features such as long range dependence, nonlinearity and heavy tails. There are numerous data sets from the fields of telecommunications, finance and economics which appear to be compatible with the assumption of heavytailed marginal distributions. Examples include file lengths, cpu time to complete a job, call holding times, interarrival times between packets in a network and lengths of o...
Chaotic Maps As Models of Packet Traffic
, 1994
"... this paper. In 2 we summarize the considerable literature on the subject along with an introduction to our approach. 3 presents results indicating the traffic characteristics that can be generated with simple piecewise linear and nonlinear maps. 4 shows how a queueing system can be represented by ..."
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Cited by 32 (0 self)
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this paper. In 2 we summarize the considerable literature on the subject along with an introduction to our approach. 3 presents results indicating the traffic characteristics that can be generated with simple piecewise linear and nonlinear maps. 4 shows how a queueing system can be represented by a 2D deterministic transformation, and outlines a potential performance analysis approach. 5 concludes this paper with a description of future directions for this work.