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64
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 ..."
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Cited by 150 (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.
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 87 (14 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...
SelfSimilarity and LongRange Dependence Through the Wavelet Lens
, 2000
"... Selfsimilar and longrange dependent processes are the most important kinds of random processes possessing scale invariance. We describe how to analyze them using the discrete wavelet transform. We have chosen a didactic approach, useful to practitioners. Focusing on the Discrete Wavelet Transform, ..."
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Cited by 81 (11 self)
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Selfsimilar and longrange dependent processes are the most important kinds of random processes possessing scale invariance. We describe how to analyze them using the discrete wavelet transform. We have chosen a didactic approach, useful to practitioners. Focusing on the Discrete Wavelet Transform, we describe the nature of the wavelet coefficients and their statistical properties. Pitfalls in understanding and key features are highlighted and we sketch some proofs to provide additional insight. The Logscale Diagram is introduced as a natural means to study scaling data and we show how it can be used to obtain unbiased semiparametric estimates of the scaling exponent. We then focus on the case of longrange dependence and address the problem of defining a lower cutoff scale corresponding to where scaling starts. We also discuss some related problems arising from the application of wavelet analysis to discrete time series. Numerical examples using many discrete time models are th...
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 65 (13 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.
A Statistical Test for the Time Constancy of Scaling Exponents
 IEEE Transactions on Signal Processing
, 1999
"... A wavelet based statistical test is described for distinguishing true time variation of the scaling exponent describing scaling behaviour, from statistical fluctuations of estimates across time of a constant exponent. The test is applicable to diverse scaling phenomena including long range dependenc ..."
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Cited by 46 (15 self)
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A wavelet based statistical test is described for distinguishing true time variation of the scaling exponent describing scaling behaviour, from statistical fluctuations of estimates across time of a constant exponent. The test is applicable to diverse scaling phenomena including long range dependence and exactly selfsimilar processes in a uniform framework, without the need for prior knowledge of the type in question. It is based on the special properties of waveletbased estimates of the scaling exponent over adjacent blocks of data, strongly motivating an idealised inference problem: the equality or otherwise of means of independent Gaussian variables with known variances. A uniformly most powerful invariant test exists for this problem and is described. A separate UMPI test is also described for when the scaling exponent undergoes a level change. The power functions of the two tests are given explicitly and compared. Using simulation the effect in practice of deviations from the ide...
A Precision Infrastructure for Active Probing
"... A highly accurate active probing measurement infrastructure is described based on GPS synchronised DAG cards as timestamping monitors, and RealTime Linux as a probe stream sender. A comparison of less accurate systems including senders based on FreeBSD and Linux is given and diverse sources of timi ..."
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Cited by 40 (9 self)
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A highly accurate active probing measurement infrastructure is described based on GPS synchronised DAG cards as timestamping monitors, and RealTime Linux as a probe stream sender. A comparison of less accurate systems including senders based on FreeBSD and Linux is given and diverse sources of timing errors are described and discussed, as is NTP synchronization. Preliminary results using the infrastructure are presented, and its application to a new approach for bottleneck bandwidth estimation outlined.
Simulation of Fractional Brownian Motion with Conditionalized Random midpoint displacement
 Advances in Performance Analysis
, 1998
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Measuring LongRange Dependence under Changing Traffic Conditions
, 1999
"... : Recent measurements of various types of network traffic have shown evidence consistent with longrange dependence and selfsimilarity. However, an alternative explanation for these measurements is nonstationarity in the data. Standard estimators of LRD parameters such as the Hurst parameter H ass ..."
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Cited by 27 (4 self)
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: Recent measurements of various types of network traffic have shown evidence consistent with longrange dependence and selfsimilarity. However, an alternative explanation for these measurements is nonstationarity in the data. Standard estimators of LRD parameters such as the Hurst parameter H assume stationarity and are susceptible to bias when this assumption does not hold. Hence LRD may be indicated by these estimators when none is present, or alternatively LRD taken to be nonstationarity. The recently developed AbryVeitch (AV) joint estimator has much better properties when a timeseries is nonstationary. In particular the effect of polynomial trends in data may be intrinsically eliminated from the estimates of LRD parameters. This paper investigates the behavior of the AV estimator when there are nonstationarities in the form of a level shift in the mean and/or the variance of a process. We examine cases where the change occurs both gradually or as a single jump discontinuit...
Robust Synchronization of Absolute and Difference Clocks over Networks
, 2009
"... We present a detailed reexamination of the problem of inexpensive yet accurate clock synchronization for networked devices. Based on an empirically validated, parsimonious abstraction of the CPU oscillator as a timing source, accessible via the TSC register in popular PC architectures, we build on ..."
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Cited by 19 (10 self)
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We present a detailed reexamination of the problem of inexpensive yet accurate clock synchronization for networked devices. Based on an empirically validated, parsimonious abstraction of the CPU oscillator as a timing source, accessible via the TSC register in popular PC architectures, we build on the key observation that the measurement of time differences, and absolute time, requires separate clocks, both at a conceptual level and practically, with distinct algorithmic, robustness, and accuracy characteristics. Combined with roundtrip time based filtering of network delays between the host and the remote time server, we define robust algorithms for the synchronization of the absolute and difference TSCclocks over a network. We demonstrate the effectiveness of the principles, and algorithms using months of real data collected using multiple servers. We give detailed performance results for a full implementation running live and unsupervised under numerous scenarios, which show very high reliability, and accuracy approaching fundamental limits due to host system noise. Our synchronization algorithms are inherently robust to many factors including packet loss, server outages, route changes, and network congestion.
An Empirical Study of the Multiscale Predictability of Network Traffic
 IN IEEE PROCEEDINGS OF HPDC
, 2003
"... Distributed applications use predictions of network traffic to sustain their performance by adapting their behavior. The timescale of interest is applicationdependent and thus it is natural to ask how predictability depends on the resolution, or degree of smoothing, of the network traffic signal. To ..."
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Cited by 18 (1 self)
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Distributed applications use predictions of network traffic to sustain their performance by adapting their behavior. The timescale of interest is applicationdependent and thus it is natural to ask how predictability depends on the resolution, or degree of smoothing, of the network traffic signal. To help answer this question we empirically study the onestepahead predictability, measured by the ratio of mean squared error to signal variance, of network traffic at different resolutions. A onestepahead prediction at a coarse resolution is a prediction of the average behavior over a long interval. We apply a wide range of linear and nonlinear time series models to a large number of packet traces, generating different resolution views of the traces through two methods: the simple binning approach used by several extant network measurement tools, and by waveletbased approximations. The waveletbased approach is a natural way to provide multiscale prediction to applications. We find that predictability seems to be highly situational in practiceit varies widely from trace to trace. Unexpectedly, predictability does not always increase as the signal is smoothed. Half of the time there is a sweet spot at which the ratio is minimized and predictability is clearly the best. Also surprisingly, predictors that can capture nonstationarity and nonlinearity provide benefits only at very coarse resolutions.