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110
Testing the Gaussian approximation of aggregate traffic
, 2002
"... We search for methods or tools to detect whether the 1dimensional marginal distribution of traffic increments of aggregate TCPtraffic satisfy the hypothesis of approximate normality. Gaussian approximation requires a high level of aggregation in both "vertical" (source aggregation) and & ..."
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Cited by 45 (1 self)
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We search for methods or tools to detect whether the 1dimensional marginal distribution of traffic increments of aggregate TCPtraffic satisfy the hypothesis of approximate normality. Gaussian approximation requires a high level of aggregation in both "vertical" (source aggregation) and "horizontal" (time scale) directions. We discuss these different concepts of aggregation first separately, with an example from real data traffic, and show how to rule out cases where the level of aggregation will not be sufficient. Gaussian approximation is then quantified with the square of the linear correlation coefficient in normalquantile plots. We propose an elementary method based on this correlation test, by looking at the behavior of the test statistic for different sample sizes, and show positive and negative examples from the example data. We use this method to look for the first time scale, where the Gaussian approximation is plausible with the example data, and then we look how much more vertical aggregation would be needed for smaller time scales in order to obtain a reasonable approximation by normal distribution.
Convergence of Scaled Renewal Processes and a Packet Arrival Model
 Bernoulli
"... We study the superposition process of a class of independent renewal processes with longrange dependence. It is known that under two different scalings in time and space either fractional Brownian motion or a stable Levy process may arise in the rescaling asymptotic limit. It is shown here that in ..."
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Cited by 33 (5 self)
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We study the superposition process of a class of independent renewal processes with longrange dependence. It is known that under two different scalings in time and space either fractional Brownian motion or a stable Levy process may arise in the rescaling asymptotic limit. It is shown here that in a third, intermediate scaling regime a new limit process appears, which is neither Gaussian nor stable. The new limit process is characterized by its cumulant generating function and some of its properties are discussed.
Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach
, 2004
"... It has become common practice to use heavytailed distributions in order to describe the variations in time and space of network traffic workloads. The asymptotic behavior of these workloads is complex; different limit processes emerge depending on the specifics of the work arrival structure and the ..."
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Cited by 33 (5 self)
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It has become common practice to use heavytailed distributions in order to describe the variations in time and space of network traffic workloads. The asymptotic behavior of these workloads is complex; different limit processes emerge depending on the specifics of the work arrival structure and the nature of the asymptotic scaling. We focus on two variants of the infinite source Poisson model and provide a coherent and unified presentation of the scaling theory by using integral representations. This allows us to understand physically why the various limit processes arise. 1
LongRange Dependence and HeavyTail Modeling for Teletraffic Data
 IEEE Signal Processing Magazine
, 2002
"... Analysis and modeling of computer network traffic is a daunting task considering the amount of available data. This is quite obvious when considering the spatial dimension of the problem, since the number of interacting computers, gateways and switches can easily reach several thousands, even in a L ..."
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Cited by 32 (3 self)
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Analysis and modeling of computer network traffic is a daunting task considering the amount of available data. This is quite obvious when considering the spatial dimension of the problem, since the number of interacting computers, gateways and switches can easily reach several thousands, even in a Local Area Network (LAN) setting. This is also true for the time dimension: W. Willinger and V. Paxson in [42] cite the figures of 439 million packets and 89 gigabytes of data for a single week record of the activity of a university gateway in 1995. The complexity of the problem further increases when considering Wide Area Network (WAN) data [28]. In light of the above, it is clear that a notion of importance for modern network engineering is that of invariants, i.e. characteristics that are observed with some reproducibility and independently of the precise settings of the network under consideration. In this tutorial paper, we focus on two such invariants related to the time d...
Simulation methods for linear fractional stable motion and FARIMA using the fast Fourier transform
 Fractals
, 2004
"... ..."
Recent advances in ARCH modelling
 In: Teyssière G. and Kriman A. (eds): Long Memory in Economics
, 2006
"... Econometric modelling of financial data received a broad interest in the last 20 years and the literature on ARCH and related models is vast. Starting with the path breaking works by Engle (1982) and Bollerslev (1986), one of the most popular models became the Generalized AutoRegressive Conditionall ..."
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Cited by 25 (0 self)
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Econometric modelling of financial data received a broad interest in the last 20 years and the literature on ARCH and related models is vast. Starting with the path breaking works by Engle (1982) and Bollerslev (1986), one of the most popular models became the Generalized AutoRegressive Conditionally
LASS: a tool for the local analysis of selfsimilarity
 COMPUT. STATIST. DATA ANAL
, 2004
"... The Hurst parameter H characterizes the degree of longrange dependence (and asymptotic selfsimilarity) in stationary time series. Many methods have been developed for the estimation of H from data. In practice, however, the classical estimation techniques can be severely affected by nonstationary ..."
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Cited by 23 (7 self)
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The Hurst parameter H characterizes the degree of longrange dependence (and asymptotic selfsimilarity) in stationary time series. Many methods have been developed for the estimation of H from data. In practice, however, the classical estimation techniques can be severely affected by nonstationary artifacts in the time series. In fact, the assumption that the data can be modeled by a stationary process with a single Hurst exponent H may be unrealistic. We focus on practical issues associated with the detection of longrange dependence in Internet traffic data and develop two tools designed to address some of these issues. The first is an animation tool which is used to visualize the local dependence structure. The second is a statistical tool for the local analysis of selfsimilarity (LASS). The LASS tool is designed to handle time series that have longrange dependence and are long enough that some parts are essentially stationary, while others exhibit nonstationarity, which are either deterministic or stochastic in nature. The tool uses wavelets to analyze the local dependence structure in the data over a set of windows. It can be used to visualize local deviations from selfsimilar, longrange dependence scaling and to provide reliable local estimates of the Hurst exponents. The tool, which is illustrated by using a trace of Internet traffic measurements, can also be applied to economic time series. We also develop a medianbased wavelet spectrum which can be used to obtain robust local or global estimates of the the Hurst parameter that are less susceptible to local nonstationarity. We make the software tools freely available and describe their use in an appendix.
Exact asymptotics for fluid queues fed by multiple heavytailed onoff flows
 Ann. Appl. Probab
"... We consider a fluid queue fed by multiple On–Off flows with heavytailed (regularly varying) On periods. Under fairly mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a reduced system. The reduced system consists of a “dominant ” subset of the flows, ..."
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Cited by 22 (10 self)
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We consider a fluid queue fed by multiple On–Off flows with heavytailed (regularly varying) On periods. Under fairly mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a reduced system. The reduced system consists of a “dominant ” subset of the flows, with the original service rate subtracted by the mean rate of the other flows. We describe how a dominant set may be determined from a simple knapsack formulation. The dominant set consists of a “minimally critical ” set of On– Off flows with regularly varying On periods. In case the dominant set contains just a single On–Off flow, the exact asymptotics for the reduced system follow from known results. For the case of several On–Off flows, we exploit a powerful intuitive argument to obtain the exact asymptotics. Combined with the reducedload equivalence, the results for the reduced system provide a characterization of the tail of the workload distribution for a wide range of traffic scenarios.
Scaling limits for random fields with longrange dependence. Institut MittagLeffler, The Royal Swedish Academy of Sciences
, 2005
"... This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density λ of the sets grows to infinity and the mean volume ρ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show tha ..."
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Cited by 20 (4 self)
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This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density λ of the sets grows to infinity and the mean volume ρ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which λ and ρ are scaled. If λ grows much faster than ρ shrinks, the limit is Gaussian with longrange dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable. 1. Introduction. Fractional Brownian
2009a) Fundamental properties of stochastic volatility models
"... We collect some of the probabilistic properties of a strictly stationary stochastic volatility process. These include properties about mixing, covariances and correlations, moments, and tail behavior. We also study properties of the autocovariance and autocorrelation functions of stochastic volati ..."
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Cited by 19 (11 self)
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We collect some of the probabilistic properties of a strictly stationary stochastic volatility process. These include properties about mixing, covariances and correlations, moments, and tail behavior. We also study properties of the autocovariance and autocorrelation functions of stochastic volatility processes and its powers as well as the asymptotic theory of the corresponding sample versions of these functions. In comparison with the GARCH model (see Lindner [26]) the stochastic volatility model has a much simpler probabilistic structure which contributes to its popularity. 1 The model We consider a stochastic volatility process (Xt)t∈Z given by the equations Xt = σt Zt, t ∈ Z, (1) where (σt)t∈Z is a strictly stationary sequence of positive random variables which is independent of the iid noise sequence (Zt)t∈Z.3 We refer to (σt)t∈Z as the volatility sequence. Following the tradition in time series analysis, we index the stationary sequences (Xt), (Zt), (σt) by the set Z of the integers. For practical purposes, one would consider e.g., the sequence (Xt)t∈N corresponding to observations at the times t = 1, 2,.... 3 It is common to assume the additional standardization conditions EZt = 0 and var(Zt) = 1. These conditions are important for example in order to avoid identification problems for the parameters of the model. In most parts of this article, these additional conditions are not needed. Moreover, in Sections 4 and 5 we will also consider results when var(Z) = ∞ or EZ  =∞.