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38
On the Selfsimilar Nature of Ethernet Traffic (Extended Version)
, 1994
"... We demonstrate that Ethernet LAN traffic is statistically selfsimilar, that none of the commonly used traffic models is able to capture this fractallike behavior, that such behavior has serious implications for the design, control, and analysis of highspeed, cellbased networks, and that aggrega ..."
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Cited by 2209 (47 self)
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We demonstrate that Ethernet LAN traffic is statistically selfsimilar, that none of the commonly used traffic models is able to capture this fractallike behavior, that such behavior has serious implications for the design, control, and analysis of highspeed, cellbased networks, and that aggregating streams of such traffic typically intensifies the selfsimilarity (“burstiness”) instead of smoothing it. Our conclusions are supported by a rigorous statistical analysis of hundreds of millions of high quality Ethernet traffic measurements collected between 1989 and 1992, coupled with a discussion of the underlying mathematical and statistical properties of selfsimilarity and their relationship with actual network behavior. We also present traffic models based on selfsimilar stochastic processes that provide simple, accurate, and realistic descriptions of traffic scenarios expected during BISDN deployment.
Stochastic Calculus for Fractional Brownian Motion. I: Theory
 SIAM J. Control optim
, 1999
"... This paperd=R+3# es some of the results in [5] for a stochastic calculus for a fractional Brownian motion with the Hurst parameter in the interval (1/2, 1). Two stochastic integrals aredeg=z with explicit expressions for their first two moments. Multipleand iterated integrals of a fractional Browini ..."
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Cited by 186 (17 self)
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This paperd=R+3# es some of the results in [5] for a stochastic calculus for a fractional Brownian motion with the Hurst parameter in the interval (1/2, 1). Two stochastic integrals aredeg=z with explicit expressions for their first two moments. Multipleand iterated integrals of a fractional Browinian motion aredeg+I and various properties of these integrals are given. A square integrable functional on a probability space of a fractional Brownian motion isexpressed as an infinite series of multiple integrals. 1I ntroduction Fractional Brownian motion is a family of Gaussian processes that areind==z by the Hurst parameter H in the interval (0, 1). These processes were introdog+ by Kolmogorov [10]. The first application of these processes wasmad by Hurst [7], [8] whoused them to mo dg the longterm storage capacity of reservoirs alongthe Nile River. MandRWg,+ [12]used these processes to mo some economic time seriesand most recently these processes have beenused to mo dg telecommunication...
Itô’s Formula with respect to Fractional Brownian Motion and its Application
 Journal of Applied Mathematics and Stochastic Analysis
, 1996
"... Fractional Brownian motion (FBM) with Hurst index 1/2 < H < 1 is not a semimartingale. Consequently, the standard It calculus is not available for stochastic integrals with respect to FBM as an integrator if 1/2 < H < 1. In this paper we derive a version of It’s formula for fractional Br ..."
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Cited by 47 (0 self)
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Fractional Brownian motion (FBM) with Hurst index 1/2 < H < 1 is not a semimartingale. Consequently, the standard It calculus is not available for stochastic integrals with respect to FBM as an integrator if 1/2 < H < 1. In this paper we derive a version of It’s formula for fractional Brownian motion. Then, as an application, we propose and study a fractional Brownian Scholes stochastic model which includes the standard BlackScholes model as a special case and is able to account for long range dependence in modeling the price of a risky asset. This article is dedicated to the memory of Roland L. Dobrushin.
Performance Decay In A Single Server Exponential Queueing Model With Long Range Dependence
 Operations Research
, 1996
"... . We discuss how long range dependence can influence the characteristics of a single server queue. We take the analogue of the G/M/1 queue except that the input stream is altered to exhibit long range dependence. The equilibrium queue size and equilibrium waiting time distributions each have heavy t ..."
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Cited by 15 (4 self)
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. We discuss how long range dependence can influence the characteristics of a single server queue. We take the analogue of the G/M/1 queue except that the input stream is altered to exhibit long range dependence. The equilibrium queue size and equilibrium waiting time distributions each have heavy tails. By suitably selecting the parameters of the inputs, the queue size or waiting time can be made to possess infinite variance and even infinite mean. Some simulations dramatically illustrate the potential for undetected long range dependence to significantly alter the queueing behavior compared to what is anticipated with traditional inputs. 1. Introduction. Long range dependence is a property of stationary time series models whose current state has a strong dependency on the remote past. Definitions vary from author to author but a commonly accepted definition in covariance stationary time series is that a process fXng has long range dependence if 1 X j=1 jcorr(X 0 ; X j )j = 1 (cf...
LONG RANGE DEPENDENCE
"... Abstract. The notion of long range dependence is discussed from a variety of points of view, and a new approach is suggested. A number of related topics is also discussed, including connections with nonstationary processes, with ergodic theory, selfsimilar processes and fractionally differenced pr ..."
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Cited by 14 (1 self)
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Abstract. The notion of long range dependence is discussed from a variety of points of view, and a new approach is suggested. A number of related topics is also discussed, including connections with nonstationary processes, with ergodic theory, selfsimilar processes and fractionally differenced processes, heavy tails and light tails, limit theorems and large deviations. 1.
Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion
, 1998
"... .  The problem of optimal filtering is investigated in a continuous time linear Gaussian system where the signal is a fixed random variable and the noise driving the observation process is a fractional Brownian motion with Hurst parameter H 2 (1=2; 1). Closed form expressions are derived both for ..."
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Cited by 10 (0 self)
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.  The problem of optimal filtering is investigated in a continuous time linear Gaussian system where the signal is a fixed random variable and the noise driving the observation process is a fractional Brownian motion with Hurst parameter H 2 (1=2; 1). Closed form expressions are derived both for the optimal filter and the variance of the filtering error. Then an application to the determination of the best linear unbiased estimator in a related parameter estimation problem is discussed. Finally integral transformations which change a fractional Brownian motion to martingales are identified and an elementary approach to a Girsanov type formula is developed which shows that the estimator is in fact the maximum likelihood estimator. Key Words and Phrases: Fractional Brownian motion; Optimal filter; Best linear unbiased estimator; Maximum likelihood estimator. AMS 1991 Subject Classification: Primary 60G35, 62M20; Secondary 60G15, 93E11. 1. Introduction.  It is now widely accepted t...
Fluid Queues, Leaky Buckets, OnOff Processes and Teletraffic Modeling with Highly Variable and Correlated Inputs
 in SelfSimilar Network Traffic and Performance
, 1998
"... INTRODUCTION There now exist several large teletraffic data sets exhibiting nonstandard features incompatible with classical assumtions of short range dependence and rapidly decreasing tails. For instance, it is worth exploring the variety of data catalogued at the ITA web site www.acm.org/sigcomm ..."
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Cited by 6 (1 self)
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INTRODUCTION There now exist several large teletraffic data sets exhibiting nonstandard features incompatible with classical assumtions of short range dependence and rapidly decreasing tails. For instance, it is worth exploring the variety of data catalogued at the ITA web site www.acm.org/sigcomm/ITA/. These data sets exhibit the phenomena of heavytailed marginal distributions and long range dependence. Tails can be so heavy that only infinite variance models are possible (eg, [43]), and sometimes, as in file size data, even first moments are infinite. See [1]. Heavy tails have been fit to file lengths ([1], [9],[10]) cpu time to complete a job, call holding times, interarrival times between packets in a network ([39]), lengths of on/off cycles ([43], [42]). Other areas where heavy tails abound are finance and economics ([12], [13], [20], [6], [7]) and insurance analysis ([30], [32]). Of course, long range dependence was originially consider
Approach to Stochastic Integration for Fractional Brownian Motion in a Hilbert Space
"... A Hilb alued stochastic integration is defined for an integrator that is acylindrical fractional Brownian motion in a Hilbert space. Since the integrator is not a semimartingale for the fractional Brownian motions considered, a di#erent definition of integration is required. Both deterministic and s ..."
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Cited by 6 (1 self)
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A Hilb alued stochastic integration is defined for an integrator that is acylindrical fractional Brownian motion in a Hilbert space. Since the integrator is not a semimartingale for the fractional Brownian motions considered, a di#erent definition of integration is required. Both deterministic and stochastic operatoralued integrands are used. The approach to integration has an analogue with Skorokhod integrals for Brownian motion by the basic use of a derivative of some functionals of Brownianmotion. An Ito formula is given for some processes obtained by this stochastic integration. 1 Int7 duct91 Fractional Brownian motion is afamil# of Gaussian processes that are indexed by the Hurst parameter H (0, 1). In a finitedimensional Eucl#nsio space these processes were introduced by Kol#6q=fi9 v [10] and some properties of these processes were given byMandel#94# and van Ness [13]. Hurst [8], [9] used this approach to describe the l#efi term capacity of reservoirs al#sfi the Nil# River which was theinitial indication that these processescoul# be used as model# of physical phenomena. Mandel#na. [12] used these processes to model some economic data and, most recentl# , these processes have been noted for model# oftel#66q7 unication tra#c (e.g., [11]). To enhance theanal#q7# and theappl#kT4fi9T4 y of these processes, a stochastic cal#icfiT has been devel#fi ed in recent years for these processes in finite dimensional spaces (e.g., [1], [3], [4]). The stochasticcal#icfi# given here uses a di#erent approach than the one used in [1], [3] or [4]. Since afractional Brownian motion, for H #=1/2, not a semimartingal## it is necessary to define a stochasticcal#icfi#6 These processes have a sel#kfi9SkqTfil# y inprobabil#k yl aw and, for H (1/2, 1), al##6 range dependence property d...
Groupbased estimation of missing hydrological data. I. Approach and general
, 2000
"... Abstract In this first paper in a set of two, the problem of estimating missing segments in streamflow records is described. The group approach, different from the traditional singlevalued approach, is proposed and explained. The approach perceives the hydrological data as sequence of groups rather ..."
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Cited by 2 (2 self)
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Abstract In this first paper in a set of two, the problem of estimating missing segments in streamflow records is described. The group approach, different from the traditional singlevalued approach, is proposed and explained. The approach perceives the hydrological data as sequence of groups rather than singlevalued observations. The techniques suggested to handle the group approach are regression, time series analysis, partitioning modelling, and artificial neural networks. Pertinent literature is reviewed and background material is used to support the group approach. Implementation and comparisons of models ' performance are deferred to the second paper. L'approche de groupe pour l'estimation des données hydrologiques manquantes: I. Présentation et méthodologie Résumé Dans ce premier de deux papiers, nous décrivons le problème de l'estimation de suites de données manquantes dans les archives de débits. Nous présentons et expliquons l'approche de groupe, différente des approches traditionnelles focalisées sur l'estimation de valeurs singulières. Cette nouvelle approche conçoit les données hydrologiques comme des suites de groupes plutôt que comme des suites d'observations singulières. Les techniques susceptibles de la servir sont: la régression, l'analyse des séries chronologiques, la segmentation et les réseaux de neurones artificiels. Nous présentons une revue de littérature d'où nous avons tiré des arguments en faveur de la promotion de l'approche de groupe. L'implementation et l'évaluation de l'approche de groupe font l'objet du second papier.
A Long Memory Count Data Time Series Model for Financial Application ∗
"... This paper concerns modelling the long memory property in a count data framework. The introduced models are applied to high frequency transactions data for two stock series. The unconditional and conditional first and second moments are given. The CLS and FGLS estimators are discussed. The evidence ..."
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Cited by 2 (0 self)
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This paper concerns modelling the long memory property in a count data framework. The introduced models are applied to high frequency transactions data for two stock series. The unconditional and conditional first and second moments are given. The CLS and FGLS estimators are discussed. The evidence of long memory is found in the AstraZeneca series, while the Ericsson B series indicates a process that has a mean reversion property.