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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 740 (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...
Proof of a Fundamental Result in SelfSimilar Traffic Modeling
 COMPUTER COMMUNICATION REVIEW
, 1997
"... We state and prove the following key mathematical result in selfsimilar traffic modeling: the superposition of many ON/OFF sources (also known as packet trains) with strictly alternating ON and OFFperiods and whose ONperiods or OFFperiods exhibit the Noah Effect (i.e., have high variability or ..."
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Cited by 287 (9 self)
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We state and prove the following key mathematical result in selfsimilar traffic modeling: the superposition of many ON/OFF sources (also known as packet trains) with strictly alternating ON and OFFperiods and whose ONperiods or OFFperiods exhibit the Noah Effect (i.e., have high variability or infinite variance) can produce aggregate network traffic that exhibits the Joseph Effect (i.e., is selfsimilar or longrange dependent). There is, moreover, a simple relation between the parameters describing the intensities of the Noah Effect (high variability) and the Joseph Effect (selfsimilarity). This provides a simple physical explanation for the presence of selfsimilar traffic patterns in modern highspeed network traffic that is consistent with traffic measurements at the source level. We illustrate how this mathematical result can be combined with modern highperformance computing capabilities to yield a simple and efficient lineartime algorithm for generating selfsimilar traf...
Is Network Traffic Approximated By Stable Lévy Motion Or Fractional Brownian Motion?
, 1999
"... Cumulative broadband network traffic is often thought to be well modelled by fractional Brownian motion. However, some traffic measurements do not show an agreement with the Gaussian marginal distribution assumption. We show that if connection rates are modest relative to heavy tailed connection le ..."
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Cited by 112 (12 self)
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Cumulative broadband network traffic is often thought to be well modelled by fractional Brownian motion. However, some traffic measurements do not show an agreement with the Gaussian marginal distribution assumption. We show that if connection rates are modest relative to heavy tailed connection length distribution tails, then stable L'evy motion is a sensible approximation to cumulative traffic over a time period. If connection rates are large relative to heavy tailed connection length distribution tails, then FBM is the appropriate approximation. The results are framed as limit theorems for a sequence of cumulative input processes whose connection rates are varying in such a way as to remove or induce long range dependence.
Inverting Sampled Traffic
 In Proceedings of the 3rd ACM SIGCOMM conference on Internet measurement
, 2003
"... Routers have the ability to output statistics about packets and flows of packets that traverse them. Since however the generation of detailed tra#c statistics does not scale well with link speed, increasingly routers and measurement boxes implement sampling strategies at the packet level. In this pa ..."
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Cited by 104 (4 self)
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Routers have the ability to output statistics about packets and flows of packets that traverse them. Since however the generation of detailed tra#c statistics does not scale well with link speed, increasingly routers and measurement boxes implement sampling strategies at the packet level. In this paper we study both theoretically and practically what information about the original tra#c can be inferred when sampling, or `thinning', is performed at the packet level. While basic packet level characteristics such as first order statistics can be fairly directly recovered, other aspects require more attention. We focus mainly on the spectral density, a second order statistic, and the distribution of the number of packets per flow, showing how both can be exactly recovered, in theory. We then show in detail why in practice this cannot be done using the traditional packet based sampling, even for high sampling rate. We introduce an alternative flow based thinning, where practical inversion is possible even at arbitrarily low sampling rate. We also investigate the theory and practice of fitting the parameters of a Poisson cluster process, modelling the full packet tra#c, from sampled data.
Linear functionals of eigenvalues of random matrices
 Trans. Amer. Math. Soc
, 2001
"... Abstract. Let Mn be a random n × n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n →∞. By Fourier analysis, this result ..."
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Cited by 98 (8 self)
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Abstract. Let Mn be a random n × n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of Mn to converge to a Gaussian limit as n →∞. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of Mn. For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of Mn are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices. 1.
Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process
, 1998
"... The asymptotic theory for the sample autocorrelations and extremes of a GARCH(1; 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to one, i.e. when one is close to an infinite variance marginal distribution. This situation has been ..."
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Cited by 91 (20 self)
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The asymptotic theory for the sample autocorrelations and extremes of a GARCH(1; 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to one, i.e. when one is close to an infinite variance marginal distribution. This situation has been observed for various financial logreturn series and led to the introduction of the IGARCH model. In such a situation the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values, and the sample autocorrelations of the squared time series have nondegenerate limit distributions. We discuss the consequences for a foreign exchange rate series. AMS 1991 Subject Classification: Primary: 62P20 Secondary: 90A20 60G55 60J10 62F10 62F12 62G30 62M10 Key Words and Phrases. GARCH, sample autocorrelations, stochastic recurrence equation, Pareto tail, extremes, extremal index, point processes, foreign exchange rates 1 Introduc...
Stable Laws and Domains of Attraction in Free Probability Theory
, 1999
"... In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite di#erent. Our wor ..."
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Cited by 87 (1 self)
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In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite di#erent. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.
Ruin probabilities and overshoots for general Lévy insurance risk processes
 ANN. APPL. PROBAB
, 2004
"... We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to − ∞ a.s. and the positive tail of the Lévy measure, or of the l ..."
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Cited by 82 (26 self)
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We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to − ∞ a.s. and the positive tail of the Lévy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Klüppelberg [Stochastic Process. Appl. 64 (1996) 103–125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207–226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewaltype properties of Lévy processes.
Asymptotic results for multiplexing subexponential onoff processes
 Advances in Applied Probability
, 1998
"... Consider an aggregate arrival process AN obtained by multiplexing N OnOff processes with exponential Off periods of rate λ and subexponential On periods τon. As N goes to infinity, with λN → Λ, AN approaches an M/G/ ∞ type process. Both for finite and infinite N, we obtain the asymptotic characteri ..."
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Cited by 81 (20 self)
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Consider an aggregate arrival process AN obtained by multiplexing N OnOff processes with exponential Off periods of rate λ and subexponential On periods τon. As N goes to infinity, with λN → Λ, AN approaches an M/G/ ∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/ ∞ arrival process A ∞ t and capacity c. When On periods are regularly varying (with noninteger exponent), we derive a precise asymptotic behavior of the queue length random variable QP t observed at the beginning of the arrival process activity periods P[Q P t +ρ−c> x] ∼ Λr P[τ c−ρ x/(r+ρ−c) on> u]du x → ∞, where ρ = EA ∞ t < c; r (c ≤ r) is the rate at which the fluid is arriving during an On period. The asymptotic (time average) queuedistributionlower boundis obtained undermoregeneral assumptions on On periods than regular variation. In addition, we analyze a queueing system in which one OnOff process, whose On period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate Eet. This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value Eet.