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58
Wavelet Analysis of Long Range Dependent Traffic
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1998
"... A Wavelet based tool for the analysis of long range dependence is introduced and a related semiparametric estimator of the Hurst parameter. The estimator is shown to be unbiased under very general conditions, and efficient under Gaussian assumptions. It can be implemented very efficiently allowing t ..."
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Cited by 268 (22 self)
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A Wavelet based tool for the analysis of long range dependence is introduced and a related semiparametric estimator of the Hurst parameter. 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 long range 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 vs multifractality is also performed, and a preliminary study of the stationarity with respect to the Hurst parameter and deterministic trends.
On the Relevance of LongRange Dependence in Network Traffic
, 1996
"... There is much experimental evidence that network traffic processes exhibit ubiquitous properties of selfsimilarity and long range dependence (LRD), i.e., of correlations over a wide range of time scales. However, there is still considerable debate about how to model such processes and about their im ..."
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Cited by 186 (1 self)
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There is much experimental evidence that network traffic processes exhibit ubiquitous properties of selfsimilarity and long range dependence (LRD), i.e., of correlations over a wide range of time scales. However, there is still considerable debate about how to model such processes and about their impact on network and application performance. In this paper, we argue that much recent modeling work has failed to consider the impact of two important parameters, namely the finite range of time scales of interest in performance evaluation and prediction problems, and the firstorder statistics such as the marginal distribution of the process.
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 152 (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.
The Importance of LongRange Dependence of VBR Video Traffic in ATM Traffic Engineering: Myths and Realities
 IN PROC. ACM SIGCOMM '96
, 1996
"... There has been a growing concern about the potential impact of longterm correlations (secondorder statistic) in variablebitrate (VBR) video traffic on ATM buffer dimensioning. Previous studies have shown that video traffic exhibits longrange dependence (LRD) (Hurst parameter large than 0.5). We ..."
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Cited by 144 (9 self)
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There has been a growing concern about the potential impact of longterm correlations (secondorder statistic) in variablebitrate (VBR) video traffic on ATM buffer dimensioning. Previous studies have shown that video traffic exhibits longrange dependence (LRD) (Hurst parameter large than 0.5). We investigate the practical implications of LRD in the context of realistic ATM traffic engineering by studying ATM multiplexers of VBR video sources over a range of desirable cell loss rates and buffer sizes (maximum delays). Using results based on large deviations theory, we introduce the notion of Critical Time Scale (CTS). For a given buffer size, link capacity, and the marginal distribution of frame size, the CTS of a VBR video source is defined as the number of frame correlations that contribute to the cell loss rate. In other words, secondorder behavior at the time scale beyond the CTS does not significantly affect the network performance. We show that whether the video source model i...
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 78 (18 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.
A Poisson Limit for Buffer Overflow Probabilities
 in Proceedings of IEEE INFOCOM
, 2002
"... Abstract — A key criterion in the design of highspeed networks is the probability that the buffer content exceeds a given threshold. We consider Ò independent identical traffic sources modelled as point processes, which are fed into a link with speed proportional to Ò. Under fairly general assumpti ..."
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Cited by 54 (1 self)
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Abstract — A key criterion in the design of highspeed networks is the probability that the buffer content exceeds a given threshold. We consider Ò independent identical traffic sources modelled as point processes, which are fed into a link with speed proportional to Ò. Under fairly general assumptions on the input processes we show that the steady state probability of the buffer content exceeding a threshold � � tends to the corresponding probability assuming Poisson input processes. We verify the assumptions for a large class of longrange dependent sources commonly used to model data traffic. Our results show that with superposition, significant multiplexing gains can be achieved for even smaller buffers than suggested by previous results, which consider Ç Ò buffer size. Moreover, simulations show that for realistic values of the exceedance probability and moderate utilisations, convergence to the Poisson limit takes place at reasonable values of the number of sources superposed. This is particularly relevant for highspeed networks in which the cost of highspeed memory is significant. Keywords—Longrange dependence, overflow probability, Poisson limit, heavy tails, point processes, multiplexing.
Abry, Cluster processes, a natural language for network traffic
 IEEE Transactions on Signal Processing, special issue ‘‘Signal Processing in Networking’’ 51
"... Abstract—We introduce a new approach to the modeling of network traffic, consisting of a semiexperimental methodology combining models with data and a class of point processes (cluster models) to represent the process of packet arrivals in a physically meaningful way. Wavelets are used to examine s ..."
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Cited by 52 (15 self)
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Abstract—We introduce a new approach to the modeling of network traffic, consisting of a semiexperimental methodology combining models with data and a class of point processes (cluster models) to represent the process of packet arrivals in a physically meaningful way. Wavelets are used to examine secondorder statistics, and particular attention is paid to the modeling of longrange dependence and to the question of scale invariance at small scales. We analyze in depth the properties of several large traces of packet data and determine unambiguously the influence of network variables such as the arrival patterns, durations, and volumes of transport control protocol (TCP) flows and internal flow structure. We show that sessionlevel modeling is not relevant at the packet level. Our findings naturally suggest the use of cluster models. We define a class where TCP flows are directly modeled, and each model parameter has a direct meaning in network terms, allowing the model to be used to predict traffic properties as networks and traffic evolve. The class has the key advantage of being mathematically tractable, in particular, its spectrum is known and can be readily calculated, its wavelet spectrum deduced, interarrival distributions can be obtained, and it can be simulated in a straightforward way. The model reproduces the main secondorder features, and results are compared against a simple black box point process alternative. Discrepancies with the model are discussed and explained, and enhancements are outlined. The elephant and mice view of traffic flows is revisited in the light of our findings. Index Terms—Internet data, longrange dependence, multifractals, point processes, scaling, time series analysis, traffic modeling, wavelets. I.
Multiplicative Multifractal Modeling of LongRangeDependent (LRD) Traffic in Computer Communications Networks
 Proceedings ICC'99
, 1999
"... Source traffic streams as well as aggregated traffic flows often exhibit longrangedependent (LRD) properties. In this work, we model traffic streams using multiplicative multifractal processes. We develop two type of models, the multifractal point processes and multifractal counting processes. We d ..."
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Cited by 25 (9 self)
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Source traffic streams as well as aggregated traffic flows often exhibit longrangedependent (LRD) properties. In this work, we model traffic streams using multiplicative multifractal processes. We develop two type of models, the multifractal point processes and multifractal counting processes. We demonstrate our model to effectively track the behavior exhibited by the system driven by the actual traffic processes. We also study the superposition of LRD flows. We prove that the superposition of a finite number of multiplicative multifractal traffic streams results asymptotically in another multifractal stream. Furthermore we demonstrate numerically that the superimposed process can be effectively modeled by an ideal multiplicative process.
Multiplexing OnOff Sources with Subexponential On Periods: Part II
, 1997
"... We consider an aggregate arrival process A N obtained by multiplexing N OnOff sources with exponential Off periods of rate λ and generally distributed On periods τ on. When N goes to infinity, with λN → Λ, AN approaches an M/G/ ∞ type process. For a fluid queue with the limiting M/G/ ∞ arrivals A ∞ ..."
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Cited by 25 (6 self)
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We consider an aggregate arrival process A N obtained by multiplexing N OnOff sources with exponential Off periods of rate λ and generally distributed On periods τ on. When N goes to infinity, with λN → Λ, AN approaches an M/G/ ∞ type process. For a fluid queue with the limiting M/G/ ∞ arrivals A ∞ t, regularly varying On periods with noninteger exponent, and capacity c, we obtain 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> x] ∼ Λ r+ρ−c P[τ c−ρ on> u]du x → ∞, x/(r+ρ−c) where ρ = EA ∞ t < c; r (c ≤ r) is the rate at which the fluid is arriving during an On period. (In particular, when P[τ on> x] ∼ x −α,1 < α < 2, the above formula applies to the socalled longrange dependent OnOff sources.) Based on this asymptotic result and the results from a companion paper we suggest a computationally efficient approximation for the case of finitely many longtailed OnOff sources. The accuracy of this approximation is verified with extensive simulation experiments.
Subexponential loss rates in a GI/GI/1 queue with applications
 QUEUEING SYSTEMS 33
, 1999
"... Consider a single server queue with i.i.d. arrival and service processes, {A, An, n � 0} and {C, Cn, n � 0}, respectively, and a finite buffer B. The queue content process {Q B n, n � 0} is recursively defined as Q B n+1 = min((Q B n + An+1 − Cn+1) +, B), q + = max(0, q). When E(A − C) < 0, and A ..."
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Cited by 23 (4 self)
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Consider a single server queue with i.i.d. arrival and service processes, {A, An, n � 0} and {C, Cn, n � 0}, respectively, and a finite buffer B. The queue content process {Q B n, n � 0} is recursively defined as Q B n+1 = min((Q B n + An+1 − Cn+1) +, B), q + = max(0, q). When E(A − C) < 0, and A has a subexponential distribution, we show that the stationary expected loss rate for this queue E(Q B n + An+1 − Cn+1 − B) + has the following explicit asymptotic characterization: E(Q B n + An+1 − Cn+1 − B) + ∼ E(A − B) + as B →∞, independently of the server process Cn. For a fluid queue with capacity c, M/G/ ∞ arrival process At, characterized by intermediately regularly varying on periods τ on, which arrive with Poisson rate Λ, the average loss rate λ B loss satisfies λ B loss ∼ Λ E(τ on η − B) + as B →∞, where η = r + ρ − c, ρ = EAt <c; r (c � r) is the rate at which the fluid is arriving during an on period. Accuracy of the above asymptotic relations is verified with extensive numerical and simulation experiments. These explicit formulas have potential application in designing communication networks that will carry traffic with longtailed characteristics, e.g., Internet data services.