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Data networks as cascades: Investigating the multifractal nature of Internet WAN traffic
, 1998
"... In apparent contrast to the welldocumented selfsimilar (i.e., monofractal) scaling behavior of measured LAN traffic, recent studies have suggested that measured TCP/IP and ATM WAN traffic exhibits more complex scaling behavior, consistent with multifractals. To bring multifractals into the realm o ..."
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Cited by 205 (14 self)
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In apparent contrast to the welldocumented selfsimilar (i.e., monofractal) scaling behavior of measured LAN traffic, recent studies have suggested that measured TCP/IP and ATM WAN traffic exhibits more complex scaling behavior, consistent with multifractals. To bring multifractals into the realm of networking, this paper provides a simple construction based on cascades (also known as multiplicative processes) that is motivated by the protocol hierarchy of IP data networks. The cascade framework allows for a plausible physical explanation of the observed multifractal scaling behavior of data traffic and suggests that the underlying multiplicative structure is a traffic invariant for WAN traffic that coexists with selfsimilarity. In particular, cascades allow us to refine the previously observed selfsimilar nature of data traffic to account for local irregularities in WAN traffic that are typically associated with networking mechanisms operating on small time scales, such as TCP flo...
A multifractal wavelet model with application to TCP network traffic
 IEEE TRANS. INFORM. THEORY
, 1999
"... In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the mo ..."
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Cited by 193 (32 self)
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In this paper, we develop a new multiscale modeling framework for characterizing positivevalued data with longrangedependent correlations (1=f noise). Using the Haar wavelet transform and a special multiplicative structure on the wavelet and scaling coefficients to ensure positive results, the model provides a rapid O(N) cascade algorithm for synthesizing Npoint data sets. We study both the secondorder and multifractal properties of the model, the latter after a tutorial overview of multifractal analysis. We derive a scheme for matching the model to real data observations and, to demonstrate its effectiveness, apply the model to network traffic synthesis. The flexibility and accuracy of the model and fitting procedure result in a close fit to the real data statistics (variancetime plots and moment scaling) and queuing behavior. Although for illustrative purposes we focus on applications in network traffic modeling, the multifractal wavelet model could be useful in a number of other areas involving positive data, including image processing, finance, and geophysics.
The Changing Nature of Network Traffic: Scaling Phenomena
 Computer Communication Review
, 1998
"... In this paper, we report on some preliminary results from an indepth, waveletbased analysis of a set of highquality, packetlevel traffic measurements, collected over the last 67 years from a number of different widearea networks (WANs). We first validate and confirm an earlier finding, origina ..."
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Cited by 138 (18 self)
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In this paper, we report on some preliminary results from an indepth, waveletbased analysis of a set of highquality, packetlevel traffic measurements, collected over the last 67 years from a number of different widearea networks (WANs). We first validate and confirm an earlier finding, originally due to Paxson and Floyd [14], that actual WAN traffic is consistent with statistical selfsimilarity for sufficiently large time scales. We then relate this largetime scaling phenomenon to the empirically observed characteristics of WAN traffic at the level of individual connections or applications. In particular, we present here original results about a detailed statistical analysis of Websession characteristics, and report on an intriguing scaling property of measured WAN traffic at the transport layer (i.e., number of TCP connection arrivals per time unit). This scaling property of WAN traffic at the TCP layer was absent in the preWeb period but has become ubiquitous in today's WWW...
Scaling Analysis of Conservative Cascades, With Applications to Network Traffic
, 1999
"... Recent studies have demonstrated that measured widearea network traffic such as Internet traffic exhibits locally complex irregularities, consistent with multifractal behavior. It has also been shown that the observed multifractal structure becomes most apparent when analyzing measured network tr ..."
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Cited by 52 (7 self)
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Recent studies have demonstrated that measured widearea network traffic such as Internet traffic exhibits locally complex irregularities, consistent with multifractal behavior. It has also been shown that the observed multifractal structure becomes most apparent when analyzing measured network traffic at a particular layer in the welldefined protocol hierarchy that characterizes modern data networks, namely the transport or TCP layer. To investigate this new scaling phenomenon associated with the dynamics of measured network traffic over small time scales, we consider a class of multiplicative processes, the socalled conservative cascades, that serves as a cascade paradigm for and is motivated by the networking application. We present a waveletbased time/scale analysis of these cascades to determine rigorously their global and local scaling behavior. In particular, we prove that for the class of multifractals generated by these conservative cascades the multifractal formal...
LogInfinitely Divisible Multifractal Processes
, 2002
"... We define a large class of multifractal random measures and processes with arbitrary loginfinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined lognormal Multifractal Random Walk processes (MRW) [33, 3] and the logPoisso ..."
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Cited by 40 (5 self)
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We define a large class of multifractal random measures and processes with arbitrary loginfinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined lognormal Multifractal Random Walk processes (MRW) [33, 3] and the logPoisson “product of cynlindrical pulses” [7]. Their construction involves some “continuous stochastic multiplication” [36] from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non degeneracy, convergence of the moments and multifractal scaling.
Multifractal Processes
, 1999
"... This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and sel ..."
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Cited by 32 (6 self)
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This paper has two main objectives. First, it develops the multifractal formalism in a context suitable for both, measures and functions, deterministic as well as random, thereby emphasizing an intuitive approach. Second, it carefully discusses several examples, such as the binomial cascades and selfsimilar processes with a special eye on the use of wavelets. Particular attention is given to a novel class of multifractal processes which combine the attractive features of cascades and selfsimilar processes. Statistical properties of estimators as well as modelling issues are addressed.
LongRange Dependence and Data Network Traffic
, 2001
"... This is an overview of a relatively recent application of longrange dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in highspeed data networks such as the Internet. We demonstrate that this new application area off ..."
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Cited by 29 (1 self)
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This is an overview of a relatively recent application of longrange dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in highspeed data networks such as the Internet. We demonstrate that this new application area offers unique opportunities for significantly advancing our understanding of LRD and related phenomena. These advances are made possible by moving beyond the conventional approaches associated with the widespread "blackbox" perspective of traditional time series analysis and exploiting instead the physical mechanisms that exist in the networking context and that are intimately tied to the observed characteristics of measured network traffic. In order to describe this complexity we provide a basic understanding of the design, architecture and operations of data networks, including a description of the TCP/IP protocols used in today's Internet. LRD is observed in the large scale behavior of the data traffic and we provide a physical explanation for its presence. LRD tends to be caused by user and application characteristics and has little to do with the network itself. The network affects mostly small time scales, and this is why a rudimentary understanding of the main protocols is important. We illustrate why multifractals may be relevant for describing some aspects of the highly irregular traffic behavior over small time scales. We distinguish between a timedomain and waveletdomain approach to analyzing the small time scale dynamics and discuss why the waveletdomain approach appears to be better suited than the timedomain approach for identifying features in measured traffic (e.g., relatively regular traffic patterns over certain time scales) that have a direct networking interpretation (e....
Multifractal products of stochastic processes: Part II, 2003. Under construction
"... some basic properties ..."
Combining multifractal additive and multiplicative chaos
 COMMUN. MATH. PHYS
, 2005
"... In this work, we study the new class of multifractal measures, which combines additive and multiplicative chaos, defined by νγ,σ = X b j≥1 −jγ j2 X 0 ≤ k ≤bj µ([kb −1 −j, (k + 1)b −j)) σ δkb−j (γ ≥ 0, σ ≥ 1), where µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2. The singularities ..."
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Cited by 20 (12 self)
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In this work, we study the new class of multifractal measures, which combines additive and multiplicative chaos, defined by νγ,σ = X b j≥1 −jγ j2 X 0 ≤ k ≤bj µ([kb −1 −j, (k + 1)b −j)) σ δkb−j (γ ≥ 0, σ ≥ 1), where µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2. The singularities analysis of the measures νγ,σ involves new results on the mass distribution of µ when µ describes large classes of multifractal measures. These results generalize ubiquity theorems associated with the Lebesgue measure. Under suitable assumptions on µ, the multifractal spectrum of νγ,σ is linear on [0, hγ,σ] for some critical value hγ,σ, and then it is strictly concave on the right of hγ,σ, and deduced from the one of µ by an affine transformation. This untypical shape is the result of the combination between Dirac masses and atomless multifractal measures. These measures satisfy multifractal formalisms. These measures open interesting perspectives in modeling discontinuous phenomena.
The singularity spectrum of Lévy processes in multifractal time
, 2007
"... The interest for multifractal stochastic processes is mainly motivated by the need for accurate models in the study of the variability of wild signals. These locally irregular signals come from physical phenomena such as fully developed turbulence, TCP Internet traffic, variations of financial price ..."
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Cited by 18 (12 self)
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The interest for multifractal stochastic processes is mainly motivated by the need for accurate models in the study of the variability of wild signals. These locally irregular signals come from physical phenomena such as fully developed turbulence, TCP Internet traffic, variations of financial prices, or heart beats.