## A Multifractal Wavelet Model with Application to Network Traffic (1998)

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### BibTeX

@MISC{Riedi98amultifractal,

author = {Rudolf H. Riedi and Matthew S. Crouse and Vinay J. Ribeiro and Richard G. Baraniuk},

title = {A Multifractal Wavelet Model with Application to Network Traffic},

year = {1998}

}

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### Abstract

In this paper, we develop a new multiscale modeling framework for characterizing positive-valued data with long-range-dependent 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 N-point data sets. We study both the second-order 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 (variance-time 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 oth...

### Citations

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Citation Context ...strong dependencies in higher order moments. For instance, many real-world data sets exhibit strong dependencies in the energy of the wavelet coefficients, corresponding to fourth-order cross-moments =-=[40]-=-, [41]. E. Related Work Constructions similar to the MWM were developed earlier in [42] and [43]. A similar multiplicative model for wavelet coefficients has been developed in [44] and [45], where it ... |

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Citation Context ...bserved at, say, a gateway. With this model, heavy-tailed ON periods lead to LRD similar to that observed in actual traffic. Convincing modeling results have made a strong case for this point of view =-=[74]-=-, [75]. However, ON/OFF models are accurate only in the limit of large time scales (seconds and longer), and they do not account for the actual queuing and multiplexing occurring in the network. A com... |

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Citation Context ... differences of the and . The wavelet transform closely approximates the Karhunen–Loève transform for fBm and fGn [11]–[13]. This fact has been leveraged into efficient approximate fBm and fGn models =-=[14]-=-: we posit that the wavelet coefficients are simply independent zero-mean Gaussian random variables with power-law decaying variance , with for fBm and for fGn. Unfortunately, despite their great simp... |

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Citation Context ...have proved wellsuited to modeling nonhomogeneous phenomena [17], [18]. More recently, the multifractal nature of network traffic has been demonstrated convincingly, first in [19] and subsequently in =-=[20]-=- and [21]. The beauty of the multifractal formalism has motivated considerable research effort in mathematics [22]–[32]; however, few multifractal data models have been developed to date. In the most ... |

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Citation Context ...lent to its autocorrelation function rB(t� s) := IE[B(t)B(s)] having the form 2 rB(t� s) = jtj 2 2H + jsj 2H ;jt; sj 2H (9) or its (generalized) power spectral density behaving as ;B(f) / jfj ;(2H+1) =-=[12]-=-. It follows from (9) that fGn has an autocorrelation function rG[ ]= 2 2 j tj2H j +1j 2H + j ; 1j 2H ; 2j j 2H : (10) As with fBm, fGn has a discrete-time power spectrum that behaves as ;G(f) /jfj ;(... |

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Citation Context ...ltiplicative construction of the MWM process is reminiscent of the binomial measure, a classical multifractal process. Multifractals were first introduced to model dissipation of energy in turbulence =-=[15]-=-, [16] and have proved wellsuited to modeling nonhomogeneous phenomena [17], [18]. More recently, the multifractal nature of network traffic has been demonstrated convincingly, first in [19] and subse... |

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Citation Context ...re, a classical multifractal process. Multifractals were first introduced to model dissipation of energy in turbulence [15], [16] and have proved wellsuited to modeling nonhomogeneous phenomena [17], =-=[18]-=-. More recently, the multifractal nature of network traffic has been demonstrated convincingly, first in [19] and subsequently in [20] and [21]. The beauty of the multifractal formalism has motivated ... |

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Citation Context ...et us set where , i.e., (88) (89)RIEDI et al.: MULTIFRACTAL WAVELET MODEL WITH APPLICATION TO NETWORK TRAFFIC 1013 Depending on the context, is called the partition function or the free energy [15], =-=[78]-=-, [79]. Again, we have added a factor for convenience. A closer look at (48) reveals that it actually shows that . As a matter of fact, it is proven in [32] and [55] that Lemma 5: For every path of (9... |

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Citation Context ...erformance can be studied through analytical means [6], [63]–[67]. In cases where theoretical analysis is intractable, models are routinely used to synthesize test data traces for simulation purposes =-=[68]-=-. Here, computational efficiency of the synthesis becomes as important as the accuracy. We begin with some historical remarks. Although LRD models have long been known to characterize a variety of phe... |

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Citation Context ...) process. When the Hurst parameter , fGn exhibits LRD. samples of fGn can be simulated exactly via direct Cholesky factorization ( computational complexity) [4] or Levinson’s recursion ( complexity) =-=[8]-=-. These costs can become overbearing, especially in networking applications where often . For such problems, approximate synthesis techniques ( complexity) based on wavelets have been developed. The d... |

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Citation Context ...orology, hydrology, and soil science [1, 2]. In signal and image processing, fractals have been applied in elds such as computer graphics, texture modeling, image compression, and pattern recognition =-=[3, 4]-=-. Fractal models have made a major impact in the area of communications recently, particularly in the area of computer data networks. As the work of Leland et al. [5] and subsequent studies have demon... |

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Citation Context ...euing behavior of traffic is important because of its influence on network management algorithms, such as connection admission control, that strive to support certain quality of service (QoS) demands =-=[71]-=-, [72]. The presence of LRD in traffic has been shown to significantly affect queuing performance [65]. For stationary traffic with only short-range dependence (SRD), classical queuing results for Mar... |

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Citation Context ..., or nature of many phenomena has led to exciting breakthroughs in a variety of scientific disciplines, including physics, chemistry, astronomy, biology, meteorology, hydrology, and soil science [1], =-=[2]-=-. In signal and image processing, fractals have been applied in fields such as computer graphics, texture modeling, image compression, and pattern recognition [3], [4]. Fractal models have made a majo... |

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Citation Context ...Large Deviation Principle (LDP): it measures how frequently or how likely the observed deviates from the “expected value” . We will elaborate on this, especially the use of a theorem of Gärtner–Ellis =-=[59]-=- toward an improvement of (50) in Appendix A (cf. Theorems 6 and 9). B. Numerical Estimation of For the MWM, we have , and the sum in (47) becomes (52) if is chosen as usual to satisfy . This holds fo... |

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Citation Context ...ssion, and pattern recognition [3, 4]. Fractal models have made a major impact in the area of communications recently, particularly in the area of computer data networks. As the work of Leland et al. =-=[5]-=- and subsequent studies have demonstrated, network tra c loads exhibit fractal properties such as self-similarity, burstiness, and long-range dependence (LRD). Inadequately modeled by Manuscript recei... |

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Citation Context ...In addition, the variance-time plots of Fig. 7 depict an excellent match of the correlation structure. 11 10Since we characterize traffic interarrival times, our result does not conflict with that of =-=[69]-=-, which concluded that the bytes-per-time and packets-pertime of the August 1989 Bellcore traces were not multifractal. Multifractal scaling of similar quality over five decades has been reported for ... |

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Citation Context ...7, where we present an application of the MWM framework to computer network tra c modeling. 5.1 The MWM is a binomial cascade The MWM extends the simple, classical multifractal | the binomial measure =-=[22, 53, 54, 57]-=- | in a natural fashion. This measure is most conveniently constructed iteratively through a so-called cascade structure, whence it is often addressed as a binomial cascade. As we willshow, its distri... |

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Citation Context ... function or the free energy [15], [78], [79]. Again, we have added a factor for convenience. A closer look at (48) reveals that it actually shows that . As a matter of fact, it is proven in [32] and =-=[55]-=- that Lemma 5: For every path of (90) As an immediate consequence, the function is concave and thus continuous and almost everywhere differentiable. It is instructive to see how the quick and dirty ar... |

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Citation Context ...s: Besov spaces are also useful for analyzing the regularity of functions, especially since an elegant description of these regularity spaces in terms of wavelet coefficients has become available. In =-=[81]-=- it is shown that the norm of the Besov space of a process with wavelet coefficients is equivalent to (102) Roughly speaking, this norm measures the smoothness of order in , where is an additional par... |

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Citation Context ...cative construction of the MWM process is reminiscent of the binomial measure, a classical multifractal process. Multifractals were first introduced to model dissipation of energy in turbulence [15], =-=[16]-=- and have proved wellsuited to modeling nonhomogeneous phenomena [17], [18]. More recently, the multifractal nature of network traffic has been demonstrated convincingly, first in [19] and subsequentl... |

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Citation Context ... present an application of the MWM framework to computer network traffic modeling. A. The MWM is a Binomial Cascade The MWM extends the simple, classical multifractal—the binomial measure [22], [53], =-=[54]-=-, [57]—in a natural fashion. This measure is most conveniently constructed iteratively through a so-called cascade structure, where it is often addressed as a binomial cascade. As we will show, its di... |

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Citation Context ...t the wavelet coefficients of for any -supported mother wavelet are distributed as with independent of and distributed as . So, it follows that (56) holds also in this setting with given by (65) (see =-=[51]-=- for a similar result on deterministic cascades). Choosing a compactly supported wavelet with enough regularity we find, using Lemma 7, that an MWM signal with identically distributed multipliers is i... |

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Citation Context ...gale arguments. Also of considerable importance is the possibility to go beyond the binary structure imposed by the Haar wavelet system and to introduce randomness in the geometry of the construction =-=[24]-=-, [25] and—as a particular case—wide sense stationarity in the signal. To describe such systems is, however, beyond the scope of this paper. VI. MULTIFRACTAL ANALYSIS OF THE MWM So far we have noted t... |

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Citation Context ...urbulence [15], [16] and have proved wellsuited to modeling nonhomogeneous phenomena [17], [18]. More recently, the multifractal nature of network traffic has been demonstrated convincingly, first in =-=[19]-=- and subsequently in [20] and [21]. The beauty of the multifractal formalism has motivated considerable research effort in mathematics [22]–[32]; however, few multifractal data models have been develo... |

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Citation Context ...ents such as bursts (small ). Fig. 9 reveals a rich multifractal spectrum. In contrast, fBm has a trivial spectrum consisting only of one point indicating that it has the same “burstiness” everywhere =-=[70]-=-. 2) Synthetic Data: Having established the LRD and multifractal characteristics of the pAug trace, we will next model these properties using the MWM. To train the MWM, we use the approach outlined in... |

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Citation Context ...h), zero 3-D volume, but finite positive 2-D volume (area); its dimension is two after all. Generalizing, we say that has at least dimension since is positive, i.e., . A complete argument is given in =-=[23]-=- and [55]. B. Deterministic Cascade with Variable Multipliers Let us now generalize slightly by allowing the almost sure multipliers to depend on scale: for all almost surely, where . Let us assume,1... |

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Citation Context ... mathematical studies on cascades considered the case of independent with [22]. These results have been generalized to conservative cascades [54] and [52], and to more general invariant measures [24]–=-=[26]-=-. Here, we present the argument of Falconer [24]. Essentially, there are two difficulties to deal with. First, the auxiliary measures are now random, and we have to ensure their existence. Second, as ... |

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Citation Context ...ree that are suppressed in scale . In fact, the consist simply of scaled sums and differences of the and . The wavelet transform closely approximates the Karhunen–Loève transform for fBm and fGn [11]–=-=[13]-=-. This fact has been leveraged into efficient approximate fBm and fGn models [14]: we posit that the wavelet coefficients are simply independent zero-mean Gaussian random variables with power-law deca... |

33 |
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Citation Context ...network traffic has been demonstrated convincingly, first in [19] and subsequently in [20] and [21]. The beauty of the multifractal formalism has motivated considerable research effort in mathematics =-=[22]-=-–[32]; however, few multifractal data models have been developed to date. In the most simple terms, multifractals possess a local smoothness that depends on in an erratic way. Equiv-994 IEEE TRANSACT... |

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Citation Context ...anding of the convergence of the algorithm, properties of the marginal distributions, advantages over monofractal fGn models, and a range of possible refinements and extensions [15], [16], [22]–[32], =-=[49]-=-–[57]. The theory of cascades comes with a dedicated set of tools for analysis, both theoretical and numerical, that we will outline in the next two sections (see Appendixes A and B for more details).... |