Numerical Methods of Multifractal Analysis in Information Communication Systems and Networks

Numerical Methods of Multifractal Analysis in Information Communication Systems and Networks

Oleg I. Sheluhin (Moscow Technical University of Communications and Informatics, Russia) and Artem V. Garmashev (Moscow Technical University of Communications and Informatics, Russia)
DOI: 10.4018/978-1-4666-2208-1.ch002
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Abstract

In this chapter, the main principles of the theory of fractals and multifractals are stated. A singularity spectrum is introduced for the random telecommunication traffic, concepts of fractal dimensions and scaling functions, and methods used in their determination by means of Wavelet Transform Modulus Maxima (WTMM) are proposed. Algorithm development methods for estimating multifractal spectrum are presented. A method based on multifractal data analysis at network layer level by means of WTMM is proposed for the detection of traffic anomalies in computer and telecommunication networks. The chapter also introduces WTMM as the informative indicator to exploit the distinction of fractal dimensions on various parts of a given dataset. A novel approach based on the use of multifractal spectrum parameters is proposed for estimating queuing performance for the generalized multifractal traffic on the input of a buffering device. It is shown that the multifractal character of traffic has significant impact on queuing performance characteristics.
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Theory Of Fractals And Multifractals

The term "fractal” was used for the first time in Benoît Mandelbrot’s work (Mandelbrot, 1982). The word fractal is derived from the Latin fractus meaning "fractured" or "broken." Mandelbrot used the term "fractals" for geometric objects that have strongly fragmented shape and can possess the property of self-similarity. It is possible to generalize the concept of fractal to any object (image, speech, telecommunication traffic, etc.) some parameters of which are remain invariant with change in scale or time. Thus, the principal property of such objects (i.e. self-similarity) implies that at augmentation, its parts are similar (in some specified sense) to its total shape.

The property of exact self-similarity is a characteristic of the regular fractals only. If an element of randomness is to be included in the algorithm of their creation instead of the determined method of construction (as it happens, for example, in many processes of diffusion growth of clusters, voltage failure, etc.), then the so-called incidental fractals appear. Their basic difference from regular ones is that the property of self-similarity holds true only after a corresponding averaging on the base of all statistically independent realizations of the object. For quantitative description of fractals, a single value is enough - a fractal dimension (Hausdorff dimension) or the index of scaling which is determined as follows

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