Generation of Gaussian Self-similar series via Wavelets for Use in Traffic Simulations

Measurements have shown that network traffic has fractal properties such as self-similarity and long memory or long-range dependence. Long memory is characterized by the existence of a pole at the origin of the power spectrum density function (1/f shape). It was also noticed that traffic may present short-range dependence at some time scales. The use of a “realistic” aggregated network traffic generator, one that synthesizes fractal time series, is fundamental to the validation of traffic control algorithms. In this article, the synthesis of approximate realizations of a kind of self-similar random process named fractional Gaussian noise is done via wavelet transform. The proposed method is also capable of synthesizing Gaussian time series with more generic spectra than 1/f, that is, time series that also have short-range dependence. The generation is done in two stages. The first one generates an approximate realization of fractional Gaussian noise via discrete Wavelet transform. The second one introduces short-range dependence through IIR (Infinite Impulse Response) filtering at the output of the first stage. A detailed characterization of the resulting series was done, using statistical moments of first, second, third and fourth orders, as well as specific statistical tests for self-similar series. It was verified that the Whittle estimator of the Hurst parameter is more robust than the periodogram method for series that simultaneously present short-range and long-range dependence.