Multivariate fractional Levy motion and its applications

Since the beginning of the 1990s, many empirical studies of real telecoomunication systems traffic have been conducted. It was found that traffic has some specific properties, which are different from common voice traffic, namely, it has the properties of self-similarity and long-range dependence and the distribution of loading size from one source has heavy tails. Some new models have been constructed, where these features were captured. Brownian fractional motion and $\alpha$-stable Levy motion are the well-known examples. But both of these models do not have all of the above properties. More complicated models have been proposed using some combination of these ones. In particular, the authors have proposed a variant of univariate fractional Levy motion. This paper considers a multivariate analog of fractional Levy motion. This process is multivariate fractional Brownian motion with random change of time, where random change of time is Levy motion with one-sided stable distributions. The properties of this process are investigated and it is proven that it is self-similar and has stationary increments. Next, it is shown that the coordinates of one-dimensional sections of this process have the distributions which are not stable. But asymptotic of tails for these distributions is the same as for the stable ones. This model is applied to analyze heterogeneous traffic and to get a lower asymptotic bound of the probability of overflow of at least one buffer. There are other possible applications.