Heavy-tailed distributions

In the engineering point of view, the self-similarity means that the traffic had similar statistical properties at a range of timescales: milliseconds, seconds, minutes, hours, even days and weeks. The merging of traffic streams, as in a statistical multiplexer, does not result in smoothing of traffic, in other words, burst data streams that are multiplexed tend to produce a bursty aggregate stream. Correspondingly, this traffic property can be described in the mathematical terms that the traffic has a long range dependence, which can be characterized that the correlation function of the traffic process is a heavy-tailed distribution.

A distribution is heavy-tailed if

$$P [X \gt x] \sim x ^{-\alpha}$$

as $x\rightarrow \infty$, where $0<\alpha <2$. One of the simplest heavy-tailed distributions is the Pareto distribution. The Pareto distribution is power-law over its entire range; its probability density function is given by

$$p(x) = \alpha{}k^{\alpha}x^{-\alpha-1}$$

where $\alpha\gt$, k>0, and $x\geq{}k$. Its distribution function has the form

$$F (x) = P [X \leq{}x] = 1-{(k/x)}^{\alpha}$$

The parameter k represents the smallest possible value of the random variable.

1.

if $\alpha{}\leq{}2$, the distribution has infinite variance.

2.

if $\alpha{}\leq{}1$, then the distribution has also infinite mean.

Thus, as depicted in Figure 1, with $\alpha{}$ decreases, a large portion of the probability mass is present in the tail of the distribution. In practical terms, a random variable that follows a heavy-tailed distribution can be extremely large with non negligible probability.

The strict mathematics indicates that there are self-similar process that are not long-range dependent, and vice versa. However, by the restriction $1<\alpha{}<2$ in the definition, self-similarity implies long-range dependence, and vice versa.

  

Figure 1: Pareto and Exponential Probability Density Functions