2.3 Heavy tailed distributions

A class of distributions that is often used to capture the characteristics of highly-variable stochastic processes, i.e., more variable than the exponential distribution, is the class of heavy-tailed distributions.

Definition A distribution is heavy-tailed if its complementary cumulative distribution (CCDF), often referred to as the tail, $F^c(t) = 1 - F(t)$, where $F(t)$ is the CDF, decays slower than exponentially, i.e., there is some $\gamma >0$ such that

$$\lim_{t \to \infty} \exp(\gamma t) F^c (t) \to \infty.$$ (2.4)

A typical heavy-tailed distribution is power-tailed if ${\mathbf{F}}^c (t) \sim \alpha t^{-\beta}$ as $t \to \infty$ for constants $\alpha >0$ and $\beta >0$.

Definition A distribution has short tail if its CCDF $F^c(t)$, decays exponentially or faster, i.e., there is some $\gamma >0$ such that

$$\lim_{t \to \infty} \exp(\gamma t) F^c (t) \to 0.$$ (2.5)

A typical short-tailed distribution is exponentially-tailed if ${\mathbf{F}}^c (t) \sim \alpha e^{-\beta t}$ as $t \to \infty$ for constants $\alpha >0$ and $\beta >0$.

In the literature, different definitions of heavy-tailed like distributions exist. For a more detailed classification of heavy-tailed distributions and their properties refer to [95] and references therein. Throughout this dissertation, we refer to a distribution as heavy-tailed if its coefficient of variation is larger than the one of the exponential distribution. Note that we distinguish between the unbounded possible values of a distribution function and the bounded possible values in a data set by referring to high variability in a data set as long-tailed behavior and in a distribution function as heavy-tailed behavior.

The Pareto distribution with $F^c_{Pareto(\beta)}(t) = t^{-\beta}$ for $\beta >0$ is a classic case of a distribution exhibiting power-tailed behavior in the entire range of its parameters. The Weibull distribution with $F^c_{Weibull(c,a)}(t) = e^{-(t/a)^c}$ for $c<1$ and $a > 0$ is heavy-tailed, but not power-tailed.

Simulation of heavy-tailed distributions for estimation of steady-state measures is not easy, as the simulation must run exceptionally long in order to capture the effect of the distribution tail, i.e., the rare events, which even with small probability of occurrence can significantly affect the system performance. Heavy-tailed distributions generally have high coefficient of variation, while power-tailed distributions can have infinite moments. For example, in a power-tailed distribution if $0< \beta < 1$ the mean is infinite, while if $1< \beta < 2$ the mean of the distribution is finite but the variance is infinite. The infinite mean or variance in a power-tailed distribution increases the complexity of their analysis. The highly variable behavior in data sets or distribution functions can be accurately approximated by PH distributions (PH distributions are discussed later in this chapter), which are tractable and can be analyzed using the matrix-analytic methodology.