Long-tail Traffic

Long-tail Traffic

This article covers a range of tools from different disciplines that may be used in the important science of determining the probability of rare events.

The terms long-range dependent, self-similar and heavy-tailed are very close in meaning. Differences in nomenclature hint at the origins and application fields of the terms. These are somewhat different but closely related phenomena.

A long-tailed or heavy-tailed probability distribution is one that assigns relatively high probabilities to regions far from the mean or median. A more formal mathematical definition is given below. In the context of teletraffic engineering a number of quantities of interest have been shown to have a long-tailed distribution. For example, if we consider the sizes of files transferred from a web-server, then, to a good degree of accuracy, the distribution is heavy-tailed, that is, there are a large number of small files transferred but, crucially, the number of very large files transferred remains a major component of the volume downloaded.

Many processes are technically long-range dependent but not self-similar. The differences between these two phenomena are subtle. Heavy-tailed refers to a probability distribution, and long-range dependent refers to a property of a time series and so these should be used with care and a distinction should be made. The terms are distinct although superpositions of samples from heavy-tailed distributions aggregate to form long-range dependent time series.

Additionally there is Brownian motion which is self-similar but not long-range dependent.