A long-tailed or heavy-tailed 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, several 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.
Overview The design of robust and reliable networks and network services has become an increasingly challenging task in today's Internet world. To achieve this goal, understanding the characteristics of Internet traffic plays a more and more critical role. Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic. Self-similar Ethernet traffic exhibits dependencies over a long range of time scales. This is to be contrasted with telephone traffic, which is Poisson in its arrival and departure process. With many time-series if the series is averaged then the data begins to look smoother. However, with self-similar data, one is confronted with traces that are spiky and bursty, even at large scales. Such behaviour is caused by strong dependence in the data: large values tend to come in clusters, and clusters of clusters, etc. This can have far-reaching consequences for network performance. Heavy-tail distributions have been observed in many natural phenomena, including both physical and sociological phenomena. Mandelbrot established the use of heavy-tail distributions to model real-world fractal phenomena, e.g., stock markets, earthquakes, and the weather. Ethernet, WWW, SS7, TCP, FTP, TELNET and VBR video (digitised video of the type that is transmitted over ATM networks) traffic is self-similar. Self-similarity in packetised data networks can be caused by the distribution of file sizes, human interactions and/or Ethernet dynamics. Self-similar and long-range dependent characteristics in computer networks present a fundamentally different set of problems to people doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.
Short-range dependence vs. long-range dependence Long-range and short-range dependent processes are characterised by their autocovariance functions. In short-range dependent processes, the coupling between values at different times decreases rapidly as the time difference increases.
The sum of the autocorrelation function over all lags is finite. As the lag increases, the autocorrelation function of short-range dependent processes decays quickly. In long-range processes, the correlations at longer time scales are more significant.
The area under the autocorrelation function summed over all lags is infinite. The decay of the autocorrelation function is often assumed to have the specific functional form,
ρ ( k ) ∼ k − α {\displaystyle \rho (k)\sim k^{-\alpha }}
where ρ(k) is the autocorrelation function at a lag k, α is a parameter in the interval (0,1) and the ~ means asymptotically proportional to as k approaches infinity.
Long-range dependence as a consequence of mathematical convergence Such power law scaling of the autocorrelation function can be shown to be biconditionally related to a power law relationship between the variance and the mean, when evaluated from sequences by the method of expanding bins. This variance-to-mean power law is an inherent feature of a family of statistical distributions called the Tweedie exponential dispersion models. Much as the central limit theorem explains how certain types of random data converge towards the form of a normal distribution there exists a related theorem, the Tweedie convergence theorem that explains how other types of random data will converge towards the form of these Tweedie distributions, and consequently express both the variance to mean power law and a power law decay in their autocorrelation functions.
The Poisson distribution and traffic Before the heavy-tail distribution is introduced mathematically, the memoryless Poisson distribution, used to model traditional telephony networks, is briefly reviewed below. For more details, see the article on the Poisson distribution. Assuming pure-chance arrivals and pure-chance terminations leads to the following:
The number of call arrivals in a given time has a Poisson distribution, i.e.:
P ( a ) = ( μ a a ! ) e − μ , {\displaystyle P(a)=\left({\frac {\mu ^{a}}{a!}}\right)e^{-\mu },}
where a is the number of call arrivals and μ {\displaystyle \mu } is the mean number of call arrivals in time T. For this reason, pure-chance traffic is also known as Poisson traffic.
The number of call departures in a given time also has a Poisson distribution, i.e.:
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