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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.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, 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.: where a is the number of call arrivals and 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