Self-similar processes are types of stochastic processes that exhibit the phenomenon of self-similarity. A self-similar phenomenon behaves the same when viewed at different degrees of magnification, or different scales on a dimension (space or time). Self-similar processes can sometimes be described using heavy-tailed distributions, also known as long-tailed distributions. Examples of such processes include traffic processes, such as packet inter-arrival times and burst lengths. Self-similar processes can exhibit long-range dependency. 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. In traditional Poisson traffic, the short-term fluctuations would average out, and a graph covering a large amount of time would approach a constant value. Heavy-tailed distributions have been observed in many natural phenomena including both physical and sociological phenomena. Mandelbrot established the use of heavy-tailed distributions to model real-world fractal phenomena, e.g. Stock markets, earthquakes, climate, 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doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity. The Poisson distribution Before the heavy-tailed distribution is introduced mathematically, the Poisson process with a memoryless waiting-time distribution, used to model (among many things) traditional telephony networks, is briefly reviewed below. 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 in time T, 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 a given time, also has a