networks which carry long-tail traffic. Previous analytic work done in Internet studies adopted assumptions such as exponentially-distributed packet inter-arrivals, and conclusions reached under such assumptions may be misleading or incorrect in the presence of heavy-tailed distributions. It has for long been realised that efficient and accurate modelling of various real world phenomena needs to incorporate the fact that observations made on different scales each carry essential information. In most simple terms, representing data on large scales by its mean is often useful (such as an average income or an average number of clients per day) but can be inappropriate (e.g. in the context of buffering or waiting queues). With the convergence of voice and data, the future multi-service network will be based on packetised traffic, and models which accurately reflect the nature of long-tail traffic will be required to develop, design and dimension future multi-service networks. We seek an equivalent to the Erlang model for circuit switched networks. There is not an abundance of heavy-tailed models with rich sets of accompanying data fitting techniques. A clear model for fractal traffic has not yet emerged, nor is there any definite direction towards a clear model. Deriving mathematical models which accurately represent long-tail traffic is a fertile area of research. Gaussian models, even long-range dependent Gaussian models, are unable to accurately model current Internet traffic. Classical models of time series such as Poisson and finite Markov processes rely heavily on the assumption of independence, or at least weak dependence. Poisson and Markov related processes have, however, been used with some success. Nonlinear methods are used for producing packet traffic models which can replicate both short-range and long-range dependent streams. A number of models have been proposed for the task of modelling long-tail traffic. These include the following: Fractional ARIMA Fractional Brownian motion Iterated Chaotic Maps Infinite Markov Modulated Processes Poisson Pareto Burst Processes (PPBP) Markov Modulated Poisson Processes (MMPP) Multi-fractal models Matrix models Wavelet Modelling Tweedie distributions No unanimity exists about which of the competing models is appropriate, but the Poisson Pareto Burst Process (PPBP), which is an M/G/ process, is perhaps the most successful model to date. It is demonstrated to satisfy the basic requirements of a simple, but accurate, model of long-tail traffic. Finally, results from simulations using -stable stochastic processes for modelling traffic in broadband networks are presented. The simulations are compared to a variety of empirical data (Ethernet,