set, e.g., arrival times, fills the space in which it is embedded [11−13]. As an example, the so-called correlation dimension associated with a measure, known as correlation integral, is an appropriate tool to characterize the behavior of self-similar sets.
6.2.2 Impacts of Fractal Nature of Packet Traffic
analyzed the loss processes in simulations driven by Ethernet traffic traces. The study has relied on correlation analysis for different data sets. It has measured the burstiness of the loss process using the fractional correlation dimension. The study has shown that when the packet loss occurs, it occurs at much higher rates than the long term rates, and hence there will be a considerably more impact on the applications than that indicated by the long term rate. In addition, other fractal parameters such as the Hurst parameter are also applicable to the loss process. Please see Erramili et al.,  for further details.
The presence of fractal properties in actual arrival, service time, and QoS processes may serve as a motivation for the development of the fractal queuing to analyze the performance implications of the processes with long range dependence. One possibility is that if fractal properties impact performance indirectly by biasing the long term traffic measurements, then they can be counted on to transform inputs to conventional queuing models. The direct analysis of models that use fractal characterizations as the input is another possibility, although the lack of a Markovian structure makes such models extremely difficult to analyze. There are, however, three promising approaches: the first one is based on a self-similar stochastic model, specifically, fractal Brownian motion, the next one is based on dynamical system approach using chaos theory, and the last one based on the neural networks theory. While the first two approaches are only mentioned briefly here, the last one is the main focus of this chapter and will be discussed in detail.