The observation that traffic in packet-switched networks shows self-similar and long-range dependent behaviour goes back to local area Ethernet investigations made by Willinger et al. . In the sequel, a lot of empiric studies have been made that confirm the self-similar nature of packet-switched traffic in local and wide area networks and for different kinds of applications . While exact or asymptotic self-similarity does not always exactly fit to observed empiric traffic samples, it still comes closer to empiric observations than Poisson-based traffic models do. In particular, the effects of burstiness at different time scales and long-range dependence (see below) cannot be modelled by Poisson traffic models.
Mathematically speaking, let be the number of packets traversing a link during the time interval Then the properties of the sequence with
are investigated. and are assumed to be realizations of stationary stochastic processes and respectively. In the following, the description of the properties of self-similar time series will concentrate on the theoretical values and It should be clear, that the empirically observed number of packets must be interpreted as realizations of these stochastic processes.
For the analysis of the properties of the time series it makes sense to introduce the scaled sample
the scaled variance
where denotes the number of scaled samples (it is assumed that ) and denotes the mean
and the autocorrelation functions
The following definition is taken from .
Definition: The time series is called asymptotically second order self-similar with self-similarity parameter or Hurst parameter if the second-order statistics of converge as follows:
If equities (1)
are valid for all and not only for the limit, the time series is said to be exactly
self-similar of second order.
For the modeling of traffic in packet-switched networks, only the range is relevant. With the presented mathematical framework, the main properties of self-similar time series with within this range can be expressed as follows 
There are a variety of methods for the generation of self-similar and long-range dependent traffic for simulation experiments: