Self-similarity

The fact that a time series is self-similar manifests itself in a number of different ways, among them [Leland 1994]

1. `burstiness' across a large range of time scales  
2. slowly decaying autocorrelations, implying long-range dependence
3. sample variance decays more slowly than the reciprocal of the sample size
4. a pole in the spectral density function as

Item 1 above is illustrated in figure 5. In this figure, an average age waveform obtained from a network with 60% load is shown at four different time scales. In the lowest panel, is plotted against the number of token passes. In subsequent graphs, the averaged over 10, 100 and 1000 passes respectively has been plotted. Note that, contrary to what would be expected if were white noise-like, the aggregated waveforms become more bursty as the level of aggregation increases.

  
Figure 5: Increasing burstiness with increasing aggregation in an M = 100 network with 60% load. The average packet age is plotted versus time, on four different time scales. There are 1000 points in each plot. In the bottom graph, is plotted against n; in the second to bottom graph, 10 successive values of are averaged to give a single point, and so on. Counterintuitively, the graphs become more `bursty' as averaging over longer time periods takes place.