Self-Similarity

Self-Similarity and fractal are notions pioneered by Benoit B. Mandelbrot [7], who describe the phenomenon where a certain property of an object, such as a natural image, is preserved with respect to scaling in space or in time. The empirical studies in the early 1990s at Bell core that resulted in the finding that Ethernet LAN traffic is self-similar or fractal in nature serves as a reminder that new discoveries do not necessarily require new mathematical concepts, or novel statistical methodologies, or for that case, new networking technologies. Instead, the aspect of discovery often lies in applying a well-known mathematical concept (e.g., self-similar processes) in a new context (e.g., networking) [2]. Network traffic that is bursty on many or all timescales can be described statistically using the notion of self-similarity, that is, the traffic relational structure remains unchanged at varying timescales.

There are a number of different, not equivalent, definitions of self-similarity. The standard one states that a continuous-time process $Y = {Y (t), t \geq 0}$ is self-similar (with self-similarity parameter H) if it satisfies the condition:

 

$$Y(t) \stackrel{d}{=} a ^{-H}Y{(at)}, \forall t \geq 0, \forall a\gt,0<H<1$$ (1)

where the equality is in the sense of finite-dimensional distributions. While a process Y satisfying 1 can never be stationary (stationary requires $Y (t)\stackrel{d}{=} Y (at)$), Y is typically assumed to have stationary increments.

A second definition of self­similarity, more appropriate in the context of standard time series theory, involves a stationary sequence $X = {X(i), i\geq 1}$. Let

 

$$X^{(m)}(k) =\frac{1}{m} \sum_{i=(k-1)m+1}^{km}X(i), k=1,2,\ldots$$ (2)

be the corresponding aggregatd sequence with level of aggregation m, obtained by dividing the orignial series X into non-overlapping blocks of size m and averaging over each block. The index k labels the block. If X is the increment process of a self-similar process Y defined in 1, that is, X(i) = Y (i+1) -Y (i), then for all integers m,

 

A stationary sequence $X = {X(i), i\geq 1}$ is called exactly self-similar if it satisfies 3 for all aggregation levels m. This second definition of self-similarity is closely related to the first, with $mX^{(m)}(\cdot)$ corresponding to $Y (a\cdot)$.

A stationary sequence $X(i), i \geq 1$ is said to be asymptotically self-similar if 3 holds as $m \rightarrow \infty$. Similarly, we call a covariance-stationary sequence $X(i), i \geq 1$ exactly second-order self-similar or asymptotically second-order self-similar if m^1-HX^(m) has the same variance and autocorrelation as X, for all m, or as $m \rightarrow \infty$.