There is an extensive study showing that most network traffic flows [i.e., generated by http, ftp, variable-bit-rate (VBR) video applications, etc.] can be characterized by self-similarity and long-range dependence (LRD) (see [WTE96] for an extensive reference list).

Figure A.1 illustrates the scaling behavior of LRD traffic in comparison with that of a short-range dependent (SRD) traffic such as the one based on the Poisson process.

Figure A.1: Scaling behavior of LRD and SRD traffic flows.

Consider a cumulative process Y( t) with stationary increments, and let X _t be its incremental process:

(A.1)

For example, Y( t) can represent the number of bytes arriving up to time t, and X _t can represent the number of bytes arriving in 1 unit of time.

The proces is an aggregated process of X _t if

(A.2)

Process X _t is said to be self-similar if X _t is indistinguishable from . This is a very restrictive definition, especially considering the stochastic nature of the network traffic. Usually, second-order self-similarity is considered for the purposes of traffic description: autocovariance functions of the original and aggregated processes should have the same values.

Let

(A.3)

and

(A.4)

Then the process X _t is exactly second-order self-similar if

(A.5)

and asymptotically second-order self-similar if

(A.6)

A convenient measure of a process s distributional self-similarity is its Hurst parameter H. A process is self-similar with parameter H (0 < H