The paper defines self-similarity for vector processes by employing the discrete-time continuous-dilation operation which has successfully been used previously by the authors to define 1-D discrete-time stochastic self-similar processes. To define self-similarity of vector processes, it is required to consider the cross-correlation functions between different 1-D processes as well as the autocorrelation function of each constituent 1-D process in it. System models to synthesize self-similar vector processes are constructed based on the definition. With these systems, it is possible to generate self-similar vector processes from white noise inputs. An important aspect of the proposed models is that they can be used to synthesize various types of self-similar vector processes by choosing proper parameters. Additionally, the paper presents evidence of vector self-similarity in two-channel wireless LAN data and applies the aforementioned systems to simulate the corresponding network traffic traces.
The authors have previously studied simulation of network traffic traces using a fractional pole-zero system derived from a discrete-time dilation or scaling operation based on warping transforms that convert discrete-time frequency to a continuous-time frequency. However, such a model can be used to analyze and simulate only stationary discrete-time self- similar signals. This paper propounds a more general model for discrete-time linear scale-invariant (DLSI) systems built on linear kernel approaches. It is shown that this model can synthesize self-similar processes for simulating non-stationary network traffic.
Discrete-time linear systems that possess scale-invariance properties even in the presence of continuous dilation were proposed by Zhao and Rao. The paper presents results of subsequent investigation characterizing self-similarity properties of discrete-time signals synthesized by these systems. It is shown that white noise inputs to these linear scale invariant systems produce self-similar outputs regardless of the marginal distribution of the noise. We investigate this with different types of inputs and in most instances the outputs are fractional Gaussian and self-similar. This is confirmed by generating the fractional Gaussian noise from the fractional Brownian motion and comparing its characteristics with the system output. For heavy tailed input distributions, the output is also heavy-tailed and self-similar. It is also shown that it is possible to synthesize statistically self-similar signals whose self-similarity parameters are consistent with those observed in network traffic.