Multifractal and Gaussian Fractional-Sum-Difference Models for Internet Traffic

A multifractal fractional sum-difference model (MFSD) is a monotone transformation of a Gaussian fractional sum-difference model (GFSD), the Gaussian image of the MFSD. The GFSD is the sum of two independent components: a moving sum of length two of discrete fractional Gaussian noise (fGn), and white noise. Internet packet traffic interarrival time are very well modeled by an MFSD in which the marginal distribution is Weibull; this is validated by extensive model checking for 715,665,213 measured arrival times on 3 Internet links. The simplicity of the model provides mathematical tractability. Mathematical investigations of many traffic statistics result in much insight through exact or approximate derivations. The most important insight is that the statistical properties depend fundamentally on how the relative variances of the fGn and white noise components change with changing factors such as the traffic rate and time aggregation of the traffic. The applicability of the fundamentals to the MFSD is enabled by the surprising discovery that while the transformation from the GFSD to the MFSD creates a highly nonlinear process, the form of the second moments is preserved to a very good approximation. The MFSD can be used to generate synthetic traffic for network simulation in which only the traffic rate needs to be specified.

This is a joint work with David Anderson and William S. Cleveland.