Multifractal and Gaussian fractional sum–difference models for Internet traffic

Abstract

A multifractal fractional sum–difference model (MFSD) is a monotone transformation of a Gaussian fractional sum–difference model (GFSD). 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 traffic packet interarrival times 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 three Internet links. The simplicity of the model provides a mathematical tractability that results in a foundation for understanding the statistical properties of the arrival process. The current foundation is time scaling: properties of aggregate arrivals in successive equal-length time intervals and how the properties change with the interval length. This scaling is also the basis for the widely discussed multifractal wavelet models. The MFSD provides a more fundamental foundation that is based on how changes in the fGn and white noise components result in changes in the arrival process as various factors change such as the aggregation time length or the traffic packet rate. Logistic models relate the MFSD model parameters to the packet rate, so only the rate needs to be specified in using the MFSD model to generate synthetic packet arrivals for network engineering simulation studies.