In this paper, we revisit the estimation of the size distribution of packet flows in Internet traffic through an inversion approach for several packet sampling schemes which are based on probabilistic sampling (PS). We first study the statistical properties of the previously introduced inversion estimator in its general form and make connections to the singular value decomposition. This motivates the use of a regularization technique in the estimation of the flow size distribution. More specifically, a penalized weighted least square approach is proposed. We compare theoretically the penalized estimator under simplified assumptions against the (non-penalized) inversion approach in order to explain differences in their statistical behaviors. A data study with two real traces shows that the proposed penalized estimator outperforms the inversion estimator for all sampling schemes, corroborating the theoretical analysis. This work reveals that the simplest sampling schemes based on PS, that do not work with small sampling probabilities under the inversion approach, can be used with the penalized approach. Furthermore, the penalized approach allows considering smaller packet sampling rates for all the other sampling schemes.

Traffic monitoring and estimation of flow characteristics, such as the size and duration distributions, can be problematic when the length of an observation window is constrained (e.g., due to hard network resources). Indeed, as shown in this work, sampled flows are usually affected by censoring in an observation window, which leads to biased estimators. To account for censoring, a mathematical framework that describes sampling of flows in a time window is developed. Using censoring analysis, we provide nonparametric maximum likelihood estimators for the flow duration and size distributions. The estimators are computed using the EM algorithm. Finally, the estimators are applied to an actual traffic trace, and are found to perform very well.

Multifractal random walks are defined as integrals of infinitely divisible stationary multifractal cascades with respect to fractional Brownian motion. Their key properties are studied, such as finiteness of moments and scaling, with respect to the chosen values of the self-similarity and infinite divisibility parameters. The range of these parameters is larger than that considered previously in the literature, and the cases of both exact and nonexact scale invariance are considered. Special attention is paid to various types of definitions of multifractal random walks. The resulting random walks are of interest in modeling multifractal processes whose marginals exhibit stationarity and symmetry.