Kolmogorov–Wiener Filter Weight Function for Stationary Traffic Forecasting: Polynomial and Trigonometric Solutions

Abstract

The paper is devoted to obtaining the Kolmogorov–Wiener filter weight function for stationary telecommunication traffic forecasting. The problem of telecommunication traffic forecasting is an urgent problem for telecommunications. There are a variety of traffic models and approaches to traffic forecasting, and in rather simple models traffic is considered to be a stationary random process. The Kolmogorov–Wiener filter may be used for the forecasting of stationary random processes, so it may be used for stationary traffic forecasting. However, the corresponding approach has not been sufficiently developed in the literature. Two models of stationary traffic are considered in the paper: a model where traffic is considered to be a process with a power-law structure function and a model where traffic is considered to be fractional Gaussian noise. In the case of a large amount of data, traffic may be described as a continuous random process, and the Kolmogorov–Wiener filter weight function is a solution to the corresponding Wiener–Hopf integral equation. An approximate solution multiply to the corresponding equation may be sought in the form of a truncated expansion in an orthogonal functional series. The paper contains a description of the authors’ previous results obtained on the basis of a truncated polynomial expansion and new results obtained on the basis of an expansion in trigonometric functions.

The aim of the work is to obtain an approximate solution to the Kolmogorov–Wiener filter weight function for stationary traffic forecasting in the framework of the model of a power-law structure function and the model of fractional Gaussian noise and to investigate the validity of the obtained results.

The object of the work is the Kolmogorov–Wiener filter used for the stationary telecommunication traffic forecasting.

The subject of the work is the weight function of the filter under consideration.

The scientific novelty of the work is the fact that for the first time the corresponding weight function is sought as a truncated trigonometric Fourier series. Trigonometric and polynomial solutions are compared. It is shown that trigonometric solutions are better than the polynomial ones for the model where the telecommunication traffic is considered as a process with a power-law structure function.