Parameter estimation of geometrically sampled fractional Brownian traffic

The parameter estimation of a traffic model based on the fractional Brownian motion (FBM) is studied. The model has three parameters: the mean rate m, variance parameter a and the Hurst parameter H. Explicit expressions for the maximum likelihood (ML) estimates m/spl circ/ and a/spl circ/ in terms of H are given, as well as the expression for the log likelihood function from which the estimate H/spl circ/ is obtained as the maximizing argument. A geometric sequence of sampling points, t/sub i/=/spl alpha//sup i/, is introduced, which fits neatly into the self-similar property of the process and also reduces the number of samples needed to cover several time scales. It is shown that by a proper 'descaling' the traffic process is stationary on this grid leading to a Toeplitz-type covariance matrix. Approximations for the inverted covariance matrix and its determinant are introduced. The accuracy of the estimations is studied by simulations. Comparisons with estimates obtained with linear sampling and with the wavelet-based A-V estimator show that the geometrical sampling indeed improves the accuracy of the estimate H/spl circ/ with a given number of samples.