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: mean rate m, variance parameter a and the Hurst parameter H . Explicit expressions for the maximum likelihood (ML) estimates m̂ and â in terms of H are given, as well as the expression for the loglikelihood function from which the estimate Ĥ is obtained as the maximizing argument. A geometric sequence of sampling points, ti = i , is introduced, which fits neatly to 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 Ĥ with a given number of samples.