Maximum Likelihood Estimation of the Parameters of Fractional Brownian Traffic with Geometrical Sampling

Traffic model based on the fractional Brownian motion (fBm) contains three parameters: the mean rate , variance parameter and the Hurst parameter . The estimation of these parameters by the maximum likelihood (ML) method is studied. Explicit expressions for the ML estimates and  in terms of are given, as well as the expression for the log-likelihood function from which the estimate  is obtained as the maximizing argument. A geometric sequence of sampling points, , is introduced in order to see the scaling behaviour of the traffic with fewer samples. 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 estimation algorithm is studied by simulations. Comparisons with corresponding estimates obtained with linear grid show that the geometrical sampling indeed improves the accuracy of the estimate  with a given number of samples.