Window based estimation of the intensity of a doubly stochastic Poisson process

Tasks related to advanced management of telecommunications networks, such as optimal dynamic routing or dynamic allocation of bandwidth for virtual paths in an ATM network, call for a method to estimate the current call arrival intensities. Such an estimate must be based on the observed history of the call arrival instants. In this paper, we study window type estimators with the purpose of giving a clear idea of their accuracy. In particular, the simple straight case and exponential windows are considered and applied to an example where the correlation function of the intensity process is of the exponential type - a system for which the exponential window, in fact, is the optimal window. It turns out that the accuracy is controlled by a single dimensionless traffic parameter. The achievable accuracy is little affected by the shape of the window function. It also turns out that a Bayesian approach, applied in an earlier study, gives only a slight improvement over the window based estimator. More important instead is the proper choice of the size of the window. In order to find the optimal size, one has to know some basic characteristics of the intensity process. We briefly address the problem how these can be inferred from the long term statistics of the counting process.