Estimation of the renewal function by empirical data-a Bayesian approach

Considering traffic measurements in advanced telecommunication networks. We study some basic estimation issues using nonparametric techniques. Using only a very limited number of independent observations of the interarrival times between the recorded events of interest, we propose a simple histogram-type estimate to restore the renewal function of an underlying renewal process with its unknown interarrival time distribution. Due to the limited number of the observed interarrival times the estimate is accurate just for closed, but sufficiently wide observation intervals. To provide the minimum of the mean-squared error for different closed time intervals of the estimation, the optimal number of terms of the histogram-type estimate is selected by a Bayesian principle. Furthermore, the effectiveness of the estimate is evaluated by a simulation study for several light- and heavy-tailed interarrival-time distributions including the exponential, Gamma and Weibull distribution with a shape parameter less than one.