How heavy-tailed distributions affect simulation-generated time averages

For statistical inference based on telecommunications network simulation, we examine the effect of a heavy-tailed file-size distribution whose corresponding density follows an inverse power law with exponent α + 1, where the shape parameter α is strictly between 1 and 2. Representing the session-initiation and file-transmission processes as an infinite-server queueing system with Poisson arrivals, we derive the transient conditional mean and covariance function that describes the number of active sessions as well as the steady-state counterparts of these moments. Assuming the file size (service time) for each session follows the Lomax distribution, we show that the variance of the sample mean for the time-averaged number of active sessions tends to zero as the power of 1 − α of the simulation run length. Therefore, impractically large sample-path lengths are required to achieve point estimators with acceptable levels of statistical accuracy. This study compares the accuracy of point estimators based on the Lomax distribution with those for lognormal and Weibull file-size distributions whose parameters are determined by matching their means and a selected extreme quantile with those of the Lomax. Both alternatives require shorter run lengths than the Lomax to achieve a given level of accuracy. Although the lognormal requires longer sample paths than the Weibull, it better approximates the Lomax and leads to practicable run lengths in almost all scenarios.