Approximate waiting times for queuing systems with variable long-term correlated arrival rates

Abstract

We consider waiting times in queuing systems with variable arrival rates in the presence of long-term correlations and periodic trends. We focus on a simplified model where the contributions of periodic and stochastic components could be analyzed separately, leading to queue lengths exhibiting periodic and stochastic resetting, respectively, with their effects summarized additively. We provide an approximate analytical solution that is based on the universal scaling of return interval statistics between level crossing events in long-term correlated data series. The accuracy of our results is validated explicitly by computer modeling, using both simulated data series and empirical traffic data from a network cluster hosting the World Cup ’98 web services characterized by extremely variable traffic intensity. We believe that the proposed approach could be useful to characterize the impact of long-term correlations and periodic trends in various complex systems, with prominent examples ranging from information, communication, logistic, transportation networks to climate, hydrological, as well as other natural, social and engineering systems.