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

Abstract

Modern information and telecommunication, transportation and logistic, economic and financial systems are represented by complex networks exhibiting traffic flows with spatio-temporal long-term persistence. Conventional queuing theory relies largely upon stationary models where traffic flows are assumed independent and are typically characterized by the first two moments of inter-arrival and service time distributions, leading to drastic underestimations of traffic flow delays. Here we extend a recent superstatistical approach focusing on traffic models with variable arrival rates by accounting for interdependent activity patterns on multiple network nodes. We suggest an analytical correction to the conventional stationary queue model given by the Kingman’s formula based on the calculation of aggregated inter-arrival times variability from the variabilities of arrival rates at individual nodes and cross-correlations between them. We confirm our analytical approximations by comparing with computer simulation results and large-batch empirical traffic analysis from the backbone of a major academic network. We believe that our results, in combination with recent data on the effects of long-term temporal persistence in network traffic flow, are applicable to various complex networks not limited to information and telecommunication, transportation, and logistics but also to economics and finance, rainfall and river flow dynamics, water accumulation in reservoirs, and many other research domains exhibiting spatio-temporal interdependence patterns.