Asymptotic Expansions and Integral Representations of Moments of Queue Lengths in Closed Markovian Networks

Abstract

In this paper a method for calculating the first and higher order moments of the queue lengths for each customer class at each node in a dosed Markovian queuing network is presented. The method is based on asymptotic expansions in powers of N -~ for the moments of interest. N is a parameter that reflects the size of the network. The derivation of the asymptotic expansions presented here is based on the techniques developed by us earlier to get asymptotic expansions in powers of N -a of the mean utilization of each processor by each class of customers. These expansions are valid in the "normal usage" case in which none of the processors are too heavily utilized. These expansions are particularly useful m the ease of large networks with many classes of customers, each class having many customers. With these expansions, we are able to analyze very large networks that earlier were computationally intractable. This is important, because dosed Markovian queuing networks have emerged as one of the most important tools for modeling computer systems, computer communication systems, on-line computer networks and other real-time computer-based system. Previous to the work reported here, their use had been restricted to relatively small networks since their use in large networks involved intractably large calculations.