Mean sojourn times in Markov queueing networks: Little´s formula revisited

It is commonly supposed that,L=lambdaWapplies to "almost any" queueing system withlambdasome average customer entrance rate,Lthe asymptotic expectation or time average of the number of customers in the system andWsome average of the sojourn time. This formula is studied for irreducible positive recurrent Markov queueing systems whose state vectorZconsists of entries representing queue lengths at the respective service stations; blocking, finite capacities, batch arrivals, and variable rates of arrival and service are consistent withZ. Sojourn times are defined inan augmented Markov modelY=(Z,U), where the customer marking processUdescribes the service discipline in sufficiently general terms to include most possibilities of interest. It is shown thatL=lambda Wis universally applicable, if properly interpreted to take account of state-varying entrance rates, batch arrivals, and multiple customer classes.L,lambda, andWmay each be equivalently viewed as time averages, means over a regeneration cycle, or expectations with respect to the asymptotic probability structure ofZ. Indeterminate forms ofL=lambda Ware possible within the scope of Markov queueing networks (MQN); as is shown by some examples, these may take the forminfty times 0forlambda W, or ofinfty= inftyforL=lambda W.