A Light-Traffic Approximation for Single-Class Departure Processes from Multi-Class Queues

Abstract

This paper discusses an approximation for single-class departure processes from multi-class queues: If the arrival rate of one class upon one visit to the queue is a small proportion of the total arrival rate there, then the departure process for that class from that visit should be nearly the same as the arrival process for that class for that visit. This can be regarded as a light-traffic approximation, but only the one class must be in light traffic; the overall traffic intensity of the queue need not be low. As a consequence, in a queueing network if the routing for one class is deterministic, and if the light-traffic condition applies at every queue this class visits, then the arrival and departure processes for this class at each visit to each queue should be nearly the same as its external arrival process. This approximation is explained in terms of different time scales, and is justified here by a limit theorem in a special case. There are important implications for parametric decomposition approximation techniques: the variability parameter partially characterizing the departure process at any visit to any queue of such a low-intensity class should be nearly the same as the variability parameter partially characterizing the arrival process for that class at that visit to that queue. The approximation principle in this form was recently proposed by G. Bitran and D. Tirupati while developing improved parametric-decomposition approximations for low-variability multi-class queueing networks with deterministic routing, which have important, applications in manufacturing. The approximation principle also has important implications for data networks showing how burstiness in originated traffic can pass through heavily shared network facilities where it has relatively little effect and then reappear at the destination.