Insensitivity of Queueing Networks

Abstract

A queueing process is said to be insensitive if the distribution of the number of jobs in the system depends on the service-time distribution only through its mean. In Chapter 4 we give a sample-path proof of the insensitivity of a batch- arrival G/G/1-LCFS-PR queue, in which the batch sizes and service times are allowed to be state dependent (see also Stidham and El-Taha [182]). In this chapter we present a unified approach to proving insensitivity of symmetric queues in discrete-time using the method of time reversal. We use discrete- time sample-path analysis to show that the asymptotic frequency distribution of the number of jobs in infinite-server, Erlang loss, and round-robin models, is insensitive to the asymptotic distribution of service times under weak assumptions. For the infinite-server model, our assumptions allow batch arrivals and permit batch sizes and service times to be dependent. We show that insensitivity holds if the frequency distribution of batch sizes solves a system of equations. A solution to this set of equations occurs if the batch size of arrivals at each time unit follows a Poisson distribution. Similar results hold for the Loss model with constant batch sizes and deterministic service requirements, and for the round-robin queue.