Sample-Path Insensitivity of Symmetric Queues in Discrete-Time

Muhammad El-Taha, Department of Mathematics and Statistics, University of Southern Maine

Shaler Stidham, Jr., Department of Operations Research, University of North Carolina

Ravinder Anand, Department of Statistics/Computer & Information Systems, The George Washington University

Abstract

We present a unified approach to proving insensitivity of three symmetric queues in discrete-time using the method of time reversal. A discrete-time sample-path approach is used 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. The insensitivity results are shown to extend to other symmetric queues, namely the Erlang loss queue and the modified RR model.