School Seminars and Colloquia

The asymptotic variance of departures from critically loaded queues

Centre of Excellence for Mathematics and Statistics of Complex Systems Seminar

by Yoni Nazarathy


Institution: EURANDOM, Technische Universiteit Eindhoven
Date: Fri 30th July 2010
Time: 3:15 PM
Location: Old Geology Theatre 1, University of Melbourne

Abstract: We study the departure process of the GI/G/1 queue and focus on the variance of the number of departures. This is typically a function which grows at an asymptotic linear rate with respect to time. Precise knowledge of this rate, helps obtain approximations for the distribution of departures over long intervals. It is quite easy to see that when the parameters of the system are such that the queue is under loaded, i.e. the inter-arrival times are longer than the service durations, the asymptotic variance equals that of the arrival process. Similarly, when the queue is overloaded, the asymptotic variance equals that of the service process.
Our focus is on the critically loaded case where a complete different behavior occurs: We find that the asymptotic variance is (1-2/pi) multiplied by the sum of the asymptotic variances of the arrival and service processes.

Our result originates from a classic heavy traffic limit theorem of the departure process, yet in proving this result there are several non-trivial challenges related to establishing uniform integrabillity conditions. The talk aims to focus on these aspects.

Joint work with Ahmad Al-Hanbali, Michel Mandjes and Ward Whitt.

For More Information: contact: Nathan Clisby. email n.clisby@ms.unimelb.edu.au