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Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Hans Daduna ,
Christian Malchin  and
Ryszard Szekli
Hans Daduna*
Affiliation:
Hamburg University
Christian Malchin*
Affiliation:
Hamburg University
Ryszard Szekli*
Affiliation:
Wrocław University
*
∗Postal address: Department of Mathematics, Hamburg University, Bundesstrasse 55, 20146 Hamburg, Germany.
∗Postal address: Department of Mathematics, Hamburg University, Bundesstrasse 55, 20146 Hamburg, Germany.
∗∗∗Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
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Abstract

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We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.

Keywords

Joint sojourn time distributioncycle timescustomer stationary behaviourbottleneck nodecentral limit theorem

MSC classification

Secondary: 60K25: Queueing theory 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 333 - 346
Copyright
Copyright © Applied Probability Trust 2008 

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