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On the settling time of the congested GI/G/1 queue

Published online by Cambridge University Press:  01 July 2016

George D. Stamoulis  and
John N. Tsitsiklis
George D. Stamoulis*
Affiliation:
Massachusetts Institute of Technology
John N. Tsitsiklis*
Affiliation:
Massachusetts Institute of Technology
*
∗Postal address for both authors: Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
∗Postal address for both authors: Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
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Abstract

We analyze a stable GI/G/1 queue that starts operating at time t = 0 with N0 ≠ 0 customers. First, we analyze the time required for this queue to empty for the first time. Under the assumption that both the interarrival and the service time distributions are of the exponential type, we prove that , where λ and μ are the arrival and the service rates. Furthermore, assuming in addition that the interarrival time distribution is of the non-lattice type, we show that the settling time of the queue is essentially equal to N0/(μ –λ); that is, we prove that where is the total variation distance between the distribution of the number of customers in the system at time t and its steady-state distribution. Finally, we show that there is a similarity between the queue we analyze and a simple fluid model.

Keywords

CONGESTED QUEUESETTLING TIMETRANSIENT ANALYSIS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 4 , December 1990 , pp. 929 - 956
Copyright
Copyright © Applied Probability Trust 1990 

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Footnotes

Research supported by the NSF under Grant ECS-8552419, with matching funds from Bellcore Inc. and Du Pont, and the ARO under Grant DAAL03-86-K-0171.

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