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LOWER BOUNDS FOR LRD/GI/1 QUEUES WITH SUBEXPONENTIAL SERVICE TIMES

Published online by Cambridge University Press:  22 January 2004

Cathy H. Xia ,
Zhen Liu ,
Mark S. Squillante  and
Li Zhang
Cathy H. Xia
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, E-mail: [email protected]
Zhen Liu
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, E-mail: [email protected]
Mark S. Squillante
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, E-mail: [email protected]
Li Zhang
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, E-mail: [email protected]
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Abstract

We investigate the tail distribution of the virtual waiting times in a LRD/GI/1 queue where the arrival process is long-range dependent (LRD) and the service times are independent and identically distributed (i.i.d.) random variables. We present two lower bounds on the stationary waiting time tail asymptotics, which illustrate the different dominating components that influence server performance under various conditions. In particular, we show that the tail distribution of the stationary waiting time is bounded below by that of the associated LRD/D/1 queues resulting from replacing all random service times by the mean. This shows the performance impact purely due to the long-range dependency of the arrival process. On the other hand, when the service times are subexponential, we show that the tail distribution of the stationary waiting time is bounded below by that of the corresponding D/GI/1 queue by replacing the dependent arrival process with its associated independent version. This shows the minimum performance impact due to the tail distribution of the service times. The above two lower bounds indicate that the performance of LRD/GI/1 queues will be dominated by the heavier tail of the corresponding LRD/D/1 and D/GI/1 queues. These features are further illustrated and quantified through examples and via numerous simulation experiments.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 1 , January 2004 , pp. 87 - 101
Copyright
© 2004 Cambridge University Press

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