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Time-changing and truncating K-capacity queues from one K to another

Published online by Cambridge University Press:  14 July 2016

Paul Glasserman  and
Wei-Bo Gong
Paul Glasserman
Affiliation:
AT&T Bell Laboratories
Wei-Bo Gong*
Affiliation:
University of Massachusetts, Amherst
*
∗∗ Postal address: Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, USA.
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Abstract

For , we obtain a K′- capacity queue from a K- capacity queue through a random time change and a truncation, provided arrivals are Poisson or service is exponential. In the case of an M/G/1/K queue, the time change erases service intervals that begin with more than K′ customers in the systems. This construction yields a straightforward sample path proof of Keilson's result on the proportionality of the ergodic queue length probabilities in M/G/1/K queues. The same approach proves a strengthened result for ‘detailed' state probabilities. It also reproduces a proportionality result for a vacation model, due to Keilson and Servi. A ‘dual' construction yields a different kind of proportionality for the G/M/1/K queue.

Keywords

SAMPLE PATH ANALYSISM/G/1/K QUEUEERGODIC QUEUE-LENGTH DISTRIBUTIONRANDOM TIME-CHANGE
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 3 , September 1991 , pp. 647 - 655
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

∗

Present address: Graduate School of Business, Columbia University, 403 Uris Hall, New York, NY 10027, USA.

References

[1] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New York.Google Scholar
[2] Baccelli, F. and Massey, W. A. (1989) Sample path analysis of the M/M/1 queue. J. Appl. Prob. 26, 418–422.Google Scholar
[3] Boxma, O. J. (1983) The cyclic queue with one general and one exponential server. Adv. Appl. Prob. 15, 857–873.Google Scholar
[4] Cassandras, C. G. and Strickland, S. G. (1989) On-line sensitivity analysis of Markov chains. IEEE Trans. Autom. Control 34, 76–86.Google Scholar
[5] Cohen, J. W. (1970) On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin.Google Scholar
[6] Doshi, B, (1985) A note on stochastic decomposition in a GI/G/1 queue with vacations or setup times. J. Appl. Prob. 22, 419–428.Google Scholar
[7] Ho, Y. C. and Li, S. (1988) Extensions of infinitesimal perturbation analysis. IEEE Trans. Autom. Control 33, 427–438.Google Scholar
[8] Keilson, J. (1966) The ergodic queue length distribution for queueing systems with finite capacity. J. R. Statist. Soc. B 28, 190–201.Google Scholar
[9] Keilson, J. and Servi, L. D. (1989) Blocking probability for M/G/1 vacation systems with occupancy level dependent schedules. Operat. Res. 37, 134–140.Google Scholar
[10] Neuts, M. F. (1981) Matrix Geometric Solutions in Stochastic Models. The Johns Hopkins University Press, Baltimore, MD.Google Scholar
[11] Whitt, W. (1980) Continuity of generalized semi-Markov processes. Math. Operat. Res. 5, 494–501.Google Scholar