Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T21:02:51.882Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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