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The backlog and depletion-time process for M/G/1 vacation models with exhaustive service discipline

Published online by Cambridge University Press:  14 July 2016

Julian Keilson  and
Ravi Ramaswamy
Julian Keilson*
Affiliation:
Massachusetts Institute of Technology
Ravi Ramaswamy*
Affiliation:
BGS Systems Inc.
*
∗Postal address: The Sloane School, MIT, 50 Memorial Drive, Cambridge, MA 02193, USA.
∗∗Postal address: BGS Systems Inc., 128 Technology Center, Waltham, MA 02254–9111, USA. This work was completed when both authors were at the University of Rochester.
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Abstract

The vacation model studied is an M/G/1 queueing system in which the server attends iteratively to ‘secondary' or ‘vacation' tasks at ‘primary' service completion epochs, when the primary queue is exhausted. The time-dependent and steady-state distributions of the backlog process [6] are obtained via their Laplace transforms. With this as a stepping stone, the ergodic distribution of the depletion time [5] is obtained. Two decomposition theorems that are somewhat different in character from those available in the literature [2] are demonstrated. State space methods and simple renewal-theoretic tools are employed.

Keywords

BACKLOG PROCESSDEPLETION TIMESSTATE-SPACE METHODSRENEWAL THEORY
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 2 , June 1988 , pp. 404 - 412
Copyright
Copyright © Applied Probability Trust 1988 

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References

[1] Cohen, J. W. (1969) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[2] Doshi, B. (1986) Queueing models with vacation — A survey. Queueing Systems 1, 29–66.CrossRefGoogle Scholar
[3] Keilson, J. (1963) The first passage time density for homogeneous skip-free walks on the continuum. Ann. Math. Statist. 24, 375–380.Google Scholar
[4] Keilson, J. and Servi, L. D. (1987) The dynamics of the M/G/1 vacation model. Operat. Res. To appear.Google Scholar
[5] Keilson, J. and Sumita, U. (1983) The depletion time for M/G/1 systems and a related limit theorem. Adv. Appl. Prob. 15, 420–443.Google Scholar
[6] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar