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On Finite Moments of Full Busy Periods of GI/G/c Queues

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Saeed Ghahramani  and
Ronald W. Wolff
Saeed Ghahramani*
Affiliation:
Western New England College
Ronald W. Wolff*
Affiliation:
University of California, Berkeley
*
∗Postal address: School of Arts and Sciences, Western New England College, Springfield, MA 01119-2684, USA.
∗∗Postal address: Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, USA. Email address: wolff@ieor.berkeley.edu
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Abstract

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For a GI/G/c queue, a full busy period is an interval that begins when an arrival finds c − 1 customers in the system, and ends when, for the first time after that, a departure leaves behind c − 1 customers in the system. We present a probabilistic proof of conditions for full busy periods to have finite moments. For queues that empty, this result may be deduced from results in the literature, but our proof is much easier. For queues that do not empty, our proof still applies, and this result is new.

Keywords

Full busy periodremaining busy periodfast single-serverequilibrium distribution

MSC classification

Secondary: 60K25: Queueing theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 275 - 278
Copyright
© Applied Probability Trust 2005 

References

Brumelle, S. L. (1971). Some inequalities for parallel-server queues. Operat. Res. 19, 402–413.CrossRefGoogle Scholar
Ghahramani, S. (1989). Finiteness of moments of virtual work for GI/G/c queues. J. Appl. Prob. 26, 423–425.CrossRefGoogle Scholar
Ghahramani, S. and Wolff, R. W. (1989). A new proof of finite moment conditions for GI/G/1 busy periods. Queueing Systems 4, 171–178.CrossRefGoogle Scholar
Kiefer, J. and Wolfowitz, J. (1955). On the theory of queues with many servers. Trans. Amer. Math. Soc. 78, 1–18.CrossRefGoogle Scholar
Kiefer, J. and Wolfowitz, J. (1956). On the characteristics of the general queueing process with applications to random walks. Ann. Math. Statist. 27, 147–161.CrossRefGoogle Scholar
Scheller-Wolf, A. and Sigman, K. (1997). Delay moments for FIFO GI/GI/s queues. Queueing Systems 25, 77–95.CrossRefGoogle Scholar
Sigman, K. (1988). Regeneration in tandem queues with multi-server stations. J. Appl. Prob. 25, 391–403.CrossRefGoogle Scholar
Sigman, K. (1990). One-dependent regenerative processes and queues in continuous time. Math. Operat. Res. 15, 175–189.CrossRefGoogle Scholar
Thorisson, H. (1985). The queue GI/G/k: finite moments of the cycle variables and uniform rates of convergence. Commun. Statist. Stoch. Models 1, 221–238.CrossRefGoogle Scholar
Whitt, W. (1972). Embedded renewal processes in the GI/G/s queues. J. Appl. Prob. 9, 650–658.CrossRefGoogle Scholar
Wolff, R. W. (1984). Conditions for finite ladder height and delay moments. Operat. Res. 32, 906–916.CrossRefGoogle Scholar
Wolff, R. W. (1988). Sample-path derivations of the excess, age, and spread distributions. J. Appl. Prob. 25, 432–436.CrossRefGoogle Scholar