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Sharp results on convergence rates for the distribution of GI/M/1/K queues as K tends to infinity

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Bong Dae Choi  and
Bara Kim
Bong Dae Choi*
Affiliation:
Korea University
Bara Kim*
Affiliation:
Korea University
*
∗ Postal address: Department of Mathematics, Korea University, 1, Anam-dong, Sungbuk-ku, Seoul, 136–701, Korea.
∗ Postal address: Department of Mathematics, Korea University, 1, Anam-dong, Sungbuk-ku, Seoul, 136–701, Korea.
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Abstract

In this paper, we investigate how fast the stationary distribution π(K) of an embedded Markov chain (time-stationary distribution q(K) of the GI/M/1/K queue converges to the stationary distribution π of the embedded Markov chain (time-stationary distribution q) of the GI/M/1 queue as K tends to infinity. Simonot (1997) proved certain equalities. We obtain sharper results than these by finding limit values limK→∞σ-K||π(K) - π|| and limK→∞σ-K||q(K) - q|| explicitly.

Keywords

GI/M/1/K queuedual sequencestationary measurestationary distributionconvergence rate

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1010 - 1019
Copyright
Copyright © by the Applied Probability Trust 2000 

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Footnotes

This work was supported in part by a research program from KOSEF (98-0101-02-01-3).

References

Asmussen, S. (1987). Applied Probability and Queues. John Wiley, New York.Google Scholar
Choi, B. D., Kim, B., and Wee, I.-S. (2000). Asymptotic behavior of loss probability in GI /M /1 /K queue as K tends to infinity. To appear in Queueing Systems.Google Scholar
Choi, B. D., Kim, B., Kim, J., and Wee, I.-S. (2000). Exact convergence rate for the distributions of GI /M /c /K queue as K tends to infinity. Submitted.Google Scholar
Miyazawa, M. (1990). Complementary generating functions for the M X/GI/1/k and GI/MY/1/k queues and their application to the comparison of loss probabilities J. Appl. Prob. 27, 684–692.CrossRefGoogle Scholar
Rudin, W. (1987). Real and Complex Analysis, 3rd edn. McGraw-Hill, New York.Google Scholar
Shanbhag, D. N. (1966). On a duality principle in the theory of queues. Operat. Res. 14, 947–949.Google Scholar
Simonot, F. (1997) Convergence rate for the distributions of GI /M /1 /n and M /GI /1 /n as n tends to infinity. J. Appl. Prob. 34, 1049–1060.Google Scholar
Simonot, F. (1998). A comparison of the stationary distribution of GI /M /c /n and GI /M /c. J. Appl. Prob. 35, 510–515.CrossRefGoogle Scholar