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The convexity of the mean queue size of the M/M/c queue with respect to the traffic intensity

Published online by Cambridge University Press:  14 July 2016

W. Grassmann
W. Grassmann*
Affiliation:
University of Saskatchewan
*
∗ Postal address: Department of Computational Science. University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0W0.
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Abstract

In this paper, we show that the expected number in an M/M/c queue is convex with respect to the traffic intensity. The proof is conducted by expressing the second derivative of the expected queue size as the sum of non-negative terms.

Keywords

MULTISERVER QUEUEVARYING ARRIVAL RATES
Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 4 , December 1983 , pp. 916 - 919
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research partly carried out while the author was a Visiting Scholar in the Department of Operations Research at Stanford University.

References

Cohen, M. A. and Lee, H. L. (1982) Customer allocation in a stochastic server system. Technical Report 86–6, Department of Decision Science, The Wharton School, University of Pennsylvania.Google Scholar
Dyer, M. E. and Proll, L. G. (1977) On the validity of marginal analysis for allocating servers in M/M/c queues. Management Sci. 23, 1019–1022.Google Scholar
Fox, B. L. (1966) Discrete optimization via marginal analysis. Management Sci. 13, 210–216.Google Scholar
Grassmann, W. K. (1977) The economic service rate. J. Operat. Res. Soc. 30, 149–155.Google Scholar
Grassmann, W. K. (1981) Stochastic Systems for Management. American Elsevier, New York.Google Scholar
Hillier, F. S. (1963) Economic models for industrial waiting line models. Management Sci. 10, 119–130.Google Scholar
Lee, H. L. and Cohen, M. A. (1983) A note on the convexity of performance measures of M/M/c queueing systems. J. Appl. Prob. 20, 920–923.CrossRefGoogle Scholar
Rolfe, A. J. (1971) A note on the marginal allocation in multi-server facilities. Management Sci. 17, 656–658.Google Scholar
Rolski, T. (1981) Queues with non-stationary input stream: Ross's conjecture. Adv. Appl. Prob. 13, 603–618.Google Scholar