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On the concavity of the waiting-time distribution in some GI/G/1 queues

Published online by Cambridge University Press:  14 July 2016

R. Szekli
R. Szekli*
Affiliation:
University of Wrocχaw
*
∗Postal address: Institute of Mathematics, University of Wrocχaw, PI. Grunwaldzki 2/4, 50–384 Wrocχaw, Poland.
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Abstract

In this paper the concavity property for the distribution of a geometric random sum (geometric compound) X, + · ·· + XN is established under the assumption that Xi are i.i.d. and have a DFR distribution. From this and the fact that the actual waiting time in GI/G/1 queues can be written as a geometric random sum, the concavity of the waiting-time distribution in GI/G/1 queues with a DFR service-time distribution is derived. The DFR property of the idle-period distribution in specialized GI/G/1 queues is also mentioned.

Keywords

GI/G/1 QUEUEDFR DISTRIBUTIONGEOMETRIC COMPOUNDINGACTUAL WAITING TIME
Type
Short Communications
Information
Journal of Applied Probability , Volume 23 , Issue 2 , June 1986 , pp. 555 - 561
Copyright
Copyright © Applied Probability Trust 1986 

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References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Barlow, R. E., Marshall, A. W. and Proschan, F. (1963) Properties of probability distributions with monotone hazard rate. Ann. Math. Statist. 34, 375–389.CrossRefGoogle Scholar
Brown, M. (1980) Bounds, inequalities and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.Google Scholar
Brown, M. (1981) Further monotonicity properties for specialized renewal processes. Ann. Prob. 9, 891–895.Google Scholar
Cohen, J. W. (1969) The Single Server Queue. North Holland, Amsterdam.Google Scholar
Daley, D. J. (1984) Inequalities for a geometric random sum. Unpublished.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York.Google Scholar
Köllerström, J. (1976) Stochastic bounds for the single server queue. Math. Proc. Camb. Phil. Soc. 80, 521–525.Google Scholar
Marshall, K. T. (1968) Some inequalities in queueing. Operat. Res. 16, 651–665.CrossRefGoogle Scholar
Roberts, A. W. and Varberg, P. E. (1973) Convex Functions. Academic Press, New York.Google Scholar
Siegel, G. (1973) The stationary waiting time and other variables in single server queues with specialities at the beginning of a busy period. Zast. Mat. XIII, 465–479.Google Scholar