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On exponential bounds for the waiting-time distribution function in GI/G/1

Published online by Cambridge University Press:  14 July 2016

R. Bergmann and
D. Stoyan
R. Bergmann
Affiliation:
Karl Marx University, Leipzig, GDR
D. Stoyan
Affiliation:
Brennstoffinstitut, Freiberg, GDR
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Abstract

Exponential bounds for the stationary waiting-time distribution of the type ae–θt are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.

Keywords

GI/G/1 QUEUESWAITING-TIME DISTRIBUTIONEXPONENTIAL BOUNDSMONOTONICITYAPPROXIMATIONSNBUENWUE

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976, pp. 411 - 417
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Bergmann, R. (1974) Qualitätsunterschiede bei Wartemodellen vom Typ G/G/1. Math. Operationsforsch. Statist. 5, 709–724.CrossRefGoogle Scholar
[2] Kingman, J. F. C. (1970) Inequalities in the theory of queues. J. R. Statist. Soc. B 32, 102–110.Google Scholar
[3] Marshall, A. W. and Proschan, F. (1972) Classes of distributions applicable in replacement with renewal theory implications. Proc. 6th Berkeley Symp. Math. Statist. Prob. 1, 495–515.Google Scholar
[4] Marshall, K. T. (1968) Some inequalities in queueing. Operat. Res. 16, 651–665.Google Scholar
[5] Rolski, T. and Stoyan, D. (1974) Two classes of semi-orderings and their application in the queueing theory. Z. angew. Math. Mech. 54, 127–128.Google Scholar
[6] Ross, S. M. (1974) Bounds on the delay distribution in GI/G/1 queues. J. Appl. Prob. 11, 417–421.CrossRefGoogle Scholar
[7] RoßBerg, H. J. and Siegel, G. (1974) Die Bedeutung von Kingmans Integralungleichungen. Math. Operationsforsch. Statist. 5, 687–699.Google Scholar
[8] Stoyan, D. (1972) Monotonieeigenschaften stochastischer Modelle. Z. angew. Math. Mech. 52, 23–30.CrossRefGoogle Scholar
[9] Stoyan, D. (1972) Über einige Eigenschaften monotoner stochastischer Prozesse. Math. Nachr. 52, 21–34.CrossRefGoogle Scholar
[10] Stoyan, D. and Stoyan, H. (1974) Bounds on the mean waiting time in single server queues. Izvestija Akad. Nauk SSSR, techn. Kibernet. 1974 6, 104–106 (Russian).Google Scholar