Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T17:53:46.005Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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