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Some results on regular variation for distributions in queueing and fluctuation theory

Published online by Cambridge University Press:  14 July 2016

J. W. Cohen
J. W. Cohen*
Affiliation:
Technological University, Delft
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Abstract

For the distribution functions of the stationary actual waiting time and of the stationary virtual waiting time of the GI/G/l queueing system it is shown that the tails vary regularly at infinity if and only if the tail of the service time distribution varies regularly at infinity.

For sn the sum of n i.i.d. variables xi, i = 1, …, n it is shown that if E {x1} < 0 then the distribution of sup, s1s2, …] has a regularly varying tail at + ∞ if the tail of the distribution of x1 varies regularly at infinity and conversely, moreover varies regularly at + ∞.

In the appendix a lemma and its proof are given providing necessary and sufficient conditions for regular variation of the tail of a compound Poisson distribution.

Keywords

REGULAR VARIATIONQUEUEING THEORYFLUCTUATION THEORYACTUAL AND VIRTUAL WAITING TIMEGENERAL SERVICE TIMEGENERAL INTERARRIVALSCOMPOUND POISSON DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 2 , June 1973 , pp. 343 - 353
Copyright
Copyright © Applied Probability Trust 1973 

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References

[1] Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. II, 2nd ed. Wiley, New York.Google Scholar
[2] Cohen, J. W. (1969) The Single Server Queue. North Holland, Amsterdam.Google Scholar
[3] Callaert, H. and Cohen, J. W. (1972) A lemma on regular variation of a transient renewal function Z. Wahrscheinlichkeitsth. 24, 275–278.CrossRefGoogle Scholar
[4] Stam, A. J. (1972) The tail of the ladder height distribution. Report TW 107, Math. Inst. Univ. Groningen.Google Scholar
[5] Teugels, J. L. (1970) Regular variation of Markov renewal functions. J. London Math. Soc. 2, 179–190.Google Scholar
[6] Cohen, J. W. (1972) On the tail of the stationary waiting time distribution for the M/G/l queue. Ann. Inst. H. Poincaré Sect. B 8, No. 3, 255–263.Google Scholar
[7] Borovkov, A. A. (1970) Factorization identities and properties of the distribution of the supremum of sequential sums. Theor. Probability Appl. 15, 359–402.Google Scholar