Some results on regular variation for distributions in queueing and fluctuation theory
- 1 June 1973
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 10 (2), 343-353
- https://doi.org/10.2307/3212351
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
This publication has 3 references indexed in Scilit:
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- Regular Variation of Markov Renewal FunctionsJournal of the London Mathematical Society, 1970
- Factorization Identities and Properties of the Distribution of the Supremum of Sequential SumsTheory of Probability and Its Applications, 1970