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

  • J. W. Cohen (a1)

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 {x 1} < 0 then the distribution of sup, s 1 s 2, …] has a regularly varying tail at + ∞ if the tail of the distribution of x 1 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.

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[1] Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. II, 2nd ed. Wiley, New York.
[2] Cohen, J. W. (1969) The Single Server Queue. North Holland, Amsterdam.
[3] Callaert, H. and Cohen, J. W. (1972) A lemma on regular variation of a transient renewal function Z. Wahrscheinlichkeitsth. 24, 275278.
[4] Stam, A. J. (1972) The tail of the ladder height distribution. Report TW 107, Math. Inst. Univ. Groningen.
[5] Teugels, J. L. (1970) Regular variation of Markov renewal functions. J. London Math. Soc. 2, 179190.
[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, 255263.
[7] Borovkov, A. A. (1970) Factorization identities and properties of the distribution of the supremum of sequential sums. Theor. Probability Appl. 15, 359402.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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