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

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 s_n the sum of n i.i.d. variables x_i, 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.