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 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.

The impact of the service discipline on delay asymptotics

“…[12]). Theorem 2.1 was first proved by Cohen [25] (who in fact considered the GI/G/1 case), and subsequently extended by several authors. In particular, Pakes [49] proved that the relation P{W > x} ∼ ρ/ (1 − ρ)P{B r > x} even holds for the larger class of service requirement distributions for which the residual service requirement distribution is subexponential, cf.…”

Fluid queues and regular variation

“…Actually, the result of [6] has been extended [14] to the larger class of subexponential distributions; accordingly, one can extend Theorem 5.1 to that class of activity period distributions. D…”

“…We can now apply Theorem l of [6] for the ordinary GI/G/1 queue, which relates the tail behaviour of the waiting-time distribution W c1 /G/I (t) and that of the service time distribution Bcf/G/I (t). This theorem states that, fort -+ oo and v > 1, and with f3 denoting mean service time, and p traffic load:…”

“…For the traditional GI/G/1 queue, Cohen [6] has studied the effect of regularly varying interarrival or service time distributions on waiting-time and workload distributions. We shall exploit his main result, which states: the waiting-time distribution in the GI/G/1 queue is regularly varying of index 1 -v (with v > 1) iff the service time distribution is regularly varying of index -v. See Abate et al…”

Elastic calls in an integrated services network: the greater the call size variability the better the QoS

“…Regarding the analysis of queueing models with heavy tailed service requirement distributions, it was shown in [5] that in the GI/G/1/F IF O queue the tail of the holding time distribution is 'one degree' heavier than that of the service requirement distribution. In [20] it is proved that both tails are equally heavy in an M/G/1/P S queue.…”