Sampling at subexponential times, with queueing applications

Abstract

We study the tail asymptotics of the r.v. X(T) where fX(t)g is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We nd that the tail of X(T) is sensitive to whether or not T has a heavier or lighter tail than a Weibull distribution with tail e p x. This leads to two distinct cases, heavy-tailed and moderately heavy-tailed, but also some results for the classical light-tailed case are given. The results are applied via distributional Little's law to establish tail asymptotics for steady{state queue length in GI/GI/1 queues with subexponential service times. Further applications are given for queues with vacations, and M/G/1 busy periods.

Tails for (max, plus) recursions under subexponentiality

“…In [13], upper and lower bounds are derived, which are shown to be tight in the irreducible case (i.e., d = 1). In [10], the tail asymptotics for W have been found if is the output transition, so that W is the maximal dater; see (3).…”

“…In this case, the mappings A and B are given by the following expressions (a proof is left to the reader): (2,3,5) ζ (3,5) ζ (2,3,5) −∞ ζ (3,5) | ζ (4,5) (5) . This network belongs to the framework of [10] and hence the maps A and B automatically satisfy Assumption (M).…”

“…If max(d (2) , d (3) , d (4) ) = 0 (in which case automatically d (1) > 0 by (SE)), we still have the exact asymptotics if we assume in addition to the subexponentiality the dominated variation of F s , namely…”

“…Moreover, under certain conditions, when different inputs are multiplexed, the 'heaviest' one dominates the tail asymptotics (see [1,15,19]). From the point of view of asymptotics, the queue length behaves qualitatively different from the waiting time; see Asmussen, Klüppelberg, and Sigman [3] and Foss and Korshunov [16]. For surveys on the state of the art for this kind of asymptotics, see the special issues of Queueing Systems, 33: 1-3 (1999), and 46: 1-2 (2004).…”

Tandem queues with subexponential service times and finite buffers

“…In the single-queue case, a vast body of literature has been devoted to asymptotic results on a variety of models; see for example Pakes [22], Embrechts and Veraverbeke [16], Willekens and Teugels [26], Asmussen, Klüppelberg and Sigman [4], Jelenković and Lazar [20], Asmussen, Henriksen and Klüppelberg [3], Asmussen [2], Asmussen and Møller [5], Whitt [25], Takine [24], Borst, Boxma and Jelenković [13], Xia, Liu, Squillante and Zhang [27], Miyoshi [21], Foss and Korshunov [18] and Foss, Konstantopoulos, and Zachary [17].…”