On the integral of the workload process of the single server queue

Abstract

This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .

Tail asymptotics for the area under the excursion of a random walk with heavy-tailed increments

“…Another application area is statistical physics, see, e.g., [8] or [3] and references therein. Applications to queuing theory for the analysis of the load in Transmission Control Protocol networks and to risk theory are discussed in [2].…”

“…In the light-tailed case logarithmic asymptotics for P(A τ > x) was obtained in [10], and exact local asymptotics in [14]. Heavy-tailed asymptotics for P(A τ > x) was previously studied in [2], which considered the case when the increments of the random walk have a distribution with regularly varying tail, that is F(x) = x −α L(x), where L(x) is a slowly varying function. For α > 1 it was shown that…”

“…In passing, we note that it is doubtful that (2) holds in full. The reason is that for both τ and A τ the asymptotics (3) and (2) are no longer valid for Weibull distributions with parameter β > 1/2. The analysis for β > 1/2 involves a more complicated optimisation procedure leading to a Cramér series, and it is unlikely that the answers will be the same for the area and for the exit time.…”

“…Furthermore, using this proposition one can give an alternative proof of (1) under the assumption of the regular variation of F, which is much simpler than the original one in [2]. We first split the event {A τ > x} into two parts,…”

Large Deviation Asymptotics for Busy Periods

“…They also reveal a lacuna in [3,Theorem 4.1] where the most likely path is assumed to be piecewise linear.…”