The ν-stable Lévy Motion in Heavy-traffic Analysis of Queueing Models with Heavy-tailed Distributions

Abstract

For the GI/G/1 queueing model with heavy-tailed serviceand arrival time distributions and traffic a < 1 the limiting distribution of the contracted actual waiting time ∆(a)w has been derived for ∆(a) ↓ 0 for a ↑ 1, see [2]. In the present study we consider the workload process {vt, t > 0}, when properly scaled, i.e. ∆(a)vτ/∆1(a) for a ↑ 1 with ∆1(a) = ∆(a)(1 − a). We further consider the noise traffic nt = kt − at and the virtual backlog ht = kt − t, with kt the traffic generated in [0, t). It is shown that nt and ht, when scaled similarly as vt, have a limiting distribution for a ↑ 1. We further consider the M/GR/1 model. It is a model with instantaneous workload reduction. The arrival process is a Poisson process and the service time distribution and that of the workload reduction are both heavy-tailed. Of this model two variants have to be considered. The M/GR/1 model is for the present purpose, the more interesting one, and for this model the properly scaled workload-, noise trafficand virtual backlog process are shown to converge weakly when the scaling parameters tend to zero as a function of the traffic b for b ↑ 1. The limiting processes of the noise traffic and virtual backlog (properly scaled) appear to be ν-stable Lévy motions for 1 < ν < 2, ν being the index of the heavy tails. The LSTs of these limiting distributions are derived. They have the same structure as those for the GI/G/1 model. The results so far obtained lead to the introduction of the Lv1/Lv2/1 model. For ν1 = ν2 = ν, 1 < ν < 2, this is a buffer storage model of which the virtual backlog process is a Lévy motion with a negative drift -c. It is shown that for 0 < c < 1 the workload or buffer content process {vt, t ≥ 0} possesses a stationary distribution and its LST has been derived. The results of the present study are new and lead to a better understanding of the stochastics of queueing models of which the modelling distributions have heavy-tails of a type tS(t) for t → ∞, 1 < ν ≤ 2 and S(t) a slowly varying function at infinity. 1991 Mathematics Subject Classification: 60K25, 60J75, 90B22

Cite this paper

@inproceedings{Cohen1998TheL, title={The ν-stable Lévy Motion in Heavy-traffic Analysis of Queueing Models with Heavy-tailed Distributions}, author={J. W. Cohen}, year={1998} }