The v-stable Lévy motion in heavy-traffic analysis of queueing models with ::::heavy-tailed distributions

For the ${GI/G/1$ queueing model with heavy-tailed service- and arrival time distributions and traffic $a<1$ the limiting distribution of the contracted actual waiting time ${Delta (a) {bf w$ has been derived for ${Delta (a) downarrow 0$ for ${a uparrow 1 $, see [2]. In the present study we consider the workload process ${ {bf v_t , t>0 $, when properly scaled, i.e. ${Delta (a) {rm v_{tau / {Delta_1 (a)$ for ${a uparrow 1 $ with ${Delta_1 (a) = {Delta (a) (1-a)$. We further consider the noise traffic $n_t = {bf k_t -at$ and the virtual backlog ${bf h_t = {bf k_t -t$, with ${bf k_t$ the traffic generated in $[0,t)$. It is shown that $n_t$ and ${bf h_t$, when scaled similarly as ${bf v_t$, have a limiting distribution for ${a uparrow 1 $. We further consider the ${M/G_R^{(1) /1$ 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 $M/G_R /1$ model is for the present purpose, the more interesting one, and for this model the properly scaled workload-, noise traffic- and virtual backlog process are shown to converge weakly when the scaling parameters tend to zero as a function of the traffic $b$ for ${b uparrow 1 $. The limiting processes of the noise traffic and virtual backlog (properly scaled) appear to be $nu$-stable L'evy motions for $1< nu <2$, $nu$ 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 ${cal L_{v_1 / {cal L_{v_2 /1$ model. For $nu_1 = nu_2 = nu$, $1< nu <2$, this is a buffer storage model of which the virtual backlog process is a L'evy motion with a negative drift -c. It is shown that for $0<c<1$ the workload or buffer content process ${ {bf v_t , t geq 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 $t^{- {rm v {cal S (t)$ for ${t rightarrow infty$, $1< nu leq 2$ and ${cal S (t)$ a slowly varying function at infinity.