The workload ${bf v_t$ of an M/G/1 model with traffic $a < 1$ is analyzed for the case with heavy-tailed message length distributions $B(tau ) $, e.g. $1-B(tau) = O (tau^{- nu) , tainfty , 1 < nu leq 2$. It is shown that a factor ${Delta (a)$ exists with ${Delta (a) downarrow 0$ for ${a uparrow 1$ such that, whenever ${bf v_t$ is scaled by ${Delta (a)$ and time $t$ by $Delta_1 (a) = {Delta (a) (1-a)$ then ${bf w_tau (a) = {Delta (a) {bf v_{tau /Delta_1 (a)$ converges in distribution for ${a uparrow 1$ and every $tau > 0$. Proper scaling of the traffic load $k_t$, generated by the arrivals in $[ 0,t)$, leads to [ tilde{{bf w_tau = max [ {bf H (tau) , suplimits_{0 < u < tau ( {bf H (tau) - {bf H (u) )] , ; tau > 0 , ] with ${bf H (tau) = {bf N (tau) - tau$. Here ${ {bf N (tau) , tau geq 0 $ with $nu neq 2$ is $nu$-stable L'evy motion, for $nu =2$ it is Brownian motions and $tilde{{bf w_tau$ has the limiting distribution of ${bf w_tau (a)$ for ${a uparrow 1$. This relation is analogous to Reich's formula for the M/G/1 model with $a < 1$. The results obtained are generalisations of the diffusion approximation of the M/G/1 model with $B(tau )$ having a finite second moment.

CWI. Probability, Networks and Algorithms [PNA]

Cohen, J.W. (1998). Heavy-traffic theory for the heavy-tailed M/G/1 queue and v-stable Lévy noise traffic. CWI. Probability, Networks and Algorithms [PNA]. CWI.