Heavy-traffic theory for the heavy-tailed M/G/1 queue and v-stable L\''evy noise traffic

• Authors: J. W. Cohen
• Venue: Heavy-traffic theory for the heavy-tailed M/G/1 queue and v-stable L\'''evy noise traffic
• Year: 1998

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. $1-B(\tau) = \O (\tau^{- \nu}) , \tainfty , 1 0$. Proper scaling of the traffic load $\k_t$, generated by the arrivals in $[ 0,t)$, leads to \[ \tilde{{\bf w}}_\tau = \max [ {\bf H} (\tau) , \sup\limits_{0 u 0 , \] 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.