The $\nu$-stable L\''evy Motion in Heavy-traffic Analysis of Queueing Models with Heavy-tailed Distributions

• Authors: J. W. Cohen
• Venue: The $\'nu$-stable L\'''evy Motion in Heavy-traffic Analysis of Queueing Models with Heavy-tailed Distributions
• Year: 1998

For the ${GI/G/1}$ queueing model with heavy-tailed service- and arrival time distributions and traffic $a0 \}$, 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 $0c1$ the workload or buffer content process $\{ {\bf v}_t , t \geq 0 \}$ possesses a stationary distribution and its LST has been derived.