The M/G/1 fluid model with heavy-tailed message length distributions

For the $M/G/1$ fluid model the stationary distribution of the buffer content is investigated for the case that the message length distribution $B(t)$ has a Pareto-type tail, i.e. behaves as 1- {rm O (t^{-nu ) for $t rightarrow infty$ with $1< nu <2$. This buffer content distribution is closely related to the stationary waiting time distribution $W(t)$ of a stable $M/G/1$ model with service time distribution $B(t)$, in particular when the input rate $gamma$ of the messages into the buffer is not less than its output rate $c=1$. The actual waiting process of this $M/G/1$-model has an imbedded {ux_{n-process which for $gamma geq 1$ has the same probabilistic structure as the {bf omega hspace{-2mm {bf omega _n-process, the latter one being an imbedded process of the buffer content process. The relations between the stationary distributions $U(t)$ and $W(t)$ are investigated, in particular between their tail probabilities. The results obtained are quite explicit in particular for $nu = 1 frac12$. Further heavy traffic results are obtained. These results lead to a heavy traffic result for the stationary distribution of the {bf omega hspace{-2mm {bf omega _n-process and to an asymptotic for the tail probabilities of this distribution

Heavy-traffic theory for the heavy-tailed M/G/1 queue and v-stable Lévy noise traffic

The workload {bf v_t of an M/G/1 model with traffic $a 0$. 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 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

Heavy-traffic limit theorems for the haevy-tailed GI/G/1 queue

The classic $GI/G/1$ queueing model of which the tail of the service time and/or the interarrival time distribution behaves as t^{-v {S(t) for {t rightarrow infty, $1<v<2$ and {S(t) a slowly varying function at infinity, is investigated for the case that the traffic load $a$ approaches one. Heavy-traffic limit theorems are derived for the case that these tails have a similar behaviour at infinity as well as for the case that one of these tails is heavier than the other one. These theorems state that the contracted waiting time {Delta (a) {bf w, with {bf w the actual waiting time for the stable $GI/G/1$ queue and {Delta (a) the contraction coefficient, converges in distribution for {a uparrow 1 . Here {Delta (a) is that root of the contraction equation which approaches zero from above for {a uparrow 1 . The structure of this contraction equation is determined by the character of the two tails. The Laplace-Stieltjes transforms of the limiting distributions are derived. For nonsimilar tails the limiting distributions are explicitly known. For the tails of these distributions asymptotic expressions are derived and compared