2,343 research outputs found
The M/G/1 fluid model with heavy-tailed message length distributions
For the fluid model the stationary distribution of the buffer content is investigated for the case that the message length distribution has a Pareto-type tail, i.e. behaves as 1- {rm O (t^{-nu ) for with . This buffer content distribution is closely related to the stationary waiting time distribution of a stable model with service time distribution , in particular when the input rate of the messages into the buffer is not less than its output rate . The actual waiting process of this -model has an imbedded {ux_{n-process which for 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 and are investigated, in particular between their tail probabilities. The results obtained are quite explicit in particular for . 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 . Proper scaling of the traffic load , generated by the arrivals in , 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 is -stable L'evy motion, for 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 . The results obtained are generalisations of the diffusion approximation of the M/G/1 model with having a finite second moment
Heavy-traffic limit theorems for the haevy-tailed GI/G/1 queue
The classic 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, and {S(t) a slowly varying function at infinity, is investigated for the case that the traffic load 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 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
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