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.