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

• Authors: J. W. Cohen
• Venue: The M/G/1 fluid model with heavy-tailed message length distributions
• Year: 1997

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.