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

For the M=G=1 uid model the stationary distribution of the buuer content is investigated for the case that the message length distribution B(t) has a Pareto-type tail, i.e. behaves as 1 ? O(t ?) for t ! 1 with 1 < < 2. This buuer 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 of the messages into the buuer is not less than its output rate c = 1. The actual waiting process of this M=G=1-model has an imbedded un-process which for 1 has the same probabilistic structure as the ! !n-process, the latter one being an imbedded process of the buuer 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 = 1 1 2. Further heavy traac results are obtained. These results lead to a heavy traac result for the stationary distribution of the ! !n-process and to an asymptotic for the tail probabilities of this distribution.