Maxima of sojourn times in acyclic Jackson queueing networks

This article considers the asymptotic behaviour of the maximum of a sequence of sojourn times in acyclic Jackson queueing networks. We first show that the asymptotic behaviour of extreme values from a finite mixture of continuous distributions is governed by a certain dominant component in the mixture, so that the asymptotic distribution is the same type as that of extreme values from the component distribution and the possible normalizing constants of the extreme value distribution is related to those of the dominant distribution. As a special case, we observe that the extreme value distribution of a large sample from a finite mixture of Erlang distributions is asymptotically a Gumbel distribution. We relate the distribution tail of the sojourn times in an acyclic Jackson network exactly or approximately to that of a finite mixture of Erlang distributions. The asymptotic results on mixtures of sum of Erlang random variables are then applied to show that the distribution of the time it takes for a group of units to travel through an acyclic Jackson network can be effectively described by a Gumbel distribution when the traffic is light or when the units are sparsely interspersed.