Overloading a Jackson network of shared queues.

Abstract

Jackson networks with finite buffers can be designed so that buffer overflows occur with very low probability. Still, queues can be filled beyond capacity so it is of interest to study the distribution of such occurrences and, as a measure of the reliability of the system, the mean time until an overload. The approach we propose is based on a change of measure which twists the stationary Jackson network into a process for which overloads are frequent. For a Jackson network of shared queues with buffer of size $\ell - 1$, we show that the stationary overflow distribution of the twisted network can be used to determine both the limiting distribution (as $\ell\to\infty$) at the moment of overload and the mean time until overload of the original Jackson network. Our estimation for the mean time until an overload is based on an article by Iscoe and McDonald (1994a) in which the mean time until the overflow of a Jackson network is estimated by the reciprocal of the smallest eigenvalue $\Lambda(B)$ of an associated Dirichlet problem. The distinct feature of this estimate is that it comes complete with error bounds so that, unlike estimators, its accuracy can be evaluated. Of course, in practice such eigensystems, however, we obtain an estimate which we show to be asymptotically equivalent to the one proposed by Aldous (1989).