On tightness and exponential tightness in generalised Jackson networks
A. Puhalskii
TL;DR
The paper addresses proving tightness and exponential tightness of stationary queue-length distributions in generalised Jackson networks under trajectorial limits corresponding to large, normal, and moderate deviations. It develops a uniform framework that majorises multidimensional queue-length processes by one-dimensional reflected processes with negative drift, leveraging Skorohod reflection. The main contributions are conditions under which exponential tightness and large deviation-type results hold for stationary distributions, for both fixed networks and sequences indexed by $n$ with converging parameters, and a clarification of prior misapplications in this area. Together, these results provide a streamlined approach to deriving stationary convergence from trajectorial convergence in queueing networks and support subsequent quasipotential analysis in moderate deviations.
Abstract
We give uniform proofs of tightness and exponential tightness of the sequences of stationary queue lengths in generalised Jackson networks in a number of setups which concern large, normal and moderate deviations.