Light tail asymptotics in multidimensional reflecting processes for queueing networks
Motivated by classical Jackson networks and semi-martingale reflecting Brownian motion a general multi-dimensional reflecting process in discrete time on the nonnegative orthant is defined with homogeneous transition rates in the interior of that cone and on the respective faces of the boundary. The focus is on processes where the stationary distribution exists but is not explicitly accessible with the aims to obtain (1a) ``A reasonable wide class of models which incorporate some structural changes'', and (1b) ``Analytically tractable characteristics which are still useful to assess performance of models.'' While the first topic is clearly met by the investigated class of processes, for the second point the focus is put onto investigating the (multidimensional) tail asymptotic of the stationary distributions under the assumption that the tail is light in a precise meaning. The focus on these light tailed distributions is well suited for many problems in the realm of operations research and related applications. The paper first gives a very readable introduction into a bulk of (mostly) recent work on the topic with an emphasis on a common terminology which is very helpful for someone who wants to enter the field. On the other side the paper is long enough to present very detailed recent development in the field. The focus of the latter is on Markov additive processes and analytic function approaches, and surveys especially recent work of the author and his coworkers. For the models with dimension \(d\) greater than 3 many open problems are marked, and for \(d=2\) a rather complete picture is given. Special attention is paid to the class of quasi-birth-death processes.
semi-martingale reflecting Brownian motion
multidimensional moment generating function
scientific article; zbMATH DE number 6139228
scientific article