Light tail asymptotics in multidimensional reflecting processes for queueing networks
type: Item
instance of: scholarly article
title: Light tail asymptotics in multidimensional reflecting processes for queueing networks
zbMATH Open document ID: 1280.60051
author: Masakiyo Miyazawa
published in: Top
publication date: 2013-02-26T00:00:00Z
review text: 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.
reviewed by: Hans Daduna
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 90B15
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 6139228
zbMATH Keywords: queueing network
zbMATH Keywords: reflecting random walk
zbMATH Keywords: semi-martingale reflecting Brownian motion
zbMATH Keywords: stationary distribution
zbMATH Keywords: tail asymptotic
zbMATH Keywords: tail decay rate
zbMATH Keywords: light tail
zbMATH Keywords: Markov additive process
zbMATH Keywords: convergence domain
zbMATH Keywords: multidimensional moment generating function
MaRDI profile type: Publication
full work available at URL: s11750-011-0179-7
OpenAlex ID: W2086391916
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Light tail asymptotics in multidimensional reflecting processes for queueing networks (DOI)
DOI: 10.1007/S11750-011-0179-7