Finding equilibrium probabilities of QBD processes by spectral methods when eigenvalues vanish
type: Item
instance of: scholarly article
title: Finding equilibrium probabilities of QBD processes by spectral methods when eigenvalues vanish
zbMATH Open document ID: 1050.65007
author: Winfried K. Grassmann
published in: Linear Algebra and its Applications
publication date: 2004-08-04T00:00:00Z
review text: Consider spectral or eigenvalue methods for finding the equilibrium probabilities of quasi-birth-death processes (QBD processes) for the case where some eigenvalues are zero. This leads to the rather difficult problem of the analysis of continuous time Markov chains with block-structured infinitesimal generator matrices \(Q\) with multiple eigenvalues at zero. The author suggests to eliminate such eigenvalues by block elimination methods. Some of the reported resulting numerical difficulties such as instabilities can be bypassed by his algorithm. To accomplish this, the dimension of the largest Jordan block must be established, and some initial equations must be eliminated. The author also discusses the geometric and algebraic multiplicities of related eigenvalues in order to deal with the initial conditions. The theory is applied to two examples, one is dealing with a tandem queue, the other one with a shorter queue problem. Questions of numerical accuracy and computational complexity are addressed too.
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 65C50
Mathematics Subject Classification ID: 60J22
Mathematics Subject Classification ID: 60J27
Mathematics Subject Classification ID: 65C40
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 65Y20
zbMATH DE Number: 2081263
zbMATH Keywords: quasi-birth-death process
zbMATH Keywords: tandem queues
zbMATH Keywords: shorter queue
zbMATH Keywords: Markov chains
zbMATH Keywords: eigenvalues
zbMATH Keywords: equilibrium probabilities
zbMATH Keywords: spectral methods
zbMATH Keywords: block elimination
zbMATH Keywords: computational complexity
zbMATH Keywords: algorithm
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: j.laa.2003.12.013
OpenAlex ID: W2095059200
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DOI: 10.1016/J.LAA.2003.12.013
description: scientific article; zbMATH DE number 2081263