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Abstract <p>We consider a class of Markov chains, called quasi-birth-and-death (QBD) processes, for which the stationary probability distribution, when it exists, is of the matrix-geometric form. An essential step in most algorithms for computing such a distribution is the evaluation of a rate matrix <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R\)</EquationSource> <!--LobJMat2561506Luh-m1--> </InlineEquation> which is a solution of a matrix quadratic equation. In this paper, we show how the eigenvalues of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(R\)</EquationSource> <!--LobJMat2561506Luh-m2--> </InlineEquation> can be determined explicitly when the infinitesimal generator of the QBD process has a special structure. Under certain conditions, the stationary probability vector is obtained in terms of the unique eigenvalue without computing R.</p>

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A Note on an Efficient Approach for Solving Quasi-Birth-and-Death Processes Exhibiting Special Structures

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Hsing Paul Luh

摘要

Abstract

We consider a class of Markov chains, called quasi-birth-and-death (QBD) processes, for which the stationary probability distribution, when it exists, is of the matrix-geometric form. An essential step in most algorithms for computing such a distribution is the evaluation of a rate matrix \(R\) which is a solution of a matrix quadratic equation. In this paper, we show how the eigenvalues of \(R\) can be determined explicitly when the infinitesimal generator of the QBD process has a special structure. Under certain conditions, the stationary probability vector is obtained in terms of the unique eigenvalue without computing R.