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Inversion and spectral analysis of matrices arising in the analysis of Markov processes.

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Title: Inversion and spectral analysis of matrices arising in the analysis of Markov processes.
Authors: Katehakis, Michael N.1 (AUTHOR) mnk@rutgers.edu, Smit, Laurens C.2 (AUTHOR) laurens@pipe.nl, Spieksma, Floske M.2 (AUTHOR) spieksma@math.leidenuniv.nl
Source: Annals of Operations Research. Nov2025, Vol. 354 Issue 3, p1145-1170. 26p.
Subjects: Markov processes, Matrix inversion, Dynamical systems, Spectral theory, Eigenvalues, Queuing theory, Eigenvectors, Matrices (Mathematics)
Abstract: In this paper we provide a novel inversion method and algorithms for nearly tridiagonal matrices arising in the analysis of Markov processes. The method provides a fast and exact computation procedure of the inverse of the matrix that contains the coefficients of a rate matrix of the Markov processes. If the matrix is of countable size, the method provides an exact solution, independent of the truncation size. In contrast, alternative inverse techniques perform much slower and work only for finite size matrices. This leads to more efficient methods to compute the solution to a countable (finite or infinite) set of equations that occurs in queueing systems and in related fields including Markov processes, birth-and-death processes and inventory systems. Furthermore, we provide a procedure to construct the eigenvalues and eigenvectors of an arbitrary matrix, using those of an easier to analyze matrix. We apply and specialize this procedure to the matrix arising in the corresponding Markov rate matrices under consideration. [ABSTRACT FROM AUTHOR]
Copyright of Annals of Operations Research is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Inversion and spectral analysis of matrices arising in the analysis of Markov processes.
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  Data: <searchLink fieldCode="AR" term="%22Katehakis%2C+Michael+N%2E%22">Katehakis, Michael N.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> mnk@rutgers.edu</i><br /><searchLink fieldCode="AR" term="%22Smit%2C+Laurens+C%2E%22">Smit, Laurens C.</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> laurens@pipe.nl</i><br /><searchLink fieldCode="AR" term="%22Spieksma%2C+Floske+M%2E%22">Spieksma, Floske M.</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> spieksma@math.leidenuniv.nl</i>
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  Data: <searchLink fieldCode="JN" term="%22Annals+of+Operations+Research%22">Annals of Operations Research</searchLink>. Nov2025, Vol. 354 Issue 3, p1145-1170. 26p.
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  Data: <searchLink fieldCode="DE" term="%22Markov+processes%22">Markov processes</searchLink><br /><searchLink fieldCode="DE" term="%22Matrix+inversion%22">Matrix inversion</searchLink><br /><searchLink fieldCode="DE" term="%22Dynamical+systems%22">Dynamical systems</searchLink><br /><searchLink fieldCode="DE" term="%22Spectral+theory%22">Spectral theory</searchLink><br /><searchLink fieldCode="DE" term="%22Eigenvalues%22">Eigenvalues</searchLink><br /><searchLink fieldCode="DE" term="%22Queuing+theory%22">Queuing theory</searchLink><br /><searchLink fieldCode="DE" term="%22Eigenvectors%22">Eigenvectors</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink>
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  Data: In this paper we provide a novel inversion method and algorithms for nearly tridiagonal matrices arising in the analysis of Markov processes. The method provides a fast and exact computation procedure of the inverse of the matrix that contains the coefficients of a rate matrix of the Markov processes. If the matrix is of countable size, the method provides an exact solution, independent of the truncation size. In contrast, alternative inverse techniques perform much slower and work only for finite size matrices. This leads to more efficient methods to compute the solution to a countable (finite or infinite) set of equations that occurs in queueing systems and in related fields including Markov processes, birth-and-death processes and inventory systems. Furthermore, we provide a procedure to construct the eigenvalues and eigenvectors of an arbitrary matrix, using those of an easier to analyze matrix. We apply and specialize this procedure to the matrix arising in the corresponding Markov rate matrices under consideration. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Annals of Operations Research is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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      – Type: doi
        Value: 10.1007/s10479-019-03460-3
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      – Code: eng
        Text: English
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        PageCount: 26
        StartPage: 1145
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      – SubjectFull: Markov processes
        Type: general
      – SubjectFull: Matrix inversion
        Type: general
      – SubjectFull: Dynamical systems
        Type: general
      – SubjectFull: Spectral theory
        Type: general
      – SubjectFull: Eigenvalues
        Type: general
      – SubjectFull: Queuing theory
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      – SubjectFull: Eigenvectors
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      – SubjectFull: Matrices (Mathematics)
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      – TitleFull: Inversion and spectral analysis of matrices arising in the analysis of Markov processes.
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            NameFull: Smit, Laurens C.
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            NameFull: Spieksma, Floske M.
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              Text: Nov2025
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