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

  • Michael N. KatehakisEmail author
  • Laurens C. Smit
  • Floske M. Spieksma
Original Research


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.


Matrix inverse Queueing Stochastic process Eigenvalues 

Mathematics Subject Classification

Primary: 15A29 15A18 15B51 60G20 



We acknowledge support for this work from the National Science Foundation, NSF Grants CMMI-1662629, and CMMI-1450743. We are indebted to an anonymous reviewer whose constructive comments have enabled us to considerably improve the presentation.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Rutgers UniversityNew BrunswickUSA
  2. 2.Leiden UniversityLeidenThe Netherlands

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