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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
  • 13 Downloads

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.

Keywords

Matrix inverse Queueing Stochastic process Eigenvalues 

Mathematics Subject Classification

Primary: 15A29 15A18 15B51 60G20 

Notes

Acknowledgements

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.

References

  1. Adan, I. J. B. F., Boxma, O. J., Kapodistria, S., & Kulkarni, V. G. (2016). The shorter queue polling model. Annals of Operations Research, 241(1–2), 167–200.CrossRefGoogle Scholar
  2. Adan, I. J. B. F., Economou, A., & Kapodistria, S. (2009). Synchronized reneging in queueing systems with vacations. Queueing Systems, 62(1–2), 1–33.CrossRefGoogle Scholar
  3. Alexanderian, A. (2013). On continuous dependence of roots of polynomials on coefficients. Retrieved May 22, 2019, from https://aalexan3.math.ncsu.edu/articles/polyroots.pdf.
  4. Altman, E., Avrachenkov, K., & Ayesta, U. (2006). A survey on discriminatory processor sharing. Queueing Systems, 53(1–2), 53–63.CrossRefGoogle Scholar
  5. Ammar, G. S. (1996). Classical foundations of algorithms for solving positive definite Toeplitz equations. Calcolo, 33(1–2), 99–113.CrossRefGoogle Scholar
  6. Ammar, G. S., & Gragg, W. B. (1988). Superfast solution of real positive definite Toeplitz systems. SIAM Journal on Matrix Analysis and Applications, 9(1), 61–76.CrossRefGoogle Scholar
  7. Anderson, W. J. (1991). Continuous-time Markov chains: An applications-oriented approach (Vol. 7). New York, NY: Springer.CrossRefGoogle Scholar
  8. Aveklouris, A., Vlasiou, M., & Zwart, B. (2019). A stochastic resource-sharing network for electric vehicle charging. IEEE Transactions on Control of Network Systems, 6, 1050–1061.CrossRefGoogle Scholar
  9. Barrett, W. W., & Feinsilver, P. J. (1981). Inverses of banded matrices. Linear Algebra and Its Applications, 41, 111–130.CrossRefGoogle Scholar
  10. Ben-Israel, A., & Greville, T. N. E. (2003). Generalized inverses: Theory and applications (Vol. 15). Berlin: Springer.Google Scholar
  11. Breen, J., Crisostomi, E., Faizrahnemoon, M., Kirkland, S., & Shorten, R. (2018). Clustering behaviour in Markov chains with eigenvalues close to one. Linear Algebra and Its Applications, 555, 163–185.CrossRefGoogle Scholar
  12. Ertiningsih, D., Katehakis, M. N., Smit, L. C., & Spieksma, F. M. (2019). Level product form QSF processes and an analysis of queues with Coxian interarrival distribution. Naval Research Logistics (NRL), 66(1), 57–72.CrossRefGoogle Scholar
  13. Gaver, D. P., Jacobs, P. A., & Latouche, G. (1984). Finite birth-and-death models in randomly changing environments. Advances in Applied Probability, 16, 715–731.CrossRefGoogle Scholar
  14. Grassmann, W. K. (2002). Real eigenvalues of certain tridiagonal matrix polynomials, with queueing applications. Linear Algebra and Its Applications, 342(1), 93–106.CrossRefGoogle Scholar
  15. Heinig, G., & Rost, K. (1984). Algebraic methods for Toeplitz-like matrices and operators. Basel: Springer.CrossRefGoogle Scholar
  16. Hunter, J. J. (2016). The computation of key properties of Markov chains via perturbations. Linear Algebra and Its Applications, 511, 176–202.CrossRefGoogle Scholar
  17. Ikebe, Y. (1979). On inverses of Hessenberg matrices. Linear Algebra and Its Applications, 24, 93–97.CrossRefGoogle Scholar
  18. Janssen, H.-K. (1981). On the nonequilibrium phase transition in reaction–diffusion systems with an absorbing stationary state. Zeitschrift für Physik B Condensed Matter, 42(2), 151–154.CrossRefGoogle Scholar
  19. Katehakis, M. N., & Smit, L. C. (2012). A successive lumping procedure for a class of Markov chains. Probability in the Engineering and Informational Sciences, 26(4), 483–508.CrossRefGoogle Scholar
  20. Katehakis, M. N., Smit, L. C., & Spieksma, F. M. (2015). DES and RES processes and their explicit solutions. Probability in the Engineering and Informational Sciences, 29(2), 191–217.CrossRefGoogle Scholar
  21. Katehakis, M. N., Smit, L. C., & Spieksma, F. M. (2016). A comparative analysis of the successive lumping and the lattice path counting algorithms. Journal of Applied Probability, 53(1), 106–120.CrossRefGoogle Scholar
  22. Kılıç, E., & Stanica, P. (2013). The inverse of banded matrices. Journal of Computational and Applied Mathematics, 237(1), 126–135.CrossRefGoogle Scholar
  23. Latouche, G., & Ramaswami, V. (1999). Introduction to matrix analytic methods in stochastic modeling (Vol. 5). Philadelphia, PA: SIAM.CrossRefGoogle Scholar
  24. Li, H.-B., Huang, T.-Z., Liu, X.-P., & Li, H. (2010). On the inverses of general tridiagonal matrices. Linear Algebra and Its Applications, 433(5), 965–983.CrossRefGoogle Scholar
  25. Mallik, R. K. (2001). The inverse of a tridiagonal matrix. Linear Algebra and Its Applications, 325(1), 109–139.CrossRefGoogle Scholar
  26. Martinsson, P.-G., Rokhlin, V., & Tygert, M. (2005). A fast algorithm for the inversion of general Toeplitz matrices. Computers & Mathematics with Applications, 50(5), 741–752.CrossRefGoogle Scholar
  27. Meurant, G. (1992). A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM Journal on Matrix Analysis and Applications, 13(3), 707–728.CrossRefGoogle Scholar
  28. Meyer, C. D. (2000). Matrix analysis and applied linear algebra. Philadelphia: SIAM.CrossRefGoogle Scholar
  29. Ross, S. M. (2013). Applied probability models with optimization applications. Chelmsford: Courier Corporation.Google Scholar
  30. Ross, S. M. (2014). Introduction to stochastic dynamic programming. New York: Academic Press.Google Scholar
  31. Ulukus, M. Y., Güllü, R., & Örmeci, L. (2011). Admission and termination control of a two class loss system. Stochastic Models, 27(1), 2–25.CrossRefGoogle Scholar
  32. Vlasiou, M., Zhang, J., & Zwart, B. (2014). Insensitivity of proportional fairness in critically loaded bandwidth sharing networks. arXiv preprint arXiv:1411.4841.

Copyright information

© 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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