Inversion and spectral analysis of matrices arising in the analysis of Markov processes
- 13 Downloads
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.
KeywordsMatrix inverse Queueing Stochastic process Eigenvalues
Mathematics Subject ClassificationPrimary: 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.
- 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.
- Ben-Israel, A., & Greville, T. N. E. (2003). Generalized inverses: Theory and applications (Vol. 15). Berlin: Springer.Google Scholar
- Ross, S. M. (2013). Applied probability models with optimization applications. Chelmsford: Courier Corporation.Google Scholar
- Ross, S. M. (2014). Introduction to stochastic dynamic programming. New York: Academic Press.Google Scholar
- Vlasiou, M., Zhang, J., & Zwart, B. (2014). Insensitivity of proportional fairness in critically loaded bandwidth sharing networks. arXiv preprint arXiv:1411.4841.