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2016 | 4 | 1 | 151-175
Tytuł artykułu

Accurate calculations of Stationary Distributions and Mean First Passage Times in Markov Renewal Processes and Markov Chains

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This article describes an accurate procedure for computing the mean first passage times of a finite irreducible Markov chain and a Markov renewal process. The method is a refinement to the Kohlas, Zeit fur Oper Res, 30, 197–207, (1986) procedure. The technique is numerically stable in that it doesn’t involve subtractions. Algebraic expressions for the special cases of one, two, three and four states are derived.Aconsequence of the procedure is that the stationary distribution of the embedded Markov chain does not need to be derived in advance but can be found accurately from the derived mean first passage times. MatLab is utilized to carry out the computations, using some test problems from the literature.
Wydawca
Czasopismo
Rocznik
Tom
4
Numer
1
Strony
151-175
Opis fizyczny
Daty
otrzymano
2015-10-05
zaakceptowano
2016-02-14
online
2016-03-18
Twórcy
  • Department of Mathematical Sciences, School of Engineering, Computer and Mathematical Sciences, Auckland University of Technology, New Zealand
Bibliografia
  • [1] Benzi, M. A direct projection method for Markov chains. Linear Algebra Appl, 386, (2004), 27–49.
  • [2] Bini, D. A., Latouche, G. and Meini B. Numerical Methods for Structured Markov Chains, Oxford University Press, New York. (2005).
  • [3] Dayar, T. and Akar, N. Computing the moments of first passage times to a subset of states in Markov chains. SIAM J Matrix Anal Appl, 27, (2005), 396–412. [Crossref]
  • [4] Grassman,W.K., Taksar, M.I., and Heyman, D.P. Regenerative analysis and steady state distributions forMarkov chains. Oper Res, 33, (1985), 1107–1116.
  • [5] Harrod,W.J. and Plemmons, R.J. Comparison of some direct methods for computing stationary distributions ofMarkov chains. SIAM J Sci Stat Comput, 5, (1984), 463–479.
  • [6] Heyman, D.P. Further comparisons of direct methods for computing stationary distributions ofMarkov chains. SIAM J Algebra Discr, 8, (1987), 226–232. [Crossref]
  • [7] Heyman, D.P. and O’Leary, D.P.What is fundamental forMarkov chains: First Passage Times, Fundamentalmatrices, and Group Generalized Inverses, Computations with Markov Chains, Chap 10, 151–161, Ed W.J. Stewart, Springer. New York, (1995).
  • [8] Heyman, D.P. and Reeves, A. Numerical solutions of linear equations arising inMarkov chain models. ORSA J Comp, 1, (1989), 52–60.
  • [9] Hunter, J.J. Generalized inverses and their application to applied probability problems. Linear Algebra Appl, 46, (1982), 157– 198. [Crossref]
  • [10] Hunter, J.J. Mathematical Techniques of Applied Probability, Volume 2, Discrete Time Models: Techniques and Applications, Academic, New York. (1983).
  • [11] Hunter, J.J. Mixing times with applications to Markov chains, Linear Algebra Appl, 417, (2006), 108–123. [WoS]
  • [12] Kemeny, J. G. and Snell, J. L. Finite Markov chains, Springer- Verlag, New York (1983), (Original version, Princeton University Press, Princeton (1960).)
  • [13] Kohlas, J. Numerical computation of mean passage times and absorption probabilities in Markov and semi-Markov models. Zeit Oper Res, 30, (1986), 197–207.
  • [14] Meyer. C. D. Jr. The role of the group generalized inverse in the theory of Markov chains. SIAM Rev, 17, (1975), 443–464. [Crossref]
  • [15] Sheskin, T.J. A Markov partitioning algorithm for computing steady state probabilities. Oper Res, 33, (1985), 228–235.
  • [16] Stewart, W. J. Introduction to the Numerical Solution of Markov chains. Princeton University Press, Princeton. (1994).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_spma-2016-0015
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