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Use of Characteristic Roots for Solving Infinite State Markov Chains

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Computational Probability

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 24))

Abstract

In this chapter, our interest is in determining the stationary distribution of an irreducible positive recurrent Markov chain with an infinite state space. In particular, we consider the solution of such chains using roots or zeros. A root of an equation f (z) = 0 is a zero of the function f (z),and so for notational convenience we use the terms root and zero interchangeably. A natural class of chains that can be solved using roots are those with a transition matrix that has an almost Toeplitz structure. Specifically, the classes of M/G/1 type chains and G/M/1 type chains lend themselves to solution methods that utilize roots. In the M/G/1 case, it is natural to transform the stationary equations and solve for the stationary distribution using generating functions. However, in the G/M/1 case the stationary probability vector itself is given directly in terms of roots or zeros. Although our focus in this chapter is on the discrete-time case, we will show how the continuous-time case can be handled by the same techniques. The M/G/1 and G/M/1 classes can be solved using the matrix analytic method [Neuts, 1981, Neuts, 1989], and we will also discuss the relationship between the approach using roots and this method.

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Author information

Authors and Affiliations

  1. IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 10598, USA

    H. Richard Gail & Sidney L. Hantler

  2. Department of Mathemtics, The University of Michigan, Ann Arbor, MI, 48109, USA

    B. Alan Taylor

Authors
  1. H. Richard Gail
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  2. Sidney L. Hantler
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  3. B. Alan Taylor
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Editor information

Editors and Affiliations

  1. University of Saskatchewan, Canada

    Winfried K. Grassmann

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Gail, H.R., Hantler, S.L., Taylor, B.A. (2000). Use of Characteristic Roots for Solving Infinite State Markov Chains. In: Grassmann, W.K. (eds) Computational Probability. International Series in Operations Research & Management Science, vol 24. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4828-4_7

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  • DOI: https://doi.org/10.1007/978-1-4757-4828-4_7

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5100-7

  • Online ISBN: 978-1-4757-4828-4

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