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Simple Procedures For Finding Mean First Passage Times In Markov Chains

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  • JEFFREY J. HUNTER

    (Institute of Information and Mathematical Sciences, Massey University, Auckland, New Zealand)

Abstract

The derivation of mean first passage times in Markov chains involves the solution of a family of linear equations. By exploring the solution of a related set of equations, using suitable generalized inverses of the Markovian kernelI - P, wherePis the transition matrix of a finite irreducible Markov chain, we are able to derive elegant new results for finding the mean first passage times. As a by-product we derive the stationary distribution of the Markov chain without the necessity of any further computational procedures. Standard techniques in the literature, using for example Kemeny and Snell's fundamental matrixZ, require the initial derivation of the stationary distribution followed by the computation ofZ, the inverse ofI - P +eπTwhereeT= (1, 1, …, 1)andπTis the stationary probability vector. The procedures of this paper involve only the derivation of the inverse of a matrix of simple structure, based upon known characteristics of the Markov chain together with simple elementary vectors. No prior computations are required. Various possible families of matrices are explored leading to different related procedures.

Suggested Citation

  • Jeffrey J. Hunter, 2007. "Simple Procedures For Finding Mean First Passage Times In Markov Chains," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 24(06), pages 813-829.
    • Handle: RePEc:wsi:apjorx:v:24:y:2007:i:06:n:s0217595907001553
    DOI: 10.1142/S0217595907001553
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    Cited by:

    1. Xu, Jianhong, 2015. "An iterative algorithm for computing mean first passage times of Markov chains," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 372-389.
    2. Andreas C. Georgiou & Alexandra Papadopoulou & Pavlos Kolias & Haris Palikrousis & Evanthia Farmakioti, 2021. "On State Occupancies, First Passage Times and Duration in Non-Homogeneous Semi-Markov Chains," Mathematics, MDPI, vol. 9(15), pages 1-17, July.

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