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On the time a Markov chain spends in a lumped state

  • Tommy Norberg (a1)

Abstract

The sojourn time that a Markov chain spends in a subset E of its state space has a distribution that depends on the hitting distribution on E and the probabilities (resp. rates in the continuous-time case) that govern the transitions within E. In this note we characterise the set of all hitting distributions for which the sojourn time distribution is geometric (resp. exponential).

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Postal address: Department of Mathematics, Chalmers University of Technology and Göteborg University, S-41296 Göteborg, Sweden.
Email: tommy@math.chalmers.se; www: http://www.math.chalmers.se/~tommy

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Research supported in part by the Swedish Natural Science Research Council.

Partly completed while visiting the Center for Stochastic Processes, University of North Carolina at Chapel Hill.

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References

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[1] Kemeny, J. G. and Snell, J. L. (1976) Finite Markov Chains. Springer, Berlin.
[2] Ledoux, J. (1993) A necessary condition for weak lumpability in finite Markov processes. Operat. Res. Lett. 13, 165168.
[3] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore.
[4] Rubino, G. and Sericola, B. (1989) On weak lumpability in Markov chains. J. Appl. Prob. 26, 446457.
[5] Rubino, G. and Sericola, B. (1989) Sojourn times in finite Markov processes. J. Appl. Prob. 27, 744756.
[6] Rubino, G. and Sericola, B. (1991) A finite characterization of weak lumpable Markov processes. Part I: The discrete time case. Stoch. Proc. Appl. 38, 195204.
[7] Seneta, E. (1981) Non-Negative Matrices and Markov Chains. Springer, Berlin.

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On the time a Markov chain spends in a lumped state

  • Tommy Norberg (a1)

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