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On the time a Markov chain spends in a lumped state

  • Tommy Norberg (a1)


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).


Corresponding author

Postal address: Department of Mathematics, Chalmers University of Technology and Göteborg University, S-41296 Göteborg, Sweden.
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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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[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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