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On occupation times for quasi-Markov processes

Published online by Cambridge University Press:  14 July 2016

Lennart Bondesson
Lennart Bondesson*
Affiliation:
University of Umeå
*
∗Postal address: Department of Mathematical Statistics, University of Umeå, S–901 87 Umeå, Sweden.
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Abstract

In this note the joint distribution for the times in an interval [0, t] spent in the states 1, 2, ···, N in a standard quasi-Markov process of order N is considered. An expression for the Laplace transform with respect to t of the Laplace–Stieltjes transform of this joint distribution is derived.

Keywords

OCCUPATION TIMESTANDARD QUASI-MARKOV PROCESSSTANDARD REGENERATIVE PHENOMENON
Type
Short Communications
Information
Journal of Applied Probability , Volume 18 , Issue 1 , March 1981 , pp. 297 - 301
Copyright
Copyright © Applied Probability Trust 

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References

Barlow, R. E. and Hunter, L. C. (1961) Reliability analysis of a one-unit system. J. Operat. Res. 9, 200–208.CrossRefGoogle Scholar
Darling, D. A. and Kac, M. (1957) On occupation times for Markoff processes. Trans. Amer. Math. Soc. 84, 444–458.Google Scholar
Darroch, J. N. (1966) Identities for passage times with applications to recurrent events and homogeneous differential functions. J. Appl. Prob. 3, 435–444.CrossRefGoogle Scholar
Darroch, J. N. and Morris, K. (1968) Passage-time generating functions for continuous-time finite Markov chains. J. Appl. Prob. 5, 414–426.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Gibson, A. E. and Conolly, B. W. (1971) On a three-state sojourn time problem. J. Appl. Prob. 8, 716–723.Google Scholar
Good, I. J. (1961) The frequency count of a Markov chain and the transition to continuous time. Ann. Math. Statist. 32, 41–48.CrossRefGoogle Scholar
Kingman, J. F. C. (1964) The stochastic theory of regenerative events. Z. Wahrscheinlichkeitsth. 2, 180–224.Google Scholar
Kingman, J. F. C. (1972) Regenerative Phenomena. Wiley, London.Google Scholar
Kovaleva, L. M. (1970) On the occupation time in a given state for the simplest semi-Markov system (in Russian). Teor. Verojatnost. i Mat. Statist. 1, 100–107.Google Scholar
Linhart, P. B. (1967) Alternating renewal processes with applications to some single server problems. Paper read at the Fifth International Telegraphic Congress, Rockefeller University, New York, June 14–20.Google Scholar
Pedler, P. J. (1971) Occupation times for two-state Markov chains. J. Appl. Prob. 8, 381–390.Google Scholar
Ràde, L. (1976) The two-state Markov process and additional events. Amer. Math. Monthly 83, 354–356.Google Scholar
Takács, L. (1957) On certain sojourn time problems in the theory of stochastic processes. Acta Math. Acad. Sci. Hung. 8, 169–191.CrossRefGoogle Scholar
Hsia, Wei Shen (1976) The joint probability density function of the occupation time of a three-state problem. J. Appl. Prob. 13, 57–64.CrossRefGoogle Scholar