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Theory of Probability and Mathematical Statistics

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Controlled semi-Markov fields with graph-structured compact state space

Authors: H. Daduna, P. S. Knopov and R. K. Chorney
Translated by: The authors
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 69 (2003).
Journal: Theor. Probability and Math. Statist. 69 (2004), 39-53
MSC (2000): Primary 60K15, 60K35, 90C40
Published electronically: February 7, 2005
MathSciNet review: 2110903
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Abstract: We introduce locally acting distributed decision makers in the theory of semi-Markov decisions for systems for which both the domain and the action space are general and compact. Such decision makers are characterized by making decisions on the basis of the information gathered at their local neighborhood only. The state transient function of the system also is of a local structure. We consider general holding times of the systems and this results in semi-Markov properties in time. The neighborhood structure of the systems resembles in space the Markov property of spatial processes. Under some regularity assumptions, we reduce the optimal problems within the set of local strategies to the corresponding problems for deterministic Markov strategies.

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Additional Information

H. Daduna
Affiliation: Universität Hamburg, Fakultät Mathematik, Bundesstrasse 55, D–20146 Hamburg, Germany

P. S. Knopov
Affiliation: Glushkov Institute for Cybernetics, National Academy of Sciences of Ukraine, Academician Glushkov Avenue 40, Kiev–187, 03680, Ukraine

R. K. Chorney
Affiliation: Inter-Regional Academy of Personnel Management, Department of Mathematics, Frometivs’ka Street 2, Kiev–39, 03039, Ukraine

Keywords: Semi-Markov processes, Markov renewal processes, optimal control, average asymptotic reward, renewal reward processes, random fields, local strategies
Received by editor(s): January 24, 2003
Published electronically: February 7, 2005
Additional Notes: The research of the second author was partially supported by a grant from Deutsche Forschungsgemeinschaft at Hamburg University
Article copyright: © Copyright 2005 American Mathematical Society