Discrete time semi-Markov models with fuzzy state space Aleka A. Papadopoulou 1 and George M. Tsaklidis 1 Department of Mathematics Aristotle University of Thessaloniki, Thessaloniki 54124, Greece (e-mail: apapado@math.auth.gr, tsaklidi@math.auth.gr) Abstract. In the present paper, the classical semi-Markov model in discrete time is examined under the assumption of a fuzzy state space. The definition of a semi-Markov model with fuzzy states is provided. Basic equations for the interval transition probabilities are given for the homogeneous and non homogeneous case. The definitions and results for the fuzzy model are provided by means of the basic parameters of the classical semi-Markov model. Keywords: fuzzy states, semi-Markov process, discrete time. 1 Introduction In this paper we extend the classical semi-Markov model by assuming a fuzzy state space. We examine its behaviour through the concept of probability of fuzzy events. Important theoretical results and applications for semi- Markov models can be found in Cinlar (1969,1975,1975), Iosifescu-Manu (1972), Teugels (1976), Pyke and Schaufele (1964), Keilson (1969,1971), Mclean and Neuts (1967), Howard (1971), McClean (1980,1986), Janssen and De Dominicis (1984), Janssen (1986) and in Janssen and Limnios (1999). The non homogeneous semi-Markov system in discrete time was examined in Vas- siliou and Papadopoulou (1992), and the asymptotic behavior of the same model was studied in Papadopoulou and Vassiliou (1994). Fuzzy states occur essentially in two cases. First, when the states of the system cannot be pre- cisely measured and thus the states used to model the system are intrinsically fuzzy. In the second case the actual states can be exactly measured and are observable but the number of states is too large and thus the decisions cannot be practically associated with the exact states of the system. In these situa- tions the decisions are associated with fuzzy states which can be defined as fuzzy sets on the original non fuzzy state space of the system (Bhattacharyya (1998)). Fuzzy set theory was introduced by Zadeh (1965,1968) to describe situations with unsharp boundaries, partial information or vagueness such as in natural language. In fuzzy set theory subsets of Ω are replaced by func- tions f : Ω → [0, 1]. In section 2 basic equations of a semi Markov model are given. In section 3 the definition of a semi Markov model with fuzzy states