document: Fuzzy non-homogeneous Markov systems In this paper the theory of fuzzy logic and fuzzy reasoning is combined with the theory of Markov systems and the concept of a fuzzy non-homogeneous Markov system is introduced for the first time . This is an effort to deal with the uncertainty introduced in the estimation of the transition probabilities and the input probabilities in Markov systems . The asymptotic behaviour of the fuzzy Markov system and its asymptotic variability is considered and given in closed analytic form . Moreover , the asymptotically attainable structures of the system are estimated also in a closed analytic form under some realistic assumptions . The importance of this result lies in the fact that in most cases the traditional methods for estimating the probabilities can not be used due to lack of data and measurement errors . The introduction of fuzzy logic into Markov systems represents a powerful tool for taking advantage of the symbolic knowledge that the experts of the systems possess
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