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Fuzzy random renewal process and renewal reward process. (English) Zbl 1138.60057
Stochastic renewal theory is a well developed part of the theory of stochastic processes. The first author and B. Liu [Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 11, No. 5, 573–586 (2003; Zbl 1076.60516)] have discussed renewal problems where interarrival times and rewards are fuzzy variables. In the present paper, the authors consider randomness and fuzziness simultaneously. Interarrival times and rewards are modelled as fuzzy random variables. Here the authors do not use the well established Puri-Ralescu approach where results on fuzzy random renewal processes are already avaiable [see e.g. C.-M. Hwang, Fuzzy Sets Syst. 116, No. 2, 237–244 (2000; Zbl 0966.60081); E. Popova, H.-C. Wu, Eur. J. Oper. Res. 117, No. 3, 606–617 (1999; Zbl 0937.90019)]. They use the B. Liu approach [see e.g. “Uncertainty theory.” An introduction to its axiomatic foundations. Studies in Fuzziness and Soft Computing 154. (Berlin): Springer. (2004; Zbl 1072.28012)] which is in some sense a defuzzified approach which leads for example to real valued expectations for fuzzy random variables. Using this Liu approach, the results are very close to the pure probabilistic results which can also seen in this paper where results are presented on the expected renewal rate and on the expected reward.

60K05Renewal theory
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