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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.

MSC:
60K05Renewal theory
References:
[1]Asmussen S. (1987). Applied probability and queues. Now York, Wiley
[2]Birolini A. (1994). Quality and reliability of technical systems. Berlin, Spring
[3]Daley D., Vere-Jones D. (1988). An introduction to the theory of point processes. Berlin, Springer
[4]Dozzi M. Merzbach E., Schmidt V. (2001). Limit theorems for sums of random fuzzy sets. Journal of Mathematical Analysis and Applications 259, 554–565 · Zbl 0988.60005 · doi:10.1006/jmaa.2000.7428
[5]Gao J., Liu B. (2001). New primiteve chance measures of fuzzy random event. International Journal of Fuzzy systems 3, 527–531
[6]Hwang C. (2000). A theorem of renewal process for fuzzy random variables and its application. Fuzzy Sets and Systems 116, 237–244 · Zbl 0966.60081 · doi:10.1016/S0165-0114(98)00143-2
[7]Kwakernaak H. (1978). Fuzzy random variables-I. Information Sciences 15, 1–29 · Zbl 0438.60004 · doi:10.1016/0020-0255(78)90019-1
[8]Kwakernaak H. (1979). Fuzzy random variables-II. Information Sciences 17, 253–278 · Zbl 0438.60005 · doi:10.1016/0020-0255(79)90020-3
[9]Kruse R., Meyer K. (1987). Statistics with vague data. Dordrecht, D. Reidel Publishing Company
[10]Li, X., & Liu, B. (2006). New independence definition of fuzzy random variable and random fuzzy variable. World Journal of Modeling and Simulation, to be published.
[11]Liu B. (2002). Theory and practice of uncertain programming. Heidelberg, Physica-Verlag
[12]Liu B. (2001a). Fuzzy random chance-constrained programming. IEEE Transactions on Fuzzy Systems 9, 713–720 · doi:10.1109/91.963757
[13]Liu B. (2001b). Fuzzy random dependent-chance programming. IEEE Transactions on Fuzzy Systems 9, 721–726 · doi:10.1109/91.963758
[14]Liu B. (2006). A survey of credibility theory. Fuzzy optimization and decision making 5, 387–408 · Zbl 1133.90426 · doi:10.1007/s10700-006-0016-x
[15]Liu B., Liu Y. (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems 10, 445–450 · doi:10.1109/TFUZZ.2002.800692
[16]Liu Y., Liu B. (2003a). Fuzzy random variables: a scalar expected value operator. Fuzzy Optimization and Decision Making 2, 143–160 · doi:10.1023/A:1023447217758
[17]Liu Y., Liu B. (2003b). Expected value operator of random fuzzy variable and random fuzzy expected value models. International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems 11, 195–215 · Zbl 1074.90056 · doi:10.1142/S0218488503002016
[18]Popova E., Wu H. (1999) Renewal reward processes with fuzzy rewards and their applications to T-age replacement policies. European Journal of Operational Research 117, 606–617 · Zbl 0937.90019 · doi:10.1016/S0377-2217(98)00247-1
[19]Puri M., Ralescu D. (1985) The concept of normality for fuzzy random variables. Annals of probability 13, 1371–1379 · Zbl 0583.60011 · doi:10.1214/aop/1176992822
[20]Puri M., Ralescu D. (1986) Fuzzy random variables. Journal of Mathematical Analysis and Applications 114, 409–422 · Zbl 0592.60004 · doi:10.1016/0022-247X(86)90093-4
[21]Ross S. (1996) Stochastic processes. New York, John Wiley & Sons
[22]Tijms H. (1994) Stochastic modelling and analysis: A computational approach. New York, Wiley
[23]Wang G., Zhang Y. (1992). The theory of fuzzy stochastic processes. Fuzzy sets and system 51, 161–178 · Zbl 0782.60039 · doi:10.1016/0165-0114(92)90189-B
[24]Zhao R., Liu B. (2003). Renewal process with fuzzy interarrival times and rewards. International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems 11, 573–586 · Zbl 1076.60516 · doi:10.1142/S0218488503002338
[25]Zhao R., Tang W. (2006). Some properties of fuzzy random processes. IEEE Transactions on Fuzzy Systems 2, 173–179 · Zbl 05452632 · doi:10.1109/TFUZZ.2005.864088