Fuzzy random renewal process and renewal reward process |
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Authors: | Ruiqing Zhao Wansheng Tang Cheng Wang |
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Institution: | (1) Institute of Systems Engineering, Tianjin University, Tianjin, 300072, China |
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Abstract: | So far, there have been several concepts about fuzzy random variables and their expected values in literature. One of the
concepts defined by Liu and Liu (2003a) is that the fuzzy random variable is a measurable function from a probability space
to a collection of fuzzy variables and its expected value is described as a scalar number. Based on the concepts, this paper
addresses two processes—fuzzy random renewal process and fuzzy random renewal reward process. In the fuzzy random renewal
process, the interarrival times are characterized as fuzzy random variables and a fuzzy random elementary renewal theorem
on the limit value of the expected renewal rate of the process is presented. In the fuzzy random renewal reward process, both
the interarrival times and rewards are depicted as fuzzy random variables and a fuzzy random renewal reward theorem on the
limit value of the long-run expected reward per unit time is provided. The results obtained in this paper coincide with those
in stochastic case or in fuzzy case when the fuzzy random variables degenerate to random variables or to fuzzy variables. |
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Keywords: | Renewal process Renewal reward process Fuzzy random variable Fuzzy variable Stochastic renewal process |
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