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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 6, DECEMBER 2016 1637 Uncertain Random Renewal Reward Process With Application to Block Replacement Policy Kai Yao and Jian Zhou Abstract—As a mixture of uncertain variable and random vari- tend to overestimate unlikely events (according to the prospect able, uncertain random variable is an important tool to describe theory by the Nobel prize winners Kahneman and Tversky [5]), indeterminacy phenomena. In order to model the evolution of un- the belief degree usually has a much larger range than the real certain random phenomena, a concept of uncertain random pro- cess has been proposed, and an uncertain random renewal process frequency. As a result, it cannot be treated by means of proba- has been designed as an example. This paper aims to propose a bility theory. So far, many other theories have been established new type of uncertain random process, called uncertain random to deal with this problem, such as fuzzy set theory (Zadeh [6]), renewal reward process, in which the interarrival times and the re- possibility theory (Zadeh [7], Dubois and Prade [8]), the un- wards are assumed to be random variables and uncertain variables, certainty theory by Liu [9], [10], and the chance theory by Liu respectively. The chance distribution of the renewal reward pro- cess is obtained, and the reward rate is derived. A renewal reward [11], [12]. This paper mainly studies the renewal reward system theorem is veri?ed, which shows that the reward rate converges within the framework of the chance theory. in distribution to an uncertain variable derived from the random Based on an uncertain measure which satis?es normality, du- interarrival times and the uncertain rewards. As an application, ality, subadditivity, and product axioms, an uncertainty theory this paper also proposes an uncertain random block replacement was established by