Modeling renewal processes in fuzzy decision system

Under expected value of fuzzy variable and continuous Archimedean triangular norms, this paper discusses a renewal process and a renewal reward process for T-independent L-R fuzzy variables in fuzzy decision systems. First, a renewal process with T-independent L-R fuzzy interarrival times is discussed, some limit theorems on renewal variable, average renewal time, and long-term renewal rate in (fuzzy) measure are obtained, and a fuzzy elementary renewal theorem is proved for the limit of the long-term expected renewal rate. Second, a renewal reward process with T-independent L-R fuzzy interarrival times and rewards is discussed, a limit theorem on reward rate in (fuzzy) measure is derived, and a fuzzy renewal reward theorem is proved for the limit value of expected reward rate. Finally, the comparison with stochastic counterparts shows an interesting and reasonable homology in convergence mode and limit value between the results obtained in fuzzy renewal processes and the corresponding results in stochastic renewal processes, though they build on two essentially different mathematical cornerstones, possibility theory and probability theory, respectively.