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The conventional reliability theory is built on the probability assumption and the binary-state assumption [1]. It has been successfully used for solving various reliability problems. However, it is not suitable when the failure probabilities concerned are very small (e.g., 10) or when there is a lack of sufficient data. As a result, researchers have been searching for new models and new reliability theories that overcome the shortcomings of the classical probabilistic definition of reliability. Among others, we mention the works by Tanaka et al. [2], Singer [3], Onisawa [4], Cappelle and Kerre [5], Cremona and Gao [6], Utkin and Gurov [7], Cai et al [1, 8, 9], Huang [10-12], and Huang et al [13-18]. All these researchers have attempted to define reliability in terms other than the probabilistic definition. The fuzzy set concept represents a new paradigm of accounting for uncertainty. Two new assumptions in these definitions include the fuzzy-state assumption and the possibility assumption. The fuzzy state assumption indicates that the state of a piece of equipment can be represented by a fuzzy variable. The possibility assumption indicates that the reliability of a piece of equipment needs to be measured subjectively. These two new assumptions have been used in place of the conventional probability and the binary-state assumption.

It is obvious that the reliability modeling is the most important discipline of reliable engineering (Kaufmann and Gupta, 1988). Traditionally, the reliability of a system’s behavior is fully characterized in the context of probability measures. However, because of the inaccuracy and uncertainties of data, the estimation of precise values of probability becomes very difficult in many systems (Chen, 1996). In recent years, some researchers have used the fuzzy set theory (Zadeh, 1965) for fuzzy system reliability analysis.

Theory of reliability is more than fifty years old. Its basic concepts were established in the 1950s as useful tools for the analysis of complex technical systems. The rapid development of the theory of reliability was closely related to the importance of its main field of applications - military and space. For this reason the origins of the research in the area of reliability are still not well known. Ralph A. Evans, one of the founders of the IEEE Transactions on Reliability, wrote in an Editorial in this journal that all important theoretical results published in the 1960s and 1970s had been already obtained even in the 1950s, and for many years remained classified. The authors of the most important publications on reliability from those years belonged to the group of the most important scientists working in theory of probability, mathematical statistics, electronics and computer sciences.

Theory and methodology of repairable system modeling is in nature a stochastic process modeling, particularly, point process modeling. Since Ascher and Feingold [2] foundational work in repairable system modeling, many works were contributing to this research field, for example, Anderson et al [1], Cox [7], Dagpunar [9], Kijima [53], and Guo et al [27].