Study on Posbist Systems

The probability theory, in general, with the help of the dichotomous state develops the theory of reliability. Recently, the fuzzy reliability has been developed based on the concept of possibility distribution and fuzzy-state assumption. In this paper, we derive the possibility distribution function and discuss the properties of a -out-of- ( ) system based on the assumption of the possibility theory and keeping the dichotomous state of the system unchanged when the lifetime distribution is either normal, Cauchy, or exponential. A few results contrary to the conventional reliability theory are obtained. 1. Introduction It is well known that the conventional reliability theory is based on two fundamental assumptions (cf. Barlow and Proschan [1]): (a) probability assumption: the system (failure) behavior can be fully characterized in the context of probability measure; (b) binary state assumption: the meaning of system failure is defined precisely and thus at any time the system is in one of the two crisp states—fully functioning state and fully failed state. It is also called PROBIST reliability theory, since it is based on the probability and binary state assumption. Systems studied in the context of probist reliability theory are called probist systems. Although these two assumptions have been accepted in the past decades (since 1950s) and sound reasonable in extensive cases, it has been strongly argued that they are no longer the case in wide range of cases (cf. Zadeh [2, 3], Kai-Yuan et al. [4–6], and Cai [7]). As a consequence, three forms of fuzzy reliability theory are developed and discussed in the literature (cf. Cai [7]): (i) the PROFUST reliability theory—based on the probability assumption and the fuzzy state assumption, (ii) POSBIST reliability theory—based on the possibility assumption and the binary state assumption, (iii) the POSFUST reliability theory—based on the possibility assumption and the fuzzy state assumption. In this paper, we have dealt with the posbist reliability theory. To clarify the concept of fuzziness, Nahmias [8] proposed a theoretical framework based on fuzzy variable analogous to the random variable in the sample space model of the probability theory. Considering the lifetimes as fuzzy variables, Kai-Yuan et al. [5] derived the possibility distribution functions of a two-component series system and a two-component parallel system under the binary state assumption. Cross [9] considered the problem of evaluation of client-server networks as nested -out-of- systems ( ) with fuzzy reliability profiles. Huang et al. [10]