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 k-out-of-n (1 ≤ k ≤ n) 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.


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][5][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 k-out-of-n systems (1 ≤ k ≤ n) with fuzzy reliability profiles.Huang et al. [10] provided a new technique, called fault tree analysis to characterize posbist system's failure.Bhattacharjee et al. [11] dealt with the possibility distribution function of a few posbist systems.
In this paper, some of the basic concepts regarding the posbist reliability theory have been described in Section 2. In Section 3, the possibility distribution function of k-outof-n systems (1 ≤ k ≤ n) is stated.Section 4 deals with a few properties of posbist properties and compares them with the analogous results of the conventional reliability theory.A brief illustration of the component lifetimes having normal, Cauchy, or exponential fuzzy variables is given in Section 5. We conclude by some concluding remarks given in Section 6.Throughout the paper, we use

Basic Concepts of the Posbist Reliability Theory
The concept of the posbist reliability theory is introduced in Kai-Yuan et al. [5].Since the binary state assumption is reserved, the failure of a system (or component) is defined precisely and the system lifetime, denoted by X, is calculated from the instant the system starts functioning to the instant the system fails.However, since the possibility assumption is taken, the instant of system failure occurrence is characterized in the context of possibility measure.The lifetimes of a posbist system and its components are taken as nonnegative real-valued fuzzy variables.
The definitions are borrowed from Kai-Yuan et al. [5] and Nahmias [8].Nahmias [8] explores a possible axiomatic framework analogous to the sample space model of the probability theory, from which a rigorous theory of fuzziness can be constructed as given below.
Let Γ be an abstract space of generic elements γ ∈ Γ.The actual construction of Γ will depend upon the particular problem being modeled in much the same way as the construction of the sample space in probability depends upon the particular random experiment.Definition 1. Suppose that G is the discrete topology on Γ (that is, the class of all subsets of Γ).A scale σ, defined on G, satisfies the following axioms: (ii) σ(∪A α ) = sup σ(A α ) for any arbitrary collection of sets A α in G.
The triplet (Γ, G, σ) is called the pattern space.
Definition 2. A fuzzy variable X is a real-valued function defined on a pattern space (Γ, G, σ).
Definition 3. The possibility distribution function of a fuzzy variable X, denoted by μ X (x), is a mapping from R to the unit interval [0, 1] and is given by Further, X is said to be normal (respectively, convex or strictly convex) if the fuzzy set {(x, μ X (x)) : x ∈ R + } is normal (respectively, convex or strictly convex).
Definition 4. Given a pattern space (Γ, G, σ), the sets A 1 , . . ., A n ⊂ G are said to be mutually unrelated if Definition 5. Given a pattern space (Γ, G, σ), the fuzzy variables X 1 , . . ., X n are said to be mutually unrelated if the sets The lifetime of a posbist system (posbist component) is a nonnegative real-valued fuzzy variable.Definition 6. Posbist reliability R(t) of a system is the possibility that the system performs the functions properly during a predefined exposure period under a given environment; that is, ( Next, we state a lemma which will be used in proving the upcoming theorems in the next section.

Lemma 7. For any sets
where minimum of a set is defined as the elementwise minimum; that is, for We now state a lemma from Kai-Yuan et al. [5] to be used in proving the subsequent theorems.Lemma 8. Suppose X is the system lifetime defined on a possibility space (Γ, G, σ).Suppose that X is a strictly convex fuzzy variable with the possibility distribution function μ X (x) which is continuous.Then there exists a unique point, say x 0 (finite or infinite), such that

The Possibility Distribution Function of Posbist Systems
The following theorem gives the possibility distribution function of a k-out-of-n system in terms of those of the components, which generalize Kai-Yuan et al. [5] work giving the possibility distribution a parallel system with two components.
Assume that X j 's are normal unrelated and strictly convex fuzzy variables with the corresponding continuous possibility distribution function μ Xj (x), j = 1, 2, . . ., n.Let X denote the system lifetime.Then there exists a set of unique real numbers {a 1 , a 2 , . . ., a n }, a i ∈ R + for i = 1, 2, . . ., n such that the possibility distribution function of X, denoted by μ X (x), with is given by Proof.We consider the following: where Here, the suffixes of A denote the number of X i 's equal to x out of n number of X i 's and the rest are less than x.Note that in (8), represents a collection of X m ≤ x with 1 ≤ m ≤ n, and Thus, for 0 Further in (11), we find that International Journal of Quality, Statistics, and Reliability (12) is obtained by using (10) and the fact that 12) follows by using the fact that μ X (x) ≤ 1 for all x ∈ R + .Now two cases may arise.
We have to obtain the maximum of the minimum of the sets enlisted in (12).By Lemma 7, it follows that the maximum of the minimum of these sets will be the one which is the smallest subset of all the sets.The smallest subset will be the one for which {i 1 , i 2 , . . ., i k+s1 } is a permutation of {i + 1, i + 2, . . ., n}; that is, In that case, the set {μ Xi+1 (x), μ Xi+2 (x), . . ., μ Xn (x)} becomes the smallest subset in (12), thereby using Lemma 7 in (12); we conclude that μ X (x) = min μ Xj (x), j = i + 1, . . ., n . (14) This completes the proof.
Interestingly, the possibility distribution function as given in Theorem 9 for a k-out-of-n system is true only when k < n.For k = n, the theorem is stated below, which generalize Kai-Yuan et al. [5] work giving the possibility distribution a series system with two components.The proof is omitted for brevity.
Theorem 10.For a series system, let X j denote the lifetime of the jth component.Assume that X j 's are normal unrelated and strictly convex fuzzy variables with the corresponding continuous possibility distribution function μ Xj (x), j = 1, 2, . . ., n.Let X denote the system lifetime.Then there exists a set of unique real numbers {a 1 , a 2 , . . .a n }, a i ∈ R + for i = 1, 2, . . ., n, such that the possibility distribution function of X, denoted by (16)

Properties of the Posbist Systems
In this section, we study a few properties of the posbist systems and compare them with the analogous results of the conventional reliability theory.For the sake of brevity, we omit the proofs of the subsequent theorems.
Theorem 11.Assume that X j is a normal unrelated and strictly convex fuzzy variable denoting the lifetime of the jth component with corresponding continuous possibility distribution function μ Xj (x), j = 1, 2, . . ., n.Let the lifetime values of a k-out-of-n system and a (k + 1)-out-of-n system formed by the components X j , j = 1, 2, . . ., n, have the possibility distribution functions μ k:n (x) and μ k+1:n (x), respectively.Then, for where {a 1 , a 2 , . . ., a n }, a i ∈ R + , i = 1, 2, . . ., n, is a unique set of real numbers.
Corollary 12.Under the conditions stated in Theorem 11, it follows that where R k:n (x) and R k+1:n (x) denote the posbist reliability of kout-of-n and (k + 1)-out-of-n system, respectively.
Below we generalize Theorems 4.6 and 4.8 of Kai-Yuan et al. [5] where the results are proved for series and parallel systems, respectively.
Theorem 13.The reliability of a k-out-of-n system with an arbitrary number of unrelated identical components coincides with the reliability of a single component.
Proof.Let us denote the lifetime of a k-out-of-n system by X k:n having its components with lifetime represented by X i for i ∈ {1, 2, . . ., n}.Since the components are identical, we have μ Xi (x) = μ Xj (x) for 1 ≤ i / = j ≤ n.It follows from Theorem 9 and Definition 6 that μ Xk:n (x) = μ Xi (x), for i ∈ {1, 2, . . ., n}.
(19) Thus, the system reliability R k:n (x) of a k-out-of-n system is given by This completes the proof.
Remark 14.The reliability of a k-out-of-n system consisting of arbitrary number of unrelated identical components does not change due to the addition of an identical component as contrary to the conventional reliability theory.Similarly, we give the results for n-out-of-n system (series system with n components) using Theorem 10.
Theorem 15.The reliability of an n-out-of-n system of unrelated identical components coincides with the reliability of a single component.
Remark 16.The reliability of an n-out-of-n system of unrelated identical components does not change due to the addition of an identical component as contrary to the conventional reliability theory.
In conventional reliability, the reliability of a parallel system increases monotonically with the number of components contained in it, whereas the reliability of a series system decreases monotonically with the number of components contained in it.But these may not be true in case of the posbist reliability.

Illustration with a Few Examples
The change with time in the possibility distribution function and reliability function has been studied in this section when the component lifetimes are normal or Cauchy (cf.Cai [7]) as follows.
International Journal of Quality, Statistics, and Reliability (I) A fuzzy variable X is said to have the Cauchy possibility distribution function if for α > 0 and β > 0, is an even integer.Then the corresponding reliability function becomes Figure 1 shows the graph of the membership function for a k-out-of-10 system for different values of k when the component lifetimes are Cauchy fuzzy variables.
(II) A fuzzy variable X is said to have normal possibility distribution if its possibility distribution function is given by Thus, the reliability function becomes The membership function of a k-out-of-10 system when the component lifetimes are normal fuzzy variables is plotted in Figure 2 for different values of k.

Concluding Remarks
The work done in this paper on various posbist systems can be concluded in the following points.(i) The possibility distribution functions of 1-out-of-2 and 2-out-of-2 systems (given in Kai-Yuan et al. [5]) can be obtained (as special cases of the present work) from the possibility distribution functions of k-outof-n and n-out-of-n systems by using k = 1 and n = 2.
(ii) In the posbist reliability theory, the possibility distribution function and hence the reliability function of a series system, that is, n-out-of-n system, cannot be obtained from the possibility distribution function of a k-out-of-n system by using k = n, whereas in the conventional reliability theory, the reliability function of an n-out-of-n system can be obtained as a special case from the reliability function of a k-outof-n system.
(iii) From Figures 1 and 2, it is clear that the possibility distribution and hence the reliability function of a kout-of-n system are is greater than that of (k + 1)out-of-n system.This matches with the result of the conventional reliability theory.
(iv) It is also evident that the differences in the values of the possibility distribution function of a k-out-of-n system with different values of k become appreciable for large values of x.

10 Figure 1 :
Figure 1: Component lifetimes having the Cauchy possibility distribution function of k-out-of-n systems with n = 10.

Figure 2 :
Figure 2: Component lifetimes having normal possibility distribution function of k-out-of-n systems with n = 10.
+2, . . ., n}, and the rest r values come from {1, 2, . . ., i}.Since here we have to consider all possible combinations in which r of i 1 , i 2 , . . ., i k takes values from {1, 2, . . ., i}, it follows from (12) that Then the smallest subset will be the one for which k − r of i 1 , i 2 , . . ., i k is a permutation of {i +1, i