# Study on Posbist Systems

Authors: Subarna Bhattacharjee; Asok K. Nanda; S. S. Alam
Journal: International Journal of Quality, Statistics, and Reliability
 (2012)

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## Abstract

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 ak-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.


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## Body

## 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 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 Section2. In Section 3, the possibility distribution function of k-out-of-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 ℝ+=[0,∞).


## 2. 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.LetΓ 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 that𝒢 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:  (i)
σ(ϕ)=0,σ(Γ)=1;(ii)
σ(∪Aα)=supσ(Aα) for any arbitrary collection of sets Aα in 𝒢.The triplet (Γ,𝒢,σ) is called the pattern space.Definition 2.
A fuzzy variableX is a real-valued function defined on a pattern space (Γ,𝒢,σ).Definition 3.
The possibility distribution function of a fuzzy variableX, denoted by μX(x), is a mapping from ℝ to the unit interval [0,1] and is given by
(1)μX(x)=σ(γ:X(γ)=x)∀x∈ℝ.
Further, X is said to be normal (respectively, convex or strictly convex) if the fuzzy set {(x,μX(x)):x∈ℝ+} is normal (respectively, convex or strictly convex).Definition 4.
Given a pattern space(Γ,𝒢,σ), the sets A1,…,An⊂𝒢 are said to be mutually unrelated if σ(Ai1∩Ai2∩⋯∩Aik)=min(σ(Ai1),σ(Ai2),…,σ(Aik)), for 1≤i1≠i2≠⋯≠ik≤n and 1≤k≤n.Definition 5.
Given a pattern space(Γ,𝒢,σ), the fuzzy variables X1,…,Xn are said to be mutually unrelated if the sets {Xi1=x1},…,{Xik=xk} are unrelated for x1,…,xk∈ℝ whenever 1≤i1≠i2≠⋯≠ik≤n and 1≤k≤n.The lifetime of a posbist system (posbist component) is a nonnegative real-valued fuzzy variable.Definition 6.
Posbist reliabilityR(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,
(2)R(t)=σ(X>t)=supu>tμX(u),t∈ℝ+.Next, we state a lemma which will be used in proving the upcoming theorems in the next section.Lemma 7.
For any setsA1,A2,…,An with Ai⊆Ai+1⊆ℝ,i=1,2,…,n-1¯,
(3)max(min1≤i≤nAi)=minA1,
where minimum of a set is defined as the elementwise minimum; that is, for A={a1,…,an},minA=min1≤i≤nai.We now state a lemma from Kai-Yuan et al. [5] to be used in proving the subsequent theorems.Lemma 8.
SupposeX 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 μX(x) which is continuous. Then there exists a unique point, say x0 (finite or infinite), such that
(4)supu≥xμX(u)={μX(x0)ifx≤x0,μX(x)ifx>x0.


## 3. The Possibility Distribution Function of Posbist Systems

The following theorem gives the possibility distribution function of ak-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.Theorem 9.
For ak-out-of-n system (1≤k<n), let Xj denote the component lifetime of the jth component, j=1,2,…,n. Assume that Xj’s are normal unrelated and strictly convex fuzzy variables with the corresponding continuous possibility distribution function μXj(x),j=1,2,…,n. Let X denote the system lifetime. Then there exists a set of unique real numbers {a1,a2,…,an},ai∈ℝ+ for i=1,2,…,n such that the possibility distribution function of X, denoted by μX(x), with
(5)a1≤a2≤⋯≤ai<x≤ai+1≤ai+2≤⋯≤an
is given by
(6)μX(x)={min{μXj(x),j=i+1,…,n}ifn-i≥kmax[min{μXj(x),j=i+1,…,n,μl1(x),μl2(x),…,μlk-n+i(x),1≤l1<l2<⋯<lk-n+i≤iμXj(x)}]ifn-i<k.Proof.
We consider the following:(7)μX(x)=σ(X=x)=σ(eachofatleastkofX1,X2,…,Xnequalsx)=σ(⋃s=0n-kAk+s)=max0≤s≤n-k(σ(Ak+s)),
where
(8)Ak+s=⋃1≤i1≠i2≠⋯≠ik+s≤n{Xi1=x,Xi2=x,…,Xik+s=x,〈〈Xm≤x:1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉}
with 0≤s≤n-k. Here, the suffixes of A denote the number of Xi's equal to x out of n number of Xi’s and the rest are less than x. Note that in (8),
(9)〈〈Xm≤x:1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉
represents a collection of Xm≤x with 1≤m≤n, and m≠i1≠i2≠⋯≠ik+s.
Without loss of generality, we assume thata1≤a2≤⋯≤an. It is to be noted that, for p=1,2,…,n,
(10)supu≤xμXp(u)={μXp(x)ifx≤ap1ifx>ap.
Thus, for 0≤s≤n-k,(11)σ(Ak+s)=σ[⋃1≤i1≠i2≠⋯≠ik+s≤n{Xi1=x,Xi2=x,…,Xik+s=x,〈〈Xm≤x:1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉}⋃1≤i1≠i2≠⋯≠ik+s≤n{Xi1=x,Xi2=x,…,Xik+s=x,]=max1≤i1≠i2≠⋯≠ik+s≤n[σ(Xi1=x,Xi2=x,…,Xik+s=x,〈〈Xm≤x:1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉)]=max1≤i1≠i2≠⋯≠ik+s≤n[min{σ(Xi1=x),σ(Xi2=x),…,σ(Xik+s=x),〈〈σ(Xm≤x):1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉≫()σ(Xi1=x)}]=max1≤i1≠i2≠⋯≠ik+s≤n[min{〈〈supu≤xσ(Xm=u):1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉σ(Xi1=x),σ(Xi2=x),…,σ(Xik+s=x),〈〈supu≤xσ(Xm=u):1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉()max1≤i1≠i2≠⋯≠ik+s≤n}max1≤i1≠i2≠⋯≠ik+s≤n]=max1≤i1≠i2≠⋯≠ik+s≤n[〈〈supu≤xμXm(u):1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉min{〈〈supu≤xμXm(u):1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉μXi1(x),μXi2(x),…,μXik+s(x),〈〈supu≤xμXm(u):1≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉}]. Further in (11), we find that (12)σ(Ak+s)=max1≤i1≠i2≠⋯≠ik+s≤n[max0≤s≤n-k21min{max0≤s≤n-k21〈〈supu≤xμXm(u):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉μXi1(x),μXi2(x),…,μXik+s(x),〈〈supu≤xμXm(u):1≤m≤i,m≠i1≠i2≠⋯≠ik+s〉〉,〈〈supu≤xμXm(u):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉μXi1(x)}]=max1≤i1≠i2≠⋯≠ik+s≤n[min{μXi1(x),μXi2(x),…,μXik+s(x),1,〈〈μXm(x):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉μXi1(x)}]=max1≤i1≠i2≠⋯≠ik+s≤n[min{μXi1(x),μXi2(x),…,μXik+s(x),〈〈μXm(x):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉μXi1(x)}]=max0≤s≤n-k{max1≤i1≠i2≠⋯≠ik+s≤n{min{μXi1(x),μXi2(x),…,μXik+s(x),〈〈μXm(x):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik+s〉〉μXi1(x)}}max1≤i1≠i2≠⋯≠ik+s≤n}=max0≤s≤n-k{max1≤i1≠i2≠⋯≠ik≤n{min{μXi1(x),μXi2(x),…,μXik(x),〈〈μXm(x):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik〉〉μXi1(x)}},max1≤i1≠i2≠⋯≠ik+1≤n{min{μXi1(x),μXi2(x),…,μXik+1(x),〈〈μXm(x):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik+1〉〉μXi1(x)}},…,min{μX1(x),μX2(x),…,μXn(x)}max1≤i1≠i2≠…≠ik≤n}=max0≤s≤n-k{min1≤i1≠i2≠⋯≠ik≤n{μXi1(x),μXi2(x),…,μXik(x),〈〈μXm(x):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik〉〉μXi1(x)},min1≤i1≠i2≠⋯≠ik+1≤n{μXi1(x),μXi2(x),…,μXik+1(x),〈〈μXm(x):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik+1〉〉μXi1(x)},…,min{μX1(x),μX2(x),…,μXn(x)}min1≤i1≠i2≠…≠ik≤n}.(12) is obtained by using (10) and the fact that a1≤a2≤⋯≤ai<x≤ai+1≤ai+2≤⋯≤an, whereas (12) follows by using the fact that μX(x)≤1 for all x∈ℝ+. Now two cases may arise.
Case I
(
n
-
i
≥
k
). Let n-i=k+s1, for some integer s1≥0. 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 {i1,i2,…,ik+s1} is a permutation of {i+1,i+2,…,n}; that is,
(13){μXi1(x),μXi2(x),…,μXik+s1(x)}221={μXi+1(x),μXi+2(x),…,μXn(x)}.
Then {〈〈μXm(x):(i+1)≤m≤n,m≠i1≠i2≠⋯≠ik+1〉〉} is a void set. In that case, the set {μXi+1(x),μXi+2(x),…,μXn(x)} becomes the smallest subset in (12), thereby using Lemma 7 in (12); we conclude that
(14)μX(x)=min{μXj(x),j=i+1,…,n}.Case II(n-i<k).Let r=k-n+i. Then the smallest subset will be the one for which k-r of i1,i2,…,ik is a permutation of {i+1,i+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 i1,i2,…,ik takes values from {1,2,…,i}, it follows from (12) that
(15)μX(x)=max[min{μXj(x),j=i+1,…,n,μl1(x),μl2(x),…,μlk-n+i(x),1≤l1≠l2≠⋯≠lk-n+i≤iμXj(x)}].
This completes the proof.Interestingly, the possibility distribution function as given in Theorem9 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, letXj denote the lifetime of the jth component. Assume that Xj’s are normal unrelated and strictly convex fuzzy variables with the corresponding continuous possibility distribution function μXj(x), j=1,2,…,n. Let X denote the system lifetime. Then there exists a set of unique real numbers {a1,a2,…an},ai∈ℝ+ for i=1,2,…,n, such that the possibility distribution function of X, denoted by μX(x), with a1≤a2≤⋯≤ai<x≤ai+1≤ai+2≤⋯≤an is given by
(16)μX(x)={min{μXj(x):j=i,i-1,…,1}if1≤i≤n,max{μXj(x):j=1,2,…,n}ifx<a1≤a2≤⋯≤an.


## 4. 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 thatXj is a normal unrelated and strictly convex fuzzy variable denoting the lifetime of the jth component with corresponding continuous possibility distribution function μXj(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 Xj,j=1,2,…,n, have the possibility distribution functions μk:n(x) and μk+1:n(x), respectively. Then, for a1≤a2≤⋯≤ai<x≤ai+1≤ai+2≤⋯≤an, one has
(17)μk:n(x)≥μk+1:n(x),
where {a1,a2,…,an},ai∈ℝ+,i=1,2,…,n, is a unique set of real numbers.Corollary 12.
Under the conditions stated in Theorem11, it follows that
(18)Rk:n(x)≥Rk+1:n(x)forx∈ℝ+,
where Rk:n(x) and Rk+1:n(x) denote the posbist reliability of k-out-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 ak-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 ak-out-of-n system by Xk:n having its components with lifetime represented by Xi for i∈{1,2,…,n}. Since the components are identical, we have μXi(x)=μXj(x) for 1≤i≠j≤n. It follows from Theorem 9 and Definition 6 that
(19)μXk:n(x)=μXi(x),fori∈{1,2,…,n}.
Thus, the system reliability Rk:n(x) of a k-out-of-n system is given by
(20)Rk:n(x)=σ(Xk:n>x)=supu>xμXk:n(u)=supu>xμXi(u)=Ri(x).
This completes the proof.Remark 14.
The reliability of ak-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 forn-out-of-n system (series system with n components) using Theorem 10.Theorem 15.
The reliability of ann-out-of-n system of unrelated identical components coincides with the reliability of a single component.Remark 16.
The reliability of ann-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.


## 5. 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.(I) A fuzzy variableX is said to have the Cauchy possibility distribution function if
(21)μX(x)=11+α(x-a)β,
for α>0 and β>0, is an even integer. Then the corresponding reliability function becomes
(22)R(t)={1ift≤a,11+α(t-a)βift>a.
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.Figure 1
Component lifetimes having the Cauchy possibility distribution function ofk-out-of-n systems with n=10.(II) A fuzzy variableX is said to have normal possibility distribution if its possibility distribution function is given by
(23)μX(x)=exp[-k(x-a)2],k>0.
Thus, the reliability function becomes
(24)R(t)={1ift≤a,exp[-k(t-a)2]ift>a.
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.Figure 2
Component lifetimes having normal possibility distribution function ofk-out-of-n systems with n=10.


## 6. 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-out-of-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-out-of-n system.(iii)
From Figures1 and 2, it is clear that the possibility distribution and hence the reliability function of a k-out-of-n system areis 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 ak-out-of-n system with different values of k become appreciable for large values of x.



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