Reliability Analysis in Uncertain Random System

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1 Reliability Analysis in Uncertain Random System Meilin Wen a,b, Rui Kang b a State Key Laboratory of Virtual Reality Technology and Systems b School of Reliability and Systems Engineering Beihang University, Beijing, , China wenmeilin@buaa.edu.cn, kangrui@buaa.edu.cn Abstract Reliability analysis of a system based on probability theory has been widely studied and used. Nevertheless, it sometimes meets with one problem that the components of a system may have only few or even no samples, so that we cannot estimate their probability distributions via statistics. Then reliability analysis of a system based on uncertainty theory has been proposed. However, in a general system, some components of the system may have enough samples while some others may have no samples, so the reliability of the system cannot be analyzed simply based on probability theory or uncertainty theory. In order to deal with systems of this type, this paper proposes a method of reliability analysis based on chance theory which is a generalization of both probability theory and uncertainty theory. In order to illustrate the method, some common systems are considered such as series system, parallel system and k-out-of-n system. Keywords: Uncertainty theory; Chance measure; Reliability; Boolean system 1 Introduction In order to describe the relationship between a system and the components consisting the system, Mine [19] first employed a Boolean function in 1959, in which the system and each component are set a value 0 or 1 according to whether they are in a failure or functioning state. Then Birnbaum et al [4] and Esary and Proschan [7] assumed each component in the system works with a given probability, and studied the system from mathematical aspect. After that, Hirsch et al [8] and Hochberg [9] considered the different intermediate performance levels between the perfectly functioning state and the complete failure state and generalized the Boolean system by means of multistate system. From then on, reliability analysis based on probability theory has got many significant achievements such as Ross [21], Koren et al [12], and Barlow and Proschan [3]. 1

2 Sometimes, the probabilities that the components work well are not known clearly. In 1975, Kaufman [11] first introduced fuzzy theory to system reliability design, proposing fuzzy reliability analysis. After that, Onisawa [20] analyzed the reliability of human-machine system, and Utkin [22] analyzed the reliability of a fuzzy repairable system. In addition, Cai et al [5, 6] studied some special fuzzy systems such as parallel and series systems involving fuzzy sets. Although reliability analysis based on probability theory and fuzzy theory finds many applications in daily life, it also comes across one problem. In many cases, we have no samples to infer the probability or the possibility of a component via statistics. In this case, we have to invite the experts to evaluate their belief degree that a component works well. In order to model the belief degree, an uncertainty theory was founded by Liu [13] in 2007, and refined by Liu [14] in Then Liu [16] defined a concept of reliability index for an uncertain system, and gave a formula to calculate the index. Following that, Wang [23] studied the structural reliability of a system based on uncertainty theory. In a general system, some components may have enough samples to ascertain their functioning probabilities, while some others may have no samples. Thus we have to combine stochastic reliability analysis and uncertain reliability analysis to deal with the system. In 2013, Liu [17, 18] proposed an uncertain random variable as a mixture of random variable and uncertain variable, and initialized a chance theory. In this paper, we aims at employing uncertain random variable to analyze the reliability of a complex system involving both randomness and uncertainty. The rest of this paper is organized as follows: In Section 2, some basic concepts about uncertain variable and uncertain random variable will be introduced. Section 3 will introduce the method to analyze the reliability of a stochastic system and an uncertain system. Then the concept of reliability index of an uncertain random system will be proposed in Section 4, and the reliability of a series system and that of a parallel system will be analyzed. A formula to calculate the reliability index of an uncertain random system will be given and illustrated by a k-out-of-n system, parallel-series system and series-parallel system in Section 5. At last, some remarks will be made in Section 6. 2 Preliminaries Probability theory is applicable when we have a large amount of samples, and uncertainty theory is applicable when we have no samples but belief degree from the experts. Chance theory, as a generalization of probability theory and uncertainty theory, is applicable for a complex system, of which some components have a large amount of samples, and some others have no samples. In this section, we will introduce some basic concepts and results about uncertainty theory and chance theory. 2

3 2.1 Uncertainty Theory Uncertainty theory was founded by Liu [13] in 2007 and refined by Liu [15] in 2010 to deal with human s belief degree. Definition 1 Let Γ be a nonempty set, and L be a σ-algebra over Γ. A set function M is called an uncertain measure if it satisfies the following three axioms, Axiom 1. M{Γ} = 1 for the universal set Γ. Axiom 2. M{Λ} + M{Λ c } = 1 for any event Λ L. Axiom 3. For every countable sequence of events Λ 1, Λ 2,, we have { } M Λ i M{Λ i }. In this case, the triple (Γ, L, M) is called an uncertainty space. Besides, in order to provide the operational law of uncertain events, another axiom named product axiom was proposed by Liu [14] in Axiom 4. Let (Γ k, L k, M k ) be uncertainty spaces for k = 1, 2, The product uncertain measure M is an uncertain measure satisfying { } M Λ k = M k {Λ k } k=1 k=1 where Λ k are arbitrarily chosen events from L k for k = 1, 2,, respectively. Definition 2 (Liu [13]) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, we have {ξ B} = {γ Γ ξ(γ) B} L. (1) Definition 3 (Liu [13]) The uncertainty distribution Φ of an uncertain variable ξ is defined by for any real number x. Φ(x) = M{ξ x} (2) If the uncertainty distribution Φ(x) of ξ has an inverse function Φ 1 (α) for α (0, 1), then ξ is called a regular uncertain variable, and Φ 1 (α) is called the inverse uncertainty distribution of ξ. Inverse uncertainty distribution plays an important role in the operations of independent uncertain variables. 3

4 Definition 4 (Liu [13]) The uncertain variables ξ 1, ξ 2,, ξ n are said to be independent if { n } M (ξ i B i ) = M {ξ i B i } (3) for any Borel sets B 1, B 2,, B n of real numbers. Theorem 1 (Liu [15]) Let ξ 1, ξ 2,, ξ n be independent regular uncertain variables with uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. Assume the function f(x 1, x 2,, x n ) is strictly increasing with respect to x 1, x 2,, x m, and strictly decreasing with respect to x m+1, x m+2,, x n. Then the uncertain variable ξ = f(ξ 1, ξ 2,, ξ n ) has an inverse uncertainty distribution Φ 1 (α) = f ( Φ 1 1 (α),, Φ 1 m (α), Φ 1 m+1 (1 α),, Φ 1 n (1 α) ). Theorem 2 (Liu [15]) Assume that ξ 1, ξ 2,, ξ n are independent Boolean uncertain variables, i.e., 1 with uncertain measure a i ξ i = (4) 0 with uncertain measure 1 a i for i = 1, 2,, n. If f is a Boolean function, then ξ = f(ξ 1, ξ 2,, ξ n ) is a Boolean uncertain variable such that f(x 1,x 2,,x n)=1 min ν i(x i ), M{ξ = 1} = if 1 f(x 1,x 2,,x n)=1 f(x 1,x 2,,x n)=0 min ν i(x i ) < 0.5 min ν i(x i ), (5) if f(x 1,x 2,,x n)=1 where x i take values either 0 or 1, and ν i are defined by a i, if x i = 1 ν i (x i ) = 1 a i, if x i = 0 min ν i(x i ) 0.5 (6) for i = 1, 2,, n, respectively. Definition 5 (Liu [13]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by E[ξ] = 2.2 Chance Theory + M{ξ r}dr 0 0 M{ξ r}dr. (7) Chance theory, as a generalization of uncertainty theory and probability theory, was founded by Liu [17, 18] to deal with a system exhibiting both randomness and uncertainty. The basic concept is the chance measure of an uncertain random event in a chance space. 4

5 Let (Γ, L, M) be an uncertainty space, and (Ω, A, Pr) be a probability space. Then (Γ, L, M) (Ω, A, Pr) = (Γ Ω, L A, M Pr) is called a chance space. Definition 6 (Liu [17]) Let (Γ, L, M) (Ω, A, Pr) be a chance space, and Θ L A be an uncertain random event. Then the chance measure Ch of Θ is defined by Ch{Θ} = 1 0 Pr{ω Ω M{γ Γ (γ, ω) Θ} r}dr. Definition 7 (Liu [17]) An uncertain random variable ξ is a measurable function from a chance space (Γ, L, M) (Ω, A, Pr) to the set of real numbers, i.e., {ξ B} = {(γ, ω) ξ(γ, ω) B} is an uncertain random event for any Borel set B. Example 1 Let τ 1, τ 2,, τ n be independent uncertain variables, and p 1, p 2,, p n be nonnegative numbers with p 1 + p p n = 1. Then ξ = is an uncertain random variable. τ 1 with probability measure p 1 τ 2 with probability measure p 2 τ n with probability measure p n (8) Definition 8 (Liu [17]) Let ξ be an uncertain random variable. Then its chance distribution is defined by Φ(x) = Ch{ξ x} (9) for any x R. Example 2 Let τ 1, τ 2,, τ n be independent uncertain variables with uncertainty distributions Υ 1, Υ 2,, Υ n, respectively. Then the uncertain random variable τ 1 with probability measure p 1 τ 2 with probability measure p 2 ξ = τ n with probability measure p n (10) has a chance distribution n Φ(x) = p i Υ i (x). (11) 5

6 Theorem 3 (Liu [18]) Let η 1, η 2,, η m be independent random variables with probability distributions Ψ 1, Ψ 2,, Ψ m, respectively, and τ 1, τ 2,, τ n be uncertain variables. Then the uncertain random variable ξ = f(η 1, η 2,, η m, τ 1, τ 2,, τ n ) (12) has a chance distribution Φ(x) = F (x; y 1, y 2,, y m )dψ 1 (y 1 )dψ 2 (y 2 ) dψ m (y m ) R m (13) where F (x; y 1, y 2,, y m ) is the uncertainty distribution of the uncertain variable f(y 1, y 2,, y m, τ 1, τ 2,, τ n ) for any real numbers y 1,, y m. 3 Reliability of Simplex System A system is called a Boolean system, if the system and its components can only be in one of two states: working or failure. Assume the system is comprised of n components, and each of them is denoted by a Boolean variable ξ i for i = 1, 2,, n, where 1, if component i works ξ i = 0, if component i fails. If ξ i is a Boolean random variable, i.e., 1 with probability measure a i ξ i = 0 with probability measure 1 a i, (14) (15) then we say ξ i is a random component with reliability a i. If ξ i is a Boolean uncertain variable, i.e., 1 with uncertain measure b i ξ i = (16) 0 with uncertain measure 1 b i, then we say ξ i is an uncertain component with reliability b i. Let a Boolean variable ξ denote the state of the system, and 1, if the system works ξ = 0, if the system fails. In order to build a bridge between the reliability of the components and that of the system, we employ a concept of structure function. A n-ary function f is called a Boolean function if it maps {0, 1} n to {0, 1}. It is usually used to model the structure of a Boolean system. (17) Definition 9 Assume that a Boolean system ξ is comprised of n components ξ 1, ξ 2,, ξ n. Boolean function f is called its structure function if Then a ξ = 1 if and only if f(ξ 1, ξ 2,, ξ n ) = 1. (18) 6

7 Obviously, when f is the structure function of the system, we also have ξ = 0 if and only if f(ξ 1, ξ 2,, ξ n ) = 0. If the components are independent random variables with reliabilities a 1, a 2,, a n, then where Pr{ξ = 1} = f(y 1,y 2,,y n)=1 a i if y i = 1 µ i (y i ) = 1 a i if y i = 0, n µ i (y i ) (i = 1, 2,, n). (19) If the components are independent uncertain variables with reliabilities b 1, b 2,, b n, then min ν i(z i ), f(z 1,z 2,,z n)=1 M{ξ = 1} = if 1 f(z 1,z 2,,z n)=1 f(z 1,z 2,,z n)=0 min ν i(z i ) < 0.5 min ν i(z i ), if f(z 1,z 2,,z n)=1 min ν i(z i ) 0.5 where b j if z j = 1 ν j (z j ) = 1 b j if z j = 0 (j = 1, 2,, n). (20) 4 Reliability of Uncertain Random System In a complex system, some components may have enough samples to estimate their probability distributions, and can be regarded as random variables, while some others may have no samples, and can only be evaluated by the experts and regarded as uncertain variables. Under this case, the system cannot be simply modeled by a stochastic system or an uncertain system. In this section, we will employ uncertain random variable to model the system, and analyze its reliability based on chance theory which is a generalization of probability theory and uncertainty theory. Definition 10 The reliability index of an uncertain random system ξ is defined as the chance measure that the system is working, i.e., Reliability = Ch{ξ = 1}. (21) If all uncertain random components degenerate to random ones, then the reliability index is the probability measure that the system is working; If all uncertain random components degenerate to uncertain ones, then the reliability index is the uncertain measure that the system is working. 7

8 Example 3 (Series System) Consider a series system containing independent random components ξ 1, ξ 2,, ξ m with reliabilities a 1, a 2,, a m, and independent uncertain components η 1, η 2,, η n with reliabilities b 1, b 2,, b n, respectively. Since the structure function is ( m ) f(ξ 1,, ξ m, η 1,, η n ) = ξ i, η j we have ( m ) Reliability = Ch ξ i η j = 1 { m } n = Pr (ξ i = 1) M (η j = 1) ( m ) = Pr{ξ i = 1} M{η j = 1} ( m ) = a i b j. Remark: If the series system degenerates to a system containing only random components ξ 1, ξ 2,, ξ m with reliabilities a 1, a 2,, a m, then m Reliability = a i. (22) If the series system degenerates to a system containing only uncertain components η 1, η 2,, η n with reliabilities b 1, b 2,, b n, then Reliability = b j. (23) Example 4 (Parallel System) Consider a parallel system containing independent random components ξ 1, ξ 2,, ξ m with reliabilities a 1, a 2,, a m, and independent uncertain components η 1, η 2,, η n with reliabilities b 1, b 2,, b n, respectively. Since the structure function is ( m ) n f(ξ 1,, ξ m, η 1,, η n ) = ξ i, η j 8

9 we have ( m ) n Reliability = Ch ξ i η j = 1 ( m ) n = 1 Ch ξ i η j = 0 { m } n = 1 Pr (ξ i = 0) M (η j = 0) ( m ) = 1 Pr{ξ i = 0} M{η j = 0} ( m ) = 1 (1 a i ) (1 b j ). Remark: If the parallel system degenerates to a system containing only random components ξ 1, ξ 2,, ξ m with reliabilities a 1, a 2,, a m, then m Reliability = 1 (1 a i ). (24) If the series system degenerates to a system containing only uncertain components η 1, η 2,, η n with reliabilities b 1, b 2,, b n, then n Reliability = 1 (1 b j ) = b j. (25) 5 Reliability Index Formula This section aims at giving a formula to calculate the reliability of a system involving both random variables and uncertain variables. Theorem 4 Assume that a Boolean system has a structure function f and contains independent random components η 1, η 2,, η m with reliabilities a 1, a 2,, a m, respectively, and independent uncertain components τ 1, τ 2,, τ n with reliabilities b 1, b 2,, b n, respectively. Then the reliability index of the uncertain random system is ( m ) Reliability = µ i (y i ) Z(y 1, y 2,, y m ) (26) (y 1,,y m) {0,1} m 9

10 where f(y 1,,y m,z 1,,z n)=1 min ν j(z j ), 1 j n if Z(y 1,, y m ) = f(y 1,,y m,z 1,,z n)=1 1 f(y 1,,y m,z 1,,z n)=0 min ν j(z j ) < j n min ν j(z j ), 1 j n (27) if f(y 1,,y m,z 1,,z n)=1 a i if y i = 1 µ i (y i ) = 1 a i if y i = 0, b j if z j = 1 ν j (z j ) = 1 b j if z j = 0 min ν j(z j ) 0.5, 1 j n (i = 1, 2,, m), (28) (j = 1, 2,, n). (29) Proof: It follows from Definition 9 of structure function and Definition 10 of reliability index that Reliability = Ch{f(η 1,, η m, τ 1,, τ n ) = 1}. By the operational law of uncertain random variables, we have ( m ) Reliability = µ i (y i ) M{f(y 1,, y m, τ 1,, τ n ) = 1}. (y 1,,y m) {0,1} m Note that the function Z(y 1,, y m ) is just M{f(y 1,, y m, τ 1,, τ n ) = 1} by Theorem 2, and we complete the proof. Example 5 (k-out-of-n System) Consider a k-out-of-n system containing independent random components ξ 1, ξ 2,, ξ m with reliabilities a 1, a 2,, a m, respectively, and independent uncertain components η 1, η 2,, η n m with reliabilities b 1, b 2,, b n m, respectively. Note that the structure function is m 1, if y i + n m z j k f(y 1, y 2,, y m, z 1, z 2,, z n m ) = m 0, if y i + n m z j < k. (30) It follows from Theorem 4 that the reliability of the uncertain random system is ( m ) Reliability = µ i (y i ) Z(y 1, y 2,, y m ) (31) (y 1,,y m) {0,1} m in which a i if y i = 1 µ i (y i ) = 1 a i if y i = 0, (32) 10

11 m n m Z(y 1, y 2,, y m ) = M y i + η j k the (k m y i )th largest value of {b 1, b 2,, b n m }, if = 1, if m y i < k m y i k. Remark: If the k-out-of-n system degenerates to a system containing only random components ξ 1, ξ 2,, ξ n with reliabilities a 1, a 2,, a n, then Reliability = y 1+ +y n k ( n ) µ i (y i ). (33) If the k-out-n system degenerates to a system containing only uncertain components η 1, η 2,, η n with reliabilities b 1, b 2,, b n, then Reliability = the kth largest value of {b 1, b 2,, b n }. (34) Example 6 (Parallel-Series System) Consider a simple parallel-series system in Figure 1 containing independent random components ξ 1, ξ 2 with reliabilities a 1, a 2, respectively, and independent uncertain components η 1, η 2 with reliabilities b 1, b 2, respectively. Note that the structure function is ξ 1 ξ 2 η 1 η 2 Figure 1: Parallel-Series System f(ξ 1, ξ 2, η 1, η 2 ) = (ξ 1 η 1 ) (ξ 2 η 2 ). It follows from Theorem 4 that the reliability index is Reliability = Ch{(ξ 1 η 1 ) (ξ 2 η 2 ) = 1} = Pr{ξ 1 = 1, ξ 2 = 1} Z(1, 1) + Pr{ξ 1 = 1, ξ 2 = 0} Z(1, 0) + Pr{ξ 1 = 0, ξ 2 = 1} Z(0, 1) + Pr{ξ 1 = 0, ξ 2 = 0} Z(0, 0) = a 1 a 2 Z(1, 1) + a 1 (1 a 2 ) Z(1, 0) + (1 a 1 )a 2 Z(0, 1) + (1 a 1 )(1 a 2 ) Z(0, 0) where Z(1, 1) = M{(1 η 1 ) (1 η 2 ) = 1} = M{1 1 = 1} = 1, Z(1, 0) = M{(1 η 1 ) (0 η 2 ) = 1} = M{1 η 2 = 1} = M{η 2 = 1} = b 2, Z(0, 1) = M{(0 η 1 ) (1 η 2 ) = 1} = M{η 1 1 = 1} = M{η 1 = 1} = b 1, Z(0, 0) = M{(0 η 1 ) (0 η 2 ) = 1} = M{η 1 η 2 = 1} = b 1 b 2. 11

12 Thus, the reliability index of the parallel-series system is Reliability = a 1 a 2 + a 1 (1 a 2 )b 2 + (1 a 1 )a 2 b 1 + (1 a 1 )(1 a 2 )(b 1 b 2 ). Example 7 (Series-Parallel System) Consider a simple series-parallel system in Figure 2 containing independent random components ξ 1, ξ 2 with reliabilities a 1, a 2, respectively, and independent uncertain components η 1, η 2 with reliabilities b 1, b 2, respectively. Note that the structure function is ξ 1 η 1 ξ 2 η 2 Figure 2: Series-Parallel System f(ξ 1, ξ 2, η 1, η 2 ) = (ξ 1 η 1 ) (ξ 2 η 2 ). It follows from Theorem 4 that the reliability index is Reliability = Ch{(ξ 1 η 1 ) (ξ 2 η 2 ) = 1} = Pr{ξ 1 = 1, ξ 2 = 1} Z(1, 1) + Pr{ξ 1 = 1, ξ 2 = 0} Z(1, 0) + Pr{ξ 1 = 0, ξ 2 = 1} Z(0, 1) + Pr{ξ 1 = 0, ξ 2 = 0} Z(0, 0) = a 1 a 2 Z(1, 1) + a 1 (1 a 2 ) Z(1, 0) + (1 a 1 )a 2 Z(0, 1) + (1 a 1 )(1 a 2 ) Z(0, 0) where Z(1, 1) = M{(1 η 1 ) (1 η 2 ) = 1} = M{η 1 η 2 = 1} = b 1 b 2, Z(1, 0) = M{(1 η 1 ) (0 η 2 ) = 1} = M{η 1 0 = 1} = M{η 1 = 1} = b 1, Z(0, 1) = M{(0 η 1 ) (1 η 2 ) = 1} = M{0 η 2 = 1} = M{η 1 = 2} = b 2, Z(0, 0) = M{(0 η 1 ) (0 η 2 ) = 1} = M{0 0 = 1} = 0. Thus, the reliability index of the series-parallel system is Reliability = a 1 a 2 (b 1 b 2 ) + a 1 (1 a 2 )b 1 + (1 a 1 )a 2 b 2. 6 Conclusion This paper mainly proposed reliability analysis for an uncertain random system. An index was given to indicate the reliability degree of the system via the structure function, and a formula was derived to calculate the index. Besides, some common systems such as k-out-of-n system, parallel-series system, and series-parallel system were used to illustrate the formula. 12

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