Reliability analysis of discrete time redundant system with imperfect switch and random uncertain lifetime

Abstract

The objective of this paper is to establish some reliability models for redundant systems based on the assumption that the conversion switches are imperfect and distribution parameters are uncertain variables. Some new concepts of random uncertain distributions associated with random uncertain variables are proposed, which are applied to redundant series-parallel systems, including cold redundant system and warm redundant system. In each type of redundant system, we consider two methods to describe the switch lifetimes: random uncertain 0-1 switch lifetime and random uncertain geometric switch lifetime. The reliability and the mean time to failure of these systems are analyzed. Some numerical examples are presented to demonstrate the proposed reliability models and perform a comparison for the system models with uncertain parameters and constant parameters.

Keywords

Redundant system Imperfect switch Uncertain variable Reliability MTTF

1 Introduction

System reliability theory takes the lifetime characteristic of a system as the main research object, which is important in product analysis. For unrepairable system, the reliability function and mean time to failure (MTTF) are the main indicators for describing the reliability characteristics. The traditional reliability theory [1 –5] has been widely and successfully used for solving various reliability problems, in which the lifetimes of the units or systems are assumed as non-negative random variables. The probability theory is the main mathematical tool to study this kind of problem.

Probability theory has a particularly wide range of applications in science and engineering, however, the application of probability theory requires that the estimated probability is close enough to the real frequency. In practical, there are usually lack of observed data for technical or some other reasons, obviously, it is not suitable to use probability theory to deal with such cases. In 1965, Zadeh [6] proposed fuzzy theory as a tool to deal with such matters, and some concepts of fuzzy sets were initiated. Thus, lots of researchers devoted to analyzing system reliability based on fuzzy theory, such as Onisawa [7], Park [8], Cai et al. [9], and so on.

Although probability theory and fuzzy theory have been extensively applied in reliability analysis, a lot of surveys showed that many uncertainty behaves were neither randomness nor fuzziness. In order to deal with human uncertainty phenomena, Liu [10] founded the uncertainty theory in 2007 and Liu [11] refined it in 2010. Numerous concepts of fundamentals were proposed by Liu in [10], such as uncertain variable for modeling the uncertain quantity, uncertain distribution for describing an uncertain variable. Furthermore, the inverse uncertainty distribution was presented to be described the relationship between uncertain measure and uncertainty distribution. As an important contribution, Liu and Ha [13] derived some useful expressions of expected value of uncertain variables. The uncertainty theory has become a new tool to describe subjective uncertainty and has a general application both in theory and engineering. For instance, uncertain programming [14], uncertain statistics [15], uncertain differential equation [16, 17], uncertain risk analysis [18], uncertainty reliability analysis [19–25 , 30], and so on.

As an application of uncertainty theory, uncertainty reliability analysis was presented by Liu [10] which was a method to deal with system reliability via uncertainty theory. Liu [20] analyzed the reliability of redundant systems and discussed that the conversion switches of systems were absolutely reliable or non-absolutely reliable. Liu et al. [21] established some fundamental mathematical models of unrepairable systems based on the assumption that the system lifetimes were uncertain variables. Gao et al. [22] illustrated some formulas for calculating the reliability index of an uncertain weighted k-out-of-n system. Zeng et al. [23] discussed the belief reliability based on uncertainty for series, parallel, k-out-n: G system. Hu et al. [24] studied the parallel-series standby system with uncertain lifetimes, and provided a simulation optimization algorithm to optimize the system model.

However, in some cases, human uncertainty and objective randomness may coexist in a complex system. In order to describe this complex phenomenon, in 2013, Liu [26] pioneered chance theory by defining uncertain random variable and chance measure to describe a system that involved both uncertainty theory and probability theory. Besides, different from uncertain random variable, a random uncertain variable was also defined in [24]. Thus, the uncertain random programming [27], uncertain random risk analysis [28] and the expected loss of uncertain random system [29] were analyzed by Liu et al. Wen and Kang [30] introduced a method of reliability analysis based on chance theory, and some common systems were used to illustrate the formula of reliability index. Gao and Yao [31] studied the importance index to model the importance extent of a component in an uncertain random reliability system. Gao et al. [32] discussed the reliability of k-out-of-n system which was assumed that the components have uncertain random lifetimes. Zhang et al. [33] introduced the belief reliability for uncertain random systems and proposed some system belief reliability formulas. In practical engineering, the lifetimes of some system units have independent and identical random distribution functions which may be determined by previous experience. However, due to the influence of external uncertain environment, there are no sample data to ascertain the distribution parameter of each unit lifetime. Thus, Cao et al. [34] considered a series of discrete time series-parallel systems and standby systems with uncertain parameters based on uncertainty theory and probability theory.

In this paper, we consider two discrete time redundant series-parallel systems with imperfect switches based on probability theory and uncertainty theory, besides, the reliability indexes of these systems are analyzed. The rest of this paper is organized as follow: In Section 2, some basic concepts and theorems of uncertain variable and random uncertain variable are given. In Section 3, some new concepts of system reliability indexes are defined, and the redundant series-parallel system with imperfect switch is illustrated. Section 4 considers a cold redundant series-parallel system that the conversion switch is non-absolutely with random uncertain 0-1 mode or random uncertain geometric mode. The warm redundant series-parallel system with imperfect switch is discussed in Section 5. Some numerical examples are illustrated in Section 6.

2 Preliminary

Let Γ be a nonempty set, and let $L$ be a σ-algebra over Γ. Each element Λ in $L$ is called an event. A set function $M$ from $L$ to [0, 1] is called an uncertain measure if it satisfies the following axioms [10]:

Axiom 1. (Normality Axiom) $M {Γ} = 1$ for the universal set Γ.

Axiom 2. (Duality Axiom) $M {Λ} + M {Λ^{c}} = 1$ for any event Λ.

Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯, we have $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$ The triplet $(Γ, L, M)$ is called an uncertain space. In addition, the product uncertain measure was defined by Liu [12] via the following product axiom:

Axiom 4. (Product Axiom) Let $(Γ_{k}, L_{k}, M_{k})$ be uncertainty spaces for k = 1, 2, ⋯ . The product uncertain measure $M$ is an uncertain measure satisfying $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}},$ where Λ_k are arbitrarily chosen events from $L_{k}$ for k = 1, 2, ⋯, respectively.

Definition 1. [10] An uncertain variable is a function ξ from an uncertainty space $(Γ, L, M)$ to the set of real numbers such that {ξ∈ B } is an event for any Borel set B.

Definition 2. [10] The uncertainty distribution Φ of an uncertain variable ξ is defined by $Φ (x) = M {ξ \leq x}$ for any real number x.

Definition 3. [10] Let ξ be an uncertain variable with regular uncertainty distribution Φ (x). Then the inverse function Φ^-1 (α) is called the inverse uncertainty distribution of ξ.

Theorem 1. [10] Let ξ₁, ξ₂, ⋯ , ξ_n be uncertain variables, and let f be a real-valued measurable function. Then f (ξ₁, ξ₂, ⋯ , ξ_n) is an uncertain variable.

Theorem 2. [13] Assume ξ₁, ξ₂, ⋯ , ξ_n are independent uncertain variables with regular uncertainty distribution Φ₁, Φ₂, ⋯ , Φ_n, respectively. If f (ξ₁, ξ₂, ⋯ , ξ_n) is strictly increasing with respect to ξ₁, ξ₂, ⋯ , ξ_m and strictly decreasing with respect to ξ_m+1, ξ_m+2, ⋯ , ξ_n, then $ξ = f (ξ_{1}, ξ_{2}, \dots, ξ_{n})$ has an expected value $\begin{matrix} E (ξ) = \int_{0}^{1} f (Φ_{1}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), \dots Φ_{n}^{- 1} (1 - α)) d α . \end{matrix}$

Uncertainty and randomness are two basic types of indeterminacy. Chance theory was pioneered by Liu [26], which is a mathematical methodology for modeling complex systems with both uncertainty and randomness.

Let $(Ω, A, Pr)$ be a probability space and let $(Γ, L,$ $M)$ be an uncertainty space. Then the product $(Ω, A,$ $Pr) \times (Γ, L, M)$ is called a chance space. Essentially, it is another triplet, $(Ω \times Γ, A \times L, Pr \times M)$ where Ω × Γ is the universal set, $A \times L$ is the product, and $Pr \times M$ is the product measure.

Definition 4. [26] A random uncertain variable is a function ξ from an uncertainty space $(Γ, L, M)$ to the set of random variables such that Pr{ ξ (γ) ∈ B } is a measurable function of γ for any Borel set B of $R$ .

Definition 5. [26] Let ξ be a random uncertain variable in chance space, and let B be a Borel set of $R$ . Then the chance of random uncertain event ξ ∈ B is defined by $\begin{matrix} Ch {ξ \in B} = \\ \int_{0}^{1} M {γ \in Γ | Pr {ω \in Ω | ξ (ω, γ) \in B} \geq r} dr . \end{matrix}$

3 Problem statement

For the redundant system in random uncertain environment, some new concepts of system reliability indexes will be defined in this section. In addition, two redundant series-parallel systems with imperfect switches and random uncertain lifetimes will be illustrated.

3.1 Reliability index

Definition 6. A discrete random uncertain variable X is said to have a random uncertain geometric distribution with uncertain parameter ξ, if it has a probability mass function of the form $P_{ξ} (k) = ξ {(1 - ξ)}^{k - 1}, k = 1, 2, \dots,$ where ξ is an uncertain variable on the uncertainty space $(Γ, L, M)$ . If X has a random uncertain geometric distribution, we write $X \sim Ge (ξ)$ .

Definition 7. A discrete random uncertain variable X is said to have a discrete random uncertain Weibull distribution if it has a probability mass function of the form $P_{η} (k) = η^{{(k - 1)}^{β}} - η^{k^{β}}, k = 1, 2, \dots,$ where η is an uncertain variable on the uncertainty space $(Γ, L, M)$ , and it is the probability of surviving the first demand, β is a shape parameter. If X has a discrete random uncertain Weibull distribution, we write $X \sim W (η, β)$ .

Definition 8. The lifetime X of a system is said to random uncertain lifetime, if X is a random uncertain variable.

Definition 9. Let the system lifetime X be a random uncertain variable having a probability distribution function with uncertain parameter ξ. For any k = 1, 2, ⋯, the system reliability at time k is defined as $\begin{matrix} R (k) = Ch {X > k} \\ = \int_{0}^{1} M {γ \in Γ | Pr {ω \in Ω | X (ω, γ) > k} \geq r} dr . \end{matrix}$ Then we denote Pr{ ω ∈ Ω|X (ω, γ) > k } as R (k, ξ), which is called uncertain reliability variable, that is $R (k) = E [Pr {ω \in Ω | X (ω, γ) > k}] = E [R (k, ξ)] .$

Definition 10. The mean time to failure of the discrete system with uncertain parameter is defined as $\begin{matrix} MTTF & = \int_{0}^{+ \infty} Ch {X > k} dk \\ = \sum_{k = 1}^{+ \infty} k \cdot E [R (k, ξ)] . \end{matrix}$

Definition 11. The random uncertain 0-1 switch lifetime is defined that the reliability of the switch is an uncertain variable ξ^S on the uncertainty space $(Γ, L, M)$ , and the probability of switch failure is 1 - ξ^S, when the switch is used each time.

Definition 12. Let the lifetime X of the switch be a random uncertain variable with random uncertain geometric distribution $X \sim Ge (ξ)$ , we say that the switch is random uncertain geometric switch lifetime.

3.2 System description

The discrete time redundant series-parallel system with uncertain parameters supported by imperfect switch is considered in this paper. There are two main classifications of redundant systems, namely cold and warm. In each type of the redundant systems, conversion switches are imperfect with random uncertain 0-1 switch lifetimes or random uncertain geometric switch lifetimes. Here, we suppose that the redundant series-parallel system is consisted of non-repairable and non-identical units and several similar switches, as shown in Fig. 1. In subsystem ij (i = 1, 2, ⋯ , n, j = 1, 2, ⋯ , m_i), the original unit U_ij starts its operation and the redundant unit enters to the idle state simultaneously. In addition, the cold redundant unit and the warm redundant unit of U_ij are denoted as $U_{ij}^{C}$ and $U_{ij}^{W}$ . In redundant series-parallel systems, there are conversion switches S_ij or K_ij which are random uncertain 0-1 switch or random uncertain geometric switch, respectively.

Fig.1

Redundant series-parallel system.

Let $X_{ij}, X_{ij}^{C}$ and $X_{ij}^{W} (i = 1, 2, \dots, n, j = 1, 2, \dots, m_{i})$ denote the lifetimes for $U_{ij}, U_{ij}^{C}$ and $U_{ij}^{W}$ , respectively, when they are in working state. Let Y_ij denote the lifetime for $U_{ij}^{W}$ in warm standby state. The unit lifetimes $X_{ij}, X_{ij}^{C}$ and $X_{ij}^{W}$ are random uncertain variables having independent distributions with random uncertain geometric distributions denoted by $Ge (ξ_{ij}), Ge (ξ_{ij}^{C})$ and $Ge (ξ_{ij}^{W})$ , respectively. The uncertain parameters $ξ_{ij}, ξ_{ij}^{C}$ and $ξ_{ij}^{W} (i = 1, 2, \dots, n, j = 1, 2, \dots, m_{i}, 0 < ξ_{ij}, ξ_{ij}^{C}, ξ_{ij}^{W} < 1)$ are independent uncertain variables defined on the uncertainty space $(Γ, L, M)$ . The lifetime Y_ij is a random uncertain variable obeying the discrete random uncertain Weibull distribution whose scale parameter is an uncertain variable η_ij (0 < η_ij < 1) on the uncertainty space $(Γ, L, M)$ , and the shape parameter is a constant value β_ij (β_ij > 1), that is $W (η_{ij}, β_{ij}) (i = 1, 2, \dots,$ n, j = 1, 2, ⋯ , m_i) .

3.3 Assumption

1. The redundant systems, units and switches at any time can only be in one of two states: functioning or failed, and they are non-repairable.

2. Lifetimes of units and switches are independent distributions with uncertain parameters.

3. All uncertain variables are mutual independence, and they have regular uncertainty distributions and inverse uncertainty distributions.

4. The switches of redundant system are imperfect. When the switch is available, switching is realized instantaneously.

5. Assume that failure of the subsystem occurs either when both original unit and standby unit fail, or as a result of failure of switch and then failure of the unit in the base connection in the moment of switch fails.

4 Cold redundant system

In the redundant system, the conversion switch is a special element, it may fail. In this section, we consider two approaches to describe the lifetimes of switches: random uncertain 0-1 switch lifetime and random uncertain geometric switch lifetime.

4.1 Cold redundant system with random uncertain 0-1 switch lifetime

In cold redundant series-parallel system with random uncertain 0-1 switch lifetime, let $X_{ij}^{S}$ denote the lifetime for the switch S_ij. When the switches are used, the reliability of the switches are independent uncertain variables $ξ_{ij}^{S} (0 < ξ_{ij}^{S} < 1, i = 1, 2, \dots, n, j =$ 1, 2, ⋯ , m_i) on the uncertainty space $(Γ, L, M)$ .

Introduce an indicator function here $I_{{X_{ij}^{S} = 1}} = {\begin{matrix} 1, & if X_{ij}^{S} is normal \\ 0, & if X_{ij}^{S} is failure . \end{matrix}$

Obviously, the lifetime of the cold redundant series-parallel system is $\begin{array}{l} X_{01}^{C} = \min { \\ \max { X _{11} + X _{11}^{C} I _{{ X _{11}^{S} = 1}}, \dots , X _{1 m_{1}} + X _{1 m_{1}}^{C} I_{{ X _{1 m_{1}}^{S} = 1}} } \\ \max { X _{21} + X _{21}^{C} I _{{ X _{21}^{S} = 1}}, \dots , X _{2 m_{2}} + X _{2 m_{2}}^{C} I_{{ X _{2 m_{2}}^{S} = 1}} } \\ \dots, \\ \max { X _{n 1} + X _{n 1}^{C} I _{{ X _{n 1}^{S} = 1 }}, \dots , X _{n m_{n}} + X _{n m_{n}}^{C} I_{{ X _{n m_{n}}^{S} = 1 }} } }} . \end{array}$

So, the uncertain reliability variable of the cold redundant series-parallel system is $\begin{matrix} R_{01}^{C} (k, ξ_{01}^{C}) = P {X_{01}^{C} > k} \\ = \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} P {X_{ij} + X_{ij}^{C} I_{{X_{ij}^{S} = 1}} \leq k}}, \end{matrix}$ (1) where $ξ_{01}^{C} = (ξ_{11}, \dots, ξ_{1 m_{1}}, \dots, ξ_{n 1}, \dots, ξ_{n m_{n}}, ξ_{11}^{C}, \dots, ξ_{1 m_{1}}^{C}, \dots, ξ_{n 1}^{C}, \dots, ξ_{n m_{n}}^{C}, ξ_{11}^{S}, \dots, ξ_{1 m_{1}}^{S}, \dots, ξ_{n 1}^{S},$ $\dots, ξ_{n m_{n}}^{S})$ .

In Equation (1), $\begin{matrix} P {X_{ij} + X_{ij}^{C} I_{{X_{ij}^{S} = 1}} \leq k} \\ = & P {X_{ij} + X_{ij}^{C} \leq k, X_{ij}^{S} = 1} + P {X_{ij} \leq k, X_{ij}^{S} = 0} \\ = & 1 - {(1 - ξ_{ij})}^{k} + ξ_{ij} ξ_{ij}^{S} \frac{{(1 - ξ_{ij})}^{k} - {(1 - ξ_{ij}^{C})}^{k}}{ξ_{ij} - ξ_{ij}^{C}} . \end{matrix}$

It follows from Theorem 2 and Definition 9 that the uncertain reliability variable $R_{01}^{C} (k, ξ_{01}^{C})$ is strictly decreasing with respect to ξ_ij and $ξ_{ij}^{C}$ , and strictly increasing with respect to $ξ_{ij}^{S}$ . The regular uncertainty distribution of ξ_ij is $Φ_{ij} (x) = M {ξ_{ij} \leq x},$ so there is an inverse uncertainty distribution $Φ_{ij}^{- 1} (α) .$ Similarly, $ξ_{ij}^{C}$ and $ξ_{ij}^{S}$ have inverse uncertainty distributions $Φ_{ij}^{- 1 (C)} (α)$ and $Φ_{ij}^{- 1 (S)} (α)$ , respectively. The reliability of the cold redundant series-parallel system is $\begin{matrix} R_{01}^{C} (k) = \\ \int_{0}^{1} \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} [1 - {(1 - Φ_{ij}^{- 1} (1 - α))}^{k} + Φ_{ij}^{- 1} (1 - α) \\ \cdot Φ_{ij}^{- 1 (S)} (α) \frac{{(1 - Φ_{ij}^{- 1} (1 - α))}^{k} - {(1 - Φ_{ij}^{- 1 (C)} (1 - α))}^{k}}{Φ_{ij}^{- 1} (1 - α) - Φ_{ij}^{- 1 (C)} (1 - α)}]} d α, \end{matrix}$ (2) the MTTF of the series-parallel system is ${MTTF}_{01}^{C} = \sum_{k = 1}^{\infty} k \cdot R_{01}^{C} (k) .$ (3)Remark 1. When the uncertain variables $ξ_{ij}, ξ_{ij}^{C}$ and $ξ_{ij}^{S}$ degenerate to constants $p_{ij}, p_{ij}^{C}$ and $p_{ij}^{S}$ , respectively, the result in Equation (2) extends to the form $\begin{matrix} R (k) = \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} [1 - {(1 - p_{ij})}^{k} + p_{ij} p_{ij}^{S} \\ \cdot \frac{{(1 - p_{ij})}^{k} - {(1 - p_{ij}^{C})}^{k}}{p_{ij} - p_{ij}^{C}}]} . \end{matrix}$

4.2 Cold redundant system with random uncertain geometric switch lifetime

Let us suppose a cold redundant system of conversion switch with random uncertain geometric switch lifetime. The lifetime of switch K_ij is denoted by $X_{ij}^{K}$ in this system. The uncertain parameter $ξ_{ij}^{K}$ of switch K_ij is defined on the uncertainty space $(Γ, TDSFTD L, TDSFTD M)$ , which is an uncertain variable with inverse uncertainty distribution $Φ_{ij}^{- 1 (K)} (α)$ .

Introduce an indicator function here $I_{{X_{ij}^{K} > X_{ij}}} = {\begin{matrix} 1, & if X_{ij}^{K} > X_{ij} \\ 0, & if X_{ij}^{K} \leq X_{ij} . \end{matrix}$

Obviously, the lifetime of the cold redundant series-parallel system is $\begin{array}{l} X_{g e o}^{C} = \min { \\ \max { X _{11} + X _{11}^{C} I_{{ X _{11}^{K} > X _{11} }}, \dots , X _{1 m _{1}} + X _{1 m _{1}}^{C} I_{{ X _{1 m _{1}}^{K} > X _{1 m _{1}} }} } , \\ \max { X _{21} + X _{21}^{C} I_{{ X _{21}^{K} > X _{21} }}, \dots , X _{2 m _{2}} + X _{2 m _{2}}^{C} I_{{ X _{2 m _{2}}^{K} > X _{2 m _{2}} }} } , \\ \dots, \\ \max { X _{n 1} + X _{n 1}^{C} I _{{ X _{n 1}^{K} > X _{n 1} }} , \dots , X _{n m _{n}} + X _{n m _{n}}^{C} I _{{ X _{n m _{n}}^{K} > X _{n m _{n}} }} } } . \end{array}$

So, the uncertain reliability variable of the cold redundant series-parallel system is $\begin{matrix} R_{geo}^{C} (k, ξ_{geo}^{C}) = P {X_{geo}^{C} > k} \\ = \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} P {X_{ij} + X_{ij}^{C} I_{{X_{ij}^{K} > X_{ij}}} \leq k}}, \end{matrix}$ (4) where $ξ_{geo}^{C} = (ξ_{11}, \dots, ξ_{1 m_{1}}, \dots, ξ_{n 1}, \dots, ξ_{n m_{n}},$ $ξ_{11}^{C}, \dots, ξ_{1 m_{1}}^{C}, \dots, ξ_{n 1}^{C}, \dots, ξ_{n m_{n}}^{C}, ξ_{11}^{K}, \dots, ξ_{1 m_{1}}^{K},$ $\dots, ξ_{n 1}^{K}, \dots, ξ_{n m_{n}}^{K}) .$

In Equation (4), $\begin{matrix} P {X_{ij} + X_{ij}^{C} I_{{X_{ij}^{K} > X_{ij}}} \leq k} \\ = & P {X_{ij} + X_{ij}^{C} \leq k, X_{ij}^{K} > X_{ij}} + P {X_{ij} \leq k, X_{ij}^{K} \leq X_{ij}} \\ = & 1 - {(1 - ξ_{ij})}^{k} - ξ_{ij} (1 - ξ_{ij}^{K}) \end{matrix}$ $\begin{matrix} \cdot \frac{{(1 - ξ_{ij}^{C})}^{k} - {(1 - ξ_{ij})}^{k} {(1 - ξ_{ij}^{K})}^{k}}{1 - ξ_{ij}^{C} - (1 - ξ_{ij}) (1 - ξ_{ij}^{K})} . \end{matrix}$

By Theorem 2 and Definition 9, the uncertain reliability variable $R_{geo}^{C} (k, ξ_{geo}^{C})$ is strictly decreasing with respect to $ξ_{ij}, ξ_{ij}^{C}$ and $ξ_{ij}^{K}$ . The reliability of the cold redundant series-parallel system is $\begin{matrix} R_{geo}^{C} (k) = \int_{0}^{1} \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} [1 - {(Ψ_{ij} (α))}^{k} - Φ_{ij}^{- 1} (1 - α) \\ Ψ_{ij}^{K} (α) \frac{{(Ψ_{ij}^{C} (α))}^{k} - {(Ψ_{ij} (α))}^{k} {(Ψ_{ij}^{K} (α))}^{k}}{Ψ_{ij}^{C} (α) - Ψ_{ij} (α) Ψ_{ij}^{K} (α)}]} d α, \end{matrix}$ (5) where $Ψ_{ij}^{*} (α) = 1 - Φ_{ij}^{- 1 (*)} (1 - α), * = C, K$ , $Ψ_{ij} (α) = 1 - Φ_{ij}^{- 1} (1 - α)$ .

The MTTF of the series-parallel system is ${MTTF}_{geo}^{C} = \sum_{k = 1}^{\infty} k \cdot R_{geo}^{C} (k) .$ (6)Remark 2. When the uncertain variables $ξ_{ij}, ξ_{ij}^{C}$ and $ξ_{ij}^{K}$ degenerate to constants $p_{ij}, p_{ij}^{C}$ and $p_{ij}^{K}$ , respectively, the result in Equation (5) extends to the form { $\begin{matrix} R (k) = & \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} [1 - {(1 - p_{ij})}^{k} + p_{ij} (1 - p_{ij}^{K}) \\ \frac{{(1 - p_{ij}^{C})}^{k} - {(1 - p_{ij})}^{k} {(1 - p_{ij}^{K})}^{k}}{1 - p_{ij}^{C} - (1 - p_{ij}) (1 - p_{ij}^{K})}]} . \end{matrix}$

5 Warm redundant system

In another case, the standby unit of the redundant system will fail after a long time event if it is in standby state, so the warm redundant system is considered.

5.1 Warm redundant system with random uncertain 0-1 switch lifetime

In the warm redundant series-parallel system with random uncertain 0-1 switch lifetime, when the switch S_ij is used, the reliability of the switches are independent uncertain variables $ξ_{ij}^{S} (0 < ξ_{ij}^{S} < 1)$ on uncertainty space $(Γ, TDSFTD L, TDSFTD M)$ .

Introduce an indicator function here $I_{{Y_{ij} > X_{ij}}} = {\begin{matrix} 1, & if Y_{ij} > X_{ij} \\ 0, & if Y_{ij} \leq X_{ij} . \end{matrix}$

Obviously, the random uncertain lifetime of the warm redundant series-parallel system is $\begin{matrix} X_{01}^{W} = & min {max {X_{11} + X_{11}^{W} I_{{Y_{11} > X_{11}}} I_{{X_{11}^{S} = 1}}, \dots, \\ X_{1 m_{1}} + X_{1 m_{1}}^{W} I_{{Y m_{1} > X_{1 m_{1}}}} I_{{X_{1 m_{1}}^{S} = 1}}}, \\ max {X_{21} + X_{21}^{W} I_{{Y_{21} > X_{21}}} I_{{X_{21}^{S} = 1}}, \dots, \\ X_{2 m_{2}} + X_{2 m_{2}}^{W} I_{{Y_{2 m_{2}} > X_{2 m_{2}}}} I_{{X_{2 m_{2}}^{S} = 1}}}, \\ \dots, \end{matrix}$ $\begin{matrix} max {X_{n 1} + X_{n 1}^{W} I_{{Y_{n 1} > X_{n 1}}} I_{{X_{n 1}^{S} = 1}}, \dots, \\ X_{n m_{n}} + X_{n m_{n}}^{W} I_{{Y_{n m_{n}} > X_{n m_{n}}}} I_{{X_{n m_{n}}^{S} = 1}}}}, \end{matrix}$

So, the uncertain reliability variable of the warm redundant series-parallel system is $\begin{matrix} R_{01}^{W} (k, ξ_{01}^{W}) = P {X_{01}^{W} > k} \\ = \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} P {X_{ij} + X_{ij}^{W} I_{{Y_{ij} > X_{ij}}} I_{{X_{ij}^{S} = 1}} \leq k}}, \end{matrix}$ (7) where $ξ_{01}^{W} = (ξ_{11}, \dots, ξ_{1 m_{1}}, \dots, ξ_{n 1}, \dots, ξ_{n m_{n}}, ξ_{11}^{W},$ $\dots, ξ_{1 m_{1}}^{W}, \dots, ξ_{n 1}^{W}, \dots, ξ_{n m_{n}}^{W}, η_{11}, \dots, η_{1 m_{1}}, \dots, η_{n 1},$ $\dots, η_{n m_{n}}, ξ_{11}^{S}, \dots, ξ_{1 m_{1}}^{S}, \dots, ξ_{n 1}^{S}, \dots, ξ_{n m_{n}}^{S}) .$

In Equation (7), $\begin{matrix} P {X_{ij} + X_{ij}^{W} I_{{Y_{ij} > X_{ij}}} I_{{X_{ij}^{S} = 1}} \leq k} \\ = & P {X_{ij} + X_{ij}^{W} I_{{Y_{ij} > X_{ij}}} \leq k, X_{ij}^{S} = 1} \\ + P {X_{ij} \leq k, X_{ij}^{S} = 0} \\ = & 1 - {(1 - ξ_{ij})}^{k} - ξ_{ij} ξ_{ij}^{S} {(1 - ξ_{ij}^{W})}^{k - 1} \\ \cdot \sum_{l = 1}^{k} [{(\frac{1 - ξ_{ij}}{1 - ξ_{ij}^{W}})}^{l - 1} η_{ij}^{l^{β_{ij}}}] . \end{matrix}$

It follows from Theorem 2 and Definition 9 that the uncertain reliability variable $R_{01}^{W} (k, ξ_{01}^{W})$ is strictly decreasing with respect to ξ_ij and $ξ_{ij}^{W}$ , and strictly increasing with respect to η_ij and $ξ_{ij}^{S}$ . In addition, let $ξ_{ij}^{W}$ and η_ij be uncertain variables with inverse uncertainty distributions $Φ_{ij}^{- 1 (W)} (α)$ and $Φ_{ij}^{- 1 (η)} (α)$ , respectively. The reliability and MTTF of the warm redundant series-parallel system are $\begin{matrix} R_{01}^{W} (k) = \int_{0}^{1} \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} [1 - {(1 - Φ_{ij}^{- 1} (1 - α))}^{k} \\ + Φ_{ij}^{- 1} (1 - α) Φ_{ij}^{- 1 (S)} (α) {(1 - Φ_{ij}^{- 1 (W)} (1 - α))}^{k - 1} \\ \cdot \sum_{l = 1}^{k} ({(\frac{1 - Φ_{ij}^{- 1} (1 - α)}{1 - Φ_{ij}^{- 1 (W)} (1 - α)})}^{l - 1} {(Φ_{ij}^{- 1 (η)} (α))}^{l^{β_{ij}}})]} d α \end{matrix}$ (8) and ${MTTF}_{01}^{W} = \sum_{k = 1}^{\infty} k \cdot R_{01}^{W} (k) .$ (9)Remark 3. When the uncertain variables $ξ_{ij}, ξ_{ij}^{S}, ξ_{ij}^{W}$ and η_ij degenerate to constants $p_{ij}, p_{ij}^{S}, p_{ij}^{W}$ and q_ij, respectively. The result in Equation (8) extends to the form $\begin{matrix} R (k) & = \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} [1 - {(1 - p_{ij})}^{k} - p_{ij} p_{ij}^{S} \\ \cdot {(1 - p_{ij}^{W})}^{k - 1} \sum_{l = 1}^{k} ({(\frac{1 - p_{ij}}{1 - p_{ij}^{W}})}^{l - 1} q_{ij}^{l^{β_{ij}}})]} . \end{matrix}$

5.2 Warm redundant system with random uncertain geometric switch lifetime

In warm redundant system with random uncertain geometric switch lifetime, the lifetime of switch K_ij follow the random uncertain geometric distribution with uncertain parameter $ξ_{ij}^{K} (0 < ξ_{ij}^{K} < 1)$ on the uncertainty space $(Γ, TDSFTD L, TDSFTD M)$ .

Obviously, the random uncertain lifetime of the warm redundant series-parallel system is $\begin{matrix} X_{geo}^{W} = \\ min {max {X_{11} + X_{11}^{W} I_{{Y_{11} > X_{11}}} I_{{X_{11}^{K} > X_{11}}}, \dots, \end{matrix}$ $\begin{matrix} X_{1 m_{1}} + X_{1 m_{1}}^{W} I_{{Y_{1 m_{1}} > X_{1 m_{1}}}} I_{{X_{1 m_{1}}^{K} > X_{1 m_{1}}}}}, \\ max {X_{21} + X_{21}^{W} I_{{Y_{21} > X_{21}}} I_{{X_{21}^{K} > X_{21}}}, \dots, \\ X_{2 m_{2}} + X_{2 m_{2}}^{W} I_{{Y_{2 m_{2}} > X_{2 m_{2}}}} I_{{X_{2 m_{2}}^{K} > X_{2 m_{2}}}}}, \\ \dots, \\ max {X_{n 1} + X_{n 1}^{W} I_{{Y_{n 1} > X_{n 1}}} I_{{X_{n 1}^{K} > X_{n 1}}}, \dots, \\ X_{n m_{n}} + X_{n m_{n}}^{W} I_{{Y_{n m_{n}} > X_{n m_{n}}}} I_{{X_{n m_{n}}^{K} > X_{n m_{n}}}}} . \end{matrix}$

So, the uncertain reliability variable of the warm redundant series-parallel system is $\begin{matrix} R_{geo}^{W} (k, ξ_{geo}^{W}) = P {X_{geo}^{W} > k} \\ = \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} P {X_{ij} + X_{ij}^{W} I_{{Y_{ij} > X_{ij}}} I_{{X_{ij}^{K} > X_{ij}}} \leq k}}, \end{matrix}$ (10) where $ξ_{geo}^{W} = (ξ_{11}, \dots, ξ_{1 m_{1}}, \dots, ξ_{n 1}, \dots, ξ_{n m_{n}}, ξ_{11}^{W},$ $\dots, ξ_{1 m_{1}}^{W}, \dots, ξ_{n 1}^{W}, \dots, ξ_{n m_{n}}^{W}, η_{11}, \dots, η_{1 m_{1}}, \dots, η_{n 1},$ $\dots, η_{n m_{n}}, ξ_{11}^{K}, \dots, ξ_{1 m_{1}}^{K}, \dots, ξ_{n 1}^{K}, \dots, ξ_{n m_{n}}^{K}) .$

In Equation (10), $\begin{matrix} P {X_{ij} + X_{ij}^{W} I_{{Y_{ij} > X_{ij}}} I_{{X_{ij}^{K} > X_{ij}}} \leq k} \\ = & P {X_{ij} + X_{ij}^{W} \leq k, Y_{ij} > X_{ij}, X_{ij}^{K} > X_{ij}} \end{matrix}$ $\begin{matrix} + P {X_{ij} \leq k, Y_{ij} > X_{ij}, X_{ij}^{K} \leq X_{ij}} \\ + P {X_{ij} \leq k, Y_{ij} \leq X_{ij}} \\ = & 1 - {(1 - ξ_{ij})}^{k} - ξ_{ij} \frac{{(1 - ξ_{ij}^{W})}^{k}}{1 - ξ_{ij}} \\ \cdot \sum_{l = 1}^{k} [{(\frac{(1 - ξ_{ij}) (1 - ξ_{ij}^{K})}{1 - ξ_{ij}^{W}})}^{l} η_{ij}^{l^{β_{ij}}}] . \end{matrix}$

By Theorem 2 and Definition 9, the uncertain reliability variable $R_{geo}^{W} (k, ξ_{geo}^{W})$ is strictly decreasing with respect to $ξ_{ij}, ξ_{ij}^{W}$ and $ξ_{ij}^{K}$ , and strictly increasing with respect to η_ij .

The reliability of the warm redundant series-parallel system is $\begin{matrix} R_{geo}^{W} (k) = \int_{0}^{1} \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} [ \end{matrix}$ $\begin{matrix} 1 - {(1 - Φ_{ij}^{- 1} (1 - α))}^{k} - \frac{Φ_{ij}^{- 1} (1 - α)}{1 - Φ_{ij}^{- 1} (1 - α)} {(Ψ_{ij}^{W} (α))}^{k} \\ \cdot \sum_{l = 1}^{k} ({(\frac{(1 - Φ_{ij}^{- 1} (1 - α)) Ψ_{ij}^{K} (α)}{Ψ_{ij}^{W} (α)})}^{l} {(Φ_{ij}^{- 1 (η)} (α))}^{l^{β_{ij}}})]} d α, \end{matrix}$ (11) where, $Ψ_{ij}^{*} (α) = 1 - Φ_{ij}^{- 1 (*)} (1 - α), * = W, K$ .

The MTTF of the series-parallel system is ${MTTF}_{geo}^{W} = \sum_{k = 1}^{\infty} k \cdot R_{geo}^{W} (k) .$ (12)

Remark 4. When the uncertain variables $ξ_{ij}, ξ_{ij}^{K}, ξ_{ij}^{W}$ and η_ij degenerate to constants $p_{ij}, p_{ij}^{K}, p_{ij}^{W}$ and q_ij, respectively. The result in Equation (11) extends to the form $\begin{matrix} R (k) = & \prod_{i = 1}^{n} {1 - \prod_{j = 1}^{m_{i}} [1 - {(1 - p_{ij})}^{k} - p_{ij} \frac{{(1 - p_{ij}^{W})}^{k}}{(1 - p_{ij})} \\ \cdot \sum_{l = 1}^{k} ({(\frac{(1 - p_{ij}) (1 - p_{ij}^{K})}{1 - p_{ij}^{W}})}^{l} {q_{ij}}^{l^{β_{ij}}})]} . \end{matrix}$

6 Numerical application

In this section, some numerical applications will be introduced to demonstrate the theoretical results.

6.1 Example 1

Consider a high pressure cold (warm) redundant system composed of power supply system A₁ and launch system A₂ in series. The A₁ consists of a power supply device 11, and the A₂ consists of launchers devices 21 and 22 connected in parallel. Let devices ij (i = 1, 2, j = 1, ⋯ , m_i, m₁ = 1, m₂ = 2) be composed of a original unit, a cold (warm) standby unit and a switch, respectively. Let the uncertain parameters $ξ_{ij}, ξ_{ij}^{C} (ξ_{ij}^{W}, ξ_{ij}^{η}), ξ_{ij}^{S}$ and $ξ_{ij}^{K}$ be zigzag uncertain variables, respectively. In addition, the values of the uncertain parameters for the lifetimes of units and switches are given in Table 1.

Table 1
The values for the high pressure cold (warm) standby system

Unit 11 21 22

ξ _ij 𝒵 (0.01, 0.015, 0.02) 𝒵 (0.01, 0.02, 0.03) 𝒵 (0.01, 0.02, 0.025)

$ξ_{ij}^{C}$ 𝒵 (0.01, 0.02, 0.03) 𝒵 (0.015, 0.02, 0.03) 𝒵 (0.015, 0.025, 0.03)

η _ij 𝒵 (0.998, 0.9985, 0.999) 𝒵 (0.99, 0.995, 0.998) 𝒵 (0, 995, 0.998, 0.999)

β _ij 3 2 3

ξ ^W 𝒵 (0.01, 0.02, 0.03) 𝒵 (0.015, 0.02, 0.03) 𝒵 (0.015, 0.025, 0.03)

$ξ_{ij}^{S}$ 𝒵 (0.98, 0.985, 0.99) 𝒵 (0.97, 0.98, 0.99) 𝒵 (0.975, 0.98, 0.985)

$ξ_{ij}^{K}$ 𝒵 (0.005, 0.01, 0.015) 𝒵 (0.01, 0.015, 0.02) 𝒵 (0.01, 0.02, 0.03)

Unit	11	21	22
ξ _ij	𝒵 (0.01, 0.015, 0.02)	𝒵 (0.01, 0.02, 0.03)	𝒵 (0.01, 0.02, 0.025)
$ξ_{ij}^{C}$	𝒵 (0.01, 0.02, 0.03)	𝒵 (0.015, 0.02, 0.03)	𝒵 (0.015, 0.025, 0.03)
η _ij	𝒵 (0.998, 0.9985, 0.999)	𝒵 (0.99, 0.995, 0.998)	𝒵 (0, 995, 0.998, 0.999)
β _ij	3	2	3
ξ ^W	𝒵 (0.01, 0.02, 0.03)	𝒵 (0.015, 0.02, 0.03)	𝒵 (0.015, 0.025, 0.03)
$ξ_{ij}^{S}$	𝒵 (0.98, 0.985, 0.99)	𝒵 (0.97, 0.98, 0.99)	𝒵 (0.975, 0.98, 0.985)
$ξ_{ij}^{K}$	𝒵 (0.005, 0.01, 0.015)	𝒵 (0.01, 0.015, 0.02)	𝒵 (0.01, 0.02, 0.03)

Applying Equations (2), (5), (8) and (11), the reliability of the high pressure cold and warm redundant systems with random uncertain 0-1 switch lifetimes and random uncertain geometric switch lifetimes are illustrated, as shown in Figs. 2 and 3. In addition, the reliability of the system with perfect switch are also shown in Figs. 2 and 3.

Fig.2

The reliability of the high pressure cold redundant system.

Fig.3

The reliability of the high pressure warm standby system.

It seems from Figs. 2 and 3 that:

1. The reliability of the high pressure cold (warm) redundant system is decreasing with respect to the time.

2. The reliability of the high pressure cold (warm) redundant system with perfect switch is obviously better than that of imperfect switch.

3. At any moment, the reliability of the high pressure cold (warm) redundant system with random uncertain 0-1 switch lifetime is better than ones with random uncertain geometric switch lifetime.

By Equations (3), (6), (9) and (12), the MTTFs of the high pressure cold (warm) redundant systems with different imperfect switches are given in Table 2. Those results can be read as in what follows,

\begin{matrix} {MTTF}_{geo}^{C} & < {MTTF}_{01}^{C}, \\ {MTTF}_{geo}^{W} & < {MTTF}_{01}^{W} . \end{matrix}

Table 2

The MTTF of the high pressure cold (warm) standby system

system	cold standby		warm standby
	0-1	geometric	0-1	geometric
MTTF	3863.14	3265.11	3636.38	2122.14

This means that the MTTF of the high pressure cold (warm) redundant system with random uncertain 0-1 switch lifetime is longer than the random uncertain geometric switch lifetime. In each type of switch lifetime, the lifetime of the high pressure cold redundant system is longer than that of the high pressure warm redundant system.

In random environment, the reliability of high pressure cold (warm) redundant systems are obtained according to Remark 1 - Remark 4, as shown in Figs. 4 and 5. The constant parameters of each unit (switch) lifetime select the expected value of Z (a, b, c). The Figs. 4 and 5 show scatter plots of the reliability of the systems with uncertain parameters and constant parameters, respectively.

Fig.4

The reliability of the high pressure cold redundant system with imperfect switch.

Fig.5

The reliability of the high pressure warm redundant system with imperfect switch.

6.2 Example 2

Let us consider a pump station cold (warm) redundant system consisting of drive equipment A₁ and control equipment A₂ in series. The drive equipment A₁ and control equipment A₂ consist of two subsystems respectively, denoted by 11, 12, 21 and 22. Let subsystems ij (i = 1, 2, j = 1, ⋯ , m_i, m₁ = 2, m₂ = 2) be composed of a original unit, a cold (warm) standby unit and a switch, respectively.

In pump station cold (warm) redundant system, let the uncertain parameters $ξ_{ij}, ξ_{ij}^{C} (ξ_{ij}^{W}, ξ_{ij}^{η}), ξ_{ij}^{S}$ and $ξ_{ij}^{K}$ be linear uncertain variables, respectively. In addition, the values of the uncertain parameters for pump station cold (warm) redundant system are given in Table 3.

Table 3
The values for the pump station cold (warm) redundant system

Unit 11 12 21 22

ξ _ij ℒ(0.01, 0.025) ℒ(0.015, 0.025) ℒ(0.01, 0.03) ℒ(0.015, 0.04)

$ξ_{ij}^{C}$ ℒ(0.01, 0.02) ℒ(0.015, 0.04) ℒ(0.02, 0.025) ℒ(0.02, 0.04)

η _ij ℒ(0.998, 0.9985) ℒ(0.98, 0.99) ℒ(0.985, 0.99) ℒ(0, 99, 0.995)

β _ij 3 4 2 2

ξ ^W ℒ(0.01, 0.02) ℒ(0.015, 0.04) ℒ(0.02, 0.025) ℒ(0.02, 0.04)

$ξ_{ij}^{S}$ ℒ(0.95, 0.98) ℒ(0.96, 0.98) ℒ(0.975, 0.985) ℒ(0.975, 0.99)

$ξ_{ij}^{K}$ ℒ(0.01, 0.03) ℒ(0.015, 0.02) ℒ(0.012, 0.02) ℒ(0.01, 0.02)

Unit	11	12	21	22
ξ _ij	ℒ(0.01, 0.025)	ℒ(0.015, 0.025)	ℒ(0.01, 0.03)	ℒ(0.015, 0.04)
$ξ_{ij}^{C}$	ℒ(0.01, 0.02)	ℒ(0.015, 0.04)	ℒ(0.02, 0.025)	ℒ(0.02, 0.04)
η _ij	ℒ(0.998, 0.9985)	ℒ(0.98, 0.99)	ℒ(0.985, 0.99)	ℒ(0, 99, 0.995)
β _ij	3	4	2	2
ξ ^W	ℒ(0.01, 0.02)	ℒ(0.015, 0.04)	ℒ(0.02, 0.025)	ℒ(0.02, 0.04)
$ξ_{ij}^{S}$	ℒ(0.95, 0.98)	ℒ(0.96, 0.98)	ℒ(0.975, 0.985)	ℒ(0.975, 0.99)
$ξ_{ij}^{K}$	ℒ(0.01, 0.03)	ℒ(0.015, 0.02)	ℒ(0.012, 0.02)	ℒ(0.01, 0.02)

For pump station cold and warm redundant systems with random uncertain 0-1 switch lifetimes and random uncertain geometric switch lifetimes, the reliability could be calculated by Equations (2), (5), (8) and (11), as shown in Figs. 6 and 7. The reliability of the pump station cold (warm) redundant system with perfect switch is also obtained. From Figs. 6 and 7, we can see that the reliability of the pump station cold and warm redundant systems are indeed affected by the imperfect switches. The Figs. 6 and 7 show that the reliability of the pump station cold and warm redundant systems for random uncertain 0-1 switches are greater than ones for random uncertain geometric switches.

Fig.6

The reliability of the pump station cold redundant system.

Fig.7

The reliability of the pump station warm redundant system.

The MTTFs of the pump station cold and warm redundant systems with different imperfect switches are obtained by Equations (3), (6), (9) and (12), as show in Table 4. The results in Table 4 show that the obtained MTTFs of the pump station cold and warm redundant systems for random uncertain 0-1 switches are longer than ones for random uncertain geometric switches.

Table 4

The MTTF of the pump station cold (warm) redundant system

system	cold standby		warm standby
	0-1	geometric	0-1	geometric
MTTF	5263.27	3561.76	3939.27	2773.62

Compared with traditional reliability analysis, the reliability of pump station cold (warm) redundant systems with imperfect switches are obtained according to Remark 1 - Remark 4, as shown in Figs. 8 and 9. The constant parameters of each unit (switch) lifetime select the expected value of Ł (a, b). The Fig. 8 shows that the reliability of pump station cold redundant system has weak sensitivity to the assumption of uncertain parameters. The Fig. 9 shows that the reliability of pump station warm redundant system has strong sensitivity to the assumption of uncertain parameters.

Fig.8

The reliability of the pump station cold redundant system with imperfect switch.

Fig.9

The reliability of the pump station warm redundant system with imperfect switch.

7 Conclusion

This paper employed uncertainty theory and probability theory to deal with the problem of discrete time system reliability. The concepts of random uncertain geometric distribution, discrete random uncertain Weibull distribution, random uncertain 0-1 switch lifetime and random uncertain geometric switch lifetime are proposed. Based on these definitions, the discrete time redundant series-parallel systems with imperfect switches and random uncertain lifetimes were established, some formulas of the reliability indexes of such redundant systems were derived. Finally, some numerical examples were given to illustrate the redundant system models. Moreover, from the viewpoint of control theory, the complexity, robustness and convergence of the system were also important in practical application. In future research, we can focus on the complexity, robustness and convergence analysis of the discrete random uncertain control system based on uncertainty theory and probability theory.

Footnotes

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (No.11601469), the Science Research Project of Education Department of Hebei Province (No. ZD2017079) and the Natural Science Foundation of Hebei Province (no.A2018203088), People’s and Republic of China.

References

Barlow

R.E.

and Proschan

, Mathematical Theory of Reliability, Wiley, New York, 1965.

Cyril

and Qlivier

, A Survey on Discrete Lifetime Distribution, International Journal of Reliability Quality and Safety Engineering 10(01) (2003), 69–98.

Papageorgiou

and Kokolakis

, A Two-unit General Parallel System with (n-2) Cold Standbys Analytic and Simulation Approach, European Journal of Operational Research 176(2) (2007), 1016–1032.

Papageorgiou

and Kokolakis

, Reliability Analysis of a Two-Unit General Parallel System with (n-2) Warm Standbys, European Journal of Operational Research 201(3) (2010), 821–827.

Goel

L.R.

and Gupta

, Analysis of a Two-Unit Standby System with Three Modes and Imperfect Switching Device, Microelectronics Reliability 24(3) (1984), 425–429.

Zadeh

L.A.

, Fuzzy Sets, Information and Control 8(3) (1965)338–353.

Onisawa

, Fuzzy Theory in Reliability Analysis, Fuzzy Sets and Systems 30(3) (1989), 361–363.

Park

K.S.

, Fuzzy Apportionment of System Reliability, IEEE Transactions on Reliability R-36(1) (2009)129–132.

Cai

, Wen

and Zhang

, Fuzzy Variables as A Basis for a Theory of Fuzzy Reliability in the Possibility Context, Fuzzy Sets and Systems 42(2) (1991), 145–172.

10.

Liu

, Uncertainty Theory, 2nd edn, Springer-Verlag, Berlin, 2007.

11.

Liu

, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.

12.

Liu

, Some Research Problems in Uncertainty Theory, Journal of Uncertain Systems 3(1) (2009), 3–10.

13.

Liu

and Ha

, Expected Value of Function of Uncertain Variables, Journal of Uncertain Systems 4(3) (2010), 181–186.

14.

Liu

, Theory and Practice of Uncertain Programming, 2nd edn, Springer-Verlag, Berlin, 2009.

15.

Gao

, Gao

and Yang

, Analysis of Order Statistics of Uncertain Variables, Journal of Uncertainty Analysis and Applications 3(1) (2015), 1–7.

16.

Yao

, Uncertain Differential Equation with Jumps, Soft Computing 19(7) (2015), 2063–2069.

17.

Yang

, Ni

and Zhang

, Stability in Inverse Distribution for Uncertain Differential Equations, Journal of Intelligent and Fuzzy Systems 32(3) (2017), 2051–2059.

18.

Liu

, Uncertain Risk Analysis and Uncertain Reliability Analysis, Journal of Uncertain Systems 4(3) (2010), 163–170.

19.

, Kang

, Wen

and Zhang

, Belief Reliability Distribution Based on Maximum Entropy Principle, IEEE Access 6 (2018), 1577–1582.

20.

Liu

, Reliability Analysis of Redundant System with Uncertain Lifetimes, Information 16(2) (2013), 881–887.

21.

Liu

, Li

and Xiong

, Reliability Analysis of Unrepairable Systems with Uncertain Lifetimes, International Journal of Security and Its Applications 9(12) (2015), 289–298.

22.

Gao

, Yao

, Zhou

and Ke

, Reliability Analysis of Uncertain Weighted k-out-of-n Systems, IEEE Transactions on Fuzzy Systems 26(5) (2018), 2663–2671.

23.

Zeng

, Kang

, Wen

and Zio

, Uncertainty Theory as a Basis for Belief Reliability, Information Sciences 429 (2018), 26–36.

24.

, Huang

, Wang

G.F.

and Tian

, Redundancy Optimization of an Uncertain Parallel-Series System with Warm Standby Elements, Complexity 2018 Article ID 3154360, 10 pages, 2018, https://doi.org/10.1155/2018/3154360

25.

Liu

, Ma

, Qu

and Li

, Reliability Mathematical Models of Repairable Systems With Uncertain Lifetimes and Repair Times, IEEE Access 6 (2018), 71285–71295.

26.

Liu

, Uncertain Random Variables: A Mixture of Uncertainty and Randomness, Soft Computing 17(4) (2013), 625–634.

27.

Liu

, Uncertain Random Programming with Applications, Fuzzy Optimization and Decision Making 12(2) (2013), 153–169.

28.

Liu

and Ralescu

D.A.

, Risk Index in Uncertain Random Risk Analysis, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 22(04) (2014), 491–504.

29.

Liu

and Ralescu

D.A.

, Expected Loss of Uncertain Random System, Soft Computing 22(17) (2017), 5573–5578.

30.

Wen

and Kang

, Reliability Analysis in Uncertain Random System, Fuzzy Optimization and Decision Making 15(4) (2016), 491–506.

31.

Gao

and Yao

, Importance Index of Components in Uncertain Random Systems, Knowledge-Based Systems 109 (2016), 208–217.

32.

Gao

, Sun

and Ralescu

D.A.

, Order Statistics of Uncertain Random Variables with Application to k-out-of-n System, Fuzzy Optimization and Decision Making 16(2) (2017), 159–181.

33.

Zhang

, R. Kang and M. Wen, Belief Reliability for Uncertain Random Systems, IEEE Transactions on Fuzzy Systems 26(6) (2018), 3605–3614.

34.

Cao

, Hu

and Li

, Reliability Analysis of Discrete Time Series-parallel Systems with Uncertain Parameters, Journal of Ambient Intelligence and Humanized Computing 10(7) (2019), 2657–2668.