Reliability analysis for devices subject to competing failure processes based on chance theory

Applied Mathematical Modelling 75 (2019) 398–413

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Reliability analysis for devices subject to competing failure processes based on chance theory Baoliang Liu∗, Zhiqiang Zhang, Yanqing Wen College of Mathematics and Statistics, Shanxi Datong University, Datong, Shanxi 037009, China

a r t i c l e

i n f o

Article history: Received 20 August 2018 Revised 15 May 2019 Accepted 21 May 2019 Available online 27 May 2019 Keywords: Competing failure processes Shock models Uncertainty theory Uncertain variable Uncertain distribution

a b s t r a c t This paper studies the reliability for devices subject to independent competing failure processes of degradation and shocks in an uncertain random environment. The continuous degradation is governed by an uncertain process, and external shocks arrive according to an uncertain random renewal reward process, in which the inter-arrival times of shocks and the shock sizes are assumed to be random variables and uncertain variables, respectively. The device reliability is defined as the chance measure that the uncertain degradation signals do not exceed a soft failure threshold L, and the uncertain random shocks do not cause the device failure. The device reliability is obtained by employing chance theory under four different shock patterns. Finally, a case study on a gas insulated transmission line is carried out to show the implementation of the proposed model. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Device failures are often the competing result of internal wear degradation and external environmental conditions. The degradation analysis has attracted considerable attention over the past few decades in reliability field. Generally, there are two types of degradation models: stochastic models and path models [1]. The existing stochastic degradation models were mainly based on a Gaussian process, Gamma process, Brownian motion process [2]. The external factors can be regarded as shocks attacking the devices. For example, extreme temperature, voltage and vibrations can affect device failure behavior by accelerating the degradation process. Typically, there are five different types of random shock models: (i) extreme shock model: a device fails when the size of any shock is beyond a specified threshold value, (ii) cumulative shock model: when the accumulated damage of shocks is beyond a critical level, a device fails, (iii) m-shock model: a device experiences failure after m shocks that are greater than a critical level are recorded, (iv) run shock model: failure occurs when there is a run of n consecutive shocks that are greater than a threshold level, and (v) δ -shock model: a device experiences failure when the inter-arrival time of two sequential shocks is less than a threshold δ , see, e.g., [3–7]. In engineering practice, devices are commonly subject to the competing failure process of wear degradation and external shocks. Reliability modeling for devices with multiple independent competing failure processes has been studied by several authors, e.g., [8–12]. For devices experiencing multiple dependent competing failure processes (MDCFP), Peng et al. [13] developed reliability models considering extreme and cumulative random shock models. Rafiee et al. [14] investigated reliability models for devices that experience MDCFP with changing degradation rate, and four different random shock patterns were considered. Wang et al. [15] considered the reliability assessment of aging structures subjected to gradual and

∗

Corresponding author. E-mail addresses: [email protected] (B. Liu), [email protected] (Z. Zhang).

https://doi.org/10.1016/j.apm.2019.05.036 0307-904X/© 2019 Elsevier Inc. All rights reserved.

B. Liu, Z. Zhang and Y. Wen / Applied Mathematical Modelling 75 (2019) 398–413

399

shock deteriorations. Raifiee et al. [16] studied condition-based maintenance for repairable deteriorating devices subject to a generalized mixed shock model. Shafiee and Finkelstein [17] investigated an optimal age-based group maintenance policy for multi-unit degrading systems. Qiu and Cui [18] presented the reliability formulas based on two-stage degradation process with extreme shocks. Other representative works on MDCFP models can be found in [19–21]. The above probability theory based multiple competing failure processes models are suitable for the situations when the degradation information is available, i.e., there are a large amount of historical failure data. So the estimated probability distribution function is close enough to the long-run cumulative frequency according to the law of large numbers. However, in engineering practice, only few failure data can be obtained for highly reliable devices, i.e., only partial information is available due to the expensive tested samples and the limited testing resources. For example, the range of a newly developed missile, the bridge strength and so on. In this case, the law of large numbers is no longer applicable, so the probability theory based models are not suitable to present the epistemic uncertainty caused by small samples. Then we have to invite domain experts to provide empirical data. It is obvious that it is not suitable to employ random variable to deal with empirical data. In order to deal with empirical data, fuzzy theory was proposed by Zedeh [22] in 1965. However, some paradox always exists in the models established on fuzzy theory, because fuzzy theory does not meet the law of excluded middle. More explanation about uncertainty theory and fuzzy theory can be referred to Liu [23]. To better quantify the epistemic uncertainty due to empirical data, Liu [24] in 2007 firstly proposed uncertainty theory to address the human uncertainty arising from the belief degree. It possesses the property of normality, duality, subadditivity and product axioms in mathematics [25]. To further study the dynamic evolution of uncertain phenomena with time, the uncertain process was introduced by Liu [26] in 2008. Later, Liu [27] invented a Liu process that is an uncertain stationary independent increment process with normal increments. Yao and Chen [28] and Yao [29] studied the uncertain differential equations driven by the Liu process. Nowadays, the uncertainty theory has gained wide applications in various fields such as uncertain reliability analysis, uncertain risk analysis and uncertain statistics analysis, and so on, and is playing significant roles both theoretically and practically. So far extensive literature can be found on uncertain reliability analysis, see Refs. [30–35]. Thus, uncertainty theory provides a rigorous mathematical foundation to measure the reliability of products that only few samples or no samples are available. Furthermore, Liu [36] concluded that uncertainty theory is better than probability theory to deal with uncertain problem by analyzing urn problems. However, in real engineering applications, complex systems are usually consist of different types of components, in which some components are innovative products with few failure data, while others components may be mature products with a lot of failure data. In order to deal with complex systems involving both random and uncertain factors, Liu [37,38] introduced an uncertain random variable and a chance theory. As applications of the chance theory, Yao and Zhou [39] investigated a new type of renewal reward process, in which the inter-arrival times and the rewards were assumed to be random variables and uncertain variables, respectively. Yao and Zhou [40] subsequently studied another new type of renewal reward process, which has uncertain inter-arrival times and random rewards. The first hitting time of uncertain random renewal process was discussed in Yao [41] considering random inter-arrival times and uncertain rewards, and the results were applied to the insurance risk process. Wen and Kang [42] presented a method of reliability analysis based on chance theory. To our best knowledge, most existing degradation models consider only one category of factors i.e., uncertain factor or random factor. Degradation models involving both uncertain and random factors have not been discussed yet so far. To bridge this gap, we shall present reliability models for devices subject to independent competing failure processes of uncertain continuous degradation and uncertain random external shocks. The two independent failure processes are soft failure due to continuous degradation which is governed by an uncertain process, and hard failure due to uncertain random external shocks. The inter-arrival times of shocks and the shock sizes are assumed to be random variables and uncertain variables, respectively. Thus, the model in the paper involves both aleatory and epistemic uncertainties. Four different shock patterns that can cause the device hard failure are considered: (i) the extreme shock model: when the first shock damage size above a hard failure threshold D is recorded; (ii) the cumulative shock model: when the cumulative damage size of the uncertain random shocks is larger than a hard failure threshold D; (iii) the δ -shock model: the inter-arrival time of two sequential shocks is less than a threshold δ ; and (iv) the m-shock model: when m shocks damage size above a hard failure threshold D are recorded. The device fails when the uncertain degradation process exceeds the soft failure threshold level L or when the uncertain random shocks cause the device failure, or hard failure occurs. The rest of the paper is organized as follows. In Section 2, the reliability models for two independent failure processes are developed. In Section 3, the formulas of device reliability for four different shock patterns are given, respectively. A case study on a gas insulated transmission line are performed in Section 4. Finally, the conclusions are summarized in Section 5. Acronyms and notations MDCFP multiple Dependent Competing Failure Processes LDM Liu process based degradation model WDM Wiener process based degradation model M { •} uncertain measure Ch{•} chance measure C( t) Liu process X( t) continuous uncertain degradation μ( t ) deterministic degradation function

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σ

diffusion coefficients hard failure threshold soft failure threshold δ inter-arrival time threshold of two sequential shocks x TL 0 lifetime of the device before soft failure F¯ (t ) survival function of the LDM N( t) number of shocks that have arrived by time t Xk random interval time of the k − 1th and the kth shocks Yk uncertain magnitude of the kth shock load S( t) cumulative damage size of the shocks at time t R( t) device reliability function of the LDM λ arrival rate of random shocks W( t) Wiener process F¯WDM (t ) survival function of the WDM RWDM (t) device reliability function of the WDM D L

2. Modeling and assumptions In order to help readers to understand this paper easily, we first introduce two concepts on uncertainty theory and chance theory formally. Definition 1. [27] The uncertain variables ξ 1 ,ξ 2 ,…, ξ n are said to be independent if n

n

i=1

i=1

M{ ∩ {ξi ∈ Bi }} = ∧ M{ξi ∈ Bi },

(1)

for any Borel sets B1 ,B2 ,…, Bn . Let ( , L, M) be an uncertain space, and (, A, P) be a probability space. Then

( , L, M ) × (, A, P ) = ( × , L × A, M × P ),

(2)

is called a chance space. Definition 2. [37] Let ( , L, M) × (, A, P) be a chance space, and  ∈ L × A be an uncertain random event. Then, the chance measure Ch of  is defined by

Ch{} =



1 0

P {ω ∈ |M{γ ∈  |(γ , ω ) ∈  } ≥ r }dr.

(3)

Details about the uncertainty theory and chance theory can be referred to [24,37,38]. In the following, we consider a device that experiences two independent competing failure processes: soft failure due to continuous uncertain degradation and hard failure due to uncertain random shocks. The device fails when the continuous uncertain degradation process exceeds the soft failure threshold level L or when hard failure occurs. The dependence analysis of these two competing failure processes is another interesting topic, e.g., [15]. The MDCFP considering both uncertain and random factors is left to our future study. 2.1. Modeling of soft failure due to continuous uncertain degradation In the traditional degradation-based reliability analysis, the Wiener process is one of the widely used stochastic processes. However, almost all sample paths of the Wiener process are continuous but non-Lipchitz functions [43]. So it is inappropriate to describe the degradation process by the Wiener process. Since almost all sample paths of the Liu process are Lipchitz continuous functions [43], it can be a better option to describe the degradation process by the Liu process. The continuous uncertain degradation X(t) for the device follows a given uncertain process

X (t ) = μ(t ) + σ C (t ), or dX (t ) = μ (t )dt + σ dC (t ),

(4)

where the deterministic degradation function μ(t) is a main degradation part, the σ C(t) is an uncertain part, and C(t) is a Liu process, i.e., C(0) = 0, μ (t), and σ are the drift and diffusion coefficients, and the continuous function μ(t) ≥ μ(0) = X(0) = x0 , without loss of generality, we assume that x0 ≥ 0 and L > x0 . Soft failure occurs when the continuous uncertain degradation process exceeds the soft failure threshold level L. The soft failure of the device occurs once the uncertain degradation process X(t) is greater than the given threshold value x x L. Let TL 0 = inf{t : X (t ) > L|X (0 ) = x0 }, then the lifetime of the device before soft failure is TL 0 , i.e., the first passage time of X(t) with respect to L. Then the uncertain measure of no soft failure by time t is

F¯ (t ) = M{max X (s ) ≤ L} = M{TLx0 > t }. 0≤s≤t

(5)

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401

Theorem 1. For the continuous uncertain degradation differential equation dX(t) = μ (t)dt + σ dC(t) with X(0) = x0 has an α -path

Xtα = μ(t ) + |σ | −1 (α )t, where

−1

(6)

is the inverse standard normal uncertain distribution.

Proof. Based on the Yao–Chen formula [28], the α -path Xtα solves the following ordinary differential equation

dXtα = μ (t )dt + |σ | −1 (α )dt,

(7)

Taking integral on the above equation, we have

Xtα = μ(t ) + |σ | −1 (α )t.

(8)

The proof is complete.  It follows from Yao–Chen formula [28] that X(t) has an inverse uncertain distribution t−1 (α ) = Xtα . Lemma 1 (Liu [34]). Let Xt be a sample-continuous independent increment process with continuous uncertainty distribution t (x). Then the first hitting time τ z that Xt reaches the level z has an uncertainty distribution,



γ (s ) =

1 − inf t (z ),

if z > X0 ;

sup t (z ),

if

0≤t≤s

(9)

z < X0 .

0≤t≤s

Theorem 2. The continuous uncertain degradation process X(t) = μ(t) + σ C(t) for the device with X(0) = x0 has a survival function

F¯ (x ) = inf t (L ),

(10)

0≤t≤x

where t (x) is the uncertain distribution of X(t). Proof. By Lemma 1, the uncertain distribution function of the first hitting time, i.e., the uncertain distribution function of x the first passage time, TL 0 , is given as follows





F (x ) = M TLx0 ≤ x = 1 − inf t (L ),

(11)

0≤t≤x

By using the duality axiom, we obtain





F¯ (x ) = M TLx0 > x = 1 − F (x ) = inf t (L ).

(12)

0≤t≤x

The proof is complete.  2.2. Modeling of hard failure due to uncertain random shocks The shocks arrive the device according to a renewal process. Let N(t) be the number of shocks that have arrived by time t, Xk be the random interval time of the k − 1th and the kth shocks. Let Yk be the uncertain magnitude of the kth shock  (t )  (t ) load, then S(t ) = N Y be the cumulative damage size of the shocks at time t. Obviously, S(t ) = N Y is an uncertain k=1 k k=1 k random renewal reward process. We suppose the independent and identically distributed random interval times, Xk ,k = 1, 2, …, have a probability distribution function ϕ , and the shock sizes, Yk ,k = 1, 2, …, are independent and identically distributed uncertain variables with an uncertain distribution φ . In the present paper, we consider four different shock patterns that can cause hard failures. 3. Reliability analysis of the device Because the device involves both random factors and uncertain factors, the device reliability is defined as the chance measure that the uncertain degradation signals do not exceed a threshold value L, and the uncertain random shocks do not cause the device fails, or hard failure does not occur, by time t. 3.1. Case 1: extreme shock model In the extreme shock model, hard failure occurs when the first shock damage size above a hard failure threshold D is recorded. Theorem 3. If the continuous uncertain degradation process X(t) for the device follows the uncertain differential equation dX(t) = μ (t)dt + σ dC(t) or X(t) = μ(t) + σ C(t), μ(t) ≥ μ(0) = X(0) = x0 for t ≥ 0. The hard failure process is governed by an uncer (t ) tain random renewal reward process S(t ) = N Y , and the shock patter is an extreme shock model, then the device reliability k=1 k is

R(t ) = P {N (t ) = 0}F¯ (t ) +

∞  k=1





P {N (t ) = k} F¯ (t ) ∧ ϕ (D ) .

(13)

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Proof. It follows from Definition 2 of chance measure that

R(t ) = Ch{no failure happens by time t }        1 N (t ) Nt (ω )



x0 x0 = Ch TL > t, P ω M TL > t, (Yk < D ) = (Yk < D ) ≥ r dr.



0

k=0

(14)

k=0

Since Nt (ω) can only take nonnegative integer values for each ω ∈ , the events

    Nt (ω )

x0 ω M TL > t, (Yk < D ) ≥ r k=0

and

(15)

     ∞ k

ω (Nt (ω ) = k ) ∩ M TLx0 > t, (Yi < D ) ≥ r k=0 i=1

(16)

are equivalent for any number r ∈ [0, 1]; Thus, we have

    Nt (ω )

x0 P ω M TL > t, (Yk < D ) ≥ r k=0      ∞ k

x0 = P ω (Nt (ω ) = k ) ∩ M TL > t, (Yi < D ) ≥ r k=0 i=1      = P ω (Nt (ω ) = 0 ) ∩ M TLx0 > t ≥ r      ∞ k 

+ P ω (Nt (ω ) = k ) ∩ M TLx0 > t, (Yi < D ) ≥ r i=1 k=1      = P ω (Nt (ω ) = 0 ) ∩ M TLx0 > t ≥ r      ∞   + P ω (Nt (ω ) = k ) ∩ M TLx0 > t ∧ min M{Yi < D} ≥ r 1≤i≤k

k=1

=P

   ω (Nt (ω ) = 0 ) ∩ F¯ (t ) ≥ r +

∞ 

P

    ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ ϕ (D ) ≥ r .

(17)

k=1

according to the additivity of probability measure and Definition 1. Thus,

R(t ) =



1

P 0



+  =

+

∞ 1 0

1 0

   ω (Nt (ω ) = 0 ) ∩ F¯ (t ) ≥ r dr

=

k=1

∞  

 F¯ (t ) 0

 F¯ (t ) 0

    ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ ϕ (D ) ≥ r dr

   P ω (Nt (ω ) = 0 ) ∩ F¯ (t ) ≥ r dr

k=1

=

P

0

1

P

    ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ ϕ (D ) ≥ r dr

P {ω|Nt (ω ) = 0 }dr +

∞  F¯ (t )∧ϕ (D )  k=1

P {N (t ) = 0}dr +

= P {N (t ) = 0}F¯ (t ) +

∞  F¯ (t )∧ϕ (D )  k=1

∞ 

0

0



P {ω|Nt (ω ) = k }dr

P {N (t ) = k}dr



P {N (t ) = k} F¯ (t ) ∧ ϕ (D ) .

(18)

k=1

The proof is complete.  3.2. Case 2: cumulative shock model In the cumulative shock model, hard failure occurs when the cumulative damage size of the uncertain random shocks is larger than a hard failure threshold D.

B. Liu, Z. Zhang and Y. Wen / Applied Mathematical Modelling 75 (2019) 398–413

403

Theorem 4. If the continuous uncertain degradation process X(t) for the device follows the uncertain differential equation dX(t) = μ (t)dt + σ dC(t) or X(t) = μ(t) + σ C(t), μ(t) ≥ μ(0) = X(0) = x0 for t ≥ 0. The hard failure process is governed by an  (t ) uncertain random renewal reward process S(t ) = N Y , and the shock patter is a cumulative shock model, then the device k=1 k reliability is

R(t ) = P {N (t ) = 0}F¯ (t ) +

∞ 



P {N (t ) = k} F¯ (t ) ∧ ϕ

 D 

k=1

.

k

(19)

Proof. It follows from Definition 2 of chance measure that

R(t ) = Ch{no failure happens by time t }        1 N (t ) N (t )   = Ch TLx0 > t, Yk < D = P ω M TLx0 > t, Yk < D ≥ r dr.



0

k=0

(20)

k=0

Since Nt (ω) can only take nonnegative integer values for each ω ∈ , the events

    N (t )  x0 ω M TL > t, Yk < D ≥ r k=0

and

(21)

     ∞ k  x0 ω (Nt (ω ) = k)∩ M TL > t, Yi < D ≥ r k=0 i=0

(22)

are equivalent for any number r ∈ [0, 1]; Thus, we have

    N (t )  x0 P ω M TL > t, Yk < D ≥ r k=0      ∞ k  x0 = P ω (Nt (ω ) = k) ∩ M TL > t, Yi < D ≥ r k=0 i=1    x0 = P ω|(Nt (ω ) = 0 )∩ M{TL > t } ≥ r      ∞ k   x0 + P ω (Nt (ω ) = k ) ∩ M TL > t, Yi < D ≥ r i=1 k=1      = P ω (Nt (ω ) = 0 ) ∩ M TLx0 > t ≥ r      ∞ k    x0  + P ω|(Nt (ω ) = k )∩ M TL > t ∧ M Yi < D ≥r i=1

k=1

   D   ∞     = P ω (Nt (ω ) = 0 ) ∩ F¯ (t ) ≥ r + P ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ ϕ ≥r . k

k=1

according to the additivity of probability measure and Definition 1. Thus,

R(t ) =



   ω (Nt (ω ) = 0 ) ∩ F¯ (t ) ≥ r dr 0  1    D    ∞ + P ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ ϕ ≥ r dr k 1

P

0



k=1

   = P ω (Nt (ω ) = 0 ) ∩ F¯ (t ) ≥ r dr 0   D    ∞  1   + P ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ ϕ ≥ r dr 1

k=1

= =

 F¯ (t ) 0

 F¯ (t ) 0

k

0

P {ω|Nt (ω ) = 0 }dr +

∞  F¯ (t )∧ϕ  k=1

P {N (t ) = 0}dr +

0

∞  F¯ (t )∧ϕ ( D )  k k=1

0

( ) D k

P {ω|Nt (ω ) = k }dr

P {N (t ) = k}dr

(23)

404

B. Liu, Z. Zhang and Y. Wen / Applied Mathematical Modelling 75 (2019) 398–413

= P {N (t ) = 0}F¯ (t ) +

∞ 



P {N (t ) = k} F¯ (t ) ∧ ϕ

k=1

 D  k

.

(24)

The proof is complete.  3.3. Case 3: δ -shock model In the δ -shock model, hard failure occurs when the inter-arrival time of two sequential shocks is less than a threshold δ . Theorem 5. If the continuous uncertain degradation process X(t) for the device follows the uncertain differential equation dX(t) = μ (t)dt + σ dC(t) or X(t) = μ(t) + σ C(t), μ(t) ≥ μ(0) = X(0) = x0 for t ≥ 0. The hard failure process is governed by an  (t ) uncertain random renewal reward process S(t ) = N Y , and the shock patter is a δ -shock model, then the device reliability is k=1 k

R(t ) = P {N (t ) = 0}F¯ (t ) +

∞ 





P {N (t ) = k} F¯ (t ) ∧ (1 − φ (δ )) .

(25)

k=1

Proof. It follows from Definition 2 of chance measure that

R(t ) = Ch{no failure happens bytime t }     N (t ) Nt (ω ) 1 x0 x0 = Ch TL > t, (Xk > δ ) = 0 P ω M TL > t, (Xk > δ ) ≥ r dr. k=0

(26)

k=0

Since Nt (ω) can only take nonnegative integer values for each ω ∈ , the events

    Nt (ω )

x0 ω M TL > t, (Xk > δ ) ≥ r k=0

and

(27)

     ∞ k

x0 ω (Nt (ω ) = k ) ∩ M TL > t, (Xi > δ ) ≥ r k=0 i=1

(28)

are equivalent for any number r ∈ [0, 1]; Thus, we have

  Nt (ω )

x0 P ω M{TL > t, (Xi > δ )} ≥ r k=0     ∞ k

= P ω (Nt (ω ) = k ) ∩ (M TLx0 > t, (Xi > δ ) ≥ r ) k=0 i=1   x0 = P ω (Nt (ω ) = 0 ) ∩ (M{TL > t } ≥ r )     ∞ k 

x0 + P ω (Nt (ω ) = k ) ∩ (M TL > t, (Xi > δ ) ≥ r ) i=1 k=1   = P ω (Nt (ω ) = 0 ) ∩ (M{TLx0 > t } ≥ r )      ∞   + P ω (Nt (ω ) = k ) ∩ M TLx0 > t ∧ min M{Xi > δ} ≥ r 1≤i≤k

k=1

=P

  ω (Nt (ω ) = 0 ) ∩ (F¯ (t ) ≥ r ) +

∞ 

P

    ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ (1 − φ (δ )) ≥ r .

k=1

according to the additivity of probability measure and Definition 1. Thus,

R(t ) =



1

P 0



+  =

0

∞ 1 0

1

   ω (Nt (ω ) = 0 ) ∩ F¯ (t ) ≥ r dr

k=1

P

    ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ (1 − φ (δ ) ) ≥ r dr

   P ω (Nt (ω ) = 0 ) ∩ F¯ (t ) ≥ r dr

(29)

B. Liu, Z. Zhang and Y. Wen / Applied Mathematical Modelling 75 (2019) 398–413 ∞  

+ = =

0

 F¯ (t ) 0

P 0

k=1

 F¯ (t )

1

405

    ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ (1 − φ (δ ) ) ≥ r dr

P {ω|Nt (ω ) = 0 }dr +

∞  F¯ (t )∧(1−φ (δ ) )  k=1

P {N (t ) = 0}dr +

∞  F¯ (t )∧(1−φ (δ ) )  k=1

= P {N (t ) = 0}F¯ (t ) +

∞ 

0

0

P {ω|Nt (ω ) = k }dr

P {N (t ) = k}dr





P {N (t ) = k} F¯ (t ) ∧ (1 − φ (δ ) ) .

(30)

k=1

The proof is complete.  3.4. Case 4: m-shock model In the m-shock model, hard failure occurs when m shocks damage size above a hard failure threshold D are recorded. Theorem 6. If the continuous uncertain degradation process X(t) for the device follows the uncertain differential equation dX(t) = μ (t)dt + σ dC(t) or X(t) = μ(t) + σ C(t), μ(t) ≥ μ(0) = X(0) = x0 for t ≥ 0. The hard failure process is governed by an  (t ) uncertain random renewal reward process S(t ) = N Y , and the shock patter is a m-shock model, then the device reliability k=1 k is

R(t ) = P {N (t ) = 0}F¯ (t ) +

m −1  k 

 

  k P {N (t ) = k} F¯ (t ) ∧ (1 − ϕ (D )) ∧ ϕ (D ) l

k=1 l=0

 

∞ m −1     k + P {N (t ) = k} F¯ (t ) ∧ (1 − ϕ (D )) ∧ ϕ (D ) . l

(31)

k=m l=0

Proof. We need to consider the following three situations for obtaining the device reliability. Situation 1. N(t) = 0. In this situation, no shocks occur by time t, i.e., N(t) = 0, then

RN (t )=0 (t ) = Ch{no failure happens by time t }





= Ch TLx0 > t , N (t ) = 0  1      = P ω (Nt (ω ) = 0 ) ∩ M TLx0 > t ≥ r dr  = =

0

1

P 0

   ω (Nt (ω ) = 0 ) ∩ F¯ (t ) ≥ r dr

 F¯ (t ) 0

P {ω|Nt (ω ) = 0 }dr =

 F¯ (t ) 0

P {N (t ) = 0}dr

= P {N (t ) = 0}F¯ (t ).

(32)

Situation 2. N(t) = k, 1 ≤ k ≤ m − 1. In this situation, the number of shocks by time t is between 1 and m − 1, i.e., N(t) = k, 1 ≤ k ≤ m − 1, then

R N (t )=k

1≤k≤m−1

(t ) = Ch{no failure happens by time t }   



k l k 



k = Ch TLx0 > t , N (t ) = k, (Yi > D ), (Yi < D ) l l=0

i=1

i=l+1

     l k 1



k x0 = P ω (Nt (ω ) = k ) ∩ M TL > t, (Yi > D ), (Yi < D ) ≥ r dr l 0 i=1 l=0 i=l+1   1     k  k = P ω (Nt (ω ) = k ) ∩ M{TLx0 > t } ∧ min M{Yi > D} ∧ min M{Yi < D} ≥ r dr l k 

 

l=0

=

  k  l=0

k l

1≤i≤l

0

1

P 0

l+1≤i≤k

    ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ (1 − ϕ (D )) ∧ ϕ (D ) ≥ r dr

406

B. Liu, Z. Zhang and Y. Wen / Applied Mathematical Modelling 75 (2019) 398–413

=

   F¯ (t )∧(1−ϕ (D))∧ϕ (D) k [ ]  k P {N (t ) = k}dr l 0 l=0

 

k    k = P {N (t ) = k} F¯ (t ) ∧ (1 − ϕ (D )) ∧ ϕ (D ) , l

(33)

l=0

where l(l = 0, 1, …, k) denotes the number of shocks greater than D. Since the total number of shocks k ≤ m − 1, the all possible combinations of l out of k must be considered. The shock sizes, Yk ,k = 1, 2, …, are independent and identically distributed, then the formula remains unchanged for each of the combinations. Situation 3. N(t) = k, k ≥ m. In this situation, the number of shocks by time t is greater than or equal to m, i.e., N(t) = k, k ≥ m, then

R N (t )=k (t ) = Ch{no failure happens by time t } k≥m

=

m −1  l=0



 



l k



k Ch TLx0 > t , N (t ) = k, (Yi > D ), (Yi < D ) l i=1

i=l+1

     l k 1



k x0 = P ω (Nt (ω ) = k ) ∩ M TL > t, (Yi > D ), (Yi < D ) ≥ r dr l 0 i=1 l=0 i=l+1   1     m −1  k = P ω (Nt (ω ) = k ) ∩ M{TLx0 > t } ∧ min M{Yi > D} ∧ min M{Yi < D} ≥ r dr l m −1 

 

  m−1 =

 l=0

k l

 l=0

=

m −1  l=0

k l

 

1

P 0

  m−1 =

1≤i≤l

0

l=0

l+1≤i≤k

    ω (Nt (ω ) = k ) ∩ F¯ (t ) ∧ (1 − ϕ (D )) ∧ ϕ (D ) ≥ r dr

[F¯ (t )∧(1−ϕ (D ))∧ϕ (D )]

0

P {N (t ) = k}dr

  k P {N (t ) = k} F¯ (t ) ∧ (1 − ϕ (D )) ∧ ϕ (D ) , l

(34)

where l(l = 0, 1, …, k) denotes the number of shocks greater than D. Since the total number of shocks k ≥ m, the all possible combinations of l out of m − 1 must be considered. The shock sizes, Yk ,k = 1, 2, …, are independent and identically distributed, then the formula remains unchanged for each of the combinations. Now by summing them all, the device reliability is derived to be

R(t ) = P {N (t ) = 0}F¯ (t ) +

m −1  k=1

= P {N (t ) = 0}F¯ (t ) +

 

R N (t )=k

1≤k≤m−1

m −1  k  k=1 l=0

 

(t ) +

∞  k=m

R N (t )=k (t ) k≥m

  k P {N (t ) = k} F¯ (t ) ∧ (1 − ϕ (D )) ∧ ϕ (D ) l

∞ m −1     k + P {N (t ) = k} F¯ (t ) ∧ (1 − ϕ (D )) ∧ ϕ (D ) . l

(35)

k=m l=0

The proof is complete.  Let T be the lifetime of the device, then the mean time to first failure of the device is ET =

 +∞ 0

Ch{T ≥ t }dt =

 +∞ 0

R(t )dt .

4. Numerical examples: a gas insulated transmission line application 4.1. Background This section demonstrates the applicability of the proposed uncertain random failure model by considering a gas insulated transmission line. As a new advanced underground technology, the gas insulated transmission line has been rapidly developed to satisfy the increasing demand for electricity delivery over long distances. In a failure survey of the Norwegian power transmission system, among 180 failures recorded during 36 years, 43% of them are due to internal flashover, and 57% of them are caused by external shocks [44]. Besides, from many gas insulated transmission line applications in China, including Jinping-I Hydropower Station and Hainan nuclear power unit, it is found that the gas insulated transmission line

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Table 1 Model parameter values. Parameters

Values

Sources

α β σ λ

2 0.2 0.3 6.1 3 2980 0.1 1 1

Hao and Yang [45] Hao and Yang [45] Hao and Yang [45] Hao and Yang [45] Hao and Yang [45] Wang et al. [46] Wang et al. [46] Assumption Assumption

L D

δ

a b

experiences both partial discharge (hard failure) and air leakage (soft failure) during its lifetime [45]. These failures not only cause tremendous economic losses, but also lead to severe safety hazards. Therefore, reliability analysis is of great importance for gas insulated transmission line in engineering practice to reduce the failure risk and support decision making. In the engineering practice, the degradation and shock parameters of gas insulated transmission lines are difficult to obtain due to technical problems. In such case, it is reasonable to adopt uncertain process to model the failure process of gas insulated transmission lines. 4.1.1. Continuous internal degradation process From the viewpoint of the Stress–Strength model, a soft failure of the gas insulated transmission line happens once the air leakage amount exceeds the threshold level L, and the air inside is not enough to keep excellent arc-quenching properties. The continuous uncertain degradation level at time t is X(t) = μ(t) + σ C(t) where σ is the diffusion parameter, μ(t) is the nonlinear drift. An exponential function μ(t) = α (eβ t − 1) is commonly used to model the non-linear drift [45]. 4.1.2. Shock damages In addition to the internal degradation process, gas insulated transmission line also subject to mechanical damages due to partial discharge. Partial discharge within the gas insulated transmission line is generally caused by defects like metallic particles, and a partial discharge with voltage exceeding the flashover voltage will result in a breakdown. These external factors will significantly accelerate the degradation speed. In engineering practice, there exist different shock effect patterns. The extreme shock model, cumulative shock model, δ -shock model and m-shock model are commonly utilized in theoretical analysis. This section conducts reliability analysis with respect to the four different shock patterns. The uncertain  (t ) random shocks renewal reward process is modelled by S(t ) = N Y . The shocks inter-arrival times, Xk , k = 1, 2, …, have k=1 k an exponential probability distribution with parameter λ. In this case, the renewal process N(t) is a Poisson process with intensity λ, thus

P {N (t ) = k} =

(λt )k k!

e−λt , k = 0, 1, 2, . . . .

(36)

The shock sizes, Yk , k = 1, 2, …, have a lognormal uncertain variables LOGN(a, b), and the hard failure threshold level is denoted by D. Parameters concerning the degradation process of the gas insulated transmission line including α , β , σ , λ and L are provided by the manufacturers of the Suzhou–Nantong gas insulated transmission line project in China [45], as shown in Table 1. The hard failure thresholds D and δ are obtained from the distribution fitting results in [46]. Parameters a and b are assumed for illustration. 4.2. Reliability analysis for Case 1 We firstly conduct a numerical analysis for the reliability under the case of extreme shock model. Device reliability function can be calculated respectively based on Theorem 3. In this case, the formulae of device reliability can be simplified as





R(t ) = e−λt F¯ (t ) + (F¯ (t ) ∧ ϕ (D ))(eλt − 1 ) .

(37)

Fig. 1 shows the reliability function when the soft failure threshold L increases from 3 to 9 with step size 3. It can be observed that there exist a change point for the device reliability for different values of L. The reliability decreases slowly before the change point but decreases rapidly after the change point. An explanation for this phenomenon is that the probability of hard failure caused by shocks is low before the change point. With the number of shocks increases, the probability of shock magnitude exceeding the hard failure threshold becomes larger. Therefore device reliability characterizes a sharp decrease after the change point. As can be observed in Fig. 1, the soft failure threshold L affects the device reliability significantly. By increasing L from 3 to 9, R(t) increases significantly and shifts to the right. Therefore, devices with a larger soft failure threshold L have a better reliability performance. Furthermore Fig. 1 indicates that when L increases from 3 to

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Fig. 1. Sensitivity analysis of R(t) on L for Case 1.

Fig. 2. Sensitivity analysis of R(t) on D for Case 2.

9, the change point becomes larger. This is because when the soft failure threshold increases the accumulative damage of external shocks on device degradation decreases and it requires a relatively longer time for the appearance of change point. 4.3. Reliability analysis for Case 2 We then investigated the device reliability under the case of cumulative shock model. In this case, the formulae of device reliability can be simplified as



R(t ) = e

−λt



F¯ (t ) +

 D  ∞  (λt )k  ¯ . F (t ) ∧ ϕ k! k

(38)

k=1

By Eq. (38) we can get the device reliability function when t varies from 0 to 12 gradually, as shown in Fig. 2. For different values of D, the device reliability remains a high level in time interval (0, 5) and decreases sharply when t is larger than 5. An explanation for this phenomenon is that the cumulative effect of external shocks on device degradation is not obvious in time interval (0, 5) and with the accumulating damage caused by shocks the device reliability decreases at a greater rate. Comparing Figs. 1 and 2 reveals that in time interval (0, 5), the reliability performance in Case 1 is worse than that in Case 2. This is due to the fact that under the current parameter setting, the probability of a hard failure in Case 1 (extreme shock model) is larger than that in Case 2 (cumulative shock model). Furthermore, as can be observed in Fig. 2,

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Fig. 3. Sensitivity analysis of R(t) on δ for Case 3.

the hard failure threshold D affects the device reliability in a slight way since the soft failure is the main cause of device failure. By increasing D from 10 to 30, R(t) shifts to the right. Therefore, devices with a larger hard failure threshold D have a better reliability performance because the device resistance of external shocks is improved and the probability of hard failure is reduced. 4.4. Reliability analysis for Case 3 This section investigated the device reliability under the case of δ shock model. In this case, the formulae of device reliability can be presented as



R(t ) = e−λt F¯ (t ) + (F¯ (t ) ∧ e−λδ )(eλt − 1 )



(39)

When t varies from 0 to 12 gradually, we get the device reliability function, R(t), as shown in Fig. 3. Comparing the reliability performance in the four cases, device reliability in the case of δ shock model is the lowest since the probability of hard failure is the largest compared with the other 3 cases. It is interesting to note that in the case of the reliability decreases sharply when t exceeds 6. This is due to the fact that the shock effects system reliability more significantly in time interval (0,6) and the cumulative effect of degradation is more obvious after the time point 6. The sensitivity analysis of δ on device reliability is analyzed. As can be observed in Fig. 3, when δ increases from 0.3 to 0.9, R(t) shifts to the right, which implies a better reliability performance for a smaller value of δ because a smaller value of δ implies a smaller probability of hard failure. 4.5. Reliability analysis for Case 4 In the case of m-shock model, the formulae of device reliability can be simplified as

      −1  k ∞ m −1 k k    m ( λ t) ( λ t) k k + . F¯ (t ) + F¯ (t ) ∧ (1 − ϕ (D )) ∧ ϕ (D )



R(t ) = e−λt



k=1 l=0

l

k!

k=m l=0

l

k!

(40)

Fig. 4 indicates the reliability function in the case of m-shock model. The comparison of Figs. 1 and 4 indicates that, the reliability performance in m-shock model is better than that in extreme shock model. This is because the probability of a hard failure in Case 4 is much smaller than that in Case 1. From Fig. 4, it can be observed that the device reliability shows a high sensitivity to the soft failure threshold L. Increasing L causes R(t) to shift to right. This implies that devices with a larger L have a better reliability. Furthermore we can see that device reliability is affected by L in a more significant way in Case 4 than that in Case 1. This is because the probability of hard failure in Case 4 is smaller, and soft failure plays a more significant role in device reliability. Thus improving the soft failure threshold in Case 4 can increase device reliability more significantly. 4.6. Comparison between the proposed reliability evaluation method and existing method This section compares the reliability obtained by the Liu process based degradation model (LDM) and the Wiener process based degradation model (WDM). The Wiener process based degradation model is built up on the probability theory. In this

410

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Fig. 4. Sensitivity analysis of R(t) on L for Case 4.

Fig. 5. Reliability functions under the LMD and WMD models in Case 1.

model, the continuous internal degradation process can be modeled as follows:

X (t ) = μ(t ) + σ W (t ), μ(t ) = α (eβ t − 1 ),

(41)

where W(t) is a standard Wiener process. Thus, the survival function for the continuous internal degradation process is denoted as (see Ref. [1])

 F¯WDM (t ) =

L − μ(t ) √ σ t



 − exp

2[μ(t ) − x0 ](L − x0 ) σ 2t



 ×



−L − μ(t ) + 2x0 , √ σ t

(42)

√ x t2 where (x ) = 1/ 2π −∞ e− 2 dt is the distribution function of a standard normal distribution. The external shocks process in the WDM, {N(t), t ≥ 0}, is a Poisson process with intensity λ, and the shock sizes, Yk ,k = 1, 2, …, are independent and identically distributed random variables with a normally distributed, i.e., Yk ∼ N(μ0 ,σ 0 2 ). There-

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411

Fig. 6. Reliability functions under the LMD and WMD models in Case 2.

Fig. 7. Reliability functions under the LMD and WMD models in Case 3.

fore, the formulae of device reliability in Cases 1–4 can be presented as follows

   μ  ⎧ RWDM (t ) = F¯WDM (t ) exp −λt 1 − D− , ⎪ σ ⎪ ⎪ $ ⎪  % ⎪ + ∞  ⎪ D−kμ (λt )k ⎪ ¯ RWDM (t ) = FWDM (t ) exp(−λt ) 1 + √kσ , ⎪ k! ⎪ ⎪ k=1 ⎪ ⎨ RWDM (t ) = F¯WDM (t ) exp [−λt (1 − exp(−λδ ) )], ⎧     l   k−l ⎫ ⎪ m −1  k ⎪ k (λt )k ⎪ ⎪ ⎪ D−kμ D−kμ ⎪ ⎪ ⎪ √ √ 1 + 1 − ⎨ ⎬ ⎪ k! kσ kσ ⎪ l ⎪ k =1 l=0 ¯   ⎪ R ( t ) = ( t ) exp ( − λ t ) F   l   k−l . WDM WDM ⎪ ∞ m −1 k ⎪  ⎪ ⎪ ⎪ D−kμ D−kμ (λt )k ⎪ ⎪ ⎩ √ √ + 1 − ⎩ ⎭ k! kσ kσ l

(43)

k=m l=0

The values of α , β , σ , λ, L, D and δ are given in Table 1 and we assume μ0 = 100 and σ 0 = 0.001. Under the current parameter setting, the plot of reliability functions under LDM and WDM in the four cases are given in Figs 5–8, respectively. It can be observed that employing WDM method can lead to series discrepancies in assessing reliability and safety characteristics of engineering systems whose failure data and degradation parameters are difficult to obtain. Figs. 5, 6 and 8 indicates that in Cases 1, 2, 4 there exists a change point before which the reliability functions obtained by LDM is lower than that by WDM. This implies that employing the WDM method will overestimate the reliability before the change point

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Fig. 8. Reliability functions under the LMD and WMD models in Case 4.

and underestimate the reliability after the change point. In Case 3, the reliability obtained by the LDM is higher than that by WDM and great difference is shown. Therefore in Case 3 the reliability will be underestimated if WDM method is used. 5. Conclusions This paper develops reliability models for devices that experiencing multiple independent competing failure processes in an uncertain random environment based on chance theory. The continuous degradation satisfies an uncertain differential equation, and the shock model is governed by an uncertain random shocks renewal reward process. Four different shock patterns that can cause hard failure are considered: (i) the extreme shock model; (ii) the cumulative shock model; (iii) the δ -shock model; and (iv) the m-shock model. We obtain formulas to calculate the device reliability by employing chance theory. Some numerical examples for these four cases are also given to illustrate the results obtained in this paper. Sensitivity analysis is also studied to the developed reliability models. The numerical results show that the reliability performance in the extreme shock model is worse than that in the cumulative shock model and m-shock model. Moreover, the device reliability in the case of δ -shock model is the lowest compared with the other 3 cases. The comparison results show that the Wiener process based degradation model (WDM) may overestimate or underestimate the reliability than that in the proposed Liu process based degradation model (LDM). These analysis results indicate that the proposed LMD is a more appropriate option for engineering systems whose failure data and degradation parameters are difficult to obtain. The current research can be extended along several lines. It is worthwhile to study the reliability for devices experiencing multiple dependent competing failure processes in an uncertain environment or in an uncertain random environment. Furthermore, the reliability analysis considering changing failure threshold is another research topic. Finally, the modelling of complex dependent failure behavior in multi-devices is more practical. Acknowledgment This research is supported by the National Natural Science of China under Grants No. 71601101 and 71631001, the Doctoral Scientific Research Foundation of Shanxi Datong University No. 2015-B-06. The authors are very grateful to comments and suggestions of the editors and two anonymous referees that improved the previous version of the paper. We also express our thanks to Qingan Qiu from Beijing Institute of Technology and Shuigui Kang from Shanxi Datong University for their effort in helping with the revision of the paper. References [1] [2] [3] [4]

L.R. Cui, J.B. Huang, Y. Li, Degradation models with wiener diffusion processes under calibrations, IEEE Trans. Reliab. 65 (2016) 613–623. C. Park, W.J. Padgett, Stochastic degradation models with several accelerating variables, IEEE Trans. Reliab. 55 (2006) 379–389. S. Eryilmaz, δ -shock model based on Polya process and its optimal replacement policy, Eur. J. Oper. Res. 263(2017):690–697. M.M. Yu, Y.H. Tang, W.Q. Wu, J. Zhou, Optimal order-replacement policy for a phase-type geometric process model with extreme shocks, Appl. Math. Model. 38 (2014) 4323–4332. [5] S.H. Hao, J. Yang, X.B. Ma, Y. Zhao, Reliability modeling for mutually dependent competing failure processes due to degradation and random shocks, Appl. Math. Model. 51 (2017) 232–249. [6] F. Mallor, E. Omey, Shocks, runs and random sums, J. Appl. Probab. 38 (2001) 438–448. [7] Z.H. Li, B.S. Huang, G.J. Wang, Life distribution and its properties of shock models under random shocks, J. Lanzh. Univ. 35 (1999) 1–7.

B. Liu, Z. Zhang and Y. Wen / Applied Mathematical Modelling 75 (2019) 398–413

413

[8] Y.H. Chien, S.H. Sheu, Z.G. Zhang, E. Love, An extended optimal replacement model of systems subject to shocks, Eur. J. Oper. Res. 175 (1) (2006) 399–412. [9] W Huang, R.G. Askin, A generalized SSI reliability model considering stochastic loading and strength aging degradation, IEEE Trans. Reliab. 53 (2004) 77–82. [10] W. Li, H. Pham, An inspection-maintenance model for systems with multiple competing processes, IEEE Trans. Reliab. 54 (2005) 318–327. [11] G.J. Wang, Y.L. Zhang, A shock model with two-type failures and optimal replacement policy, Int. J. Syst. Sci. 36 (2005) 209–214. [12] Q.A. Qiu, L.R. Cui, H.D. Gao, H. Yi, Optimal allocation of units in sequential probability series systems, Reliab. Eng. Syst. Saf. 169 (2018) 351–363. [13] H. Peng, Q.M. Feng, D.W. Coit, Reliability and maintenance modeling for systems subject to multiple dependent competing failure processes, IIE Trans. 43 (2011) 12–22. [14] K. Rafiee, Q.M. Feng, D.W. Coit, Reliability modeling for dependent competing failure processes with changing degradation rate, IIE Trans. 46 (2014) 483–496. [15] C. Wang, H. Zhang, Q.W. Li, Reliability assessment of aging structures subjected to gradual and shock deteriorations, Reliab. Eng. Syst. Saf. 161 (2017) 78–86. [16] K. Rafiee, Q. Feng, D.W. Coit, Condition-based maintenance for repairable deteriorating systems subject to a generalized mixed shock model, IEEE Trans. Reliab. 64 (2015) 1164–1174. [17] M. Shafiee, M. Finkelstein, An optimal age-based group maintenance policy for multi-unit degrading systems, Reliab. Eng. Syst. Saf. 134 (2015) 230–238. [18] Q.A. Qiu, L.R. Cui, Reliability evaluation based on a dependent two-stage failure process with competing failures, Appl. Math. Model. 64 (2018) 699–712. [19] L. Jiang, Q.M. Feng, D.W. Coit, Modeling zoned shock effects on stochastic degradation in dependent failure processes, IIE Trans. 47 (2015) 460–470. [20] L. Jiang, Q.M. Feng, D.W. Coit, Reliability and maintenance modeling for dependent competing failure processes with shifting failure thresholds, IEEE Trans. Reliab. 61 (2012) 932–948. [21] S. Song, D.W. Coit, Q. Feng, H. Peng, Reliability analysis for multiple-component systems subject to multiple dependent competing failure processes, IEEE Trans. Reliab. 63 (2014) 331–345. [22] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353. [23] B. Liu, Why is there a need for uncertainty theory? J. Uncert. Syst. 6 (2011) 3–10. [24] B. Liu, Uncertainty Theory, second ed, Springer-Verlag, Berlin, Germany, 2007. [25] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, Germany, 2010. [26] B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncert. Syst. 2 (1) (2008) 3–16. [27] B. Liu, Some research problems in uncertainty theory. J. Uncert. Syst. 3(2009): 3–10. [28] K. Yao, X.W. Chen, A numerical method of solving uncertain differential equations, J. Intell. Fuzzy Syst. 25 (2013) 825–832. [29] K. Yao, Extreme value and integral of solution of uncertain differential equation, J. Uncert. Anal. Appl. 1 (2013) Article 2. [30] J.W. Gao, K. Yao, J. Zhou, H. Ke, Reliability Analysis of Uncertain Weighted K-Out-Of-N Systems, IEEE Transactions on Fussy Systems, 2018, doi:10.1109/ TFUZZ.2018.2806365. [31] C.X. Zhang, C.R. Guo, Uncertain block replacement policy with no replacement at failure, J. Intell. Fuzzy Syst. 27 (2014) 1991–1997. [32] H. Ke, K. Yao, Block replacement policy with uncertain lifetime, Reliab. Eng. Syst. Saf. 148 (2016) 119–124. [33] T.P. Zu, R. Kang, M.L. Wen, Q.Y. Zhang, Belief Reliability Distribution Based on Maximum Entropy Principle, IEEE Access, 2017, doi:10.1109/ACCESS.2017. 2779475. [34] B. Liu, Extreme value theorems of uncertain process with application to insurance risk model, Soft Comput. 17 (2013) 549–556. [35] K. Yao, J. Zhou, Ruin time of uncertain insurance risk process, IEEE Trans. Fuzzy Syst. 26 (2018) 19–28. [36] B. Liu, Uncertain Urn Problems and Ellsberg Experiment, Soft Computing, 2018 https://doi.org/10.10 07/s0 050 0- 018- 3306- 7. [37] Y.H. Liu, Uncertain random variables: a mixture of uncertainty and randomness, Soft Comput. 17 (2013) 119–124. [38] Y.H. Liu, Uncertain random programming with applications, Fuzzy Optim. Decis. Mak. 12 (2013) 153–169. [39] K. Yao, J. Zhou, Uncertain random renewal reward process with application to block replacement policy, IEEE Trans. Fuzzy Syst. 24 (2016) 1637–1647. [40] K. Yao, J. Zhou, Renewal reward process with uncertain interarrival times and random rewards, IEEE Trans. Fuzzy Syst. 26(2018): 1757–1762. [41] K. Yao, First hitting time of uncertain random renewal reward process and its application in insurance risk process, Soft Comput. (2018), doi:10.1007/ s0 050 0- 018- 3024- 1. [42] M.L. Wen, R. Kang, Reliability analysis in uncertain random system, Fuzzy Optim. Decis. Mak. 15 (2016) 491–506. [43] B.D. Liu, Toward uncertain finance theory, J. Uncert. Anal. Appl. 1 (2013) 1–15. [44] M. Istad, M. Runde, Thirty-six years of service experience with a national population of gas-insulated substations, IEEE Trans. Power Delivery 25 (2010) 2448–2454. [45] S.H. Hao, J. Yang, Dependent competing failure modeling for the GIL subject to partial discharge and air leakage with random degradation initiation time, IEEE Trans. Reliab. (2018), doi:10.1109/TR.2018.2875819. [46] Q. Wang, Z.Y. He, S. Lin, Z.Y. Li, Failure modeling and maintenance decision for GIS equipment subject to degradation and shocks, IEEE Trans. Power Deliv. 32 (2017) 1079–1088.