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Reliability modeling for degradation-shock dependence systems with multiple species of shocks
Accepted Manuscript
Reliability modeling for degradation-shock dependence systems with multiple species of shocks Hongda Gao , Lirong Cui , Qingan Qiu PII: DOI: Reference:
S0951-8320(18)30320-X https://doi.org/10.1016/j.ress.2018.12.011 RESS 6332
To appear in:
Reliability Engineering and System Safety
Received date: Revised date: Accepted date:
19 March 2018 14 November 2018 15 December 2018
Please cite this article as: Hongda Gao , Lirong Cui , Qingan Qiu , Reliability modeling for degradation-shock dependence systems with multiple species of shocks, Reliability Engineering and System Safety (2018), doi: https://doi.org/10.1016/j.ress.2018.12.011
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Highlights Degradation-shock dependences are considered by two SEPs; The general path model and the Wiener process model are applied to the dependent competing failure processes; Analytical results of the reliability indexes are achieved for models 1 and 2; Simulations are adopted to obtain the system reliability and lifetime pdf for models 3
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and 4.
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Reliability modeling for degradation-shock dependence systems with multiple species of shocks Hongda Gao*, Lirong Cui, Qingan Qiu School of Management and Economics, Beijing Institute of Technology, Beijing, China, 100081
Abstract: In this research, reliability models are developed for systems or devices to
dependent
competing
soft
and
hard
failure
processes
with
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subject
degradation-shock dependences. The new models in this paper extend previous researches by considering novel shock effect patterns (SEPs) resulting from multiple species of external shocks. A soft failure occurs when overall degradation level exceeds the soft failure threshold. Meanwhile, a hard failure occurs when the
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transmitted shock size exceeds the hard failure threshold. The dependences between the soft and hard failure processes are considered in terms of two different SEPs, degradation level and degradation rate. For SEP I, the shocks can increase the degradation level. Thus, the soft failure is indicated by the overall degradation level
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comprising of the internal continuous degradation and the additional damage caused by shocks. For SEP II, the shocks can increase the degradation rate of the soft failure
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process. Specifically, in this research four reliability models focusing on different stochastic processes and SEPs are developed. Closed-form reliability formulas are
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derived for the general path models, while simulation methods are adopted to get the reliability for the Wiener process models. Finally, numerical examples are given to
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illustrate the models and the approaches. Keywords: dependent competing failure; degradation; two SEPs; reliability;
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simulation
1. Introduction Nowadays plenty of highly reliable systems and devices appear with the
development of production and manufacturing technology. Failure data is increasingly difficult to collect. Thus, conventional reliability methods based on failure data are sometime not workable. As a result, degradation-based reliability modeling and *
Corresponding author. Email address: [email protected] (H. Gao).
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analysis [1, 2] have attracted much attention in recent years. Meanwhile advanced sensor technology has improved the convenience of prognostic health management and degradation data collection [3]. Degradation analysis method is becoming an effective alternative to solve issues resulting from insufficient failure data. It is known that stochastic processes are commonly used in degradation modeling because of their applicability in the description of stochastic volatility. Ye and Xie [4] gave a detailed
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review on the degradation models in literature both in theory and application. Specifically, Zuo et al. [5] used a general path model to fit parameters of the model as functions of time. Based on Gamma process model, Pan and Balakrishnan [6] assessed the reliability of products for the degradation processes with multiple performance characteristics. Park and Padgett [7] built a Wiener process model to fit
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the fatigue data of metal. Gao et al. [8] developed a multi-phase Wiener process model and derived the system reliability function and lifetime pdf. In most of the above literature, single failure mode was considered. The degradation processes referred in the above researches are called soft failure processes in which the systems’
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performance decreases gradually and the lifetimes can be described by using the first passage time (FPT) to the failure threshold.
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Apart from the soft failure process, due to complex operation environments, many industrial systems and devices experience a hard failure process [9] which is often
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modeled by random shocks. Fortunately, nowadays the shocks could be well tracked and recorded by various inspect equipment. The multiple failure processes are usually
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competing [10, 11] in which the hard failure is caused by shocks and the soft failure is caused by degradation. A well-known engineering example of the competing failure
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processes is a Micro-Electro-Mechanical system (MEMS) [12] subject to wear (or soft failure) and shock (or hard failure) processes, which is also called a degradation-threshold-shock model which considers both the degradation and shock effects [13]. Lin et al. [14] developed a multi-state physics model for reliability assessment by introducing the semi-Markov and random shock processes. Kong et al. [15] constructed a two-phase degradation process model by considering an abrupt jump at the change point caused by shocks. For hard failure processes, there exist several kinds of failure modes in literature [16, 17], mainly including extreme shock
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mode, cumulative shock mode, k-shock mode, consecutive-k shock mode, and δ-shock mode. It is worth mentioning that the dependencies among the multiple failure processes [18, 19] present a challenging issue which has also been investigated by several researchers. It should be noted that the independence assumption was adopted in many researches [20, 21]; however, it is not realistic. Furthermore, there were also
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many studies focusing on the dependence between the soft and hard failure processes. Specifically, Wang and Pham [22] used a time-scaled covariate factor to modulate the dependent relationship between random shocks and degradation processes. Peng et al. [23] proposed reliability models for systems subject to multiple dependent competing failure processes. Fan et al. [24] built a degradation-shock dependence model in
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which the shocks have different effects on the system’s failure behavior according to their classification. Ye et al. [25] considered the dependence between the soft and hard failure by assuming that the destructive probability of the shocks depends on the remaining lifetime. To improve system reliability and safety, a sensor-based
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calibration method is proposed [26] by combining the competing failure model. For the related maintenance and optimization models, Wang and Pham [27] studied the
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multi-objective maintenance optimization for a dependent competing failure model. In reality, the magnitude of the external shocks may have positive effects on the
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degradation level. In terms of that, Song et al. [28] analyzed the reliability of a multiple-component series competing failure system. In Jiang et al. [ 29 ], the
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dependence relationship lies in that the failure threshold is assumed to be changing due to shocks. In addition, Zhao et al. [30] considered the relationship between the
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degradation rate and load stress levels. However, some assumptions in the above literature are not realistic. In the above literature, although that many researches have considered the dependence competing models, the competing failure models were usually described by considering a single type of shocks for the hard failure process. What’s more, reliability analysis based on the Wiener process has not got enough attention, which can describe the non-monotonic degradation. On the aspect of shock species, few researches considered the multiple shock processes which widely exist in the real world. Due to the complexity of the operation
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environment of systems or products, the species of external shocks are always various, such as temperature, humidity, pressure, audio or ultrasonic, etc. This requires us to consider multiple shock processes rather than regarding the shocks belonging to the exclusive type. Hence, in this paper we develop reliability models for degradation systems experiencing multiple species of stochastic shocks, which is a novelty of this research.
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Furthermore, on the aspect of the shock effects, previous researches mainly focused on direct damage imposed to the degradation level of the system resulting from external shocks. In terms of many applications, for instance, in a real example of the MEMS reported in Tanner and Dugger [12], the external shocks can influence the degradation process by different SEPs. For instance, the degradation rate can be
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changed by shocks and it is often assumed to be fixed or follow a distribution with the determined parameters [31]. This kind of shock effect has also been reported in many studies. Specifically, the degradation process can be separated into different phases and the degradation rate is changing [32] due to shocks. Gao and Cui [33] studied the
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shock effect and get the system reliability by using simulation methods based on general path model. Based on Gamma process model, Saassouh et al. [34] studied a
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deteriorating system with random changes of the degradation rate. Fouladirad and Grall [35] developed a condition-based maintenance model by assuming that the
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degradation rate changes at an unknown time point. Rafiee et al. [36] considered a fluctuant degradation rate model based on different shock models.
In this paper,
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novel shock effects will be considered and the system reliability indexes will be derived for the multi-phase degradation process.
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Based on the analysis above, the shocks may have two aspects of effects on the degradation process. One lies in the occurrence of the hard failure and the other lies in the change of the behavior of the soft failure process. In this paper, we strive to build a novel dependence relation between the shocks and the influence imposed to the degradation process. This research mainly studies the dynamic change of the degradation process which is affected by shocks. Not only the abrupt change of the degradation level but also the changes of the degradation rate contributed by the external shocks are considered. To sum up, four dependent competing failure models
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are developed in this paper which focus on the following aspects: 1) Exploring a system subject to dependent competing failures including both the hard failure and soft failure; 2) Modeling the dependency relation between the hard failure and soft failure by two different SEPs; 3) Focusing on both the general path model and the Wiener process model and the system reliability indexes will be derived, respectively. The reminder of the article is organized as follows. System description and basic
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model assumptions are presented in Section 2. The two competing failure models based on the general path under SEPs I and II respectively are developed in Section 3, and analytical solutions of the system reliability and lifetime distribution are derived. Similarly, in Section 4, the two competing failure models based on the Wiener process under the two SEPs are constructed. Then simulation procedures are proposed to
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determine the reliability indexes. In Section 5, numerical examples and simulations are implemented to illustrate the proposed models. Finally, conclusions and future work are discussed in Section 6.
failure threshold for the soft failure process
N (t )
total shock times by time t
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DS
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Notation
arrival rate of the HPP transmitted shock magnitude of the j th shock for the hard
Wj
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failure process damage for j 1, 2,
n
, N (t )
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number of the shock event species
, An
shock event species
p1 , p2 ,
, pn
transform coefficients of shocks for the hard failure process
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A1 , A2 ,
WA1 ,WA2 ,
,WAn
failure process of all the shock species hard failure threshold
DH S A1 , S A2 ,
magnitudes that affect the transmitted shock size for the soft
, S An
magnitudes that affect the transmitted shock size for the hard failure process of all the shock species
1 , 2 , , n
transform coefficients of all shock species under SEP I
Yj
additional damage to the soft failure process of the j th shock 6
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random shock effect
YH (t )
overall shock damage to the soft failure process by time t
initial degradation value for the general path model
initial degradation rate for the general path model
X g , (t )
overall degradation level for Model 1
j
degradation rate in the j th phase for Model 2
1, 2 , , n
transform coefficients of all the shock species under SEP II
X g , (t )
overall degradation level for Model 2
x0
initial degradation value for the Wiener process model
initial degradation rate for the Wiener process model
diffusion coefficient for the Wiener process model
X W , (t )
overall degradation level for Model 3
j
degradation rate in the j th phase for Model 4
XW , (t )
overall degradation level for Model 4
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Y
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Acronyms
probability density function
cdf
cumulative distribution function
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pdf
independent and identically distributed
FPT HPP
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MEMS SEP
first passage time
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iid
homogeneous Poisson process micro-electro-mechanical system shock effect pattern
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2. System description and model assumptions 2.1. Motivation example In this section, we present an engineering example of a degradation system subject
to both the soft and hard failure processes which motivates the research of this paper. Take a MEMS [12] as example which is designed to machine, manufacture, and control micro materials. The system functions normally under certain working conditions. The dominant failure mechanism is wear of contacting surfaces. Due to
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the variability of the environment, the working conditions are fluctuant which can be regarded as stochastic shocks. An extreme shock over a specific threshold of the normal condition comprising several species, for instance, local pressure, velocity, voltage, temperature, humidity, and audio, can cause a shut-down of the operation system. In addition, those factors can influence the inner mechanism of wear by either one of the two aspects as follows:
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(1) A shock can increase the wear level abruptly when it arrives. It should be pointed out that the influence coefficients of the shocks to wear are different in terms of different shock species.
(2) Either, the shocks can change the wear rate of the MEMS. For instance, sharp
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wear, i.e., higher wear rate, is observed when the environment of the micro-engine becomes dry from humid. Also, the increase of contact pressure or interfacial velocity could lead to a rapid wear rate from a mild one because of severe of the working conditions.
Consequently, the system is subject to multiple competing failure processes. The
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above example can be described by reliability models for degradation systems subject to dependent competing soft and hard failure processes. Hence in this paper, we
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develop novel reliability models and analyze corresponding reliability indexes for systems as described above.
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2.2. Model assumptions
In this paper we consider a system subject to competing soft and hard failure
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processes, i.e., the system fails when either of the two failure modes occurs. The assumptions adopted in this paper are summarized as follows.
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1. There exist n species of external shocks, A1 , A2 ,
, An for each shock. W j
represents the transmitted shock size of the j th shock for the hard failure process damage and it is a random variable for j 1, 2,
. WAi represents the random
shock magnitude of shock Ai and pi is the transmission parameter that links the shock size to the hard failure process damage for i 1, 2,
, n . The arrival of the
shocks is described by a HPP with the arrival rate . 2. The hard failure occurs when the transmitted shock magnitude W j exceeds the 8
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hard failure threshold DH . S Ai represents the random shock magnitude of shock type Ai and i is the transmission parameter that links the shock size to the soft failure process damage for i 1, 2,
, n . The soft failure occurs when the overall
degradation of the system is beyond its threshold DS .
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3. The degradation mechanism of the system can be influenced by shocks through either one of the two patterns. For SEP I, shocks can cause abrupt increase of the degradation level. For SEP II, shocks can increase the degradation rate of the soft failure process.
4. The internal degradation under the general path model is denoted by
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X g (t )= t , where and are initial degradation value and the degradation
rate, respectively. The internal degradation under the Wiener process for the non-monotonous degradation system is XW (t ) x0 t B(t ) , where x0 is the initial degradation value, and are the drift and diffusion coefficients,
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respectively, and B (t ) is a standard Brownian motion.
5. The overall degradation level based on the general path model and the Wiener
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process model under SEP i (i I, II) are denoted by X g ,i (t ) and X W ,i (t ) . The failure processes for the degradation system under SEPs I and II are shown in
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Figs. 1 and 2, respectively. 1 , 2 ,
, N (t ) are a set of realized shock time points. DS
is the soft failure threshold of the system while DH is the hard failure threshold.
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W j ( j 1, 2, ) represents the transmitted shock magnitudes of the j th shock
which are iid random variables. Y j ( j 1, 2, ) is the additional degradation
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increments. The lifetime of the system is T which is described by the FPT when either of the soft and hard failure occurs.
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Degradation
DS
Y3
Y2
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Y1 0
T
Shock magnitude
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DH
W2
W1 0
t
1
W3
2
3
t
Fig. 1. Dependent competing failure processes of systems under SEP I.
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Degradation
DS
phase3
phase 2
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phase1
0
T
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Shock magnitude
DH
W1 0
t
1
W2
2
t
Fig. 2. Dependent competing failure processes of systems under SEP II.
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3. Reliability analysis for competing failure systems based on a general path model The reliability models are developed and then the system reliabilities are derived in this research. As illustrated in Section 2, the point of SEP I lies in the increase of degradation level while that of SEP II lies in the increase of degradation rate of the soft failure processes. In this section, for a degradation system whose soft failure
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process is governed by the general path, models 1 and 2 are developed by considering different SEPs and then the corresponding system reliabilities are obtained. 3.1. Model 1: system reliability of the general path model under SEP I 3.1.1. Hard failure caused by shocks
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Let W j represents the transmitted shock magnitude for the hard failure process damage caused by the j th shock and it is a random variable, where j 1, 2, the assumption, the magnitudes of shocks W1 ,W2 ,
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shock, we can express W j as follows,
Wj p1WA1 p2WA2
. A1 , A2 ,
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where W j are iid for j 1, 2,
are independent. For the j th
pnWAn ,
(1)
, An represent the existing shock species
,WAn are the random magnitudes of the shock
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when a shock arrives. WA1 ,WA2 ,
. As
events. Suppose that the magnitude of a shock is normally distributed, i.e.,
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WAi ~ N ( Ai , A2i ) , thus the load W j of the j th shock follows a normal distribution, n
n
i 1
i 1
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i.e., W j ~ N ( pi Ai , pi2 A2i ) . DH is the constant hard failure threshold. Thus, the probability that the system survives from the applied stress of the j th shock is n P(W j DH )= DH pi Ai i 1
where (t )
t
n
p i 1
2 i
2 Ai
,
(2)
1 1 exp( x 2 )dx is the cdf of a standard normal distribution. 2 2
N (t ) is a point process representing the overall shock times arrived by time t . 11
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Thus, in terms of total probability formula, the probability that the system survives from the external shocks [23] can be given by N (t )
(W j DH ) N (t ) 1) P( N (t ) 1) P( N (t ) 0)
PH (t )=P( j 1
m
m 1
j 1
= P (
(3)
(W j DH )) P( N (t ) m) P( N (t ) 0),
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where PH (t ) is also the probability that no hard failure occurs by time t . Due to that W j ( j 1, 2, ) are iid random variables, Eq. (3) can be expressed by
PH (t ) ( P(W j DH )) m P( N (t ) m) P( N (t ) 0). m 1
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Based on Eq. (2), we have n PH (t )= DH pi Ai m0 i 1
(4)
m
p P( N (t ) m). i 1 n
2 i
2 Ai
(5)
3.1.2. Soft failure caused by degradation and shocks
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A soft failure occurs when the overall degradation exceeds soft failure threshold
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DS . As shown in Fig. 1, the overall degradation level is the sum of the degradation due to the continuous degradation and instantaneous damage caused by shocks. The
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general degradation path, X g (t ) , is used to describe the degradation level, X g (t )= t ,
(6)
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where and represent the initial degradation value and the random degradation
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rate, respectively. In this model we adopt a normal distribution for the degradation 2 rate with ~ N ( g , g ) .
Let Y j represents the increased shock magnitude for the soft failure process
damage caused by the j th shock and it is a random variable. A linearly transmitted model is proposed to describe the degradation increment caused by a shock which can be expressed as
Yj Y 1S A1 2 S A2 12
n S An ,
(7)
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where Y represents random effect independent on the shock damage, which is generally regarded as a normal distribution. Here we assume that Y ~ N ( Y , Y2 ) . S Ai represents the shock magnitude related to the soft failure process damage of
shock type Ai and it is also a random variable for i 1, 2,
, n are transform parameters between the shock
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S Ai ~ N ( Ai , A2i ) . 1 , 2 ,
, n . Here we assume that
magnitude and the degradation damage imposed to the soft failure process. So we can get n
n
i 1
i 1
Y j ~ N ( Y i Ai , Y2 i2 A2i ),
(8)
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where the values of the transform parameters can be estimated based on degradation data, life testing, engineering judgment, etc. The following steps in this section are aiming to get the probability that the system survives from the soft failure risk [23]. The accumulated degradation damage by time t can be expressed as (9)
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N (t ) Y j , if N (t ) 1, YH (t ) j 1 0, if N (t ) 0.
Hence in terms of Eqns. (6)-(9), the overall degradation level, X g , (t ) , under SEP I
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is
X g , (t ) X g (t ) YH (t ) ,
(10)
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Then, the probability of the overall degradation by time t being less than
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threshold DS , can be derived as
P X g , (t ) DS P X g (t ) YH (t ) DS | P( N (t ) m) P( N (t ) m).
(11)
m0
Generally, if we consider G ( x, t ) as the cdf of X g (t ) , fYk ( y ) as the k -fold
convolution of Y j , Eq. (11) can be derived as DS P X g , (t ) DS G ( DS u , t ) fYm (u )du P ( N (t ) m). m0 0
(12)
Suppose that the shocks follow a HPP with arrival rate . Then the probability of 13
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the system surviving from the soft failure process can be expressed as n D ( t m ( )) m S g Y i Ai i 1 exp(t )(t ) . (13) P X g , (t ) DS n m! m0 g2t 2 m( Y2 i2 A2i ) i 1
3.1.3. System reliability under SEP I
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The system fails when either one of the failure modes occurs. The separate analysis of the hard and soft failure processes was executed in Sections 3.1.1 and 3.1.2. Therefore the system lifetime can be described by the FPT when either of the soft or hard failure occurs. Combing the soft and hard failure process, the system reliability
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for Model 1 can be expressed as follows, N (t ) Rmodel1 (t ) P X g , (t ) DS , (W j DH ) | P( N (t ) m) P( N (t ) m) m 1 j 1
(14)
P X g (t ) DS P( N (t ) 0).
As the assumptions, the hard failure damage W j and the additional degradation
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level Y j for the soft failure process are independent. Thus, under condition
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P( N (t ) m) , the events that the system survives from the soft and hard failures are independent. Hence we have
Rmodel1 (t ) P X g , (t ) DS | P( N (t ) m) P(W j DH ) P( N (t ) m)
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m 1
m
P X g (t ) DS P( N (t ) 0),
(15)
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where P(W j DH ) is expressed in Eq. (2). By combining Eqns. (5) and (13), Eq. (15) can be expressed as follows,
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Rmodel1 (t )
n n D ( t m ( ) D pi Ai S g Y i Ai H i 1 i 1 n n m0 g2t 2 m( Y2 i2 A2i ) pi2 A2i i 1 i 1
m
m (16) exp(t )(t ) , m!
where m! represents the factorial of m . Until now we have got the system reliability by combining the soft and hard failure processes for Model 1. The analytical solution of the system reliability has been shown in Eq. (16). 14
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Therefore, the pdf of the lifetime variable can be obtained by taking the negative derivation of the reliability function, i.e., n D pi Ai H i 1 f model1 (t ) n m0 pi2 A2i i 1
dRmodel1 (t ) dt
, as follows,
m
m m 1 exp(t ) t m!
(17)
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g2tA(t , m) A(t , m) A(t , m) g t (m t ) , 3 B(t , m) B(t , m) B(t , m) B (t , m) n
where the defined functions are given by A(t , m) DS ( g t m( Y i Ai ) i 1
n
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and B(t , m) g2t 2 m( Y2 i2 A2i ) . i 1
In industry, besides the system reliability and the lifetime pdf, failure rate function is also an intuitive and convincing measure for reliability engineers. We can get the
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failure rate function r (t ) for t 0 by
f (t ) , R(t )
(18)
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r (t )
(17).
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where the expression of f (t ) and R (t ) are have been expressed in Eq. (16) and
3.2. Model 2: system reliability of the general path model under SEP II
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In this section, we analyze the reliability for the degradation system governed by the general path model under SEP II. The degradation mechanism of the system can
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be changed by the external shocks. Specifically, SEP II is proposed to describe the effect in which the degradation rate may increase randomly by shocks. Different types of shocks can cause different increments through their specific transform coefficients. Once a shock arrives, the degradation rate renews a time. The degradation process can be separated into multiple dependent phases as described in Fig. 2. The number of phases by time t is determined by the number of shocks. 3.2.1. Soft failure process caused by degradation and shocks
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As illustrated in Section 2, the degradation rate is increasing in different phases. For the general path model, denote j as a random variable representing the degradation rate in the j th phase for j 1, 2,
, N (t ) . In this model, the linear
dependence mode under SEP II is proposed as follows,
j 1 j 1S A 2 S A 1
2
n S An ,
(19)
, N (t ) and i represents the transform coefficient of shock event
Ai for i 1, 2,
, n which is determined by the physical property of the shocks.
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where j 1, 2,
2 According to Eq. (6), naturally there is 1 ~ N ( g , g ) representing the initial
degradation rate. In terms of Eq. (19), we have n
n
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j ~ N ( g ( j 1) i A , g2 ( j 1) i2 A2 ) . i
i 1
i 1
i
(20)
Hence, the overall degradation level under SEP II can be expressed as
X g , (t ) 1T1 2 (T2 T1 ) N (t )
N (t ) (TN ( t ) TN ( t ) 1 ) N ( t ) 1 (t TN ( t ) )
i (Ti Ti 1 ) N (t ) 1 (t TN ( t ) ),
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i 1
(21)
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where Ti Ti 1 represents the time interval of successive shocks. 3.2.2. System reliability under SEP II
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In this section, we derive the system reliability for the general path model under SEP II. Eq. (21) gives the expression of overall degradation level, X g , (t ) , which
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doesn’t possess a closed-form distribution. Now we derive the system reliability by proposing a conditional distribution method. First we consider the event of N (t ) m ,
, m , of
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i.e., m shocks arrive by time t , with an occurrence time sequence, 1 , 2 ,
the realized shocks. Naturally we can get the distribution of X g , (t ) conditioned on above shock sequence, X g , (t 1 , 2 , , m ) m
n
n
j 1
i 1
i 1
~ N ( ( g ( j 1) i Ai )( j j 1 ) ( g m i Ai )(t m ), m
n
n
j 1
i 1
i 1
(22)
( g2 ( j 1)( i2 A2i ))( j j 1 )2 ( g2 m i2 A2i )(t m )2 ), where m 1, 2,
representing the shock number. In this case, the conditional 16
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survival probability of no soft failure occurring by time t can be obtained by
P( X g , (t 1 , 2 ,
, m ) DS )
m n n DS ( g ( j 1) i Ai )( j j 1 ) (t m )( g m i Ai ) j 1 i 1 i 1 . (23) m n n ( g2 ( j 1) i 2 A 2 )( j j 1 ) 2 ( g2 m i 2 A2 )(t m ) 2 i i j 1 i 1 i 1
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In this research, the shocks are assumed to follow a HPP with arrival rate . By applying the law of order statistics, the joint conditional pdf of the arriving time sequence for m(m 1) ordered shocks is
,Tm
(1 , 2 ,
, m t ,
(24)
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fT1 ,T2 ,
m! , 0 1 , 2 , , m | N (t ) m) t k others. 0,
The shock magnitudes imposed to the hard failure process are iid random variables. The probability that the system survives from the applied stress of the j th shock have been given in Eq. (2). By combing the hard and soft failure process based on Eqns. (2), (19)-(22), we can derive the system reliability function as follows, (25)
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N (t ) Rmodel2 (t ) P X g , (t ) DS , (W j DH ) | P( N (t ) m) m 1 j 1 P( N (t ) m) P( X g (t ) DS ) P( N (t ) 0).
Under the condition of P( N (t ) m) , the soft and the hard failure processes are
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independent, so we have
m
m 1
j 1
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Rm odel2 (t ) P X g , (t ) DS | P ( N (t ) m) P (
(W j DH )) P ( N (t ) m)
(26)
P ( X g (t ) DS ) P ( N (t ) 0).
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By using the conditional distribution method proposed in this section and the
results of Eqns. (2) and (22), Eq. (26) can be expressed as follows,
17
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Rmodel2 (t ) t m 1 0
m t
f
T1 ,T2 , ,Tm
( 1 , 2 ,
, m ) P( X S g , (t 1 , 2 ,
, m ) Ds ) d 1d 2
0
( P(W j DH )) m P N (t ) m P( X g (t ) DS ) P( N (t ) 0) t m 1 0
m t
m!
t
m
P( X S g , (t 1 , 2 ,
, m ) Ds )d 1d 2
0
where P( X g , (t 1 , 2 ,
(27)
d m
m
m exp(t )(t ) ( DS ( g t ) ) exp(t ), m! g
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n DH pi Ai i 1 n pi2 A2i i 1
d m
, m ) Ds ) is expressed in Eq. (23).
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In Eq. (27), Rmodel2 (t ) is difficult to compute directly since there is the multiple integral in it. In order to address this, we propose a Monte Carlo multiple integral method which is mainly used to solve the multiple integral. Before solving Eq. (27), Fig. 3 is first given to describe the procedure of computing a general multiple b1
b2
a1
a2
bs
as
f ( x1 x2
xs )dx1dx2
dxs , where f ( x1 x2
M
integral
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function and s is the dimension of it.
18
xs ) is an integrable
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Begin s
Sum 0, i 1, Mul (b j a j ) j 1
Set the value of sample variable N
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Sample s randomly distributed point series u1 , u2 , , us in intervals [a1 , b1 ],[a2 , b2 ], ,[as , bs ], respectively
Compute the value V of the integral function f (u1 , u2 , , us ),
i i 1
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Sum Sum V Mul
No
i N?
Yes
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Compute the approximate value 1 Sum of the integral by N
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End
Fig. 3. Flow chart for computing a general multiple integral.
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Then, in terms of the above flow chart for computing a general multiple integral, a simulation procedure is proposed to derive the numerical solution of term t m t
m!
0
m
P( X g , (t 1 , 2 ,
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t
, m ) Ds )d1d 2
d m for m 1, 2,
as follows,
0
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Simulation procedure 1 Step 1: Assign specific values of t , m , and other parameters used in Eq. (23), then denote
t m t
G(t , m) 0
m!
t
m
P( X g , (t 1 , 2 ,
, m ) Ds )d1d 2
d m .
For
0
i 1 to N , perform the following Monte Carlo runs (Steps 2-4);
Step 2: Generate m random numbers by U (0, t ) , then sort them as 1 , 2 ,
, m in
order of size from small to large; Step 3: Based on 1 , 2 ,
, m generated in Step 2, calculate the value of the integral 19
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function by Gi (t , m)
m! P( X S g , (t | 1 , 2 , tm
, m ) Ds ) according to Eq.
(20), and the volume of the integral domain by VS t m ; Step 4: If i N , we have completed all iterations of the simulation. Get the 1 N
N
G (t , m)V i 1
i
S
1 , where m!
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approximate value of the integration by G (t , m)
m! represents the possible permutation number of 1 , 2 ,
End simulation procedure 1
, m .
It should be pointed out that, in Eq. (27), m ranges from 0 to infinite, so we can select m from 0 to a relatively large number when the probability of such number of
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shocks occurring by t is negligible. Hence system reliability, Rmodel2 (t ) , can be given by using Eq. (27) and simulation procedure 1.
4. Reliability analysis for competing failure systems based on a Wiener process model
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In many cases, a non-monotone process can provide a good description of the
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system’s behavior. A system could cross a threshold and then return below as a result of a reduced intensity of use. A Wiener process can characterize a non-monotonic degradation process. However, the previous work has mainly focused on the general
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process model [33] and the degradation rate remains the same in different stages [28,
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29]. Thus, in this section, we consider two Wiener process models for the system with a non-monotone degradation subject to competing soft and hard failure processes
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under SEPs I and II. Specifically, let X W (t ) represent the internal degradation level. A Wiener diffusion
process can be expressed by
XW (t ) x0 t B(t ),
(28)
where x0 is the initial degradation value, is the random drift parameter which are used to represent the degradation rate, is the constant diffusion parameter and
B (t ) is a standard Brownian motion. For a degradation process model, the lifetime of the system can be indicated by 20
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using the FPT which is described as T inf{t : X (t ) DS } , where DS is the predetermined failure threshold. Then the system reliability with initial value x0 and failure threshold DS can be expressed as R(t | x0 , DS ) P{T t | x0 , DS } . 4.1. Model 3: system reliability under SEP I As commonly used by researchers [37, 38], in this model is assumed to follow a normal distribution by ~ N ( W , W2 ) , and is assumed to be a constant. Then XW , (t ) XW (t ) YH (t ) ,
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according to SEP I, the overall degradation level can be expressed as
(29)
where X W (t ) is expressed in Eq. (28), and YH (t ) is the additional shock magnitude by time t which is expressed in Eq. (9).
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The soft failure occurs when the degradation level exceeds threshold DS . The system fails when either of the soft or the hard failures occurs. Then the system reliability can be expressed as follows,
Rmodel3 (t ) P max X W , ( z ) DS , YH (t ) DH 0 z t
M
N (t ) P max X W , ( z ) DS , W j DH N (t ) 1 P( N (t ) 1) 0 z t j 1
(30)
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P max X W ( z ) DS P( N (t ) 0). 0 z t
The degradation process is separated into several phases by shocks. Thus the
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well-known inverse Gaussian distribution of system lifetime for the Wiener process
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model is not available. Thus Eq. (30) is hard to be solved by analytical methods. Thus, In order to get the system reliability of the model, simulation method is adopted for this model.
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The first step of the simulation method is to obtain sufficient failure data of the
degradation system. As illustrated in Section 4, first we express the system reliability by using the FPT as follows,
Rmodel3 (t ) P T W , min TS
W
,
: TS
W
,
inf t : X W (t ) YH (t ) DS ,
TH : TH min j : W j DH , j 0,1,
, N (t ) t ,
(31)
where TW , is the FPT variable of the system for getting a failure data. j is the
21
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arrived time point series of the shocks when the transmitted shock damage W j is larger than the hard failure threshold DH . Then by Eq. (31), simulation procedure for deriving the system reliability of Model 3 is described as follows: Simulation procedure 2
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Step 1: Assign step size t , iteration times M , parameter values of x0 , W , W ,
, , DS , n , and pi , i , DA , A , A for i 1, 2, , n ; i
i
i
Step 2: For i 1:1: M , assign j 1 , t t , 0 0 , then run Steps 3-6;
Step 3: Generate a shock time interval based on arrival rate of the HPP of the shocks and assign j j 1 . Then let j j 1 and N (t ) j . Next go
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to Step 4;
Step 4: If t j , calculate X W , (t ) based on Eq. (29), then go to Step 5; Else if t j , generate the shock magnitude W j for the hard failure process damage according to Eq. (1). Calculate YH ( j ) , X W ( j ) and XW , ( j )
M
based on Eqns. (9), (28), and (29), then go to Step 6;
Step 5: If XW , (t ) DS , assign t t t , then go back to Step 4;
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Else if XW , (t ) DS , get a failure sample t and store it into array A , then go back to Step 2;
Step 6: If XW , ( j ) DS and W j DH , go to Step 3;
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Else if XW , ( j ) DS or W j DH , get a failure sample t and store it into array A , then go back to Step 2;
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Step 7: If i M , all iterations of the simulation have been completed. Calculate
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system reliability by the formulation R(t ) length(find( A t )) M and the empirical pdf of the system lifetime by using the histogram method.
End simulation procedure 2 Thus, the system reliability for Model 3 can be obtained by utilizing simulation
procedure 2. Numerical examples are given in Section 4 to illustrate the procedure. 4.2. Model 4: system reliability under SEP II Model 4 considers a system subject to dependent competing soft and hard failure processes based on the Wiener process under SEP II. The Wiener diffusion process is 22
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expressed in Eq. (28). As illustrated in Section 2, the degradation rate is increased by shocks under SEP II. In this section, denote j as the degradation rate in the j th phase. The dependence between degradation rate and shocks is modeled as follows,
j+1 j 1S A 2 S A 1
n S An ,
2
(32)
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j 1, 2, , N (t ) . We have 1 ~ N (W ,W2 ) representing the initial
where
degradation rate as shown in subsection 4.1, i is the transform coefficient and S Ai is the shock magnitude of Ai and I A1 is the indicator function of event Ai which is defined in Eq. (1) for j 1, 2,
, N (t ) .
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By using Eqns. (1) and (32), we can get that n
n
j ~ N ( W ( j 1) i A , W2 ( j 1) i2 A2 ), i
i 1
where j 1, 2,
i 1
i
(33)
, N (t ) . Then the overall degradation level of the Wiener process
N (t ) (TN (t ) TN (t ) 1 ) N ( t ) 1 (t TN ( t ) ) B(t ) (34)
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x0 1 (T1 ) 2 (T2 T1 )
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model, XW , (t ) , under SEP II is expressed as X W , (t ) N (t )
j (T j T j 1 ) N (t )1 (t TN (t ) ) B(t ), j 1
PT
where Tj Tj 1 represents the random time interval for the ( j 1) th and j th shocks.
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The failure process of the system is competing for the hard and soft failure
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processes. By using the total probability formula, the system reliability can be expressed as
Rmodel4 (t ) P(max X W , (u ) DS , m 1
0u t
N (t )
(W j DH )) | P( N (t ) m) P( N (t ) m)
j 1
P(max X W (u ) DS ) P( N (t ) 0) 0u t
m P max X W , (u ) DS | P N (t ) m P (W j DH ) P N (t ) m 0u t m 1 j 1 P(max X W (u ) DS ) P( N (t ) 0).
0u t
23
(35)
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Although the system reliability, Rmodel4 (t ) , is described in Eq. (35), its analytical solution is not easy to achieve due to the complexity of max X W , (u ) . Thus, similar 0u t
to Model 3, for this model simulation method is adopted to derive the system reliability which can be expressed by using the FPT method,
Rmodel4 (t ) P T W , min TS
W
,
: TS
W
,
inf t : X SW , (t ) DS , , N (t ) t.
(36)
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TH : TH min j : W j DH , j 0,1,
where j is defined similar to that in Eq. (30), and TW , represents the FPT variable of the system to reach a system failure. Then, the simulation procedure for
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deriving the system reliability of Model 4 can be described as follows: Simulation procedure 3
Step 1: Assign step size t , iteration times M , parameter values of x0 , W , W ,
, , DS , n , and pi , i , DA , A , A for i 1, 2, , n ; i
i
i
Step 2: For i 1:1: M , assign j 1 , t t , 0 0 , then run Steps 3-6;
M
Step 3: Generate a shock time interval based on arrival rate of the HPP of the
to Step 4;
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shocks and assign j j 1 . Then let j j 1 and N (t ) j . Next go Step 4: If t j , calculate XW , (t ) based on Eq. (34), then go to Step 5;
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Else if t j , generate the shock magnitude W j for the failure process damage according to Eq. (1). Calculate X W , ( j ) based on Eq. (34), then go
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to Step 6;
Step 5: If XW , (t ) DS , assign t t t , then go to Step 4;
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Else if XW , (t ) DS , get a failure sample t and store it into array A , then go back to Step 2;
Step 6: If XW , ( j ) DS and W j DH , go to Step 3; Else if XW , ( j ) DS or W j DH , get a failure sample t and store it into array A , then go back to Step 2; Step 7: If i M , all iterations of the simulation have been completed. Calculate system reliability by the formulation R(t ) length(find( A t )) M and the empirical pdf of the system lifetime by using the histogram method. 24
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End simulation procedure 3 Apart from the extreme shock mode used in the four proposed models, other shock modes could also be studied such as the cumulative shock model, consecutive- k shock model and -shock model. Furthermore, the reliability formulas and numerical examples could also be yielded which are omitted here.
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5. Numerical examples and simulations Based on the examples reported by Tanner and Dugger [12], Peng et al. [23], and Rafiee et al. [36], some parameters and assumptions are extended to illustrate the models proposed in this paper. Meanwhile, simulations and sensitivity analysis corresponding to the models are implemented as well.
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5.1. Illustrated examples for models 1 and 2
Table 1. Parameter values for reliability analysis of the MEMS.
Parameter
Value
Source
DS
1.25 103
0
~ N ( g , g2 )
8.4823 109 ,
Tanner and Dugger [12],
6.0016 1010
Peng et al. [23]
Y 0.9 104 ,
Song et al. [28]
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M
Tanner and Dugger [12]
Y ~ N ( Y , Y2 )
Tanner and Dugger [12]
PT
Y 2.1105
2.5 105
Tanner and Dugger [12]
n
2
Assumption
DH
3.5 105
Assumption
( p1 , p2 )
(0.18, 0.25)
Assumption
(1 , 2 )
(0.042,0.075)
Assumption
( 1 , 2 )
(0.5 105 ,0.3 105 )
Assumption
( A1 , A2 )
(1.5 104 ,2.5 105 )
Assumption
( A1 , A2 )
(2 105 ,2.5 106 )
Assumption
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A commonly used application example on the MEMS is adopted for models 1 and 2 25
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in this paper. Consider a MEMS consisting of orthogonal linear comb drive actuators that are mechanically connected to a rotating gear [12]. The device experiences two dependent competing failure processes, i.e., soft failure due to wear and hard failure of debris caused by shocks. The micro-engine system is used in a load-sharing redundant system wherein there exist multiple types of shocks, such as pressure, temperature, and humidity. All the shocks could increase the degradation level or the
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degradation rate. So the system can be described by a competing dependent soft and hard failure processes based on the general path model under SEPs I and II.
The parameters used for models 1 and 2 of the example are provided in Table 1. According to Eqns. (16) and (17), the system reliability for Model 1, Rmodel1 (t ) , and
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the pdf of the lifetime can be obtained. Moreover, the system failure rate is given based on Eq. (18). Specifically, the sensitivity analysis of Rmodel1 (t ) , lifetime pdf and failure rate function on the shock arrival rate and the soft failure threshold DS are shown in Figs. 4, 5, and 6. For the left sub-graphs in the above figures, we adopt
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DS 1.25e3 ; meanwhile, for the right sub-graphs, we adopt 3.5e5 . In order to
verify the analytical results, simulation results are also provided with the same
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parameter values in the corresponding figures. In the simulations, 30,000 failure data is collected. Then, the system reliability and lifetime curves are got based on the
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failure data. In terms of the results in Figs. 4 and 5, it is found that the simulation results are in line with the analytical ones.
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CE
R(t )
R(t )
2.5e5 3.5e5 DS 1.75e3 DS 1.25e3
t
t
Fig. 4. Sensitivity analysis of reliability on and DS for Model 1.
Based on the curves of the reliability indexes, some valuable results can be got for
26
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reliability engineers. Fig. 4 indicates that both the shock arrival rate and the soft failure thresholds DS have effects on the system reliability. By the left sub-graph in Fig. 4, the larger of is, the higher of the system reliability will be. By the right sub-graph in Fig. 4, we know that the soft failure threshold DS has no much effect on the system reliability at the beginning. This is because that the system fails mainly caused by the hard failure but not the soft failure at the beginning stage of the
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degradation. With time goes on, the failure may be caused by both the two failure modes. The system reliability could be improved when the failure threshold increases or the shock arrival rate decreases. f (t )
2.5e5
t
M
3.5e5
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f (t )
DS 1.25e3
DS 1.75e3
t
ED
Fig. 5. Sensitivity analysis of lifetime pdf on and DS for Model 1.
The pdf is the deviation of the reliability function which reflects the change rate of
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reliability. By Fig. 5, we know that the peak point are various for different parameters. In this example, we find that the larger of or the smaller of DS is, the earlier will
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the peak points occur.
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r (t )
r (t ) 3.5e5 2.5e5
DS 3.5e3 DS 2.5e3
t
t
Fig. 6. Analytical results of sensitivity analysis of the system failure rate on and DS for 27
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Model 1.
Fig. 6 shows the failure rate curves of the system, which is valuable for reliability engineers. For the left sub-graph, the failure rate is always greater than that when the shock arrival rate increases. For the right sub-graph, there is an intersection for the failure rate curves. R(t )
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R(t )
2.5e5 3.5e5
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DS 1.75e3 DS 1.25e3
t
t
Fig. 7. Sensitivity analysis of the system reliability on and DS for Model 2.
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PT
ED
M
f (t )
t
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Fig. 8. Simulation results of sensitivity analysis of lifetime pdf on , DH , and DS for Model 2.
For Model 2, the results of system reliability, Rmodel2 (t ) , is shown in Fig. 7
according to Eqns. (27). When computing the multiple integral in Eq. (27) by using simulation procedure 1, the adopted largest integer m is 50, and for each value of
m , we adopt the sample number N 10, 000 . Also, in order to verify the analytical results, simulation results of the reliability are presented in Fig. 7. Similar to Model 1, 28
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M 30, 000 degradation paths of the system are simulated to get the failure data. However, due to the complexity of the reliability function Rmodel2 (t ) , it is not easy to get the closed form of the lifetime pdf; thus, simulation result is achieved which is plotted and the sensitivity analysis on , DH , and DS is shown in Fig. 8. In terms of the sensitivity analysis, similar results of the parameter effects can be found as those in Model 1. Both the shock arrival rate and the failure threshold have
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great effect on the system reliability indexes. In conclusion, the results also remind us to improve the system reliability by reducing the shock frequency in operation and increasing the failure threshold of reliability systems. 5.2. Illustrated examples for models 3 and 4
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Models 3 and 4 are developed based on the Wiener process under SEPs I and II. Due to the lack of real data for Wiener process based degradation system, numerical results of the system reliability and lifetime pdf are given by simulation. Corresponding simulation procedures are described in Section 4. Suppose that the
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common parameters for models 3 and 4 are as follows: x0 0 , n 2 , p1 0.25 ,
p2 0.15 , A1 4 , A1 3 , A2 3 , and A2 1.5 . Specific parameters adopted
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in Model 3 are W 0.4 , W 0.1 , 1 0.1 and 2 0.075 . By simulation procedure 2 proposed in Section 4, sensitivity analysis of the reliability and lifetime
PT
pdf on , DS , and have been presented in Figs. 9 and 10. It is critical to get sufficient failure date when executing reliability assessment for a specific degradation
CE
system. In terms of the simulation procedures given in Section 4, FPT method is applied to get the failure data for each of the simulated path. In this example,
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M 30, 000 degradation paths of the system are simulated through which the system
reliability and the lifetime pdf are obtained. The shocks increase the degradation level for Model 3 while the shocks increase
the degradation rate for Model 4. Based on the sensitivity analysis, the results indicate that the failure thresholds DS , the shock arrival rate , and the diffusion parameter
all have significant effects on the reliability indexes. The system reliability will increase with the increase of DS while it decreases with the increase of . In addition, with the increase of , the fluctuation of the degradation will increase; thus, 29
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the variance of the system lifetime will decrease and this can be reflected by the pdf curve in Fig. 10. These facts coincide well with reality. Also, the sensitivity analysis can also illustrate the proposed simulation procedures.
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R (t )
t
Fig. 9. Simulation results of sensitivity analysis of the system reliability for Model 3.
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PT
ED
M
f (t )
t
AC
Fig. 10. Simulation results of sensitivity analysis of the lifetime pdf for Model 3.
For Model 4, the adopted specific parameters are W 0.4 , W 0.03 , 1 0.01 ,
and 2 0.025 . The system reliability and the lifetime pdf can be obtained by executing simulation procedure 3. The results of sensitivity analysis are also presented in Figs. 11 and 12, respectively. Similarly, in Model 4, M 30, 000 failure data of the degradation system are collected. In terms of the sensitivity analysis, similar results of the parameter effects can be found as Model 3. The shock arrival rate, the
30
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failure threshold and the diffusion coefficient all have obvious effects on the system reliability and lifetime pdf which are omitted to be described here.
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R(t )
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t
Fig. 11. Simulation results of the system reliability curves for Model 4.
PT
ED
M
f (t )
CE
t
Fig. 12. Simulation results of sensitivity analysis of the lifetime pdf for Model 4.
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6. Conclusions and future work In this research, four reliability models are proposed for systems subject to
competing soft and hard failure processes with degradation-shock dependences. Motivated by the real example of a MEMS, these models are natural extensions of the conventional degradation-threshold-shock ones. For models 1 and 2 based on the general path model, analytical solutions of system reliabilities are presented. For models 3 and 4 based on the Wiener process model, corresponding reliability indexes 31
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are also obtained by using simulation methods. Sensitivity analysis is presented on the system reliability of some parameters. As the results illustrated, there are several ways to enhance the system reliability, such as attempting to decrease the frequency of shocks and increase the failure threshold, i.e., improve the resistibility to degradation caused by internal deterioration and shock damages. The main contributions of the paper are as follows. First, multiple specifies of shocks are considered in the shock
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process which is more realistic in practice compared to the existing work on the shock-degradation models. Second, in addition to the commonly used general process model, Wiener process model is used to describe the system degradation which has intensive application in degradation modeling. Third, two novel SEPs on the
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dependence between the soft and hard failure processes are proposed which can clearly explain the internal fluctuation mechanism influenced by the external environment. Hence, this research is meaningful for making maintenance decisions to improve the system reliability.
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In addition, more extensions of these models can be developed. First, the changing failure threshold models might be considered for systems subject to multiple
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dependent competing failure processes. Second, different SEPs could be taken into account at the same time due to the complexity of devices or systems in reality. In
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addition, other kinds of stochastic process models are worth studying, such as the Gamma process models and the inverse Gaussian process models.
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Acknowledgement
This work is supported by the NSF of China Grants 71631001. Hongda Gao is also
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supported by China Scholarship Council to study in Penn State University as a visiting scholar in 2017-2018. We thank Prof. Tim Johnson for his linguistic assistance during the preparation of this manuscript. References [1] Lu CJ, Meeker WQ. Using degradation measures to estimate a time-to-failure distribution. Technometrics, 1993, 35(2):161-174.
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Business Media, 2007. [17] Wang GJ, Zhang YL. A shock model with two-type failures and optimal replacement policy. International Journal of Systems Science, 2005, 36(4):209-214.
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Reliability modeling for degradation-shock dependence systems with multiple species of shocks Hongda Gao , Lirong Cui , Qingan Qiu PII: DOI: Reference:
S0951-8320(18)30320-X https://doi.org/10.1016/j.ress.2018.12.011 RESS 6332
To appear in:
Reliability Engineering and System Safety
Received date: Revised date: Accepted date:
19 March 2018 14 November 2018 15 December 2018
Please cite this article as: Hongda Gao , Lirong Cui , Qingan Qiu , Reliability modeling for degradation-shock dependence systems with multiple species of shocks, Reliability Engineering and System Safety (2018), doi: https://doi.org/10.1016/j.ress.2018.12.011
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Highlights Degradation-shock dependences are considered by two SEPs; The general path model and the Wiener process model are applied to the dependent competing failure processes; Analytical results of the reliability indexes are achieved for models 1 and 2; Simulations are adopted to obtain the system reliability and lifetime pdf for models 3
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and 4.
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Reliability modeling for degradation-shock dependence systems with multiple species of shocks Hongda Gao*, Lirong Cui, Qingan Qiu School of Management and Economics, Beijing Institute of Technology, Beijing, China, 100081
Abstract: In this research, reliability models are developed for systems or devices to
dependent
competing
soft
and
hard
failure
processes
with
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subject
degradation-shock dependences. The new models in this paper extend previous researches by considering novel shock effect patterns (SEPs) resulting from multiple species of external shocks. A soft failure occurs when overall degradation level exceeds the soft failure threshold. Meanwhile, a hard failure occurs when the
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transmitted shock size exceeds the hard failure threshold. The dependences between the soft and hard failure processes are considered in terms of two different SEPs, degradation level and degradation rate. For SEP I, the shocks can increase the degradation level. Thus, the soft failure is indicated by the overall degradation level
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comprising of the internal continuous degradation and the additional damage caused by shocks. For SEP II, the shocks can increase the degradation rate of the soft failure
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process. Specifically, in this research four reliability models focusing on different stochastic processes and SEPs are developed. Closed-form reliability formulas are
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derived for the general path models, while simulation methods are adopted to get the reliability for the Wiener process models. Finally, numerical examples are given to
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illustrate the models and the approaches. Keywords: dependent competing failure; degradation; two SEPs; reliability;
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simulation
1. Introduction Nowadays plenty of highly reliable systems and devices appear with the
development of production and manufacturing technology. Failure data is increasingly difficult to collect. Thus, conventional reliability methods based on failure data are sometime not workable. As a result, degradation-based reliability modeling and *
Corresponding author. Email address: [email protected] (H. Gao).
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analysis [1, 2] have attracted much attention in recent years. Meanwhile advanced sensor technology has improved the convenience of prognostic health management and degradation data collection [3]. Degradation analysis method is becoming an effective alternative to solve issues resulting from insufficient failure data. It is known that stochastic processes are commonly used in degradation modeling because of their applicability in the description of stochastic volatility. Ye and Xie [4] gave a detailed
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review on the degradation models in literature both in theory and application. Specifically, Zuo et al. [5] used a general path model to fit parameters of the model as functions of time. Based on Gamma process model, Pan and Balakrishnan [6] assessed the reliability of products for the degradation processes with multiple performance characteristics. Park and Padgett [7] built a Wiener process model to fit
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the fatigue data of metal. Gao et al. [8] developed a multi-phase Wiener process model and derived the system reliability function and lifetime pdf. In most of the above literature, single failure mode was considered. The degradation processes referred in the above researches are called soft failure processes in which the systems’
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performance decreases gradually and the lifetimes can be described by using the first passage time (FPT) to the failure threshold.
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Apart from the soft failure process, due to complex operation environments, many industrial systems and devices experience a hard failure process [9] which is often
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modeled by random shocks. Fortunately, nowadays the shocks could be well tracked and recorded by various inspect equipment. The multiple failure processes are usually
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competing [10, 11] in which the hard failure is caused by shocks and the soft failure is caused by degradation. A well-known engineering example of the competing failure
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processes is a Micro-Electro-Mechanical system (MEMS) [12] subject to wear (or soft failure) and shock (or hard failure) processes, which is also called a degradation-threshold-shock model which considers both the degradation and shock effects [13]. Lin et al. [14] developed a multi-state physics model for reliability assessment by introducing the semi-Markov and random shock processes. Kong et al. [15] constructed a two-phase degradation process model by considering an abrupt jump at the change point caused by shocks. For hard failure processes, there exist several kinds of failure modes in literature [16, 17], mainly including extreme shock
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mode, cumulative shock mode, k-shock mode, consecutive-k shock mode, and δ-shock mode. It is worth mentioning that the dependencies among the multiple failure processes [18, 19] present a challenging issue which has also been investigated by several researchers. It should be noted that the independence assumption was adopted in many researches [20, 21]; however, it is not realistic. Furthermore, there were also
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many studies focusing on the dependence between the soft and hard failure processes. Specifically, Wang and Pham [22] used a time-scaled covariate factor to modulate the dependent relationship between random shocks and degradation processes. Peng et al. [23] proposed reliability models for systems subject to multiple dependent competing failure processes. Fan et al. [24] built a degradation-shock dependence model in
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which the shocks have different effects on the system’s failure behavior according to their classification. Ye et al. [25] considered the dependence between the soft and hard failure by assuming that the destructive probability of the shocks depends on the remaining lifetime. To improve system reliability and safety, a sensor-based
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calibration method is proposed [26] by combining the competing failure model. For the related maintenance and optimization models, Wang and Pham [27] studied the
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multi-objective maintenance optimization for a dependent competing failure model. In reality, the magnitude of the external shocks may have positive effects on the
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degradation level. In terms of that, Song et al. [28] analyzed the reliability of a multiple-component series competing failure system. In Jiang et al. [ 29 ], the
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dependence relationship lies in that the failure threshold is assumed to be changing due to shocks. In addition, Zhao et al. [30] considered the relationship between the
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degradation rate and load stress levels. However, some assumptions in the above literature are not realistic. In the above literature, although that many researches have considered the dependence competing models, the competing failure models were usually described by considering a single type of shocks for the hard failure process. What’s more, reliability analysis based on the Wiener process has not got enough attention, which can describe the non-monotonic degradation. On the aspect of shock species, few researches considered the multiple shock processes which widely exist in the real world. Due to the complexity of the operation
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environment of systems or products, the species of external shocks are always various, such as temperature, humidity, pressure, audio or ultrasonic, etc. This requires us to consider multiple shock processes rather than regarding the shocks belonging to the exclusive type. Hence, in this paper we develop reliability models for degradation systems experiencing multiple species of stochastic shocks, which is a novelty of this research.
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Furthermore, on the aspect of the shock effects, previous researches mainly focused on direct damage imposed to the degradation level of the system resulting from external shocks. In terms of many applications, for instance, in a real example of the MEMS reported in Tanner and Dugger [12], the external shocks can influence the degradation process by different SEPs. For instance, the degradation rate can be
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changed by shocks and it is often assumed to be fixed or follow a distribution with the determined parameters [31]. This kind of shock effect has also been reported in many studies. Specifically, the degradation process can be separated into different phases and the degradation rate is changing [32] due to shocks. Gao and Cui [33] studied the
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shock effect and get the system reliability by using simulation methods based on general path model. Based on Gamma process model, Saassouh et al. [34] studied a
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deteriorating system with random changes of the degradation rate. Fouladirad and Grall [35] developed a condition-based maintenance model by assuming that the
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degradation rate changes at an unknown time point. Rafiee et al. [36] considered a fluctuant degradation rate model based on different shock models.
In this paper,
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novel shock effects will be considered and the system reliability indexes will be derived for the multi-phase degradation process.
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Based on the analysis above, the shocks may have two aspects of effects on the degradation process. One lies in the occurrence of the hard failure and the other lies in the change of the behavior of the soft failure process. In this paper, we strive to build a novel dependence relation between the shocks and the influence imposed to the degradation process. This research mainly studies the dynamic change of the degradation process which is affected by shocks. Not only the abrupt change of the degradation level but also the changes of the degradation rate contributed by the external shocks are considered. To sum up, four dependent competing failure models
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are developed in this paper which focus on the following aspects: 1) Exploring a system subject to dependent competing failures including both the hard failure and soft failure; 2) Modeling the dependency relation between the hard failure and soft failure by two different SEPs; 3) Focusing on both the general path model and the Wiener process model and the system reliability indexes will be derived, respectively. The reminder of the article is organized as follows. System description and basic
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model assumptions are presented in Section 2. The two competing failure models based on the general path under SEPs I and II respectively are developed in Section 3, and analytical solutions of the system reliability and lifetime distribution are derived. Similarly, in Section 4, the two competing failure models based on the Wiener process under the two SEPs are constructed. Then simulation procedures are proposed to
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determine the reliability indexes. In Section 5, numerical examples and simulations are implemented to illustrate the proposed models. Finally, conclusions and future work are discussed in Section 6.
failure threshold for the soft failure process
N (t )
total shock times by time t
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DS
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Notation
arrival rate of the HPP transmitted shock magnitude of the j th shock for the hard
Wj
PT
failure process damage for j 1, 2,
n
, N (t )
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number of the shock event species
, An
shock event species
p1 , p2 ,
, pn
transform coefficients of shocks for the hard failure process
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A1 , A2 ,
WA1 ,WA2 ,
,WAn
failure process of all the shock species hard failure threshold
DH S A1 , S A2 ,
magnitudes that affect the transmitted shock size for the soft
, S An
magnitudes that affect the transmitted shock size for the hard failure process of all the shock species
1 , 2 , , n
transform coefficients of all shock species under SEP I
Yj
additional damage to the soft failure process of the j th shock 6
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random shock effect
YH (t )
overall shock damage to the soft failure process by time t
initial degradation value for the general path model
initial degradation rate for the general path model
X g , (t )
overall degradation level for Model 1
j
degradation rate in the j th phase for Model 2
1, 2 , , n
transform coefficients of all the shock species under SEP II
X g , (t )
overall degradation level for Model 2
x0
initial degradation value for the Wiener process model
initial degradation rate for the Wiener process model
diffusion coefficient for the Wiener process model
X W , (t )
overall degradation level for Model 3
j
degradation rate in the j th phase for Model 4
XW , (t )
overall degradation level for Model 4
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Y
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Acronyms
probability density function
cdf
cumulative distribution function
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independent and identically distributed
FPT HPP
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MEMS SEP
first passage time
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iid
homogeneous Poisson process micro-electro-mechanical system shock effect pattern
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2. System description and model assumptions 2.1. Motivation example In this section, we present an engineering example of a degradation system subject
to both the soft and hard failure processes which motivates the research of this paper. Take a MEMS [12] as example which is designed to machine, manufacture, and control micro materials. The system functions normally under certain working conditions. The dominant failure mechanism is wear of contacting surfaces. Due to
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the variability of the environment, the working conditions are fluctuant which can be regarded as stochastic shocks. An extreme shock over a specific threshold of the normal condition comprising several species, for instance, local pressure, velocity, voltage, temperature, humidity, and audio, can cause a shut-down of the operation system. In addition, those factors can influence the inner mechanism of wear by either one of the two aspects as follows:
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(1) A shock can increase the wear level abruptly when it arrives. It should be pointed out that the influence coefficients of the shocks to wear are different in terms of different shock species.
(2) Either, the shocks can change the wear rate of the MEMS. For instance, sharp
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wear, i.e., higher wear rate, is observed when the environment of the micro-engine becomes dry from humid. Also, the increase of contact pressure or interfacial velocity could lead to a rapid wear rate from a mild one because of severe of the working conditions.
Consequently, the system is subject to multiple competing failure processes. The
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above example can be described by reliability models for degradation systems subject to dependent competing soft and hard failure processes. Hence in this paper, we
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develop novel reliability models and analyze corresponding reliability indexes for systems as described above.
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2.2. Model assumptions
In this paper we consider a system subject to competing soft and hard failure
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processes, i.e., the system fails when either of the two failure modes occurs. The assumptions adopted in this paper are summarized as follows.
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1. There exist n species of external shocks, A1 , A2 ,
, An for each shock. W j
represents the transmitted shock size of the j th shock for the hard failure process damage and it is a random variable for j 1, 2,
. WAi represents the random
shock magnitude of shock Ai and pi is the transmission parameter that links the shock size to the hard failure process damage for i 1, 2,
, n . The arrival of the
shocks is described by a HPP with the arrival rate . 2. The hard failure occurs when the transmitted shock magnitude W j exceeds the 8
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hard failure threshold DH . S Ai represents the random shock magnitude of shock type Ai and i is the transmission parameter that links the shock size to the soft failure process damage for i 1, 2,
, n . The soft failure occurs when the overall
degradation of the system is beyond its threshold DS .
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3. The degradation mechanism of the system can be influenced by shocks through either one of the two patterns. For SEP I, shocks can cause abrupt increase of the degradation level. For SEP II, shocks can increase the degradation rate of the soft failure process.
4. The internal degradation under the general path model is denoted by
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X g (t )= t , where and are initial degradation value and the degradation
rate, respectively. The internal degradation under the Wiener process for the non-monotonous degradation system is XW (t ) x0 t B(t ) , where x0 is the initial degradation value, and are the drift and diffusion coefficients,
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respectively, and B (t ) is a standard Brownian motion.
5. The overall degradation level based on the general path model and the Wiener
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process model under SEP i (i I, II) are denoted by X g ,i (t ) and X W ,i (t ) . The failure processes for the degradation system under SEPs I and II are shown in
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Figs. 1 and 2, respectively. 1 , 2 ,
, N (t ) are a set of realized shock time points. DS
is the soft failure threshold of the system while DH is the hard failure threshold.
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W j ( j 1, 2, ) represents the transmitted shock magnitudes of the j th shock
which are iid random variables. Y j ( j 1, 2, ) is the additional degradation
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increments. The lifetime of the system is T which is described by the FPT when either of the soft and hard failure occurs.
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Degradation
DS
Y3
Y2
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Y1 0
T
Shock magnitude
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DH
W2
W1 0
t
1
W3
2
3
t
Fig. 1. Dependent competing failure processes of systems under SEP I.
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Degradation
DS
phase3
phase 2
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phase1
0
T
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Shock magnitude
DH
W1 0
t
1
W2
2
t
Fig. 2. Dependent competing failure processes of systems under SEP II.
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3. Reliability analysis for competing failure systems based on a general path model The reliability models are developed and then the system reliabilities are derived in this research. As illustrated in Section 2, the point of SEP I lies in the increase of degradation level while that of SEP II lies in the increase of degradation rate of the soft failure processes. In this section, for a degradation system whose soft failure
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process is governed by the general path, models 1 and 2 are developed by considering different SEPs and then the corresponding system reliabilities are obtained. 3.1. Model 1: system reliability of the general path model under SEP I 3.1.1. Hard failure caused by shocks
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Let W j represents the transmitted shock magnitude for the hard failure process damage caused by the j th shock and it is a random variable, where j 1, 2, the assumption, the magnitudes of shocks W1 ,W2 ,
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shock, we can express W j as follows,
Wj p1WA1 p2WA2
. A1 , A2 ,
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where W j are iid for j 1, 2,
are independent. For the j th
pnWAn ,
(1)
, An represent the existing shock species
,WAn are the random magnitudes of the shock
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when a shock arrives. WA1 ,WA2 ,
. As
events. Suppose that the magnitude of a shock is normally distributed, i.e.,
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WAi ~ N ( Ai , A2i ) , thus the load W j of the j th shock follows a normal distribution, n
n
i 1
i 1
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i.e., W j ~ N ( pi Ai , pi2 A2i ) . DH is the constant hard failure threshold. Thus, the probability that the system survives from the applied stress of the j th shock is n P(W j DH )= DH pi Ai i 1
where (t )
t
n
p i 1
2 i
2 Ai
,
(2)
1 1 exp( x 2 )dx is the cdf of a standard normal distribution. 2 2
N (t ) is a point process representing the overall shock times arrived by time t . 11
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Thus, in terms of total probability formula, the probability that the system survives from the external shocks [23] can be given by N (t )
(W j DH ) N (t ) 1) P( N (t ) 1) P( N (t ) 0)
PH (t )=P( j 1
m
m 1
j 1
= P (
(3)
(W j DH )) P( N (t ) m) P( N (t ) 0),
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where PH (t ) is also the probability that no hard failure occurs by time t . Due to that W j ( j 1, 2, ) are iid random variables, Eq. (3) can be expressed by
PH (t ) ( P(W j DH )) m P( N (t ) m) P( N (t ) 0). m 1
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Based on Eq. (2), we have n PH (t )= DH pi Ai m0 i 1
(4)
m
p P( N (t ) m). i 1 n
2 i
2 Ai
(5)
3.1.2. Soft failure caused by degradation and shocks
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A soft failure occurs when the overall degradation exceeds soft failure threshold
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DS . As shown in Fig. 1, the overall degradation level is the sum of the degradation due to the continuous degradation and instantaneous damage caused by shocks. The
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general degradation path, X g (t ) , is used to describe the degradation level, X g (t )= t ,
(6)
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where and represent the initial degradation value and the random degradation
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rate, respectively. In this model we adopt a normal distribution for the degradation 2 rate with ~ N ( g , g ) .
Let Y j represents the increased shock magnitude for the soft failure process
damage caused by the j th shock and it is a random variable. A linearly transmitted model is proposed to describe the degradation increment caused by a shock which can be expressed as
Yj Y 1S A1 2 S A2 12
n S An ,
(7)
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where Y represents random effect independent on the shock damage, which is generally regarded as a normal distribution. Here we assume that Y ~ N ( Y , Y2 ) . S Ai represents the shock magnitude related to the soft failure process damage of
shock type Ai and it is also a random variable for i 1, 2,
, n are transform parameters between the shock
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S Ai ~ N ( Ai , A2i ) . 1 , 2 ,
, n . Here we assume that
magnitude and the degradation damage imposed to the soft failure process. So we can get n
n
i 1
i 1
Y j ~ N ( Y i Ai , Y2 i2 A2i ),
(8)
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where the values of the transform parameters can be estimated based on degradation data, life testing, engineering judgment, etc. The following steps in this section are aiming to get the probability that the system survives from the soft failure risk [23]. The accumulated degradation damage by time t can be expressed as (9)
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N (t ) Y j , if N (t ) 1, YH (t ) j 1 0, if N (t ) 0.
Hence in terms of Eqns. (6)-(9), the overall degradation level, X g , (t ) , under SEP I
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is
X g , (t ) X g (t ) YH (t ) ,
(10)
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Then, the probability of the overall degradation by time t being less than
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threshold DS , can be derived as
P X g , (t ) DS P X g (t ) YH (t ) DS | P( N (t ) m) P( N (t ) m).
(11)
m0
Generally, if we consider G ( x, t ) as the cdf of X g (t ) , fYk ( y ) as the k -fold
convolution of Y j , Eq. (11) can be derived as DS P X g , (t ) DS G ( DS u , t ) fYm (u )du P ( N (t ) m). m0 0
(12)
Suppose that the shocks follow a HPP with arrival rate . Then the probability of 13
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the system surviving from the soft failure process can be expressed as n D ( t m ( )) m S g Y i Ai i 1 exp(t )(t ) . (13) P X g , (t ) DS n m! m0 g2t 2 m( Y2 i2 A2i ) i 1
3.1.3. System reliability under SEP I
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The system fails when either one of the failure modes occurs. The separate analysis of the hard and soft failure processes was executed in Sections 3.1.1 and 3.1.2. Therefore the system lifetime can be described by the FPT when either of the soft or hard failure occurs. Combing the soft and hard failure process, the system reliability
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for Model 1 can be expressed as follows, N (t ) Rmodel1 (t ) P X g , (t ) DS , (W j DH ) | P( N (t ) m) P( N (t ) m) m 1 j 1
(14)
P X g (t ) DS P( N (t ) 0).
As the assumptions, the hard failure damage W j and the additional degradation
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level Y j for the soft failure process are independent. Thus, under condition
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P( N (t ) m) , the events that the system survives from the soft and hard failures are independent. Hence we have
Rmodel1 (t ) P X g , (t ) DS | P( N (t ) m) P(W j DH ) P( N (t ) m)
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m 1
m
P X g (t ) DS P( N (t ) 0),
(15)
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where P(W j DH ) is expressed in Eq. (2). By combining Eqns. (5) and (13), Eq. (15) can be expressed as follows,
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Rmodel1 (t )
n n D ( t m ( ) D pi Ai S g Y i Ai H i 1 i 1 n n m0 g2t 2 m( Y2 i2 A2i ) pi2 A2i i 1 i 1
m
m (16) exp(t )(t ) , m!
where m! represents the factorial of m . Until now we have got the system reliability by combining the soft and hard failure processes for Model 1. The analytical solution of the system reliability has been shown in Eq. (16). 14
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Therefore, the pdf of the lifetime variable can be obtained by taking the negative derivation of the reliability function, i.e., n D pi Ai H i 1 f model1 (t ) n m0 pi2 A2i i 1
dRmodel1 (t ) dt
, as follows,
m
m m 1 exp(t ) t m!
(17)
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g2tA(t , m) A(t , m) A(t , m) g t (m t ) , 3 B(t , m) B(t , m) B(t , m) B (t , m) n
where the defined functions are given by A(t , m) DS ( g t m( Y i Ai ) i 1
n
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and B(t , m) g2t 2 m( Y2 i2 A2i ) . i 1
In industry, besides the system reliability and the lifetime pdf, failure rate function is also an intuitive and convincing measure for reliability engineers. We can get the
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failure rate function r (t ) for t 0 by
f (t ) , R(t )
(18)
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r (t )
(17).
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where the expression of f (t ) and R (t ) are have been expressed in Eq. (16) and
3.2. Model 2: system reliability of the general path model under SEP II
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In this section, we analyze the reliability for the degradation system governed by the general path model under SEP II. The degradation mechanism of the system can
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be changed by the external shocks. Specifically, SEP II is proposed to describe the effect in which the degradation rate may increase randomly by shocks. Different types of shocks can cause different increments through their specific transform coefficients. Once a shock arrives, the degradation rate renews a time. The degradation process can be separated into multiple dependent phases as described in Fig. 2. The number of phases by time t is determined by the number of shocks. 3.2.1. Soft failure process caused by degradation and shocks
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As illustrated in Section 2, the degradation rate is increasing in different phases. For the general path model, denote j as a random variable representing the degradation rate in the j th phase for j 1, 2,
, N (t ) . In this model, the linear
dependence mode under SEP II is proposed as follows,
j 1 j 1S A 2 S A 1
2
n S An ,
(19)
, N (t ) and i represents the transform coefficient of shock event
Ai for i 1, 2,
, n which is determined by the physical property of the shocks.
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where j 1, 2,
2 According to Eq. (6), naturally there is 1 ~ N ( g , g ) representing the initial
degradation rate. In terms of Eq. (19), we have n
n
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j ~ N ( g ( j 1) i A , g2 ( j 1) i2 A2 ) . i
i 1
i 1
i
(20)
Hence, the overall degradation level under SEP II can be expressed as
X g , (t ) 1T1 2 (T2 T1 ) N (t )
N (t ) (TN ( t ) TN ( t ) 1 ) N ( t ) 1 (t TN ( t ) )
i (Ti Ti 1 ) N (t ) 1 (t TN ( t ) ),
M
i 1
(21)
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where Ti Ti 1 represents the time interval of successive shocks. 3.2.2. System reliability under SEP II
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In this section, we derive the system reliability for the general path model under SEP II. Eq. (21) gives the expression of overall degradation level, X g , (t ) , which
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doesn’t possess a closed-form distribution. Now we derive the system reliability by proposing a conditional distribution method. First we consider the event of N (t ) m ,
, m , of
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i.e., m shocks arrive by time t , with an occurrence time sequence, 1 , 2 ,
the realized shocks. Naturally we can get the distribution of X g , (t ) conditioned on above shock sequence, X g , (t 1 , 2 , , m ) m
n
n
j 1
i 1
i 1
~ N ( ( g ( j 1) i Ai )( j j 1 ) ( g m i Ai )(t m ), m
n
n
j 1
i 1
i 1
(22)
( g2 ( j 1)( i2 A2i ))( j j 1 )2 ( g2 m i2 A2i )(t m )2 ), where m 1, 2,
representing the shock number. In this case, the conditional 16
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survival probability of no soft failure occurring by time t can be obtained by
P( X g , (t 1 , 2 ,
, m ) DS )
m n n DS ( g ( j 1) i Ai )( j j 1 ) (t m )( g m i Ai ) j 1 i 1 i 1 . (23) m n n ( g2 ( j 1) i 2 A 2 )( j j 1 ) 2 ( g2 m i 2 A2 )(t m ) 2 i i j 1 i 1 i 1
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In this research, the shocks are assumed to follow a HPP with arrival rate . By applying the law of order statistics, the joint conditional pdf of the arriving time sequence for m(m 1) ordered shocks is
,Tm
(1 , 2 ,
, m t ,
(24)
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fT1 ,T2 ,
m! , 0 1 , 2 , , m | N (t ) m) t k others. 0,
The shock magnitudes imposed to the hard failure process are iid random variables. The probability that the system survives from the applied stress of the j th shock have been given in Eq. (2). By combing the hard and soft failure process based on Eqns. (2), (19)-(22), we can derive the system reliability function as follows, (25)
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N (t ) Rmodel2 (t ) P X g , (t ) DS , (W j DH ) | P( N (t ) m) m 1 j 1 P( N (t ) m) P( X g (t ) DS ) P( N (t ) 0).
Under the condition of P( N (t ) m) , the soft and the hard failure processes are
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independent, so we have
m
m 1
j 1
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Rm odel2 (t ) P X g , (t ) DS | P ( N (t ) m) P (
(W j DH )) P ( N (t ) m)
(26)
P ( X g (t ) DS ) P ( N (t ) 0).
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By using the conditional distribution method proposed in this section and the
results of Eqns. (2) and (22), Eq. (26) can be expressed as follows,
17
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Rmodel2 (t ) t m 1 0
m t
f
T1 ,T2 , ,Tm
( 1 , 2 ,
, m ) P( X S g , (t 1 , 2 ,
, m ) Ds ) d 1d 2
0
( P(W j DH )) m P N (t ) m P( X g (t ) DS ) P( N (t ) 0) t m 1 0
m t
m!
t
m
P( X S g , (t 1 , 2 ,
, m ) Ds )d 1d 2
0
where P( X g , (t 1 , 2 ,
(27)
d m
m
m exp(t )(t ) ( DS ( g t ) ) exp(t ), m! g
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n DH pi Ai i 1 n pi2 A2i i 1
d m
, m ) Ds ) is expressed in Eq. (23).
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In Eq. (27), Rmodel2 (t ) is difficult to compute directly since there is the multiple integral in it. In order to address this, we propose a Monte Carlo multiple integral method which is mainly used to solve the multiple integral. Before solving Eq. (27), Fig. 3 is first given to describe the procedure of computing a general multiple b1
b2
a1
a2
bs
as
f ( x1 x2
xs )dx1dx2
dxs , where f ( x1 x2
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integral
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function and s is the dimension of it.
18
xs ) is an integrable
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Begin s
Sum 0, i 1, Mul (b j a j ) j 1
Set the value of sample variable N
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Sample s randomly distributed point series u1 , u2 , , us in intervals [a1 , b1 ],[a2 , b2 ], ,[as , bs ], respectively
Compute the value V of the integral function f (u1 , u2 , , us ),
i i 1
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Sum Sum V Mul
No
i N?
Yes
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Compute the approximate value 1 Sum of the integral by N
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End
Fig. 3. Flow chart for computing a general multiple integral.
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Then, in terms of the above flow chart for computing a general multiple integral, a simulation procedure is proposed to derive the numerical solution of term t m t
m!
0
m
P( X g , (t 1 , 2 ,
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t
, m ) Ds )d1d 2
d m for m 1, 2,
as follows,
0
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Simulation procedure 1 Step 1: Assign specific values of t , m , and other parameters used in Eq. (23), then denote
t m t
G(t , m) 0
m!
t
m
P( X g , (t 1 , 2 ,
, m ) Ds )d1d 2
d m .
For
0
i 1 to N , perform the following Monte Carlo runs (Steps 2-4);
Step 2: Generate m random numbers by U (0, t ) , then sort them as 1 , 2 ,
, m in
order of size from small to large; Step 3: Based on 1 , 2 ,
, m generated in Step 2, calculate the value of the integral 19
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function by Gi (t , m)
m! P( X S g , (t | 1 , 2 , tm
, m ) Ds ) according to Eq.
(20), and the volume of the integral domain by VS t m ; Step 4: If i N , we have completed all iterations of the simulation. Get the 1 N
N
G (t , m)V i 1
i
S
1 , where m!
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approximate value of the integration by G (t , m)
m! represents the possible permutation number of 1 , 2 ,
End simulation procedure 1
, m .
It should be pointed out that, in Eq. (27), m ranges from 0 to infinite, so we can select m from 0 to a relatively large number when the probability of such number of
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shocks occurring by t is negligible. Hence system reliability, Rmodel2 (t ) , can be given by using Eq. (27) and simulation procedure 1.
4. Reliability analysis for competing failure systems based on a Wiener process model
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In many cases, a non-monotone process can provide a good description of the
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system’s behavior. A system could cross a threshold and then return below as a result of a reduced intensity of use. A Wiener process can characterize a non-monotonic degradation process. However, the previous work has mainly focused on the general
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process model [33] and the degradation rate remains the same in different stages [28,
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29]. Thus, in this section, we consider two Wiener process models for the system with a non-monotone degradation subject to competing soft and hard failure processes
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under SEPs I and II. Specifically, let X W (t ) represent the internal degradation level. A Wiener diffusion
process can be expressed by
XW (t ) x0 t B(t ),
(28)
where x0 is the initial degradation value, is the random drift parameter which are used to represent the degradation rate, is the constant diffusion parameter and
B (t ) is a standard Brownian motion. For a degradation process model, the lifetime of the system can be indicated by 20
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using the FPT which is described as T inf{t : X (t ) DS } , where DS is the predetermined failure threshold. Then the system reliability with initial value x0 and failure threshold DS can be expressed as R(t | x0 , DS ) P{T t | x0 , DS } . 4.1. Model 3: system reliability under SEP I As commonly used by researchers [37, 38], in this model is assumed to follow a normal distribution by ~ N ( W , W2 ) , and is assumed to be a constant. Then XW , (t ) XW (t ) YH (t ) ,
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according to SEP I, the overall degradation level can be expressed as
(29)
where X W (t ) is expressed in Eq. (28), and YH (t ) is the additional shock magnitude by time t which is expressed in Eq. (9).
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The soft failure occurs when the degradation level exceeds threshold DS . The system fails when either of the soft or the hard failures occurs. Then the system reliability can be expressed as follows,
Rmodel3 (t ) P max X W , ( z ) DS , YH (t ) DH 0 z t
M
N (t ) P max X W , ( z ) DS , W j DH N (t ) 1 P( N (t ) 1) 0 z t j 1
(30)
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P max X W ( z ) DS P( N (t ) 0). 0 z t
The degradation process is separated into several phases by shocks. Thus the
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well-known inverse Gaussian distribution of system lifetime for the Wiener process
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model is not available. Thus Eq. (30) is hard to be solved by analytical methods. Thus, In order to get the system reliability of the model, simulation method is adopted for this model.
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The first step of the simulation method is to obtain sufficient failure data of the
degradation system. As illustrated in Section 4, first we express the system reliability by using the FPT as follows,
Rmodel3 (t ) P T W , min TS
W
,
: TS
W
,
inf t : X W (t ) YH (t ) DS ,
TH : TH min j : W j DH , j 0,1,
, N (t ) t ,
(31)
where TW , is the FPT variable of the system for getting a failure data. j is the
21
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arrived time point series of the shocks when the transmitted shock damage W j is larger than the hard failure threshold DH . Then by Eq. (31), simulation procedure for deriving the system reliability of Model 3 is described as follows: Simulation procedure 2
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Step 1: Assign step size t , iteration times M , parameter values of x0 , W , W ,
, , DS , n , and pi , i , DA , A , A for i 1, 2, , n ; i
i
i
Step 2: For i 1:1: M , assign j 1 , t t , 0 0 , then run Steps 3-6;
Step 3: Generate a shock time interval based on arrival rate of the HPP of the shocks and assign j j 1 . Then let j j 1 and N (t ) j . Next go
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to Step 4;
Step 4: If t j , calculate X W , (t ) based on Eq. (29), then go to Step 5; Else if t j , generate the shock magnitude W j for the hard failure process damage according to Eq. (1). Calculate YH ( j ) , X W ( j ) and XW , ( j )
M
based on Eqns. (9), (28), and (29), then go to Step 6;
Step 5: If XW , (t ) DS , assign t t t , then go back to Step 4;
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Else if XW , (t ) DS , get a failure sample t and store it into array A , then go back to Step 2;
Step 6: If XW , ( j ) DS and W j DH , go to Step 3;
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Else if XW , ( j ) DS or W j DH , get a failure sample t and store it into array A , then go back to Step 2;
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Step 7: If i M , all iterations of the simulation have been completed. Calculate
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system reliability by the formulation R(t ) length(find( A t )) M and the empirical pdf of the system lifetime by using the histogram method.
End simulation procedure 2 Thus, the system reliability for Model 3 can be obtained by utilizing simulation
procedure 2. Numerical examples are given in Section 4 to illustrate the procedure. 4.2. Model 4: system reliability under SEP II Model 4 considers a system subject to dependent competing soft and hard failure processes based on the Wiener process under SEP II. The Wiener diffusion process is 22
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expressed in Eq. (28). As illustrated in Section 2, the degradation rate is increased by shocks under SEP II. In this section, denote j as the degradation rate in the j th phase. The dependence between degradation rate and shocks is modeled as follows,
j+1 j 1S A 2 S A 1
n S An ,
2
(32)
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j 1, 2, , N (t ) . We have 1 ~ N (W ,W2 ) representing the initial
where
degradation rate as shown in subsection 4.1, i is the transform coefficient and S Ai is the shock magnitude of Ai and I A1 is the indicator function of event Ai which is defined in Eq. (1) for j 1, 2,
, N (t ) .
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By using Eqns. (1) and (32), we can get that n
n
j ~ N ( W ( j 1) i A , W2 ( j 1) i2 A2 ), i
i 1
where j 1, 2,
i 1
i
(33)
, N (t ) . Then the overall degradation level of the Wiener process
N (t ) (TN (t ) TN (t ) 1 ) N ( t ) 1 (t TN ( t ) ) B(t ) (34)
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x0 1 (T1 ) 2 (T2 T1 )
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model, XW , (t ) , under SEP II is expressed as X W , (t ) N (t )
j (T j T j 1 ) N (t )1 (t TN (t ) ) B(t ), j 1
PT
where Tj Tj 1 represents the random time interval for the ( j 1) th and j th shocks.
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The failure process of the system is competing for the hard and soft failure
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processes. By using the total probability formula, the system reliability can be expressed as
Rmodel4 (t ) P(max X W , (u ) DS , m 1
0u t
N (t )
(W j DH )) | P( N (t ) m) P( N (t ) m)
j 1
P(max X W (u ) DS ) P( N (t ) 0) 0u t
m P max X W , (u ) DS | P N (t ) m P (W j DH ) P N (t ) m 0u t m 1 j 1 P(max X W (u ) DS ) P( N (t ) 0).
0u t
23
(35)
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Although the system reliability, Rmodel4 (t ) , is described in Eq. (35), its analytical solution is not easy to achieve due to the complexity of max X W , (u ) . Thus, similar 0u t
to Model 3, for this model simulation method is adopted to derive the system reliability which can be expressed by using the FPT method,
Rmodel4 (t ) P T W , min TS
W
,
: TS
W
,
inf t : X SW , (t ) DS , , N (t ) t.
(36)
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TH : TH min j : W j DH , j 0,1,
where j is defined similar to that in Eq. (30), and TW , represents the FPT variable of the system to reach a system failure. Then, the simulation procedure for
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deriving the system reliability of Model 4 can be described as follows: Simulation procedure 3
Step 1: Assign step size t , iteration times M , parameter values of x0 , W , W ,
, , DS , n , and pi , i , DA , A , A for i 1, 2, , n ; i
i
i
Step 2: For i 1:1: M , assign j 1 , t t , 0 0 , then run Steps 3-6;
M
Step 3: Generate a shock time interval based on arrival rate of the HPP of the
to Step 4;
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shocks and assign j j 1 . Then let j j 1 and N (t ) j . Next go Step 4: If t j , calculate XW , (t ) based on Eq. (34), then go to Step 5;
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Else if t j , generate the shock magnitude W j for the failure process damage according to Eq. (1). Calculate X W , ( j ) based on Eq. (34), then go
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to Step 6;
Step 5: If XW , (t ) DS , assign t t t , then go to Step 4;
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Else if XW , (t ) DS , get a failure sample t and store it into array A , then go back to Step 2;
Step 6: If XW , ( j ) DS and W j DH , go to Step 3; Else if XW , ( j ) DS or W j DH , get a failure sample t and store it into array A , then go back to Step 2; Step 7: If i M , all iterations of the simulation have been completed. Calculate system reliability by the formulation R(t ) length(find( A t )) M and the empirical pdf of the system lifetime by using the histogram method. 24
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End simulation procedure 3 Apart from the extreme shock mode used in the four proposed models, other shock modes could also be studied such as the cumulative shock model, consecutive- k shock model and -shock model. Furthermore, the reliability formulas and numerical examples could also be yielded which are omitted here.
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5. Numerical examples and simulations Based on the examples reported by Tanner and Dugger [12], Peng et al. [23], and Rafiee et al. [36], some parameters and assumptions are extended to illustrate the models proposed in this paper. Meanwhile, simulations and sensitivity analysis corresponding to the models are implemented as well.
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5.1. Illustrated examples for models 1 and 2
Table 1. Parameter values for reliability analysis of the MEMS.
Parameter
Value
Source
DS
1.25 103
0
~ N ( g , g2 )
8.4823 109 ,
Tanner and Dugger [12],
6.0016 1010
Peng et al. [23]
Y 0.9 104 ,
Song et al. [28]
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M
Tanner and Dugger [12]
Y ~ N ( Y , Y2 )
Tanner and Dugger [12]
PT
Y 2.1105
2.5 105
Tanner and Dugger [12]
n
2
Assumption
DH
3.5 105
Assumption
( p1 , p2 )
(0.18, 0.25)
Assumption
(1 , 2 )
(0.042,0.075)
Assumption
( 1 , 2 )
(0.5 105 ,0.3 105 )
Assumption
( A1 , A2 )
(1.5 104 ,2.5 105 )
Assumption
( A1 , A2 )
(2 105 ,2.5 106 )
Assumption
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A commonly used application example on the MEMS is adopted for models 1 and 2 25
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in this paper. Consider a MEMS consisting of orthogonal linear comb drive actuators that are mechanically connected to a rotating gear [12]. The device experiences two dependent competing failure processes, i.e., soft failure due to wear and hard failure of debris caused by shocks. The micro-engine system is used in a load-sharing redundant system wherein there exist multiple types of shocks, such as pressure, temperature, and humidity. All the shocks could increase the degradation level or the
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degradation rate. So the system can be described by a competing dependent soft and hard failure processes based on the general path model under SEPs I and II.
The parameters used for models 1 and 2 of the example are provided in Table 1. According to Eqns. (16) and (17), the system reliability for Model 1, Rmodel1 (t ) , and
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the pdf of the lifetime can be obtained. Moreover, the system failure rate is given based on Eq. (18). Specifically, the sensitivity analysis of Rmodel1 (t ) , lifetime pdf and failure rate function on the shock arrival rate and the soft failure threshold DS are shown in Figs. 4, 5, and 6. For the left sub-graphs in the above figures, we adopt
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DS 1.25e3 ; meanwhile, for the right sub-graphs, we adopt 3.5e5 . In order to
verify the analytical results, simulation results are also provided with the same
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parameter values in the corresponding figures. In the simulations, 30,000 failure data is collected. Then, the system reliability and lifetime curves are got based on the
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failure data. In terms of the results in Figs. 4 and 5, it is found that the simulation results are in line with the analytical ones.
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CE
R(t )
R(t )
2.5e5 3.5e5 DS 1.75e3 DS 1.25e3
t
t
Fig. 4. Sensitivity analysis of reliability on and DS for Model 1.
Based on the curves of the reliability indexes, some valuable results can be got for
26
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reliability engineers. Fig. 4 indicates that both the shock arrival rate and the soft failure thresholds DS have effects on the system reliability. By the left sub-graph in Fig. 4, the larger of is, the higher of the system reliability will be. By the right sub-graph in Fig. 4, we know that the soft failure threshold DS has no much effect on the system reliability at the beginning. This is because that the system fails mainly caused by the hard failure but not the soft failure at the beginning stage of the
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degradation. With time goes on, the failure may be caused by both the two failure modes. The system reliability could be improved when the failure threshold increases or the shock arrival rate decreases. f (t )
2.5e5
t
M
3.5e5
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f (t )
DS 1.25e3
DS 1.75e3
t
ED
Fig. 5. Sensitivity analysis of lifetime pdf on and DS for Model 1.
The pdf is the deviation of the reliability function which reflects the change rate of
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reliability. By Fig. 5, we know that the peak point are various for different parameters. In this example, we find that the larger of or the smaller of DS is, the earlier will
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the peak points occur.
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r (t )
r (t ) 3.5e5 2.5e5
DS 3.5e3 DS 2.5e3
t
t
Fig. 6. Analytical results of sensitivity analysis of the system failure rate on and DS for 27
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Model 1.
Fig. 6 shows the failure rate curves of the system, which is valuable for reliability engineers. For the left sub-graph, the failure rate is always greater than that when the shock arrival rate increases. For the right sub-graph, there is an intersection for the failure rate curves. R(t )
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R(t )
2.5e5 3.5e5
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DS 1.75e3 DS 1.25e3
t
t
Fig. 7. Sensitivity analysis of the system reliability on and DS for Model 2.
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PT
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M
f (t )
t
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Fig. 8. Simulation results of sensitivity analysis of lifetime pdf on , DH , and DS for Model 2.
For Model 2, the results of system reliability, Rmodel2 (t ) , is shown in Fig. 7
according to Eqns. (27). When computing the multiple integral in Eq. (27) by using simulation procedure 1, the adopted largest integer m is 50, and for each value of
m , we adopt the sample number N 10, 000 . Also, in order to verify the analytical results, simulation results of the reliability are presented in Fig. 7. Similar to Model 1, 28
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M 30, 000 degradation paths of the system are simulated to get the failure data. However, due to the complexity of the reliability function Rmodel2 (t ) , it is not easy to get the closed form of the lifetime pdf; thus, simulation result is achieved which is plotted and the sensitivity analysis on , DH , and DS is shown in Fig. 8. In terms of the sensitivity analysis, similar results of the parameter effects can be found as those in Model 1. Both the shock arrival rate and the failure threshold have
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great effect on the system reliability indexes. In conclusion, the results also remind us to improve the system reliability by reducing the shock frequency in operation and increasing the failure threshold of reliability systems. 5.2. Illustrated examples for models 3 and 4
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Models 3 and 4 are developed based on the Wiener process under SEPs I and II. Due to the lack of real data for Wiener process based degradation system, numerical results of the system reliability and lifetime pdf are given by simulation. Corresponding simulation procedures are described in Section 4. Suppose that the
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common parameters for models 3 and 4 are as follows: x0 0 , n 2 , p1 0.25 ,
p2 0.15 , A1 4 , A1 3 , A2 3 , and A2 1.5 . Specific parameters adopted
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in Model 3 are W 0.4 , W 0.1 , 1 0.1 and 2 0.075 . By simulation procedure 2 proposed in Section 4, sensitivity analysis of the reliability and lifetime
PT
pdf on , DS , and have been presented in Figs. 9 and 10. It is critical to get sufficient failure date when executing reliability assessment for a specific degradation
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system. In terms of the simulation procedures given in Section 4, FPT method is applied to get the failure data for each of the simulated path. In this example,
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M 30, 000 degradation paths of the system are simulated through which the system
reliability and the lifetime pdf are obtained. The shocks increase the degradation level for Model 3 while the shocks increase
the degradation rate for Model 4. Based on the sensitivity analysis, the results indicate that the failure thresholds DS , the shock arrival rate , and the diffusion parameter
all have significant effects on the reliability indexes. The system reliability will increase with the increase of DS while it decreases with the increase of . In addition, with the increase of , the fluctuation of the degradation will increase; thus, 29
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the variance of the system lifetime will decrease and this can be reflected by the pdf curve in Fig. 10. These facts coincide well with reality. Also, the sensitivity analysis can also illustrate the proposed simulation procedures.
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R (t )
t
Fig. 9. Simulation results of sensitivity analysis of the system reliability for Model 3.
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PT
ED
M
f (t )
t
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Fig. 10. Simulation results of sensitivity analysis of the lifetime pdf for Model 3.
For Model 4, the adopted specific parameters are W 0.4 , W 0.03 , 1 0.01 ,
and 2 0.025 . The system reliability and the lifetime pdf can be obtained by executing simulation procedure 3. The results of sensitivity analysis are also presented in Figs. 11 and 12, respectively. Similarly, in Model 4, M 30, 000 failure data of the degradation system are collected. In terms of the sensitivity analysis, similar results of the parameter effects can be found as Model 3. The shock arrival rate, the
30
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failure threshold and the diffusion coefficient all have obvious effects on the system reliability and lifetime pdf which are omitted to be described here.
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R(t )
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t
Fig. 11. Simulation results of the system reliability curves for Model 4.
PT
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M
f (t )
CE
t
Fig. 12. Simulation results of sensitivity analysis of the lifetime pdf for Model 4.
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6. Conclusions and future work In this research, four reliability models are proposed for systems subject to
competing soft and hard failure processes with degradation-shock dependences. Motivated by the real example of a MEMS, these models are natural extensions of the conventional degradation-threshold-shock ones. For models 1 and 2 based on the general path model, analytical solutions of system reliabilities are presented. For models 3 and 4 based on the Wiener process model, corresponding reliability indexes 31
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are also obtained by using simulation methods. Sensitivity analysis is presented on the system reliability of some parameters. As the results illustrated, there are several ways to enhance the system reliability, such as attempting to decrease the frequency of shocks and increase the failure threshold, i.e., improve the resistibility to degradation caused by internal deterioration and shock damages. The main contributions of the paper are as follows. First, multiple specifies of shocks are considered in the shock
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process which is more realistic in practice compared to the existing work on the shock-degradation models. Second, in addition to the commonly used general process model, Wiener process model is used to describe the system degradation which has intensive application in degradation modeling. Third, two novel SEPs on the
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dependence between the soft and hard failure processes are proposed which can clearly explain the internal fluctuation mechanism influenced by the external environment. Hence, this research is meaningful for making maintenance decisions to improve the system reliability.
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In addition, more extensions of these models can be developed. First, the changing failure threshold models might be considered for systems subject to multiple
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dependent competing failure processes. Second, different SEPs could be taken into account at the same time due to the complexity of devices or systems in reality. In
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addition, other kinds of stochastic process models are worth studying, such as the Gamma process models and the inverse Gaussian process models.
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Acknowledgement
This work is supported by the NSF of China Grants 71631001. Hongda Gao is also
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supported by China Scholarship Council to study in Penn State University as a visiting scholar in 2017-2018. We thank Prof. Tim Johnson for his linguistic assistance during the preparation of this manuscript. References [1] Lu CJ, Meeker WQ. Using degradation measures to estimate a time-to-failure distribution. Technometrics, 1993, 35(2):161-174.
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[2] Yuan XX, Pandey MD. A nonlinear mixed-effects model for degradation data obtained from in-service inspections. Reliability Engineering & System Safety, 2009, 94(2):509-519. [3] Kaiser KA, Gebraeel NZ. Predictive maintenance management using sensor-based degradation models. IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, 2009, 39(4):840-849.
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[4] Ye ZS, Xie M. Stochastic modelling and analysis of degradation for highly reliable products. Applied Stochastic Models in Business and Industry, 2015, 31(1):16-32.
[ 5 ] Zuo MJ, Jiang RY, Yam RCM. Approaches for reliability modeling of continuous-state devices. IEEE Transactions on Reliability, 1999, 48(1):9-18.
[6] Pan ZQ, Balakrishnan N. Reliability modeling of degradation of products with
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multiple performance characteristics based on Gamma processes. Reliability Engineering & System Safety, 2011, 96(8):949-957. [7] Park C, Padgett WJ. New cumulative damage models for failure using stochastic processes as initial damage. IEEE Transactions on Reliability, 2005, 54(3):530-540. [8] Gao HD, Cui LR, Kong DJ. Reliability analysis for a Wiener degradation process
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