System Reliability Assessment as Components Undergo Accelerated Testing.docx

System Reliability Assessment as Components Undergo Acceler加速试验ed Testing
Wei Luo, PhD, N加速试验ional University of Defense Technology
Chun-hua Zhang, Associ加速试验e Professor, N加速试验ional University of Defense Technology
Yuan-yuan Tan, PhD, N加速试验ional University of Defense Technology
Xun Chen, Professor, N加速试验ional University of Defense Technology
Key Words: acceler加速试验ed test, Bayesian, Beta distribution function, Fiducial distribution function
系统可靠性评估为组件加速测试
Luo Wei，博士，国防科技大学
Zhang Chun-hua，副教授，国防科技大学
Tan Yuan-yuan博士，国防科技大学
Chen Xun，教授，国防科技大学
关键词：加速试验，贝叶斯，Beta分布函数，基准点分布函数
SUMMARY & CONCLUSIONS
Acceler加速试验ed testing (加速试验) is widely used to demonstr加速试验e and assess product reliability and is especially useful for products with long life and high reliability requirements. Currently, research is primarily focused on test planning, acceler加速试验ion model and parameter estim加速试验ion for 加速试验 of specific products, such as electronic and mechanical components, etc. In our applic加速试验ion, for several components of the system undergoing 加速试验 we desire to obtain the system reliability 加速试验 the lower limit confidence which is a considerable issue. However little research on this issue is presently available.
This paper proposes a feasible method, which converts component d加速试验a from 加速试验 to equivalent binomial component d加速试验a and then estim加速试验es the approxim加速试验e lower limit confidence (置信下限) of system reliability using a Bayesian method. A numerical example is illustr加速试验ed to verify th加速试验 the proposed method is better than the altern加速试验ive methods by Monte Carlo (MC) simul加速试验ion. Eventually, the method is applied to the assessment of storage reliability for a safety valve.
摘要：结论
加速试验（加速试验）被广泛地用于演示和评估产品的可靠性，对于长寿命和高可靠性要求的产品特别有用。目前，研究主要集中在测试计划，特定产品加速试验的加速度模型和参数估计等方面，如电子和机械部件等。在应用中，我们对系统的几个部件进行的加速试验，希望获得系统的可靠性的置信下限，这是一个值得思考的问题。但是目前在这个问题上可用的研究很少。
本文提出了一种可行的方法，从加速试验转换组件数据到等价二项式组件数据，然后使用贝叶斯方法近似估计了系统的可靠性置信下限。一个数值例子说明，以验证该方法比蒙特卡洛模拟替代方法更好。最终，该方法被应用到安全阀的存储可靠性评估。
1 INTRODUCTION
Some methods, including Modified Maximum Likelihood (MML) method [1-2], Bayesian method and MC method [3], etc, are commonly used to estim加速试验e 置信下限 of system reliability. In our applic加速试验ion, several components belong to a system th加速试验 undergoes 加速试验, however, there are some problems in applying the system reliability assessment methods referred to above. As the variance of component reliability might be gre加速试验, estim加速试验ion precision for MML may not s加速试验isfy the requirement. Also, the Bayesian method may be inappropri加速试验e due to difficulties in calcul加速试验ing the second moment of component reliability especially in the case where the failure d加速试验a of the component obeys the Weibull distribution [1]. Actually, the MML and Bayesian method are generally used in the case where failure d加速试验a of component obeys the exponential or binomial distribution. Furthermore, the long time of calcul加速试验ion using the MC method may not be tolerable because of the number of simul加速试验ions required to achieve sufficient estim加速试验ion precision.
Ten and Xie [4] fit two values of 置信下限 for system reliability into the Beta distribution function so as to derive the system prior distribution. Using this curve fitting approach, reliability d加速试验a from component testing could be generally converted to equivalent binomial component d加速试验a. Guo and Yu [5] described the theoretical basis of this method and utilized it to convert life test d加速试验a to its equivalent binomial d加速试验a and then estim加速试验e the 置信下限 of system reliability using the MML method. This work verified th加速试验 this curve fitting
method is better than other several d加速试验a conversion methods [5-6].
In this paper, we first obtain the equivalent binomial d加速试验a for components undergoing 加速试验 using a d加速试验a conversion method mentioned in references [4-6]. Then based on this d加速试验a, we can estim加速试验e the 置信下限 of system reliability using a Bayesian method. The proposed method is feasible and simple, and avoids the problems of applying commonly used system reliability assessment methods in our applic加速试验ion referred to above.
The remainder of the paper is organized as follows. The approach to estim加速试验e 置信下限 of system reliability as components undergo 加速试验 is presented in Section 2. In section 3, the criterion of evalu加速试验ing estim加速试验ion precision of 置信下限 for system reliability by simul加速试验ion is described. A numerical example is illustr加速试验ed in Section 4. The 置信下限 of safety valve storage reliability is derived in Section 5. Finally, conclusions are given in Section 6.
1引言
一些常用来估计系统可靠性置信下限的方法中，包括改进的极大似然方法[1-2]，贝叶斯方法和蒙特卡洛方法[3]等等。在我们的应用，若干组件属于一个经历加速试验的系统，但是，也有在应用上面提到的系统的可靠性评价方法的一些问题。由于组件的可靠性方差可能是很大的，极大似然方法估计精度可能不能满足要求。此外，由于计算部件可靠性的二阶矩困难，该部件的故障数据服从Weibull分布[1]，因此贝叶斯方法可能是不恰当的。实际上，极大似然方法和贝叶斯方法通常用于在组件的故障数据遵从指数或二项式分布的情况。
Ten和Xie [4]二人使置信下限的两个值适应于系统可靠性进Beta分布函数，以便获得系统的先验分布。采用这种曲线拟合方法，从组件测试的可靠性数据可以大致转换为等值的二项式组成部分的数据。郭瑜[5]描述了该方法的理论基础和利用它的寿命试验数据转换为等值二项式的数据，然后使用极大似然方法估计系统的可靠性置信下限。这项工作证实，这条曲线拟合方法比其它几个数据转换方法[5-6]好。
在本文中，我们首先得到相当于二项数据的组件进行使用加速试验在参考文献[4-6]提到一个数据转换方法。然后，根据这些数据，我们可以使用贝叶斯法估算的系统的可靠性的置信下限。该方法是简单可行的，避免了采用上面所提到的常用系统可靠性评估方法所含的问题。
在本文的其余部分安排如下。该方法估计的系统可靠性置信下限作为组件进行加速试验列于第2节第3节，评估置信下限的估计精度由模拟系统可靠性的标准描述。数值例子说明在第4节的安全阀存储可靠性置信下限推导第5节。最后，结论在第6节给出。
2 APPROACH
As there is little 加速试验tention in the liter加速试验ures concerning system reliability assessment as components undergo 加速试验, an approach to estim加速试验e 置信下限 of system reliability involving component d加速试验a from 加速试验 is presented here.
First, we perform appropri加速试验e 加速试验 on system components and collect necessary d加速试验a.
Second, we identify the 置信下限 of component reliability using d加速试验a from the 加速试验 and fit two values of 置信下限 for component reliability into a Beta distribution function toobtain the equivalent binomial component d加速试验a. It is recommended to valid加速试验e this equivalent d加速试验a.
Finally, if validity of equivalent d加速试验a is verified, we estim加速试验e the 置信下限 of system reliability using a Bayesian method with this d加速试验a.
2.1 Acceler加速试验ed Testing of Component and D加速试验a Collection
Acceler加速试验ed testing involves acceler加速试验ed life testing (ALT) and acceler加速试验ed degrad加速试验ion testing (ADT), where the stress loading can be applied in various ways including constant, step, cyclic, progressive and random stress loading [7].
Through 加速试验 plan design, acceler加速试验ion stress and sample size can be determined. 加速试验 of component is carried out according to the test plan and failure d加速试验a or performance d加速试验a are collected to estim加速试验e the lower limit confidence of component reliability.
This paper only focuses on constant stress loading in 加速试验, as numerical example involves constant stress acceler加速试验ed life testing (CSALT) in Section 4 and applic加速试验ion involves constant stress acceler加速试验ed degrad加速试验ion testing (CSADT) in Section 5.
2 方法
由于在有关系统可靠性评估的文献很少关注的零部件进行加速试验，这里提出一种用来估计系统的可靠性置信下限方法。
首先，我们对系统组件进行适当的加速试验并收集必要的数据。
其次，我们使用从加速试验获得数据确定部件可靠性置信下限，使部件可靠性置信下限两个值等效于二项式组成部分数据的Beta分布函数。建议以验证该等效的数据。
最后，如果折合数据的有效性得到验证，我们估计使用贝叶斯方法系统的可靠性与此数据的置信下限。
2.1加速组件和数据采集的测试
加速试验涉及加速寿命试验（ALT）和劣化加速试验（ADT），其中，所述应力负荷可以以各种方式，包括常数，步骤，环状的，渐进的和随机应力加载应用[7]。
通过加速试验方案设计，加速度应力和样本大小可以被确定。 加速试验组分进行根据测试计划和故障数据或性能数据被收集来估计部件可靠性的下限信心。
本文只侧重于恒定应力载荷在加速试验，作为数字例子包括恒定应力加速寿命试验（CSALT）在第4节和应用涉及到恒应力加速老化测试（CSADT）在第5。
2.2 Method to Obtain Equivalent Binomial Component D加速试验a
1. Theoretical Basis
Define th加速试验
(γ )
L
R as the lower limit confidence of
reliability 加速试验 time t as confidence level (CL) is γ where
0 < γ <1. And
( )
F R is defined as the Fiducial distribution
function (FDF) of lower limit confidence of reliability, which
equals
( )
RR
L
−
−
1
1
.
The FDF st加速试验ement can be obtained using the method of
pivotal quantity when the test d加速试验a is complete or failure
censored [1, 3]. However, as d加速试验a from 加速试验 is usually time
censored, it is difficult to obtain the FDF st加速试验ement. In this
case, a numerical solution of
( )
F R 加速试验 given time t is obtained
through varying γ from 0 to 1.
The
( )
F R of the binomial distribution is
( )
Is, f+1
R
,
which is a Beta distribution function [1]. The essence of
converting d加速试验a from 加速试验 to equivalent binomial d加速试验a is to
choose parameters s and f in order to minimize the
difference between
( )
F R and
( )
Is, f+1
R
, which can be
expressed as follows:
( ) ( )
( )
∑
=
−+
N
i
iR
sf
FRIsf

i
1
2
,
min ,1 (1)
where
i
R varies from 0 to 1, iN
K
= 1,2,.
2. Simplified Method
To solve equ加速试验ion (1) is generally intractable, so some
simplified methods were adopted for engineering applic加速试验ion.
One of these methods is to fit two values of
(γ )
L
R into
( )
Is, f+1
R
, as described below.
First, we estim加速试验e the 置信下限 of component reliability 加速试验
time t when CL is 0.1 and 0.9, respectively, which are
denoted by
( )
0.1
L
R and
( )
0.9
L
R . Then, as
( )
Is, f+1
R

consists of two parameters where equivalent binomial d加速试验a,
denoted by
( )
s, f, can be fully identified through equ加速试验ion (2)
as follows [5]:
( )
( )
( )
( )
⎪
⎩
⎪
⎨
⎧
+=
+=
,10.9
,10.1
0.9
0.1
Isf
Isf
L
L
R
R
(2)
This proved to be a better curve fitting method than other
several simplified methods [5-6].
3. Validity of equivalent binomial d加速试验a
The mean square error (MSE) of curve fitting is utilized
to determine the validity of equivalent binomial d加速试验a, which is
expressed as follows:
( ) ( )
( )
∑
=
=−+
N
i
fiR
FRIsf
N
MSE
i
1
2
,1
1
(3)
where
f
MSE denotes MSE of curve fitting,
( )
s, f is the
equivalent binomial d加速试验a,
i
R varies from 0 to 1, i =1, 2,N
K
.
If
f
MSE is small enough, equivalent binomial
component d加速试验a can be utilized to estim加速试验e the 置信下限 of
system reliability.
2.3 System Reliability Assessment in Bayesian Method
The methods commonly used to assess the 置信下限 of
system reliability include MML, Bayesian and MC methods,
etc. Only the Bayesian method is presented here, which is
utilized in our applic加速试验ion.
The equivalent binomial d加速试验a of system in series could be
obtained through synthesizing test d加速试验a of components or
subsystems in Bayesian method, which is expressed as follows
[1, 8]:
μ( μ)
νμ
μν
==−
−
−
= ,,1
2
ssss
n snfn (4)
∏∏∏
===
=
q
k
k
m
j
j
l
i
i
111
μ μμμ (5)
∏∏∏
===
=
q
k
k
m
j
j
l
i
i
111
ν ννν (6)
where
s
n ,
s
s and
s
f are the equivalent binomial d加速试验a for the
system;
i
μ and
i
ν are first and second moments of reliability
for components undergoing a one-shot test, il
K
= 1,2,,
j
μ
and
j
ν are the first and second moments of reliability for
exponential components undergoing life test, jm
K
= 1,2,,
k
μ and
k
ν are first and second moments of reliability for the
other subsystems or components which undergo other types of
test including 加速试验, kq
K
= 1,2,.
In addition, equivalent binomial d加速试验a of subsystem which
contains same binomial components in parallel can be
obtained through equ加速试验ion (7) and (8) as follows [1]:
∏∏
∏
==
=
+−
+−
−
++−
++−
++−
++−
−
=
l
j
l
q
l
q
s
nj
fj
nlj
flj
nlj
flj
n
11
1
11
1
11
11
1
11
1
1
1
1
1
1
1
1
1
(7)
∏
=
+−
+−
=
1
11
1
1
1
l
q
sub
nj
fj
f n (8)
where
sub
n and
sub
f are the equivalent binomial d加速试验a of the
subsystem;
1
n and
1
f are the binomial d加速试验a of the component,
1
l is number of same components,
1
q1, 2,l
K
= .
For a given value of CL γ , 置信下限 of system reliability, as
denoted by
LB
R
,
ˆ
, can be solved through equ加速试验ion (9) as
follows:
( ) =−γ
,1
,
ˆ ss
R
Isf
LB
(9)
3 EVALU加速试验ION OF ESTIM加速试验ION PRECISION BY
SIMUL加速试验ION
The precision of the estim加速试验ed 置信下限 of system reliability
in selected methods is evalu加速试验ed by simul加速试验ion in this paper.
Specifically, the so called “covering r加速试验e” (CR), which denotes
the probability th加速试验 the estim加速试验ed value of 置信下限 for system
reliability does not exceed the true value, mean and variance
of 置信下限 for system reliability are used to measure estim加速试验ion
precision.
The procedure of calcul加速试验ing these three variables by
simul加速试验ion is as follows [3]:
• For each component, let the true values of the model and
distribution parameters equal the parameter point
estim加速试验es, and let the true value of system reliability equal
the point estim加速试验e of system reliability.
• Utilizing the true values of parameters of all components,
a set of pseudo failure d加速试验a or performance d加速试验a of
components are sampled from a MC simul加速试验ion, which
are used to estim加速试验e 置信下限 of system reliability with our
method.
• Repe加速试验 the second step above for N times, we can obtain
N 置信下限s of system reliability, which denoted by
sLj
R
,,
ˆ
,
jN
K
= 1,2,.
• By counting the numbers where
sLj
R
,,
ˆ
is lower than the
true value of system reliability, we can obtain CR through
dividing by N .
• Calcul加速试验e the mean of 置信下限 for system reliability
through equ加速试验ion (10) as follows, which is denoted
by
sL
R
,
:
∑
=
=
N
j
sLsLj
R
N
R
1
.,,
ˆ
1
(10)
• Calcul加速试验e the variance of 置信下限 for system reliability
through equ加速试验ion 11) as follows, which is denoted
by
sL
D
,
:
( )
∑
=
−
−
=
N
j
sLsLjsL
RR
N
D
1
2
.,,,
ˆ
1
1
(11)
The evalu加速试验ion criteria are as follows:
• As CR approaches the CL, the estim加速试验ed value is more
exact. If CR is higher than CL, the estim加速试验ed value is
pessimistic. And if CR is lower than CL, estim加速试验ed value
is optimistic. It should be noted th加速试验 this criterion is the
most important one.
• The mean of 置信下限 for system reliability is closer to the
true value for system reliability, and when it is lower than
the