TR-2017-587R2
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Abstract—This paper presents a reliability modeling and analy-
sis framework for load-sharing systems with identical components
subject to continuous degradation. It is assumed that the
components in the system suffer from degradation through an ad-
ditive impact under increased workload caused by consecutive
failures. A log-linear link function is used to describe the relation-
ship between the degradation rate and load stress levels. By assum-
ing that the component degradation is well modeled by a step-wise
drifted Wiener process, we construct maximum likelihood esti-
mates (MLEs) for unknown parameters and related reliability
characteristics by combining analytical and numerical methods.
Approximate initial guesses are proposed to lessen the computa-
tional burden in numerical estimation. The estimated distribution
of MLE is given in the form of multivariate normal distribution
with the aid of Fisher information. Alternative confidence inter-
vals are provided by bootstrapping methods. A simulation study
with various sample sizes and inspection intervals is presented to
analyze the estimation accuracy. Finally, the proposed approach is
illustrated by track degradation data from an application exam-
ple.
Index Terms— continuous degradation, data uncertainty, load-
sharing system, maximum likelihood estimation, Wiener process.
A
CRONYMS
BS
Bootstrapping
LS
Large-sample approximation
MLE
Maximum likelihood estimation
MTTF
Mean time to failure
SE
Standard error
NOTATIONS
()
Standard Brownian motion
Number of components in each system
Failure threshold
This work was supported in part by the Research Grants Council of Hong
Kong under a theme-based project under Grant T32-101/15-R and a General
Research Fund (CityU 11203815) and in part by the National Natural Science
Foundation of China under a Key Project under Grant 71532008.
X. Zhao is with the Department of Systems Engineering and Engineering
Management, City University of Hong Kong, Kowloon, Hong Kong, and also
with the Shenzhen Research Institute, City University of Hong Kong, Shenzhen
518000, China (e-mail: xiujizhao2-c@my.cityu.edu.hk).
Number of degradation inspections for the
th failed component in the th system
Total number of observed systems
Load on each surviving component after the
(−1)th failure
()
Wiener degradation process
Lifetime of the th failed component in the
th system
∆
Inspection interval
,
Observed degradation increments and fail-
ure time
(|,)
Log-likelihood function
Drift parameter under load
Unknown parameters
Real space for unknown parameters
Diffusion parameter
Standardized stress level under workload
I. INTRODUCTION
EDUNDANCY techniques are commonly used to enhance
the reliability of various systems. Numerous existing mod-
els of reliability redundancy assume that the components are
working independently [1]. The assumption of independence
provides convenient mathematical properties and computa-
tional efficiency in reliability assessment. However, the inter-
dependence of components in redundant systems cannot be
ignored for many practical reasons. One typical scenario is that
many systems have load-sharing characteristics, i.e., the
components are subject to a shared system workload. In such
systems, component failures result in an elevated workload of
the surviving components, which typically accelerates the
failure of the whole system. Load-sharing systems are widely
applied in various industries, such as power systems and gear
systems [2], [3].
B. Liu is with the Department of Civil and Environmental Engineering, Uni-
versity of Waterloo, Waterloo, ON N2L 3G1, Canada. (b274liu@uwater-
loo.ca).
Y. Liu is with the Department of Systems Engineering and Engineering
Management, City University of Hong Kong, Kowloon, Hong Kong, and also
with the School of Automation Science and Engineering, South China Univer-
sity of Technology, Guangzhou, China (e-mail: aulyq@scut.edu.cn).
Reliability Modeling and Analysis of Load-
Sharing Systems With Continuously Degrading
Components
Xiujie Zhao, Bin Liu, Member, IEEE, and Yiqi Liu
R
TR-2017-587R2
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Load-sharing redundant systems have been intensely studied
in the literature. Shao and Lamberson [4] presented a Markov
model to analyze the -out-of- load-sharing systems. With re-
spect to such systems, numerous studies have explored the is-
sues of reliability evaluation [5]–[7], inspection scheduling [8],
maintenance optimization and system design [9].
To facilitate inspection and maintenance planning, decision
makers need to figure out the reliability characteristics of load-
sharing systems, which can be modeled by unknown parameters
that can be estimated from test or field data. Liu [10] evaluated
the reliability of load-sharing -out-of- systems where the
lifetime distributions of the components are different and arbi-
trary. Kim and Kvam [5] proposed a maximum likelihood esti-
mation (MLE) approach to systems with unknown load-sharing
rules. Kvam and Peña [11] used a nonparametric method to
make inferences of load-sharing life models. Park [12] consid-
ered a parallel load-sharing system with identical components
and derived analytical MLEs by assuming that the underlying
lifetime distribution of each component is exponential or
Weibull. In a follow-up study [13], Expectation-Maximization
(EM) algorithm was adopted to estimate the parameters for sim-
ilar systems with components of which lifetime distribution is
lognormal or normal. Aside from these, interval estimation for
the reliability of -out-of- load-sharing systems were studied
with exponential component lifetime [7]. Wang et al. [14] eval-
uated the reliability of load-sharing parallel systems by intro-
ducing the failure dependency and characterized the system dy-
namics with the semi-Markov process. However, most of these
studies have focused on the lifetime modeling of load-sharing
systems, where only shock failures were considered.
As sensor technologies advance rapidly, the degradation of
quality characteristics (QC) of many systems can be observed
and measured precisely. System degradation has been proven
to be closely associated with reliability. For systems suffering
from corrosion, wear or cumulative usage, degradation
measures provide reasonable predictions of system failures. For
some other systems, the degradation can be measured by the
performance reduction. For instance, LED lamps and LCD
monitors are deemed to have failed when the brightness falls
below a critical level. Stochastic process models and general
path models are two main types of degradation modeling ap-
proaches. The most widely used stochastic processes to model
degradation data include Wiener process [15], [16], gamma
process [17] and inverse Gaussian process [18]. Stochastic
models have clear physical explanations, making it convenient
to incorporate covariates and random effects to reflect various
properties of degradation data. For general path models, Hong
et al. [19] modeled the degradation of an organic coating in en-
vironments with dynamic covariates. In an extended work by
Xu et al. [20], nonlinear general path models with time-varying
environmental covariates were analyzed.
Although degradation-based models are considered to be su-
perior in reliability analysis, there is hardly any literature ad-
dressing the reliability of load-sharing systems with degrading
components. Ye et al. [21] proposed the cumulative workload
(CWL) to degradation failure mode to model the load-sharing
system and carried out a cost analysis. Liu et al. [22] presented
a preventive maintenance modeling approach to load-sharing
parallel systems with identical degrading components. In Liu
et al. [23], the MLE of parameters of load-sharing systems was
discussed for Wiener processes and inverse Gaussian processes.
Nevertheless, the assumption of different parameters for
different workloads adopted in Liu et al. [23] and many previ-
ous works [12], [13] makes the statistical inference less effi-
cient as the number of components in the system increases.
Moreover, to the best of our knowledge, no literature addressed
the variability of parameter estimates for degrading load-
sharing system. This study intends to fill this gap.
In this paper, we present systematic parameter estimation
procedures for parallel load-sharing systems with continuously
degrading components. First, we construct the system reliabil-
ity model and identify unknown parameters. The components
in the system are assumed either identical or heterogeneous.
Wiener process is used to model the degradation path of each
component. We assume that the system load is evenly
distributed to each working component. To reduce the number
of unknown parameters, we take advantage of a link function
that describes the relationship between the degradation rate and
the workload. Afterward, the MLEs of unknown parameters are
obtained by numerical methods. Finally, we use two methods
to quantify the uncertainty in parameter estimates. The large-
sample approximation method gives the Fisher information and
constructs the estimated joint distribution for parameter esti-
mates to allow interval estimation. Alternatively, bootstrapping
approach can generate a large sample of parameter estimates to
quantify the estimation variability nonparametrically.
The remainder of the paper is organized as follows. Section
II presents the reliability modeling of load-sharing systems with
degrading components. In Section III, the likelihood function is
formulated and the estimated distribution of unknown parame-
ters is derived. A simulated numerical example is used to illus-
trate the proposed method in Section IV. Section V presents a
case study with data from a track degradation test. Finally, Sec-
tion VI gives concluding remarks and suggestions for future
works.
II. D
ESCRIPTION AND MODELING OF LOAD-SHARING
SYSTEMS WITH DEGRADING COMPONENTS
Components in load-sharing systems generally have depend-
ent degradation paths due to the common system load imposed
upon them. At the time when the system initiates to work, the
components degrade slowly as the load on each component is
low. When the degradation levels of some components in the
system reach the critical failure threshold, these components are
deemed to have failed. In other words, they are not able to share
the system load afterward. In this situation, each surviving
component in the system has to burden heavier workload and
thereby suffer from higher degradation rates. An illustrative
degradation path of such a system is shown in Fig. 1. One typi-
cal example of such systems comes from the railway systems.
For particular areas of the track where vehicles frequently go
by, when the wear of some subsections becomes severe, the
wheels have no seamless contact on these subsections. Mean-
while, other small subsections of the track tend to suffer a
higher rate of wear afterward. Another example that has the
similar load-sharing mechanism is the wastewater treatment
system. Activated sludge process (ASP) is the most commonly
used technique to remove organic matter and nutrients (mainly
TR-2017-587R2
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nitrogen and phosphorus) in the wastewater plant [24]. During
the organic matter and nutrients degradation process, several
aerobic biological tanks are connected to each other to ensure
the treatment efficiency. However, once the treatment effi-
ciency of an aerobic biological system reaches the critical fail-
ure threshold, the corresponding component fails and it cannot
share the workload afterward. This makes other aerobic tanks
burden heavier workload and this accelerates their degradation
processes.
A. Assumptions and Wiener Degradation Models
Consider identical systems each with components con-
nected in parallel. For each system, we make the following as-
sumptions that are similar with those in Park [13]:
1. Each component is subject to continuous degradation
that can be well modeled by a Wiener process.
2. Each component is deemed to have failed when its deg-
radation level exceeds a predetermined threshold .
3. The load of the whole system is constant and is equally
distributed to each working component.
4. The degradation measures are taken periodically.
5. Component failures are self-announcing, i.e., the exact
failure time of each component can be observed.
Remark: Assumption 5 is made based on real practices in reli-
ability tests. Component failures are commonly easier to ob-
serve than degradation levels. For example, in adhesive bond
tests [25] the failures are immediately observed as the bonds
break up upon failures. However, the degradation level of ad-
hesive bonds cannot be observed continuously. Engineers need
to employ specific instruments to measure the degradation
level. Another example is from water treatment systems con-
sisting of multiple filters. When a filter degrades to the critical
level, it cannot yield the required volume of water, which can
be detected immediately. In contrast, the degradation levels of
these filters need to be revealed by inspections, which are
usually carried out periodically.
When all components in the system are working, we use a
linear Wiener process () to model the degradation process
for each component, that is
where
is the drift parameter, is the diffusion parameter,
and () is the standard Brownian motion. For any component
in the system, the lifetime follows an inverse Gaussian distribu-
tion with mean ⁄
and shape
⁄ , and the distribution
function is denoted by
(; ⁄
,
⁄ ).
B. Load-Sharing Modeling and Link Function
Since the component failure times are s-dependent in a load-
sharing system, it is inappropriate to model the degradation pro-
cess for each component independently. Under the assumption
that the components in the systems are of the same type and the
workload on each component is equal, it is reasonable to imply
that each surviving component is suffering from an equal dam-
age that leads to degradation growth at an arbitrary time.
Let
, =1,…, be the time at which the th compo-
nent in the th system fails, where
≤≤
, =
1,…,. Note that components =1,…, are ordered by the
sequence of failures. A realization of
is denoted by
.
Specifically, we assume that
≡0 and
≡0. For sim-
plicity, we denote the th component in system by component
(,) in the following contexts.
For period
−
<≤
, the workload on each compo-
nent is denoted by
. If the total load is normalized as 1, it is
straightforward that
=1 ( −+1)⁄ for =1,…,. In
other words, the last failing component experiences different
workloads throughout the lifespan of the system. Under
, we
assume that the Wiener degradation parameters for a single sur-
viving component are
and
. Some previous studies [13],
[26] have assumed an additive parameter under each load and
estimated the parameters. However, in many real parallel sys-
tems, the number of components may be relatively large, and
this approach will introduce a large number of unknown param-
eters, which deteriorates the generality and efficiency of statis-
tical inferences. As stated in Kong and Ye [7], we can resort to
establishing a link function to connect the workload and degra-
dation model parameters for degrading components in load-
sharing systems.
As stated in the literature that discussed the relationship be-
tween the Wiener degradation model and external stresses [27],
[28], it is reasonable to assume that the diffusion parameter
does not change across various workloads and environments,
i.e.,
≡. The assumption of constant diffusion parameter
has been widely validated by many real degrading products,
such as LED lamps [29] and carbon-film resistors [30]. A log-
linear link function for
is assumed as follows:
Let =(
,…,
)
, where
=(
) is the standardized
stress level under workload
, and we have 0=
<<
=1. The form of (
) varies for different types of systems
and loads, and it is noted that there are typically no unknown
parameters in (
). In reliability analysis, the log-linear link
functions are commonly used in degradation modeling and ac-
celerated tests [25], [31]. Further discussions of the log-linear
link function and (
) can be found in Appendix A.
() =
+(),
(1)
log
=
+
, =1,..,.
(2)
Fig. 1.
An illustration of degradation levels of load-sharing systems with
four
parallel components
0
5
10
15
20 25 30 35 40
time
0
2
4
6
8
10
Degradation level
Critical
level
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C. Reliability Function
We use random vector
=
,…,
to denote the
failure time of each component in an arbitrary system , where
≤≤
. The distribution function for the failure time
of the whole system is represented in a conditional manner as
follows:
= …
−
(
|
=
,…,
−
=
−
)…
(
|
=
)
(
)d
d
…d
,
(3)
where
() is the density function of
, and
(|) is
the conditional density of
for =2,…,. Specifically,
since
is the first order statistic, the distribution function of
evaluated at
=
is given by the probability that the
minimum of the first passage times (FPTs) of all degradation
processes initiating at zero is smaller than
. As the degrada-
tion processes are mutually independent between any two con-
secutive failures, we can obtain that
(
)=1−[1−
(
; ⁄ ,
⁄ )]
, and the density function of
is
(
)=
(
)
=
1−1−
(
;
⁄ ,
⁄ )
=[1−
(
;
⁄ ,
⁄ )]
−
(
;
⁄ ,
⁄ )
(4)
Further, the conditional density for
, ≥2 is given by
and
;
=
,…,
−
=
−
can be given by
;
=
,…,
−
=
−
=
;
−
=
−
−
,
−
,0, (6)
and
(;,
,,) is the density function of the truncated
normal distribution with mean and variance
, and upper
and lower bounds being and , respectively. Likewise, in (4),
the conditional density
(
|
,…,
−+
) is given by (7).
We can derive the reliability function at a given time via (3)
by ()=1−(). Numerical evaluation can be carried out
by utilizing (3)-(7). However, the evaluation is very computa-
tionally intensive due to the multiple integrals. In Appendix B,
we use an approximation-based simulation method to generate
samples of the failure time, then the reliability function can be
evaluated non-parametrically via simulated life data.
III. D
ATA MODELING AND ESTIMATION OF UNKNOWN
PARAMETERS
A. Data Modeling and Contributions to Likelihood
In this study, we assume that periodic inspections are carried
out on each surviving component for system where =
1,…,, and the inspection interval is fixed at ∆. Denote the
number of degradation inspections for component (,) by
,
then we have
=
/∆. For component (,), let
be the th degradation measurement, where =1,…,
. The
measured degradation increments for component (,) are de-
noted by ∆
=∆
,…,∆
, of which each
element is given by ∆
=
−
(−)
. Note that we
set
≡0. Let
=
−
∆ be the time to failure
since the final inspection for component (,). If
≥1, for
2≤
≤, ∆
follows normal distributions as shown in
(8). Here, we note that it is likely that more than one component
in the system fails between two particular inspection epochs. In
this paper, we assume that ∆ is relatively small so that the
chance of such cases of multiple failures is low. Additionally,
even though few such cases occurred, the normal distribution in
(8) gives a good approximation for the degradation increments.
If a dataset contains a considerable number of cases where sev-
eral failures occur in one inspection interval for one system, we
can change the mean of the normal distribution in the third case
()=Pr
≤
=
,…,
−
=
−
≈ …
;
=
,…,
−
=
−
×
…×
−+
;
=
,…,
−
=
−
×
×
,…,
−+
,
−
d
…d
−+
,
(5)
,…,
−+
,
−
=
1−∏ 1−
−
−
;−
⁄ ,−
⁄
−+
=
.
(7)
∆
~
(
∆,
∆), if 1≤ ≤
,
∆,
∆, if
(
−)
+2≤≤
,
(
−)
−
+
∆ −
(
−)
,
∆, if =
(
−)
+1.
(8)
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in (8) into a linear combination of more than two piecewise deg-
radation models with change points. The details are given in
Appendix C. For simplicity, we employ (8) to model the incre-
ments in the following context.
By utilizing the independence of non-overlapping increments
of Wiener process, we can evaluate the likelihood contribution
of ∆
conveniently by computing the product of likelihoods
contributed by ∆
for all . Furthermore, the observed in-
formation provided by component (,) also contains
. In
other words, the FPT of the degradation process to the critical
level is
. Since the FPT of a Wiener process follows the
inverse Gaussian distribution, by conditioning on observing the
last degradation measure
=, the FPT beyond the last
inspection
~(−)
⁄ ,(−)
/
, and the den-
sity function is given by
If component (,) fails before any degradation measure is
taken, i.e.,
=0, the likelihood is merely contributed by
=
.
In the model we have described, the unknown parameters can
be denoted by =(
,
,)
. Let , and be the realiza-
tions of ∆=∆
,=1,…,,=1,…,,
and
for all the components in all the systems, and
is equal to
the length of ∆
. Based on (8) and (9), we can evaluate the
total likelihood with (10).
To obtain the MLEs of the unknown parameters, we need to
rewrite the likelihood function into the log-likelihood function
as shown in (11). Since the log-likelihood function is compli-
cated and the link function is nonlinear, it is very difficult, if not
impossible, to obtain closed-form MLEs by directly taking first
derivatives of (|,). Alternatively, we resort to numeri-
cal methods to maximize the log-likelihood function. Newton
or quasi-Newton optimization methods [32] have been widely
used to solve non-linear programming problems and they can
be easily implemented in various software packages for numer-
ical analysis and optimization.
B. Initial Guesses in Parameter Estimation
The efficiency of Newton optimization method depends on
the initial guess to a great extent. A better initial point can sig-
nificantly decrease the number of iterations till convergence,
especially when the sample size is relatively large. Therefore,
we propose to make an initial guess that is reasonably close to
the MLE of to facilitate the estimation procedure. Partial ob-
servations from the complete dataset are used to rapidly gener-
ate initial guesses. Specifically, we use part of the degradation
measurements to obtain rough estimates of . First, for
;,,
,
=
(−)
2
⁄
exp
−
−+
2
.
(9)
(|,)=
−
2
⁄
exp
−
−+
2
∆
−
∆
√
∆
{≤≤
}
∆
−
i ∆
√
∆
−
+≤≤
∆
−
(
−)
−
−
∆ −
(
−)
√
∆
=
(
−)
+
=
=
=
=
.
(10)
(|,)=
log−
−
1
2
log(2)−log−
3
2
log
−
−+
2
=
=
+
−
1
2
log2−log−1
{≤≤
}
∆
−
∆
2
∆
=
−
1
−
+≤≤
∆
−
∆
2
∆
=
+1
=
(
−)
+
∆
−
(
−)
−
−
∆ −
(
−)
2
∆
.
(11)
