American Journal of Mathematical and Management
Sciences
ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/umms20
Stress-Strength Reliability Estimation of a Series
System with Cold Standby Redundancy Based on
Kumaraswamy Half-Logistic Distribution
Thomas Xavier, Joby K. Jose & Subhash C. Bagui
To cite this article: Thomas Xavier, Joby K. Jose & Subhash C. Bagui (2023): StressStrength Reliability Estimation of a Series System with Cold Standby Redundancy Based on
Kumaraswamy Half-Logistic Distribution, American Journal of Mathematical and Management
Sciences, DOI: 10.1080/01966324.2023.2213835
To link to this article: https://doi.org/10.1080/01966324.2023.2213835
Published online: 23 Jun 2023.
Submit your article to this journal
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at
https://www.tandfonline.com/action/journalInformation?journalCode=umms20
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
https://doi.org/10.1080/01966324.2023.2213835
Stress-Strength Reliability Estimation of a Series System
with Cold Standby Redundancy Based on Kumaraswamy
Half-Logistic Distribution
Thomas Xaviera
, Joby K. Joseb
, and Subhash C. Baguic
a
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Delhi, India; bDepartment of
Statistical Sciences, Kannur University, Kerala, India; cDepartment of Mathematics and Statistics,
University of West Florida, Pensacola, FL, USA
ABSTRACT
This paper deals with study and estimation of stress-strength reliability for a system where strength and stress components are connected in series with cold standby redundancy at system level. It is
supposed that the random stress and strength both follow
Kumaraswamy half-logistic distribution. In this redundant system, we
consider that there N subsystems and in each subsystem, there are
M statistically independent strength components under the impact
of M statistically independent stress components The problem of
estimation is solved in two cases. First under the assumption that
random stress and strength have common first shape parameter and
different second shape parameter and second with the assumption
that common shape parameter is known. The stress-strength reliability is estimated using maximum likelihood and Bayesian estimation
methods. Also asymptotic and Bayesian intervals for the stressstrength reliability under both the cases are constructed. Monte
Carlo simulations will be performed to compare the performance of
various methods. Finally a real life data set is analyzed to demonstrate the findings.
KEYWORDS AND
PHRASES
Cold-standby redundancy
system; Kumaraswamy
half-logistic distribution;
stress-strength model;
type-II progressive
censoring
1. Introduction
The reliability of a system is defined as the probability that the system will perform its
intended function competently given that it operates under stated environmental conditions. Suppose the random stress applied on a certain appliance be represented by Y
and the random strength to sustain the stress be represented as X. Thus a measure of
reliability of a system is given by R ¼ PrfX > Yg: The main idea was introduced by
Birnbaum (1956) and developed by Birnbaum and McCarty (1958).
The stress-strength reliability for several distributions like exponential, normal,
gamma, Weibull, Burr, generalized exponential, generalized Weibull, generalized logistic
and many more have been developed in the statistical literature by many authors, which
shows the importance of such problems. Kotz et al. (2003) provides an excellent detailed
explanation of the development of the stress-strength models up to 2003. Some of the
CONTACT Thomas Xavier
thomasmxavier@gmail.com
Statistical Institute, Delhi 110016, India.
ß 2023 Taylor & Francis Group, LLC
Theoretical Statistics and Mathematics Unit, Indian
2
T. XAVIER ET AL.
applications of the stress-strength model are discussed in Pakdaman et al. (2019).
Recently Jose et al., (2019) estimated the stress-strength reliability for Kumaraswamy
half-logistic distribution, Xavier and Jose (2021) studied stress-strength reliability for a
generalization of the half-logistic distribution.
A lot of attention has also been given to develop multicomponent stress-strength reliability. Consider a system having k statistically independent and identically distributed
strength components, subject to a common stress. Bhattacharyya and Johnson (1974)
first studied this multicomponent stress-strength system which functions when sð1 s kÞ or more components simultaneously survive. The following system corresponds
to series and parallel when s ¼ k and s ¼ 1 respectively. Xavier and Jose (2021) studied
the multicomponent stress-strength reliability based on power transformed half-logistic
model.
Standby redundancy plays an important role in inflating the system reliability, it was
first studied by Sriwastav and Kakati (1981). The use of such systems is that when one
unit in the system operates the other is kept at standby and once the operating system
fails, the standby unit starts working. Clearly this gives much higher reliability to the
system which makes it extensively adopted in many applications. For example, in military satellite, standby redundancy system can improve the lifetime of the satellite.
Standby redundancy can be applied at component or system levels. The effectiveness of
adding cold standby redundancy to a coherent system at system and component levels
is studied by Eryilmaz (2017).
The stress–strength reliability of a standby redundancy system for both stress and
strength are independent and follow different probability distributions - exponential,
gamma and Lindley were studied by Khan and Jan (2014). Ke et al., (2007) presented
the reliability and sensitivity analysis of a system with M primary units, W warm
standby units, and R unreliable service stations where warm standby units switching to
the primary state might fail.
Siju and Kumar (2017) studied the estimation of stress-strength reliability of a parallel
system with active, warm and cold standby components. Liu et al. (2018) studied the
reliability estimation of a N M-cold-standby redundancy system in a multicomponent
stress–strength model with generalized half-logistic distribution. C€
uran and Kizilaslan
(2021a) studied statistical inference of the stress-strength reliability of parallel system
with cold standby redundancy at system and component levels. C€
uran and Kizilaslan
(2021b) studied inference of the stress-strength reliability and mean remaining strength
of series system with cold standby redundancy at system and component levels. Clearly,
the stress-strength reliability estimation for series or parallel systems with cold standby
redundancy setup has not been paid much attention in the literature.
Thus, this paper aims to study the stress-strength reliability estimation of a N M
cold-standby redundancy system in a series stress-strength model based on
Kumaraswamy half-logistic distribution. The rest of the paper is organized as follows.
In section 2, we give the motivation and the N M cold-standby redundancy system
is described; also the expression for stress-strength reliability is obtained. In section 3,
we study inference of the reliability parameters with the parameter a is unknown. In
section 4, we study inference of the reliability parameters with the parameter a is
known. In section 5, the results of small sample comparisons derived from simulations
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
3
are given, and the process is illustrated using a real life example. Finally the conclusion
are given in section 6.
2. Reliability of the system
2.1. Motivation
Liu et al., (2018) studied the estimation of reliability of a multicomponent system,
named N-M-cold-standby redundancy system, based on progressive Type-II censoring
sample. Motivated by that idea, we consider a series system for both the strength and
stress components. Thus in this model each N subsystem, there are M statistically independent strength and stress components. This will also help us to consider the model
where we need to compare two components, where each of them have a series formation. For example suppose we need to study the efficiency of battery system connected
in series for two different companies, then the following model can be used given that
they follow the distribution assumption. Moreover in this study we assume that the system components come from Kumaraswamy half-logistic distribution. Thus this paper
has novelty because of problem structure and component distributions.
2.2. The system and its reliability
A random variable Z is said to follow a Kumaraswamy half-logistic distribution if the
cumulative distribution function (CDF) and the probability density function (PDF)
respectively are
(
a )h
1 ez
FðzÞ ¼ 1 1 (1)
1 þ ez
a1 "
a #h1
2ez
1 ez
1 ez
f ðzÞ ¼ ah
1
; 0z<1
(2)
1 þ ez
ð1 þ ez Þ2 1 þ ez
and f ðzÞ ¼ 0 elsewhere. Here h > 0 and a > 0 are shape parameters. When h ¼ 1 and
a ¼ 1 it reduces to standard half-logistic distribution.
In real life engineering practice, standby redundancy system plays an important role
in enhancing the reliability of a product. Such system may consist of some same subsystems having its won structure, such as, series structure, parallel structure or series-parallel hybrid structure. Our focus is a standby redundancy system with certain number of
same subsystems with series structure. Consider there are N subsystems each consisting
of M components in series, of which one is working under impact of stresses and the
remaining N 1 subsystems are standby. Thus we have a N M cold-standby redundancy system. As a cold standby redundancy system is considered, the standbys will
only be called upon when the subsystem under the impact of stress fails.
Let Zi1 , Zi2 , :::, ZiM , i ¼ 1, 2, :::, N be a set of M independent random variables, repre0
senting the strengths of M components in the ith subsystem. Let Zi10 , Zi20 , :::, ZiM
, i¼
1, 2, :::, N be another set of M independent random variables, representing the stress of
M components in the ith subsystem. Define Xi ¼ minfZi1 , Zi2 , :::, ZiM g and Yi ¼
4
T. XAVIER ET AL.
0
minfZi10 , Zi20 , :::, ZiM
g i ¼ 1, 2, :::, N: Thus the subsystems have a series structure,
X1 , X2 , :::, XN be N independent random variables representing the strength of the subsystems and let Y1 , Y2 , :::, YN be another N random variables representing stresses on
the subsystems. Then, the system reliability is given by
RNM ¼ Rð1Þ þ Rð2Þ þ ::: þ RðNÞ
(3)
RðiÞ ¼ PrfX1 < Y1 , :::, Xi1 < Yi1 , Xi > Yi g, i ¼ 1, 2, :::, N
(4)
where
The following model can be used to study the reliability of a system under stressstrength setup and would also be used to compare two independent models. Assume
0
that Zi1 , Zi2 , :::, ZiM and Zi10 , Zi20 , :::, ZiM
, i ¼ 1, 2, :::, N are independently distributed as
Kumaraswamy half-logistic (KHL) distribution with same parameter a and different
parameters hi1 , hi2 , :::, hiM , i ¼ 1, 2, :::, N and bi1 , bi2 , :::, biN respectively. Then the cumulative distribution function (CDF) of Zij and Zij0 , i ¼ 1, 2, :::, N, j ¼ 1, 2, :::, M is
(
a )hij
1 exij
; 0 zij < 1, a, hij > 0
FZij ðzij Þ ¼ 1 1 1 þ exij
8
!a 9bij
<
=
zij0
1e
; 0 yi < 1, a, bij > 0
FZ0 0 ðzij0 Þ ¼ 1 1 0
:
ij
1 þ ezij ;
The CDF for Xi , i ¼ 1, 2, :::, N is given by
HXi ðxi Þ ¼ 1 PrfXi > xi g ¼ 1 PrfminfZi1 , Zi2 , :::, ZiM g > xi g
(
a )hi
YM
1 exi
¼ 1 j¼1 ½1 Fðxi Þ ¼ 1 1 ; 0 xi < 1, a, hi > 0
1 þ exi
P
where hi ¼ M
j¼1 hij , i ¼ 1, 2, :::, N: Thus Xi , i ¼ 1, 2, :::, N follow KHL distribution
with common parameter a and different parameters h1 , h2 , :::, hN respectively.
Similarly, the CDF for Yi , i ¼ 1, 2, :::, N is given by
GYi ðyi Þ ¼ 1 PrfYi > yi g ¼ 1 PrfminfZ 0 i1 , Z0 i2 , :::, Z0 iM g > yi g
(
)bi
yi a
YM 1
e
¼ 1 j¼1 1 F 0 ðyi Þ ¼ 1 1 ; 0 xi < 1, a, bi > 0
1 þ eyi
where bi ¼
M
X
bij , i ¼ 1, 2, :::, N: Thus Yi , i ¼ 1, 2, :::, N follow KHL distribution with
j¼1
common parameter a and different parameters b1 , b2 , :::, bN respectively.
The density function of Yi , i ¼ 1, 2, :::, N is given by
a1 "
a #bi 1
2eyi
1 eyi
1 eyi
gYi ðyi Þ ¼ abi
1
1 þ eyi
ð1 þ eyi Þ2 1 þ eyi
and gðyi Þ ¼ 0, elsewhere.
(5)
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
Then we can find the reliability of each subsystem RðiÞ by writing (4) as
ð 1
ð 1
HX1 ðy1 ÞgY1 ðy1 Þdy1 :::
HXi1 ðyi1 ÞgYi1 ðyi1 Þdyi1
RðiÞ ¼
ð0 1
0
ð1 HXi ðyi ÞÞgYi ðyi Þdy1
0
¼
5
(6)
i1
Y
hj
bi
hi þ bi j¼1 hj þ bj
Hence, reliability of the system, (3) can be rewritten as
RNM ¼
N
i1
X
hj
b1
bi Y
þ
h1 þ b1 i¼2 hi þ bi j¼1 hj þ bj
(7)
To study the effect of standby components on stress-strength reliability, consider the
two plots given in Figure 1. A contour plot is also drawn. Both the line and contour
plot suggest that adding a standby component increases the system reliability, as
expected. In the contour plot we can see that when a redundant system is added, that
is, when N is increased from 1 to 2, the system reliability increases.
In Figure 1, the first plot it can be observed that for smaller values of b1 (given on
the x-axis), the system reliability with cold standby redundancy with one strength component (M ¼ 1) is higher vastly than a system which has components connected in series (for instance, say M ¼ 10). In other words, when Y or the stress random variable is
stochastically larger than X or the strength random variable, then it is preferable to
have a cold standby redundancy system with one strength component.
From the second plot it can be observed that for large values of h1 , the system reliability with cold standby redundancy is extremely higher for components connected in
series than a single strength component. Therefore, when Y or the stress random variable is stochastically smaller than X or the strength random variable, then it is preferable to have a cold standby redundancy system with strength components connected in
series (Figure 2).
3. Inference on system reliability with unknown common parameter a
In this section, the maximum likelihood and Bayesian estimate of the system reliability
is obtained when the common parameters a is unknown. The Bayesian estimates are
obtained under squared error loss function.
3.1. Maximum likelihood estimate
In this subsection, the maximum likelihood estimate (MLE) of the system reliability is
obtained. Let Xi and Yi follow KHLða, hi Þ and KHLða, bi Þ, i ¼ 1, 2, :::, N respectively.
Now our goal is to achieve the MLE of RNM under type-II progressive censoring
(PC) scheme. As RNM is a function of the unknown parameters h1 , :::, hN , b1 , :::, bN
and a, therefore first the MLE of the parameters is obtained. Let xiji and yili , ji ¼
1, 2, :::, ni , li ¼ 1, 2, :::, mi , i ¼ 1, 2, :::, N are type-II PC random samples with schemes
6
T. XAVIER ET AL.
Figure 1. Plot of values for stress-strength reliability, RNM :
Figure 2. Contour plot of values for stress-strength reliability, RNM :
fNi , ni , Di1 , :::, Dini g and fMi , mi , Si1 , :::, Sini g respectively. At the time of the first failure,
Di1 of ni 1 and Si1 of mi 1 surviving units are randomly removed, at the time of
second failure, Di2 of ni 2 Di1 and Si2 of m i 2 Si1 surviving units are removed
and so on. Under these assumptions, the likelihood function of the parameters can be
written as
Lðh1 , :::, hN , b1 , :::, bN , aÞ
N
Y
¼
gXi1 , :::, Xini ðxi1 , :::, xini Þ 1 FXi1 , :::, Xini ðxi1 , :::, xini Þ Di
i¼1
hYi1 , :::Yimi ðyi1 , :::, yimi Þ 1 HYi1 , :::Yimi ðyi1 , :::, yimi Þ Si
a1 Y
a hi ð1þDiji Þ1
ni ni N Y
Y
1 exiji
1 exiji
ni
ðahi Þ
1
/
1 þ exiji
1 þ exiji
i¼1
ji ¼1
ji ¼1
a mi a bi ð1þSili Þ1
mi Y
1 eyili Y
1 eyili
mi
1
ðabi Þ
1 þ eyili l ¼1
1 þ eyili
l ¼1
i
i
(8)
(9)
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
7
Then the log likelihood function, l ¼ ln L (say) is given by
l
X
X
ni mi N X
1 exiji
1 eyili
/
ðni þ mi Þ ln a þ ni ln hi þ mi ln bi þ a
þ
1 þ exiji
1 þ eyili
i¼1
ji ¼1
li ¼1
ni
mi
X
X
1 exiji a
1 eyili a
þ bi ð1 þ Sili Þ ln 1 þhi ð1 þ Diji Þ ln 1 1 þ exiji
1 þ eyili
ji ¼1
li ¼1
a
a
ni
mi
X
X
1 exiji
1 eyili
ln 1 ln
1
1 þ exiji
1 þ eyili
ji ¼1
l ¼1
i
(10)
^ , :::, b
^ and ^a , the following equation should be
Consequently to derive ^h 1 , :::, ^h N , b
1
N
solved.
"
a #
ni
@l
ni X
1 exiji
¼ þ
ð1 þ Diji Þ ln 1 (11)
@hi
hi ji ¼1
1 þ exiji
@l
@bi
"
a #
mi
mi X
1 eyili
¼
þ
ð1 þ Sili Þ ln 1 bi l ¼1
1 þ eyili
(12)
i
ni mi X
@l ni þ mi X
1 exiji
1 eyili
¼
þ
þ
@a
a
1 þ exiji
1 þ eyili
ji ¼1
l ¼1
i
a "
a #1
Xni
1 exiji
1 exiji
1 exiji
1
ð1 þ Diji Þ
ln
hi
ji ¼1
1 þ exiji
1 þ exiji
1 þ exiji
a "
a #1
mi
X
1 eyili
1 eyili
1 eyili
1
bi ð1 þ Sili Þ
ln
1 þ eyili
1 þ eyili
1 þ eyili
li ¼1
a "
a #1
ni X
1 exiji
1 exiji
1 exiji
1
þ
ln
1 þ exiji
1 þ exiji
1 þ exiji
ji ¼1
a "
a #1
mi X
1 eyili
1 eyili
1 eyili
1
þ
ln
1 þ eyili
1 þ eyili
1 þ eyili
l ¼1
i
(13)
8
T. XAVIER ET AL.
Using (11) and (12), we obtain
0
1
"
# 1
ni
xiji a
X
1
e
A
h^i ðaÞ ¼ ni @ ð1 þ Diji Þ ln 1 1 þ exiji
ji ¼1
0
1
"
# 1
mi
yili a
X
1
e
A
b^i ðaÞ ¼ mi @ ð1 þ Sili Þ ln 1 1 þ eyili
l ¼1
i
To derive ^
a , we solve (13) using numerical methods such as Newton-Raphson
method. After obtaining the MLEs of h1 , :::, hN , b1 , :::, bN and a, we use the invariance
property to obtain the MLE of RNM :
3.2. Interval estimation
The asymptotic confidence interval can be obtained by deriving the asymptotic distribution of RNM : For that we first obtain asymptotic distributions of h1 , :::, hN , b1 , :::, bN
and a: Consider the Fisher information matrix of ðh1 , :::, hN , b1 , :::, bN , aÞ given by
2
3
I1, 1
I1, 2Nþ1
6
7
..
..
Iðh1 , :::, hN , b1 , :::, bN , aÞ ¼ 4 ...
5
.
.
I2Nþ1, 2Nþ1
3
!
2
@
l
@
l
7
6 E
2
7
6
@h
@a
@h
1
7
6
1
7
6
..
..
..
¼ 6
7
6 . .
. 7
7
6
2
2
@ l
@ l 5
4
E
E
@a@h1
@a2
I2Nþ1, 1
2
2
where
!
ni
@2l
mi
¼ 2,E
¼ 2
@bi
hi
bi
!
!
!
2
2
@ l
@ l
@2l
E
¼E
¼0
¼E
@hi @bj
@hi @hj
@bi @bj
2 a "
a #1
ni
X
@ l
1 exiji
1 exiji
1 exiji
E
ð1 þ Diji Þ
ln
1
¼
@hi @a
1 þ exiji
1 þ exiji
1 þ exiji
ji ¼1
"
!
a a #1
mi
X
@2l
1 eyili
1 eyili
1 eyili
E
ð1 þ Sili Þ
ln
1
¼
@bi @a
1 þ eyili
1 þ eyili
1 þ eyili
l ¼1
@2l
E
@h2i
!
i
for i 6¼ j, i ¼ 1, :::, N and j ¼ 1, :::, N: Then as ni ! 1 and mi ! 1, i ¼ 1, :::, N we
have ½ðh^1 h1 Þ, :::, ðh^N hN Þ, ðb^1 bN Þ, :::, ðb^N bN Þ, ð^a aÞT ! ANðO, I 1 ðh1 , :::, hN ,
b1 , :::, bN , aÞÞ where AN denotes asymptotic normal distribution. Now to find the
asymptotic distribution of RNM , consider
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
@RNM
@hk
¼
N
X
@RðiÞ
i¼k
¼
@RNM
@bk
¼
@RNM
@a
@hk
k1
N
i1
X
Y
hj
hj
bk Y
bi
bk
þ
2
2
h
þ
b
h
þ
b
h
þ
bj
ðhk þ bk Þ j¼1 j
j
i ðhk þ bk Þ j6¼k j
i¼kþ1 i
N
X
@RðiÞ
i¼k
¼
9
@bk
k1
N
i1
X
Y
Y
hj
hj
hk
bi
hk
2
2
h
þ
b
h
þ
b
h
þ
bj
ðhk þ bk Þ j¼1 j
j
i ðhk þ bk Þ j6¼k j
i¼kþ1 i
¼0
for k ¼ 1, :::, N:
d
Then the asymptotic distribution of Rd
is given by ðR
RNM Þ NM
NM
ANð0, GT I 1 GÞ, where GT ¼
@RNM
@RNM @RNM
@RNM @RNM
@h1 , :::, @hN , @b1 , :::, @bN ,
@a
: Thus an
approximate 100ð1 aÞ% confidence interval for RNM is given by
^ NM Kc
R
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^
^ NM Þ
^
^ ðR NM Þ, R NM þ K2c r
^ ðR
r
^ NM Þ ¼ GT I 1 G and Kc is the upper c th quantile of the standard normal
^ ðR
where r
2
2
distribution.
3.3. Bayesian estimation
The aim of this section is to obtain the Bayesian estimate of R. For that consider that
the unknown parameters a, hi and bi follow gamma priors, that is, a Cða, bÞ, hi Cðai1 , bi1 Þ and bi Cðai2 , bi2 Þ where i ¼ 1, 2, :::, N: Using the observed censored samples, the joint posterior density function can be given by
pðhi , bi , ajdataÞ / Lðdatajhi , bi , aÞpi1 ðhi Þpi2 ðbi Þp3 ðaÞ
where
pi1 ðhi Þ / hiai1 1 ebi1 hi
pi1 ðbi Þ / biai2 1 ebi2 bi
pi1 ðaÞ / aa1 eba
(14)
10
T. XAVIER ET AL.
After simplifying (14) we can have the posterior densities of hi , bi and a as
8 0
19
"
# =
ni
<
xiji a
X
1
e
A
ð1 þ Diji Þ ln 1 gi1 ðhi ja, dataÞ / hiai1 þni 1 exp hi @bi1 :
;
1 þ exiji
ji ¼1
0
1
"
#
a
ni
X
1 exiji
@
A
ð1 þ Diji Þ ln 1 ) hi ja, data C ai1 þ ni , bi1 xiji
1
þ
e
j ¼1
8i 0
19
"
# =
mi
<
yili a
X
1e
A
ð1 þ Sili Þ ln 1 gi2 ðbi ja, dataÞ / biai2 þmi 1 exp bi @bi2 :
;
1 þ eyili
li ¼1
0
1
"
#
a
mi
X
1 eyili
@
A
ð1 þ Sili Þ ln 1 ) bi ja, data C ai2 þ ni , bi2 yili
1
þ
e
l ¼1
i
a ni a hi ð1þDiji Þ
ni Y
1 exiji Y
1 exiji
1
1 þ exiji ji ¼1
1 þ exiji
i¼1
ji ¼1
mi mi Y
1 eyili a Y
1 eyili a bi ð1þSili Þ a1 ba
mi
1
a e
a
1 þ eyili l ¼1
1 þ eyili
l ¼1
gðajhi , bi , dataÞ /
N Y
i
ani
i
Here it can be observed that some well-known distribution cannot be applied to
describe the posterior density of a, thus we use Metropolis-Hastings method with the
proposal distribution as normal. The algorithm can be suggested as follows in six steps.
Step 1: Set initial values ðh0i , b0i , a0 Þ and set p ¼ 1:
p1 p1
Step 2: Generate ap from gðajhi , bi , dataÞ using the Metropolis-Hastings method
p1
with proposal density as Nða , 1Þ:
ap1
Pni
x
p
1e iji
Step 3: Generate hi from C ai1 þ ni , bi1 ji ¼1 ð1 þ Diji Þ ln 1 1þexiji
where i ¼ 1, 2, :::, N:
ap1
y
Pmi
p
1e ili
Step 4: Generate bi from C ai2 þ ni , bi2 li ¼1 ð1 þ Sili Þ ln 1 1þeyili
where i ¼ 1, 2, :::, N:
P
bi Qi1 hj
1
Step 5: Compute RNM ¼ h1bþb
þ Ni¼2 hi þb
j¼1 hj þbj and set p ¼ p þ 1:
1
i
Step 6: Repeat steps 2-5 for T times.
Now the first B observations will be removed from the T observations, to reduce the
effect of starting distribution or which is called the burn-in period. Now out of the
remaining t B observations, every tenth sample is taken so as to break the dependance
among observations. Also we can have a 100ð1 cÞ% HPD credible interval of RNM
using the method of Chen and Shao (1999).
4. Inference on system reliability with known common parameter a
In this section, the maximum likelihood estimate and Bayesian estimate of the system
reliability is obtained when the common parameter a is known. The Bayesian estimates
are obtained under squared error loss function.
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
11
4.1. Maximum likelihood estimate
In this subsection, the maximum likelihood estimate (MLE) of the system reliability is
obtained. Let Zi and Yi follow KHLða, hi Þ and KHLða, bi Þ, i ¼ 1, 2, :::, N respectively.
Now our goal is to achieve the maximum likelihood estimate of RNM : Assume that
the common shape parameter a is known. Then,
0
1
"
# 1
ni
xiji a
X
1
e
A
h^i ¼ ni @ ð1 þ Diji Þ ln 1 xiji
1
þ
e
ji ¼1
0
1
"
# 1
mi
yili a
X
1
e
A
b^i ¼ mi @ ð1 þ Sili Þ ln 1 yili
1
þ
e
l ¼1
i
4.2. Interval estimation
The asymptotic confidence interval can be obtained by deriving the asymptotic distribution of RNM : For that we first obtain asymptotic distributions of h1 , :::, hN , b1 , :::, and
bN : Consider the Fisher information matrix of ðh1 , :::, hN , b1 , :::, bN Þ given by
3
2
!
2
2
@
l
@
l
7
6 E
2
7
6
@h
@h
@h
1
N
7
6
1
7
6
..
..
..
7
Iðh1 , :::, hN , b1 , :::, bN Þ ¼ 6
6
.
.
.
!
!7
7
6
6
@2l
@2l 7
5
4E
E
@b1 @h1
@b2N
where
@2l
E
@h2i
!
@2l
E
@hi @bj
!
!
@2l
mi
E
¼ 2
2
@bi
bi
!
!
@2l
@2l
¼E
¼0
¼E
@hi @hj
@bi @bj
ni
¼ 2,
hi
for i 6¼ j, i ¼ 1, :::, N and j ¼ 1, :::, N:
Then as ni ! 1 and mi ! 1, i ¼ 1, :::, N we have ½ðh^1 h1 Þ, :::, ðh^N hN Þ, ðb^1 bN Þ, :::, ðb^N bN ÞT ! ANðO, I 1 ðh1 , :::, hN , b1 , :::, bN ÞÞ where AN denotes asymptotic
^ NM RNM Þ ANð0, GT I 1 GÞ, where GT ¼
normal distribution. Thus ðR
@RNM
@RNM @RNM
@RNM
@h1 , :::, @hN , @b1 , :::, @bN
: And an approximate 100ð1 aÞ% confidence
interval for RNM is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^
^ NM Þ
^
^
^ ðR NM Þ, R NM þ Kc r
^ ðR
R NM Kc r
2
2
^ NM Þ ¼ GT I 1 G and Kc is the upper c th quantile of the standard normal
^ ðR
where r
2
2
distribution.
12
T. XAVIER ET AL.
4.3. Bayesian estimation
The aim of this section is to obtain the Bayesian estimate of RNM : For that consider
that the unknown parameters hi and bi follow gamma priors, that is, hi Cðai1 , bi1 Þ
and bi Cðai2 , bi2 Þ where i ¼ 1, 2, :::, N: Using the observed censored samples, the joint
posterior density function can be given by
pðhi , bi ja, dataÞ / Lðdatajhi , bi , aÞpi1 ðhi Þpi2 ðbi Þ
(15)
where
pi1 ðhi Þ / hiai1 1 ebi1 hi
pi1 ðbi Þ / biai2 1 ebi2 bi
After simplifying (15) we can have the posterior densities of hi , bi and a as
8 0
19
"
# =
ni
<
xiji a
X
1
e
A
gi1 ðhi ja, dataÞ / hiai1 þni 1 exp hi @bi1 ln 1 :
;
1 þ exiji
ji ¼1
0
1
"
#
a
ni
X
1 exiji
@
A
ln 1 ) hi ja, data C ai1 þ ni , bi1 xiji
1
þ
e
j ¼1
8i 0
19
"
# =
mi
<
yili a
X
1e
A
ln 1 gi2 ðbi ja, dataÞ / biai2 þmi 1 exp bi @bi2 :
;
1 þ eyili
li ¼1
0
1
"
#
a
mi
X
1 eyili
@
A
ln 1 ) bi ja, data C ai2 þ ni , bi2 yili
1
þ
e
l ¼1
i
Pni
Consider Wi ¼ ji ¼1 ln 1 x
1e iji
x
1þe iji
a
and Vi ¼
Pmi
li ¼1 ln
1
y
1e ili
y
1þe ili
a
: Then,
the Bayesian estimate of RNM can be obtained by following the idea of Liu et al.,
(2018) as
ð1 ð1
^ NM ¼
R
:::
RNM g11 ðh1 jX11 , :::, X1n1 Þg12 ðb1 jY11 , :::, Y1n1 Þ:::
0
0
gN1 ðhN jXN1 , :::, XNnN ÞgN1 ðbN jYN1 , :::, YNmN Þ dh1 :::dbN
N
i1
X
X
¼ /1 þ
/i xj
i¼2
j¼1
where
/i
and
ðbi1 Wi Þai1 þni ðbi2 Vi Þai2 þmi
i1 þ ni ÞCðai2 þ mi Þ
ð 1 ðCða
1
bi
hiai1 þni 1 eðbi1 Wi Þhi biai2 þmi 1 eðbi2 Vi Þbi dhi dbi
0
0 hi þ bi
¼
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
xi
13
ðbi1 Wi Þai1 þni ðbi2 Vi Þai2 þmi
i1 þ ni ÞCðai2 þ mi Þ
ð 1 ðCða
1
hi
hai i1 þni 1 eðbi1 Wi Þhi biai2 þmi 1 eðbi2 Vi Þbi dhi dbi
h
þ
b
i
0
0
i
¼
hi
Let ui ¼ hi þ bi and ti ¼ hi þb
, then
i
ðbi1 Wi Þai1 þni ðbi2 Vi Þai2 þmi
/i ¼
Cðai1 þ ni ÞCðai2 þ mi Þ
ð1 ð1
b V
ðb W Þu 1 1b i2Wi ti
i1
i
dui dti
ti ui ½ui ð1 ti Þai1 þni 1 ðui ti Þai2 þmi 1 e i1 i i
0 0
ð 1 ai2 þmi
ai2 þmi
1
ti
ð1 ti Þai1 þni 1
Vi
h
i dti
¼ bbi1i2W
i
Bðai1 þ ni , ai2 þ mi Þ 0 1 1 bi2 Vi t q
i
bi1 Wi
8
>
a
þ
m
b
V
i2
i
i2
i
>
>
¼1
,
<
q
bi1 Wi
¼
ai2 þmi ai2 þ mi
bi2 Vi
bi2 Vi
>
bi2 Vi
>
,
F1 ai2 þ mi , q; q þ 1; 1 <1
>
: bi1 Wi
q 2
bi1 Wi
bi1 Wi
where q ¼ ai1 þ ni þ ai2 þ mi : Similarly,
xi
¼
bi1 Wi
bi2 Vi
ai1 þni
ð 1 ai1 þni
1
t
ð1 ti Þai2 þmi 1
hi
i dti
Bðai1 þ ni , ai2 þ mi Þ 0 1 1 bi1 Wi t q
i
¼
bi2 Vi
8
>
ai1 þ ni
bi2 Vi
>
>
¼1
,
<
q
bi1 Wi a þn
bi1 Wi
>
bi1 Wi i1 i ai1 þ ni
>
,
F1 ai1 þ ni , q; q þ 1; 1 >
: bi2 Vi
q 2
bi2 Vi
bi1 Wi
<1
bi2 Vi
where v Fw ðzÞ is the generalized hypergeometric function. The generalized hypergeometric function v Fw ðzÞ is defined as
v Fw ða1 , :::, av ; b1 , :::, bw ; zÞ ¼
1
X
ða1 Þ :::ðav Þ zk
k
k
ðb1 Þk :::ðbw Þk k!
k¼0
where ai and bj with bj 6¼ 0, 1, 2, :::; i ¼ 1, 2, :::, v; j ¼ 1, 2, :::, w, : The convergence conditions and other details are available from books on special functions, see for
example Mathai and Haubould (2008).
5. Simulation study and data analysis
In this section, simulation study is considered to show the performance of the MLE and
Bayes estimates. Also the application of the proposed model is illustrated by analyzing a
real life dataset.
14
T. XAVIER ET AL.
5.1. Simulation study
Here we illustrate the performance of maximum likelihood and Bayesian estimates of
the stress-strength reliability considering the 1-2- (R12 ) and 2-2-cold-standby (R22 )
redundancy systems. We will look into the effect of redundant systems on the stressstrength reliability using Monte Carlo Simulations. The performance of the point estimators is measured in terms of biases and mean square errors (MSEs) and the performance of interval estimates is made in terms of confidence lengths (CL) or confidence
regions (CR). For all simulation results we will fix the value of a ¼ 1:5: Random samples of sizes n ¼ m ¼ 10, 20, 30 and 50 with 1000 replications are considered Note that
for computing the values of R22 , h1 and b1 are taken to be same as that while computing R12 : Also because of the flexibility of the component structure, the results in
P
Tables 1 and 2 can be extended to larger values of M given that hi ¼ M
j¼1 hij
PM
and bi ¼ j¼1 bij , i ¼ 1, 2, :::, N:
From Table 1 it can be observed that when there is one cold standby redundant component, the system reliability is higher compared to no redundant components. Also it
can be observed that the estimates of the stress-strength reliability and closer to the
population value and moreover with increase in the sample size, the corresponding
MSEs and the CLs decrease in magnitude which suggest the consistent property of the
estimators.
To illustrate the performance of the Bayesian estimates, we assume all hyperparameters as 0.0001, from the suggestions of Congdon (2001); thus obtaining Bayesian estimates under non-informative priors. As the posterior density of a is of complex form,
we use Metropolis-Hastings algorithm to generate samples, with normal distribution as
the proposal density. A total of 20,000 samples are generated; out of which first 10,000
samples are removed out to reduce the effect of starting distribution, which is called the
burn-in period. Then, out of the remaining 10,000 samples, every tenth sample is taken
so as to break the dependence among the samples; which at the end leaves us with 1000
samples. The rest of the parameters can be generated from their respective gamma conditional posterior densities. It can be observed that the estimates are close to the population value and they are consistent.
From Table 2 it can be observed that when there is one cold standby redundant component, the system reliability is higher compared to no redundant components. Also it
can be observed that with increase in the sample size, the corresponding MSEs and the
CRs decrease in magnitude. The MLEs are observed to perform better than the
Bayesian estimates under non-informative priors for the small samples.
5.2. Application
In this section, a real life dataset discussed by Bilschke and Murthy (2000) is considered.
The data set (Table 2.11, page 48) represents the force for an electronic connector and
consists of a sample of 30 leads. In a medical application, it is necessary to be able to
insert and withdraw leads to a connector with a minimum amount of force. For each
lead, force for insertion and withdrawal was recorded on each of two trials. We will
consider the minimum of two trials, thus we have a R12 setup and hence X be the
0.3636
0.5000
0.7273
0.9091
(15,2)
(7,4)
(4)
(1.5,4)
(0.5,5)
R12
0.1176
ðh1 , b1 Þ
ðn, mÞ
(10,10)
(20,20)
(30,30)
(50,50)
(10,10)
(20,20)
(30,30)
(50,50)
(10,10)
(20,20)
(30,30)
(50,50)
(10,10)
(20,20)
(30,30)
(50,50)
(10,10)
(20,20)
(30,30)
(50,50)
0.1144
0.1145
0.1161
0.1162
0.3633
0.3641
0.3649
0.3612
0.5014
0.4996
0.5011
0.4986
0.7331
0.7329
0.7305
0.7285
0.9128
0.9125
0.9117
0.9103
^R 12
MSE
0.0036
0.0017
0.0010
0.0007
0.0134
0.0061
0.0040
0.0023
0.0145
0.0070
0.0044
0.0026
0.0102
0.0047
0.0032
0.0016
0.0022
0.0010
0.0007
0.0004
Bias
0.0032
0.0031
0.0016
0.0014
0.0003
0.0005
0.0012
0.0024
0.0014
0.0004
0.0011
0.0014
0.0058
0.0056
0.0032
0.0012
0.0037
0.0034
0.0026
0.0012
Table 1. Bias, MSE and CL for MLE of R12 and R22 :
CL
0.2127
0.1540
0.1278
0.0994
0.3969
0.2873
0.2364
0.1834
0.4204
0.3039
0.2500
0.1946
0.3484
0.2522
0.2081
0.1627
0.1705
0.1231
0.1016
0.0800
(5,5)
(10,5)
(15,5)
(20,5)
(20,2)
ðh2 , b2 Þ
R22
0.9545
0.8182
0.6250
0.4909
0.1979
^R 22
0.1877
0.1916
0.1957
0.1950
0.4831
0.4847
0.4877
0.4883
0.6171
0.6211
0.6210
0.6224
0.8179
0.8220
0.8177
0.8191
0.9567
0.9563
0.9556
0.9549
Bias
0.0101
0.0062
0.0021
0.0028
0.0078
0.0062
0.0032
0.0026
0.0079
0.0039
0.0040
0.0026
0.0003
0.0039
0.0004
0.0010
0.0021
0.0017
0.0011
0.0004
MSE
0.0047
0.0023
0.0015
0.0009
0.0113
0.0055
0.0034
0.0020
0.0101
0.0048
0.0034
0.0018
0.0051
0.0023
0.0016
0.0009
0.0007
0.0003
0.0002
0.0001
CL
0.1769
0.1229
0.1012
0.0774
0.3527
0.2522
0.2072
0.1611
0.3534
0.2544
0.2089
0.1626
0.2435
0.1734
0.1449
0.1122
0.0712
0.0518
0.0430
0.0340
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
15
ðn, mÞ
(10,10)
(20,20)
(30,30)
(50,50)
(10,10)
(20,20)
(30,30)
(50,50)
(10,10)
(20,20)
(30,30)
(50,50)
(10,10)
(20,20)
(30,30)
(50,50)
(10,10)
(20,20)
(30,30)
(50,50)
R12
0.1176
0.3636
0.5000
0.7273
0.9091
ðh1 , b1 Þ
(15,2)
(7,4)
(4,4)
(1.5,4)
(0.5,5)
^R 12
0.0825
0.1241
0.1028
0.1294
0.4234
0.4238
0.3724
0.3572
0.5404
0.4608
0.5413
0.5392
0.7424
0.7490
0.7360
0.7458
0.8710
0.9630
0.8659
0.9521
Bias
0.0351
0.0065
0.0148
0.0118
0.0598
0.0601
0.0088
0.0064
0.0404
0.0392
0.0413
0.0392
0.0152
0.0217
0.0087
0.0185
0.0381
0.0539
0.0432
0.0430
MSE
0.0032
0.0019
0.0010
0.0008
0.0145
0.0100
0.0038
0.0022
0.0138
0.0073
0.0057
0.0040
0.0079
0.0043
0.0027
0.0019
0.0047
0.0031
0.0029
0.0020
Table 2. Bias, MSE and CR for Bayesian estimates of R12 and R22 :
CR
0.1687
0.1671
0.1090
0.0999
0.4057
0.3136
0.2488
0.1853
0.4240
0.2927
0.2363
0.1987
0.3336
0.2394
0.1912
0.1505
0.2197
0.1597
0.1340
0.0865
(5,5)
(10,5)
(15,5)
(20,5)
ðh2 , b2 Þ
(20,2)
0.9545
0.8182
0.6250
0.4909
R22
0.1979
^R 22
0.1466
0.1969
0.1833
0.1990
0.5429
0.5014
0.4747
0.4893
0.6000
0.5746
0.6344
0.6304
0.7960
0.8083
0.8048
0.8081
0.9218
0.9786
0.9431
0.9743
Bias
0.0513
0.0009
0.0145
0.0011
0.0520
0.0105
0.0162
0.0016
0.0250
0.0503
0.0094
0.0054
0.0222
0.0098
0.0134
0.0100
0.0328
0.0240
0.0114
0.0197
MSE
0.0066
0.0029
0.0017
0.0010
0.0123
0.0057
0.0037
0.0020
0.0100
0.0072
0.0030
0.0019
0.0055
0.0024
0.0017
0.0010
0.0024
0.0007
0.0004
0.0003
CR
0.2600
0.2102
0.1501
0.1229
0.3800
0.2903
0.2283
0.1807
0.3911
0.2623
0.2061
0.1700
0.2716
0.1859
0.1523
0.1172
0.1437
0.1362
0.0638
0.0364
16
T. XAVIER ET AL.
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
17
Table 3. Goodness of fit for data set 1.
Data set
X
Y
a
9.5
9.5
h
190.1
b
990
K-S Statistic
0.1045
0.1458
p-value
0.8985
0.5464
Figure 3. Histogram and fitted pdf for dataset 1.
Figure 4. Survival plot for dataset.
force for insertion and Y be the force for withdrawal. Thus the data values are 1.35,
1.02, 1.45, 0.935, 1.315, 1.455, 1.415, 1.42, 0.985, 1.335, 0.95, 1.225, 1.355, 1.53, 1.39,
1.16, 1.22, 1.405, 1.35, 0.98, 1.225, 1.135, 1.33, 1.2, 1.11, 1.145, 1.045, 1.05, 0.78 and 1.66
for X and 1.13, 1, 0.93, 0.76, 0.985, 0.835, 0.995, 1.115, 0.92, 1.1, 0.875, 0.835, 0.94,
1.265, 1.035, 1.085, 1.005, 0.995, 1.1, 0.985, 0.985, 1.055, 1.07, 0.94, 1, 0.985, 1.22, 0.89,
0.85 and 1.3 for Y.
First we check whether the KHL distribution fits X and Y. The Table 3 gives the
MLEs of the unknown parameters for X and Y. The Kolmogorov-Smirnov (K-S) test
statistic and the corresponding p-values are also given which suggests that one cannot
reject the hypothesis that the data are coming from KHL distribution. The histogram
with fitted lines for X and Y is given in Figure 3 and the non-parametric survival lines
and the corresponding fit is given in Figure 4.
18
T. XAVIER ET AL.
Now we consider a 1-2-cold-standby redundancy system, and obtain the probability,
R12 : The ML estimate of R12 is obtained as 0.8389 with 95% ACI as (0.7587,
0.9191). The Bayes estimate of R12 under non-informative prior is obtained as 0.8443
with 95% HPDCI as (0.7554, 0.9090). For illustration purpose of Bayes estimate under
informative prior, we consider the following values for the hyperparameters; a ¼ 3, b ¼
3, a1 ¼ 6, b1 ¼ 2, a2 ¼ 30, b2 ¼ 8: The Bayes estimate of R12 is obtained as 0.8519
with 95% HPDCI as (0.7934, 0.8984).
Now we assume that a is known and set it to a ¼ 9:5: Then the estimates of h and b
are 188.5 and 986.3 respectively. The KS-statistic for X is obtained as 0.1034 and the pvalue as 0.9053 whereas the KS-statistic for Y is obtained as 0.1444 and the p-value as
0.5584. The ML estimate of R12 is obtained as 0.8395 with 95% ACI as (0.7714,
0.9077). The Bayes estimate of R12 under non-informative prior is obtained as 0.8371
with 95% HPDCI as (0.7514, 0.8990). The Bayes estimate under the informative prior
a1 ¼ 6, b1 ¼ 2, a2 ¼ 30, b2 ¼ 8 of R12 is obtained as 0.8607 with 95% HPDCI as
(0.7934, 0.8984).
6. Conclusions
In this paper, we have considered a series system of M components, named as N M
cold standby redundancy system. We consider statistically independent and non-identically distributed M strength components, which are exposed to statistically independent
and non-identically distributed M different random stresses. Both the strength and
stress variables are assumed to follow Kumaraswamy half-logistic distribution. The following setup can be used to obtain the reliability of a stress-strength model and can
also be interpreted as comparison of two series system with cold standby redundancy.
The estimate of the stress-strength reliability is developed under maximum likelihood
and Bayesian methods and under progressive type-II censoring. The interval estimates
are also discussed. The application of the developed model is illustrated using a real life
data set.
Disclosure statement
No potential conflict of interest was reported by the author(s).
ORCID
Thomas Xavier
http://orcid.org/0000-0002-0527-6239
Joby K. Jose
http://orcid.org/0000-0003-2674-6145
Subhash C. Bagui
http://orcid.org/0000-0001-6140-5384
References
Bhattacharyya, G. K., & Johnson, R. A. (1974). Estimation of reliability in multicomponent stress
strength model. Journal of American Statistical Association, 69(348), 966–970. https://doi.org/
10.1080/01621459.1974.10480238
Bilschke, W. R., & Murthy, D. N. P. (2000). Reliability: Modeling, prediction, and optimization.
Wiley series in Probability and Statistics.
AMERICAN JOURNAL OF MATHEMATICAL AND MANAGEMENT SCIENCES
19
Birnbaum, Z. W. (1956). On a use of Mann-Whitney statistics. In Proceedings of the 3rd Berkeley
Symposium on Mathematical Statistics and Probability (Vol. 1, pp. 13–17).
Birnbaum, Z. W., & McCarty, B. C. (1958). A distribution-free upper confidence bounds for
PrfY < Xg based on independent samples of X and Y. The Annals of Mathematical Statistics,
29(2), 558–562. https://doi.org/10.1214/aoms/1177706631
Chen, M.-H., & Shao, Q.-M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1), 69–92. https://doi.org/10.2307/
1390921
Congdon, P. (2001). Bayesian statistical modelling. John Wiley and Sons.
C€
uran, G., & Kizilaslan, F. (2021a). Estimation of stress-strength reliability of a parallel system
with cold standby redundancy at component level. Istatistik: Journal of the Turkish Statistical
Association, 13(2), 74–87.
C€
uran, G., & Kizilaslan, F. (2021b). Statistical inference of the stress-strength reliability and mean
remaining strength of series system with cold standby redundancy at system and component
levels. Hacettepe Journal of Mathematics & Statistics, 50(6), 1793–1821.
Eryilmaz, S. (2017). The effectiveness of adding cold standby redundancy to a coherent system at
system and component levels. Reliability Engineering and System Safety, 165, 331–335. https://
doi.org/10.1016/j.ress.2017.04.021
Jose, J. K., Xavier, T., & Drisya, M. (2019). Estimation of stress-strength reliability using
Kumaraswamy half-logistic distribution. Journal of Probability and Statistical Science, 17(2),
141–154.
Ke, J. B., Lee, W. C., & Wang, K. H. (2007). Reliability and sensitivity analysis of a system with
multiple unreliable service stations and standby switching failures. Physica A: Statistical
Mechanics and Its Applications, 380(1), 455–469. https://doi.org/10.1016/j.physa.2007.02.095
Khan, A. H., & Jan, T. (2014). Estimation of multi-component systems reliability in stressstrength models. Journal of Modern Applied Statistical Methods, 13(2), 389–398. https://doi.org/
10.22237/jmasm/1414815600
Kotz, S., Lumelskii, Y., & Pensky, M. (2003). The stres-strength model and its generalizations:
Theory and applications. Word Scientific.
Liu, Y., Shi, Y., Bai, X., & Zhan, P. (2018). Reliability estimation of a N-M-cold-standby redundancy system in a multicomponent stress–strength model with generalized half-logistic distribution. Physica A: Statistical Mechanics and Its Applications, 490, 231–249. https://doi.org/10.
1016/j.physa.2017.08.028
Mathai, A. M., & Haubould, H. J. (2008). Special function for applied scientists. Springer-Verlag.
Pakdaman, Z., Ahmadi, J., & Doostparast, M. (2019). Signature-based approach for stress-strength
systems. Statistical Papers, 60(5), 1631–1647. https://doi.org/10.1007/s00362-017-0889-5
Siju, K. C., & Kumar, M. (2017). Estimation of stress-strength reliability of a parallel system with
active, warm and cold standby components. Journal of Industrial and Production Engineering,
34(8), 590–610. https://doi.org/10.1080/21681015.2017.1314983
Sriwastav, G., & Kakati, M. (1981). A stress-strength model with redundancy. Indian Association
of Productivity Quality and Reliability, 6(1), 21–27.
Xavier, T., & Jose, J. K. (2021). A study of stress-strength reliability using a generalization of
power transformed half-logistic distribution. Communications in Statistics - Theory and
Methods, 50(18), 4335–4351. https://doi.org/10.1080/03610926.2020.1716250
Xavier, T., & Jose, J. K. (2021). Estimation of reliability in a multicomponent stress-strength
model based on power transformed half-logistic distribution. International Journal of
Reliability, Quality and Safety Engineering, 28(02), 2150009. https://doi.org/10.1142/
S0218539321500091