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M. Kalai.............................n
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
Estimation of reliability characteristics for linear
consecutive ..-out-of-..:.. systems based on exponentiated
Weibull distribution
M. Kalaivani1,2 and R. Kannan1
�
Department of Statistics, Annamalai University, Chidambaram, Tamil Nadu, India
Department of Mathematics and Statistics, SRM Institute of Science and Technology,
Kattankulathur, Tamil Nadu, India
kalaivam1@srmist.edu.in, statkannan@gmail.com
Abstract
The focus of this paper is to estimate the reliability characteristics of a linear consecutive ..-out-of- ..:.. system with .. linearly ordered components. The components are independent and identically
distributed with exponentiated Weibull lifetimes. The system fails if and only if at least .. successive
components fail. In such a system, the reliability function and mean time to system failure are
obtained by maximum likelihood estimation method using uncensored failure observations. The
asymptotic confidence interval is determined for the reliability function. The results are obtained by
Monte Carlo simulation to compare the performance of the systems using various sample sizes and
combination of parameters. The procedure is also illustrated through a real data set.
Keywords: Maximum Likelihood Estimation (MLE), Reliability function, Mean
Time to System Failure (MTSF), Asymptotic confidence interval and
Exponentiated Weibull distribution
1. Introduction
Redundancy can be used to increase the system reliability. The most popular type of redundancy is ..-out-of-.. system structure which find the wide applications in both industry and defense systems.
The consecutive ..-out-of-.. system is a special type of redundancy in fault-tolerant systems such as
oil pipeline systems, street illumination systems, street parking, communication relay stations batch
sampling-based quality control systems, computer networks and multi-pump system in hydraulic
control system. These systems are characterized as physical or logical connections between the
system components that are arranged in line or circle. Pham [16] proposed two basic aspects that
have been used to obtain better reliability of a system. The first is to use redundancy such as parallel
system, ..-out-of-.. system and the second one is a manufactured a high reliable system product. Let .. components be linearly connected in such a way that the system fails if and only if at least ..
consecutive components fail. Figure 1 shows a linear consecutive 3-out-of-7:.. system. Whenever
the number of consecutive failures reaches 3 the signal flow is interrupted from source to sink and
the system fails. Chao et al. [3] emphasized that ....../....../../../.. system has a much higher reliability
than series system and is often cheaper than the parallel system. In this paper, we are considering
M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
....../....../../..:.. system redundancy and develop ways to obtain the maximum likelihood estimate
of reliability and MTSF of the proposed system, where the components are independent and
identically distributed (........) with exponentiated Weibull lifetimes
Figure 1: Linear consecutive 3-out-of-7:.. system
First, the consecutive ..-out-of-.. system have been studied by Kontoleon [11]. Chiang and Ni
[4] have giving special attention to the reliability of this system. Extensive review of consecutive ..-
out-of-.. and related systems can be found in Hwang [9], Derman and Ross [6], Kuo and Zuo [20]
and Eryilmaz [7]. The reliability estimation of a consecutive ..-out-of-..:.. system has received little
attention in the literature. Shi et al. [18] discussed the classical and Bayes approach to study the
performance of ..-consecutive-..-out-of-..:.. system with Burr XII components. Madhumitha and
Vijayalakshmi [12] have proposed the Bayesian estimation for reliability and mean time to system
failure for Linear (Circular) ....../../..:.. using exponential distribution. Recently, Kalaivani and
Kannan [10] estimated the reliability function and MTSF of ..-out-of-.. system using Weibull failure
time model by MLE and Bayes estimation. Demiray and Kizilaslan [5] investigated the point and
interval estimates of stress-strength reliability in a consecutive ..-out-of-..:.. system when stress and
strength variable follow the proportional hazard rate model. The reliability estimation is studied
under both classical and Bayes estimates. In reliability analysis, Weibull family of distribution is
mostly used for modeling consecutive ..-out-of-.. systems with monotone failure rates.
The exponentiated Weibull distribution introduced by Mudholkar and Srivastava [14] provides
a good fit to lifetime datasets that exhibit bathtub shaped as well as unimodal failure rates. The
performance of the product may involve high initial failure rate and possible high failure rates due
to wear out and aging, reflecting a bathtub failure rate. Pathak and Chaturvedi [15] obtained the ML
estimator of the reliability function ..(..>..) and ..(..>..) using exponentiated Weibull
distribution. Srinivasa Rao et al. [19] have estimated the multicomponent stress-strength of a system
when stress and strength follow two parameter exponentiated Weibull distribution with different
shape parameters and common scale parameter. Alghamdi and Percy [2] studied the reliability
equivalence factors of a series-parallel system with each component has an exponentiated Weibull
distribution. Mendez-Gonzalez et al. [13] analyzed the reliability of an electronic component using
exponentiated Weibull model and inverse power law.
According to Hong and Meeker [8], components and system structure determine the reliability
of the system. When the component level data is available, it can be used to estimate system
reliability. Confidence intervals (CIs) are essential to assess the statistical uncertainty in the
estimations. Yet, the best estimation of the redundant consecutive ..-out-of-.. systems would still be
of interest due to recent practical applications of the complex systems. This paper establishes the
reliability function, mean time to system failure and asymptotic confidence interval at 95% level of
significance for linear consecutive ..-out-of-..:.. system based on three parameter exponentiated
Weibull model that provides a better approach to fit monotone as well as non-monotone failure rates
which are quite common in reliability analysis.
This paper is organized as follows. In the introductory section the motivation for the present
study and brief review on ....../....../../..:.. systems. In Section 2, description of system reliability
characteristics and assumptions are given. Section 3 devoted to reliability function, mean time to
M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
failure and the asymptotic confidence interval of the proposed system. In section 4, the results based
on simulation study and real data set are illustrated. Finally, the paper ends with a concluding
remark are presented in Section 5.
2. Background of System Reliability Characteristics
Assumptions
� Each component and the system are either good or failed state
� The components of the system fail statistically independently of each other
� All component lifetimes are independently and identically distributed
� Life time of the component follows exponentiated Weibull distribution with unknown
parameters ..,.. and ..
� The system fails if and only if at least .. consecutive components fail, where 1......
Notation .. Number of components in a system .. Minimum number of consecutive components whose failures cause system
to failure ....../....../../..:.. Linear consecutive ..-out-of-..:.. ..(..) Component reliability function. All the components have ...... lifetimes ..!"(..) System reliability function of ....../....../../..:.. system .. Component mean time to failure ..!" Mean time to system failure ..@(..) MLE of ..(..) ..@!"(..) MLE of ..!"(..) ... MLE of .. ...!" MLE of ..!" .... Largest integer less than or equal to .. ......... Independent and identically distributed
A consecutive ..-out-of-.. system consists of .. linearly ordered components where the system fails if
and only if a minimum of .. components fail. This type of structure is called the linear ....../../..:..
system shortly represented by ....../....../../..:... Here it is commonly assumed that 1......, and for ..=1, the ....../....../../..:.. system becomes the series system and when ..=.., the proposed system
becomes a parallel system.
The reliability function ..!"(..) of a ....../....../../..:.. system is
..!"(..)=G& ..(..,..,..) ..#$%
%'( ..% (1)
When the components are ......... replacing .. and .. with ..(..) and 1...(..) in (1), we get the
following reliability function for ....../....../../..:.. system
..!"(..)=GG% (.1))
)'(
&
%'( ..(..,..,..)M.. ..
N (..(..))#$%*) (2)
where
M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
..=O......XP....+..+..11YR + ..1.. .. ..+.. ..1 +.... 1.. ........ ................................ ........ ....
and
..(..,..,..)=
..
.. .
..
M.. ..
N, 0.......1
G(.1))M.....+1 .. NP....... .....R +%,
-
.'( 0, , .. ..>........
and .. represents the maximum number of failed components that may exist in the system without
causing the system to fail. But ..(..,..,..) is the number of ways arranging .. failed components in a
line such that no .. or more failed components are consecutive.
Let .. be the lifetime of each component following Exponentiated Weibull Distribution (EWD) with
probability density function (pdf)
..(..,..,..,..)=...... P.. ..
R/$0`1...$123
4!a5$0..$123
4!, ..>0,..,..,..>0 (3)
where .. is scale parameter, .. and .. are shape parameters and are unknown.
Failure rate (hazard rate) function, is an important function in lifetime modeling is given by
.(..)=..(..) ..(..)=...... M.. ..
N/$0`1...$123
4!a5$0..$123
4!
1.`1...$123
4!a5 (4)
It is pertinent to note that the .(..) is:
i. Constant =..$0 if ..=..=1
ii. Increasing (decreasing) FR if ...1 and .....1 (...1 and .....1)
iii. Bathtub shaped FR if ..>1 and ....<1
iv. Upside-down bathtub shaped (unimodal) FR if ..<1 ...... ....>1
From (3), we know that the EWD includes many distributions as special cases. If ..=1, it
reduced to exponentiated exponential distribution (EED). If ..=2, it becomes exponentiated
Rayleigh distribution (ERD). If ..=1, it deduced as the standard 2-parameter Weibull distribution
(WD). The particular case for ..=2 and ..=1 is the Rayleigh distribution. If ..=1 and ..=1, it
becomes the one-parameter exponential distribution.
The component reliability for mission time .. is given by
..(..)=1.`1...$123
4!a5,..>0 (5)
and mean time to component failure is expressed as
..=f6..(..)
( ....=..gP1+1..
RG(.1))M.. ..
N..$70/
9 8
)'( (6)
The reliability function of a ....../....../../..:.. system is obtained as
..!"(..)=GG% (.1))
)'(
&
%'( ..(..,..,..)M.. ..
N h1.`1...$123
4!a5i#$%*) (7)
M. Kalaivani, R. Kanna n
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
Using ..!"(..), we have the following expression for the mean time to system failure (MTSF) of
a ....../....../../..:.. system
..!" =f6..!"(..) ....
( =GG(.1)) %
)'(
&
%'( ..(..,..,..)M.. ..
N f6(..(..))#$%*)
( ....
Now, f6[..(..)]#$%*)
( ....=f`1.h1...$123
6 4!i5a#$%*)....
(
=#G$%*)(.1):
:'( M.......+..Nf`h1...$123
6 4!i:5a
( ....
=#G$%*)(.1):
:'( M.......+..NG(.1). 9
.'( M......Nfh..$.123
6 4!i
( ....
Hence from (7), we have ..!" =.. gP1+1..
R GG% (.1))
)'(
&
%'( ..(..,..,..)M.. ..
N#G$%*)(.1):
:'( M.......+..NG(.1). 9
.'( M......N..$70/
8 (8)
where ..=m.1 ,2 ....,3 ,..�.. ........ .......... ..... ............................
3. Reliability and MTSF Estimation
In this section, we have obtained the ML estimator of ..!"(..) and the MTSF, ..!" for a ....../....../../..:..
system. Let .. units are put on test and the test ends when all the units have failed. Let ..0,..;,�,..# be
the random failure times and assume they follow an exponentiated Weibull distribution with
density function given in (3). The log likelihood function of the parameters is ..=..(..,..,..)=..........+..........+(...1)G........)..............G# P....)R/
)'0
#
)'0
+(...1)G# ......`1.exp h.P....)R/ia
)'0
Then the maximum likelihood estimator (MLE) of ..,.. and .. say ..t,..u and ..@ respectively can be
obtained by solving the following simultaneous nonlinear equations using numerical methods .... ....=.. ..+G# ......`1...$123"4!a
)'0 =0 (9)
.... ....=.. ..
+G# ........)...........
)'0 .GP....)R# /
)'0 ......P....)R+(...1)GxM....)N/......M....)N..$123"4!
1...$123"4! y #
)'0 =0 (10)
........=...+G# P....)R/
)'0 .(...1)GxM....)N/..$123"4!
1...$123"4! y #
)'0 =0 (11)
By applying the invariance property of MLE, the MLE of the reliability function and mean time
to failure of components and ....../....../../..:.. system is obtained by substituting ..t,..u ...... ..@ in (5),
(6), (7) and (8).
M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
4. Asymptotic Confidence Interval of ..!"(..)
The Fishers information matrix for ..=(..,..,..) is
..=..(..)=...
.......
........;;....; ......;...... ......;...... ........ ..;.. ....; ..;.. ........ ..;.. ........ ..;.. ........ ..;.. ....;.......
=.......0;<000 ......0;<;;; ......0;<<<<.
where
..;;=. ....;.G# P....)R/......;P....)R+
)'0 (...1).G.....
h1...$123"4!i..0...$;123"4!M....)N;/......;M....)N/
h1...$123"4!i; .....
#
)'0
..<<=......; .G# ....;
)'0 P....)R/+P.. ..
R;P..) ..R/.(...1)G.....
h1...$12"34!i..;+P.. ..
R;M....)N;/..$123"4#!
h1...$123"4!i; .....#
)'0
..0;=..;0=G# xM....)N/1M.....)..N$/1..23.."4..!M....)Ny
)'0
..0<=..<0=.GP.. ..
R..$123"4!M....)N/
1...$123"4! #
)'0
..;<=..<;=... ..
+G`1..P..) ..R/+P.. ..
# RP....)R/......P....)Ra
)'0
+G.....
h1...$12"34!i..<+P.. ..
RM....)N;/......M....)N..$;123"4!
h1...$123"4!i; .....
#
)'0
and ..0=`..$123"4!P....)R/......;P....)R/...$123"4!P....)R;/......;P....)R/a
..;=P.. ..R;..$12"34!P..) ..R;/.....;P..) ..R/..$12"34! .P.. ..
R;..$123"4!P....)R/
..<=P.. ..RP..) ..R;/......P..) ..R..$12"34! +1..P..) ..R/..$12"34! .P.. ..
RP....)R/......P....)R..$123"4!
The MLE of ..!"(..), ..@!"(..) is asymptotically normal with mean ..!"(..), and variance
..=;$%(2)=GG......!"..()..)......!"..(%..)
<
)'0
<
%'0 ..)$%0
M. Kalaivani, R. Kannann
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
where ..)$%0 is the (..,..)... element of ..(..) (see Rao[17]). Then,
..=;$%(') =h......!"..(..)i;..0$00+h......!"..(..)i;..;$;0+h......!"..(..)i;..<$<0+2......!"..(..)......!"..(..) ..0$;0+2......!"..(..)......!"..(..)..0$<0
+2......!"..(..)......!"..(..)..;$<0
where ....!"(..) .... =.......`1...$123
4!a`1...$123
4!a5. ....!"(..) .... =...P.. ..
R/..$123
4!......P.. ..
R`1...$123
4!a5$0. ....!"(..) .... =...... P.. ..
R/..$123
4!`1...$123
4!a5$0.
and .=GG% (.1))
)'(
&
%'( ..(..,..,..)M.. ..
N(.....+..)(..(..))#$%*)$0
Therefore, an asymptotic 100(1...)% confidence interval of ..!"(2) is given by ..!"(..)....@!"(..)�..@/;.
where ..@/; is the upper ../2th quantile of the standard normal distribution and ..t=$%(2) is the value of ..=$%(2) at the MLE of the parameters.
5. Simulation Study and Data Analysis
In this section, a simulation study is carried out along with the application of the ....../....../../..:.. system and a real data set to the estimate system reliability and mean time to system
failure when samples are drawn from EWD.
5.1. Simulation Study
We study some results based on Monte Carlo simulation to compare the performance of ..!"(..) ..!"
and asymptotic confidence interval using different sample sizes ..=30 and 50 for combination of
parameters (..,..,..) = (0.5,0.5,1), (0.5,1.5,1), (0.5,2.5,1),(1.5,0.5,1) and (2.5,0.5,1) and are
evaluated using R software.
i. For each combination of ..,..,.. and sample size .., we can derive the random
samples from the EWD by inverting the cumulative distribution of (3). ..=...........1...0/5..0//, ..~..(0,1)
ii. Based on the data and using (9), (10) and (11), we estimate the MLE of ..,.. and .., ..!"(..), ..!" and asymptotic confidence interval.
iii. Repeat step (i) and (ii) over 3000 times and the mean square errors for the estimators
are calculated.
iv. 3The above steps are repeated for ....../....../../..:.. system by taking ..=10 and ..= and 6. The results are presented in Table 1, 2 and 3.
M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
Table 1: Reliability Estimation of ....../....../3/10:.. system
Parameters .. ..!"(..) ..=30 ..=50
..@!"(..) MSE
Asymptotic CI ..@!"(..) MSE
Asymptotic CI
LL UL LL UL ......===100 ..55,,
0.2 0.297747 0.303069 0.018713 0.249567 0.355806 0.297385 0.011802 0.264664 0.329299
0.4 0.162046 0.176572 0.011650 0.137303 0.214943 0.169849 0.007063 0.146110 0.192815
0.6 0.100745 0.116668 0.007394 0.087137 0.145560 0.110241 0.004283 0.092532 0.127299
0.8 0.067312 0.082330 0.004848 0.059416 0.104628 0.076508 0.002686 0.062907 0.089549
1.0 0.047150 0.060606 0.003270 0.042428 0.078231 0.055435 0.001737 0.044833 0.065699 ......===101 ..55,,
0.2 0.854684 0.831226 0.008891 0.789136 0.873496 0.833757 0.005326 0.807877 0.859582
0.4 0.546503 0.537213 0.021241 0.474321 0.599168 0.531110 0.013585 0.492483 0.569434
0.6 0.281312 0.294728 0.018049 0.241666 0.346063 0.283371 0.011144 0.251503 0.314827
0.8 0.122807 0.143560 0.009532 0.109112 0.176196 0.132677 0.005272 0.112644 0.152358
1.0 0.047150 0.063644 0.003753 0.044633 0.081262 0.055754 0.001731 0.045140 0.066163 ......===...... ......,,
0.2 0.983359 0.973400 0.000761 0.962288 0.984654 0.976043 0.000371 0.969691 0.982422
0.4 0.830987 0.805319 0.010896 0.759474 0.851011 0.808428 0.006486 0.780539 0.836273
0.6 0.505575 0.493241 0.022685 0.430379 0.555143 0.490158 0.014306 0.451895 0.528157
0.8 0.197713 0.210763 0.014190 0.166860 0.253530 0.202739 0.008425 0.176460 0.228638
1.0 0.047150 0.061131 0.003794 0.042884 0.078609 0.054835 0.001680 0.044332 0.065048 ......===110 ..55,,
0.2 0.934871 0.915864 0.004256 0.889625 0.942087 0.919800 0.002230 0.904357 0.935242
0.4 0.815863 0.788815 0.013581 0.740635 0.836936 0.792504 0.007750 0.763610 0.821397
0.6 0.693397 0.667344 0.020847 0.607435 0.727154 0.669078 0.012392 0.632893 0.705262
0.8 0.582536 0.561714 0.024429 0.497174 0.626124 0.561157 0.014812 0.522036 0.600278
1.0 0.487040 0.472712 0.025109 0.407868 0.537404 0.470101 0.015345 0.430729 0.509473 ......===120 ..55,,
0.2 0.996454 0.992783 0.000106 0.989188 0.996379 0.993912 0.000045 0.992011 0.995813
0.4 0.976451 0.965417 0.001186 0.952008 0.978826 0.968151 0.000583 0.960528 0.975774
0.6 0.937420 0.918920 0.004066 0.892858 0.944981 0.922706 0.002155 0.907477 0.937936
0.8 0.883603 0.859445 0.008437 0.820951 0.897940 0.863512 0.004693 0.840678 0.886346
1.0 0.820477 0.793148 0.013325 0.744110 0.842186 0.796827 0.007664 0.767454 0.8262
M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
Table 2: Reliability Estimation of ....../....../6/10:.. system
Parameters .. ..!"(..) ..=30 ..=50
..@!"(..) MSE
Asymptotic CI ..@!"(..) MSE
Asymptotic CI
LL UL LL UL ......===100 ..55,,
0.2 0.878187 0.854031 0.009330 0.820145 0.887532 0.861369 0.005120 0.841135 0.881186
0.4 0.767127 0.742157 0.017567 0.695727 0.787805 0.750602 0.010092 0.722402 0.778081
0.6 0.676749 0.654458 0.022542 0.602294 0.705814 0.663003 0.013098 0.630975 0.694176
0.8 0.602410 0.583262 0.025178 0.528435 0.637110 0.591528 0.014681 0.557627 0.62446
1.0 0.540364 0.524086 0.026268 0.468303 0.578803 0.531901 0.015332 0.497441 0.565671 ......===101 ..55,,
0.2 0.997601 0.995090 4.75E-05 0.992582 0.997590 0.995899 0.000018 0.994557 0.997237
0.4 0.965283 0.952884 0.001772 0.937312 0.968168 0.955734 0.000925 0.946606 0.964788
0.6 0.868454 0.849576 0.008993 0.814768 0.883334 0.852847 0.005244 0.831893 0.873555
0.8 0.714275 0.698971 0.019396 0.648864 0.747063 0.701058 0.011516 0.670576 0.731087
1.0 0.540364 0.532961 0.025731 0.476643 0.586475 0.533617 0.014971 0.499182 0.567542 ......===102 ..55,,
0.2 0.999975 0.999860 0.000000 0.999752 0.999970 0.999913 0.000000 0.999869 0.999958
0.4 0.996649 0.993299 0.000084 0.990042 0.996534 0.994407 0.000033 0.992675 0.996134
0.6 0.956381 0.940101 0.002694 0.921477 0.958424 0.944680 0.001395 0.933970 0.955323
0.8 0.804664 0.779584 0.014721 0.736400 0.821867 0.786669 0.008520 0.760821 0.81225
1.0 0.540364 0.524527 0.025982 0.468467 0.579204 0.530556 0.015108 0.496113 0.564469 ......===110 ..55,,
0.2 0.999574 0.998694 0.000008 0.997905 0.999481 0.999065 0.000002 0.998704 0.999426
0.4 0.995942 0.991603 0.000156 0.987592 0.995609 0.993213 0.000050 0.991174 0.995252
0.6 0.986857 0.977439 0.000723 0.968280 0.986586 0.980672 0.000282 0.975772 0.985572
0.8 0.971944 0.957120 0.001876 0.941835 0.972381 0.961952 0.000820 0.953511 0.970392
1.0 0.951824 0.932068 0.003592 0.910394 0.953705 0.938280 0.001691 0.926041 0.950519 ......===120 ..55,,
0.2 0.999999 0.999986 0.000000 0.999974 0.999999 0.999993 0.000000 0.999989 0.999997
0.4 0.999949 0.999760 0.000000 0.999586 0.999935 0.999847 0.000000 0.999777 0.999918
0.6 0.999608 0.998782 0.000007 0.998018 0.999546 0.999126 0.000002 0.998780 0.999473
0.8 0.998523 0.996374 0.000039 0.994360 0.998388 0.997205 0.000011 0.996229 0.998182
1.0 0.996166 0.991969 0.000142 0.987952 0.995987 0.993500 0.000047 0.991461 0.99554
M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
Table 3: Estimation of mean time to system failure (MTSF)
Parameters ..=0.5, ..=0.5, ..=1 ..=0.5, ..=1.5, ..=1 ..=0.5, ..=2.5, ..=1 ..=1.5, ..=0.5, ..=1 ..=2.5, ..=0.5, ..=1 ....../....../../....:.. system
Sample size ..!" 0.233168 0.475392 0.614456 1.338123 2.431575 ..=.... ...!" 0.265047 0.483305 0.620654 1.364669 2.435709
MSE 0.0278 0.014594 0.151323 0.234167 0.460743 ..=.... ...!" 0.250012 0.473758 0.608595 1.336523 2.405559
MSE 0.014272 0.006604 0.004345 0.13133 0.264359 ....../....../../....:.. system
Sample size ..!" 2.564288 1.134486 1.051981 6.084472 8.497633 ..=.... ...!" 3.520812 1.140934 1.112734 6.36528 8.728028
MSE 373.2682 0.173304 10.29648 7.102796 9.147431 ..=.... ...!" 2.844258 1.12965 1.044834 6.247015 8.627549
MSE 7.607097 0.022023 0.007051 3.34101 4.402205
Figure 2: Reliability of ....../....../3/10:.. system Figure 3: Reliability of ....../....../6/10:.. system
According to Table 1, 2 and 3, the MSEs for estimate of reliability and average lifetime of the
system decreases as the sample size increases. The expected length of the confidence interval reduces
as sample size increases at 95% level of significance for all combinations of parameters. By
comparing the system reliabilities and MTSFs for ..=3 and ..=6 , we can see that the system
reliability and MTSF are increase when the number of consecutive failed components, .. increases
from 3 to 6 and other parameters are kept unchanged. The results are consistent with the definition
of the consecutive ..-out-of-..:.. system. Further, it is observed that the time increases the reliability
of the system declines, as expected. In addition, the length of the confidence interval of the
L..../....../6/10:.. larger than the L..../....../3/10:.. system for the all combinations of parameters.
Figure 2 and 3 show the true value of reliability in L..../....../3/10:.. and L..../....../6/10:... ..!"(..) is low when ..<1 ,..<1 and ....<1, the components of the system having initial failure rate.
On the other hand, for ..<1,..>1 and ....<1, the performance of the system is improved, the
M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
system�s components are in bath tub failure mode. When, ..<1,..>1 and ....>1, ..!"(..) declines
quickly because the components are in the increasing failure rate. If ..>1,..<1 and ....<1, the
reliability improved well when compare with first one even though the components are in the
decreasing failure rate. Suppose ..>1,..<1 and ....>1, the performance of the system is high,
because the system�s components possessing unimodal failure rate.
From Table 3, it is seen that the MSEs of MTSF of L..../....../6/10:.. is more as compare with
L..../....../3/10:.. system. This is due the number of consecutive failed components is large. The
MTTF of each component are same. Therefore, it can be concluded that the failure rate of the
distribution will affect the average failure time of the system. However, the influence is dependent
on the values of the parameters.
5.2. Data Analysis
In this section, the model proposed in (3) is applied to estimate the lifetimes of 18 electronic devices
shown in Table 4. The presented data were taken from Ahmad and Ghazal [1] as a lifetime
distribution having bathtub shaped failure rate. The MLE of ..,.., .. and their standard errors are
given below
..t=0.144884 (0.008049) ..u=5.285692 (0.275845) ..@=373.218 (9.72593)
The ML estimates of reliability and MTSF for a ....../....../../10:.. system has been evaluated
with ..=10 and ..=3 ...... 6.
Table 5: Reliability, MTSF and asymptotic confidence intervals at 95% level of significance
.. ....../....../3/10:.. system
...!" =154.3145 ....../....../6/10:.. system ...!" =289.8265 ..@!"(..)
Asymptotic CI ..@!"(..)
Asymptotic CI
LL UL LL UL
0 1 - - 1 - -
50 0.936522 0.895956 0.977088 0.999596 0.999051 1.000142
100 0.747427 0.652674 0.84218 0.991666 0.984417 0.998914
150 0.497669 0.387101 0.608237 0.95448 0.927456 0.981504
200 0.265821 0.179307 0.352334 0.858565 0.801264 0.915866
250 0.106003 0.05878 0.153227 0.686347 0.602146 0.770548
300 0.028072 0.010935 0.045209 0.457671 0.366109 0.549232
350 0.004127 0.000463 0.007791 0.236461 0.161964 0.310958
400 0.000257 -0.00011 0.00062 0.08773 0.043646 0.131815
From Table 5, it can be seen that the reliability and MTSF of the consecutive k-out-of-n the failure
rate of the distribution when time increases the reliability of the system decreases, as expected. The
system parameter .. increases the reliability and expected life time increase. In addition, the length
of the confidence interval decreases when the number of consecutive failure components decreases.
These can be seen in the simulation study results.
Table 4: Lifetime of 18 electronic devices
5 11 21 31 46 75 98 122 145 165
196 224 245 293 321 330 350 420
M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
6. Conclusions
In this paper, we have proposed a ....../....../../..:.. system which composed of .. independent and
identically distributed components having exponentiated Weibull lifetimes with three unknown
parameters and studied the reliability characteristics. This distribution has the ability to model the
non-monotonic and monotonic failure rate. The reliability and mean time to system failure are
estimated based on simulated observations by maximum likelihood estimation for various
combination of parameters. Asymptotic confidence intervals were also constructed. The MSEs of ..!"(..), MTSF and length of the confidence interval decrease as sample size increases for ....../....../3/10:.. and ....../....../6/10:.. In addition, MSEs of MTSF as well as the mean length of the
confidence interval increases as the number consecutive failed components .. increases. A real-life
data set was used to show the entire approach.
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M. Kalaivani, R. Kannan
ESTIMATION OF RELIABILITY CHARACTERISTICS FOR LINEAR
CONSECUTIVE ..-out-of-..:.. SYSTEMS BASED ON EXPONENTIATED
WEIBULL DISTRIBUTION
RT&A, No 3 (69)
Volume 17, September 2022
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