Научная статья на тему 'Estimation of reliability characteristics for linear consecutive 𝒌-out-of-𝒏: 𝑭 systems based on exponentiated Weibull distribution'

Estimation of reliability characteristics for linear consecutive 𝒌-out-of-𝒏: 𝑭 systems based on exponentiated Weibull distribution Текст научной статьи по специальности «Медицинские технологии»

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Ключевые слова
Maximum Likelihood Estimation (MLE) / Reliability function / Mean Time to System Failure (MTSF) / Asymptotic confidence interval and Exponentiated Weibull distribution

Аннотация научной статьи по медицинским технологиям, автор научной работы — M. Kalaivani, R. Kannan

The focus of this paper is to estimate the reliability characteristics of a linear consecutive 𝑘-out-of𝑛: 𝐹 system with 𝑛 linearly ordered components. The components are independent and identically distributed with exponentiated Weibull lifetimes. The system fails if and only if at least 𝑘 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.

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Текст научной работы на тему «Estimation of reliability characteristics for linear consecutive 𝒌-out-of-𝒏: 𝑭 systems based on exponentiated Weibull distribution»

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

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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):

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:'( 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

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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

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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

[15] Chaturvedi, A., and Pathak, A. (2012). Estimation of the Reliability Function for

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