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60
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
Estimation of Stress-Strength Reliability Model Using
Finite Mixture of М-Transformed Exponential
Distributions
!Adil H. Khan and 2T .R. Jan
•
P.G Department of Statistics, University of Kashmir, Srinagar, India
^mail of Corresponding author: khanadil 192@yahoo.com
2Email: drtrjan@gmaiLcom
Abstract
In this paper Stress -Strength reliability is studied where various cases have been
considered for stress (Y) and strength (X) variables viz., the strength follows finite mixture
of М-Transformed Exponential distributions and stress follows exponential, Lindley and
М-Transformed Exponential distributions. The reliability of a system and the parameters
are obtained by the method of maximum likelihood method. At the end results are
illustrated with the help of numerical evaluations and real life data.
Key words: Reliability, М-Transformed Exponential Distribution, Stress, Strength.
1. Introduction
The term "stress-strength reliability" specify the quantity R=P(Y < X), where X is a random strength
and Y random stress such that the system fails if the stress Y exceeds the strength Y. Church and
Harris imported the term stress-strength for the first time. The stress-strength reliability and its
problems for considerable distributions have been discussed by Church and Harris [1970],
Woodward and Kelley [1977], Beg and Singh [1979], Awad and Gharraf [1986], Surles and Padgett
[1998, 2001], Raqab and Kundu [2005], Mokhlis [2005], and Saragoglu et al. [2011]. Kotz et al. [2003]
have presented an inspection of all approaches and results on the stress-strength reliability in the
last four decades. Adil H. khan and T.R Jan [2014] have studied the stress-strength reliability for
two parameter Lindley distribution. Exponential distribution and Gamma distribution. And have
considered different conditions for stress and strength variables.
In the present paper, we discuss stress strength reliability and have considered that the
strength variable follows finite mixture of М-Transformed Exponential distribution and stress
variables follows finite mixture of exponential or Lindley or М-Transformed Exponential
distributions. The structural properties and uses of М-Transformed Exponential distribution as a
lifetime distribution are studied by Dinesh Kumar et al. [2017]. We debate the estimation method
for finite mixture of М-Transformed Exponential distribution by the method of maximum
likelihood estimation. The М-Transformed Exponential distribution with parameters a is defined
by its probability density function (p.d.f.)
_x_
2e oc
f(x; a) =-----------~ ; a, x > 0 (1)
a (2 — e~E^j
In the present paper, we consider three cases
90
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
1) Stress follows exponential distribution and strength follows finite mixture of M-
Transformed Exponential distribution.
2) Stress follows Lindley distribution and strength follows finite mixture of M-Transformed
Exponential distribution.
3) Stress follows finite mixture of Lindley distributions and strength follows finite mixture of
M-Transformed Exponential distributions.
4) Stress and strength both follows finite mixture of M-Transformed Exponential distribution.
2. Statistical Model
In this model we assume that random variables X (strength) and Y (stress) are independent and the
values of X and Y are non-negative. The reliability of a component with strength X and stress Y
imposed on it is given by
oo г X
R = P(X > Y) =
g(y)dy
f(x)dx
(2.1)
o L0 J
where f(x) and g(y) are pdf of strength and stress respectively.
A finite mixture of M-Transformed Exponential distributions with v components can be expressed
as
f(x) = Pifi(x) + p2f2(x) + ••• + pvfv(x) ; Pi > 0, i = 1,2,
(2.2)
3. Reliability Computations
Let X be the strength of the v-components with probability density functions fj(x); i = 1,2,..., v. The
pdf of X which follows finite mixture of M-Transformed Exponential distributions is
f,(x) = Pi /2е _T2 ; X>0, «i> 0,Pi > 0, i = l,2,...(v,Eb=iPi = 1
aJ2-e ai)
Case I: The stress Y follows exponential distribution
As Y follows exponential distribution, pdf of Y is given by
g(y) = Ае”Лу, A > 0. у > 0
For two components v = 2, and
__x_ x_
2e “i 2e “2
f(x) = px-----------7 + p2---------
а1(2-е~Щ a2(2-e~k)
As X and Y are independent then from (2), Reliability function R2 is
x J ; pj +p2 = l,a1,a2,x> 0
r
о 0
OO x
=/Ny
_x_
2e
_x_
2e a2
Pi
P1 / .ix2+P2 /
a± ( 2 - e “ij a2 (:
00 00
v mi m±x
/ ^ e +P2 Z
Zj 2mi at
m1=l m2 = 1
(2-e-t)
dxdy
m2
2m2 a2
m2x
■ e
dxdy
2 00
0 i=1 mi=1
2 00 2
i=1 m1 = l
91
(3.1)
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
For three components v = 3 , we have
X X X
2e ai 2e a2 2e a3
f(x) = Pi —;----------— + P2 —;-----------— + Рз
„,(2 -e-k) аг{г-е~Щ а,(г-е^)
; Pi + p2 + p3 = l,a1(a2(a3(x > 0
oo x
R3 = //xe-^
0 0
2e ai
Pi-
■ + p2
2e a2
+ Рз-
X
2e
ax
(2 — e a^j a2 (2 — e a2^ a3 ^2 — e a3^
dxdy
R,
Я.. v1 TTl-i £ V1 Ш2 -m2x V1
Ле y p- Z + p* Z “г +Ps z
тх = 1
2™2 a
m2=1
3 00
m3 = l
77l3
2тъа,о
m3x
■e аз
dxdy
3 00
0 i=l mj=l
R» =1~Il2^mO : IP‘ = 1
i=l т^=1 i=l
In general for v-components, f(x) = Pifi(x) + p2f2(x) + —I- pvfv(x) ; IX1 Pi = 1, we hav
v CO v
r- 2^(Г;Г+^); 2>=i
i=l т^=1 i=l
Case II: The stress Y follows two parameter Lindley distribution
As Y follows Lindley distribution, pdf of Y is given by
02
g(y) = -—- (1 + Ay)e”0y ; у > 0, в > 0, a > — в
е + л
For two components v = 2
2e ai 2e a2
f(x) = Pi—--------7-7 +P2—---------7-2 : Pi+p2 = lai,a2>x > 0
at(2 — e a2(2 — e aA
As X and Y are independent then from (2), Reliability function R2 is
RN/Aa+iyHp--
X X
2e ai 2e az
+ P2---;------r-2 dxdy
0 0
00 x
x2'U2 x 2
i± ^2 - e a^j a2{^ 2 — e a2^
- \
Я0 V mi -ZEi£ v1 m2 ,
— (l + Ay)e-0d Pl ^ +p2 2, )dxdy
m± — 1
m2=1
2m2a7
R-
-/(
. 2 00
0 + A + A0x _взс\ / v v т,- -2ЦЕ
0 +A
2 oo
i=l wij=l \o 0
2 00
r = i-VY ( 1 A9af \
2 Zj Zj ^ 2m; \т,- + 0a,- (0 + A)(m,- + 0a;)2/
, (0 + A) (mi + 0a,
1 = 1 77li = l
For three components v = 3, we have
__x_ x_ x_
2e 2e a2 2e aз
f(x) = f(x) = Pi----;--------—2 + p2------------772 + Рз
аг (2 - e “i) a2 (2 - e az)
(3.2)
(3.3)
(3.4)
a3 (2 — e “3)
; Pi + P2 + Рз — b ai> a2> cx3, x > 0
92
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
yju A
Rs = //eTI(1 + Xy)e
0 0
oo
R3=f(,
-0y
2e ai
Pi-
- + p2-
2e
- + Рз-
X
2e
ax
6 + A + A0x
0+A
0 \i=l ГП!
i=l ^i=l о
3 OO
v^1 Ш/ / 1 А0а,-
Кз = 1"11 Pi2^(m; + e<xt +
i=1 vn\=1
^2 - e a^j a2^2-e a3i^2-
f3 oo
e “з)
dxdy
+
Л0
в + Х
~{lF+9)*
xe '
dx
4y)
(6 + X)(l7li + Octi.
In general for v-components, f(x) = p^Cx) + p2f2(x) + —I- pvfv(x) ; TJ=i Pi = 1, we have
V oo
A0 at
R.=i-ZZp'S(U^+-
t,)-)
(3.5)
(3.6)
. (.в + Я)(гП( + 6at
1=1 mi=1
Special case
When A = 1, two parameter Lindley distribution reduces to one parameter Lindley distribution
and then the reliability function is given as
V oo
Qat
R. = 1-ZZp'S(U^+;
t,i2)
(0 + l)(77lj + 6ct}
i=1 m\=1
Case III: The stress Y follows mixture of two parameter Lindley distributions
As Y follows mixture of М-Transformed Exponential distributions, pdf of X and Y is given by
For two components, v = 2
X X
2e ai 2e a2
fOO = Pi —;--------------777 + P2
(3.7)
a4 (2 - e “i) a2 (2 — e “2)
; Pi + P2 — b cxi, a2, x > 0
g(y) = Рз;
0?
-(1 + Я3у)е-я^ + р4
02
е3+Яз 04 + X4
As X and Y are independent then from (2), Reliability function R2 is
oo X
R,=//|P,
0 о
(1 + Я4у)е ; p3 + p4 = 1,
_Х_ X
2е ai 2е а2
--------2 +Р2----------
а4 (2 — е “i) а2 (2 — е “2)
-04у) (
02
03 + А3
■ (1 + Л3у)е
"АзУ
+ р4
J4 т 'Ч
4 2 00 х оо
д-—^-(1 + Я4у)е Й4У I dxdy
1,2 = ZI/ /й X 2^e‘“;’p'eph;d+x,y)e-^>'dxdy
)=i+2 i=l о 0 wij = l J J
j=i+2 i=l о mj=l 4 y y
4 2 00 00 /
e-0ix |dx
R, = 1
j=i+2 i=l mi = l
4 2 00
r--XZX^
j=i+2 i=l in i = 1
■ +
+ AJ
AjQjOCi
щ + 0;^ _j_ Ay)(mi + Ojai)
(3.8)
For three components v = 3, we have
X X_ X
2e 2e а2 2e aз
f(x) = Pi--:------772 + P2----------772 + Рз
а4 (2 — e “1) а2 (2 - e “2)
«з (2 - e “з)
; Pi + P2 + Рз — b cclf a2, a3, x > 0
93
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
Al
A?
g(y) = P4 ГТ— (1 + а4у)е Л4У + p5 H— (1 + а5У)е Asy + P3
A?
A4 + a
A5 + a5
■(1 + a6y)e
-^бУ
’ A6 + a6
; p4 + p5 + Рб = 1Д4Д5Дб,у > о
X and Y are independent then from (I), Reliability function R is
X X X
Я I 2e “1 2e “2 2e “3
Pl—------------+ '-----^72+Рз-
о o \ a1 (2 — e ai ) a2 [ z — e u2 I a3\
at^2 - e a2 ^2 - e a2^ a3 ^2 - e a3^
P4A2
A4 + 0C4 v У A5 + 0C5
6 3 00 x 00
4 (1 + a4y)e XaY + 5 (1 + a5y)e Asy + 6 (1 + a6y)e ЛбУ) dxdy
A6 + a6
1,3 = Z Z/ /Pi Z 2^e‘“;’p'erh;P+x,y)e-^>'dxdy
j=i+3 i=l о 0 wij = l
4 2 00 00
‘-ZZNZ^U1
R* = 1
j=i+3 i=l о mj = 1
6 3 00 00
у у у PiPj^i Г
Zj Zj Zj 2miCCi J
j = i + 3 i = l 77li = l о
6 3 OO
^+^+^0;xe-0;x,dx
+
6j + Ay
-о-)'
dx
R,
j=i+3 i=l mj=l
Pipj^i 1
2 mi Щ + djai
+ ■
А;0;«(
(3.9)
In general for v-components,
V
f(x) = рЛ(х) + p2f2(x) + - + pvfv(x) ; ^ Pi = 1
i=l
2v
g(y) = Pv+lfv+l(y) + Pv+2^v+2(y) + - + P2vf2v(y) ’ ^ Pi = 1
i=v+l
oo X
Rv = I I (Plfl(^ + p2f2^ + + Pkfk(x)) (Pv+lfv+l(y) + Pv+2fv+2(y) + - + P2vf2v(y))dxdy
2v V 00 x 00 2
R-= Z Z//p' Z ^e^'pivh:P + xiy>‘1|J,<lxdy
)=i+v i=l о 0 Щ = 1
2v v 00 00
ш/ -Шх ( б,- + А,- + А/0/Х й А
pipi Z 2^;е 4 ' dx
j—i+v i=l о mj=l 4 J J 7
R—ZZZ
О о
j=i+v i=l vn\ = \
2v v 00
R. = 1
zzz
j=i+v i=l mi = l
PiPj™i
2™;
■ +
-ен
Ay 0y cri
mi + (#y + Ay) (ш^ + Ojai)
(3.10)
Special case
When Ay = 1, two parameter Lindley distribution reduces to one parameter Lindley distribution
and then Rk is the reliability function when X follows mixture of one parameter Lindley
distribution and Y follows mixture of one М-Transformed Exponential distribution and is given as
2v
ZZZ
j=i+v i=l mi
PiPj™i
■ +
0y (%i
2mi mi + Ojai _j_ 1)^772- + Ojai)
(3.11)
Case IV: The stress Y follows mixture of М-Transformed Exponential distributions
As Y follows mixture of М-Transformed Exponential distributions, pdf of X and Y is given by
94
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
For two components, v = 2
X X
2e ai 2e a2
f(x) = Pi —;-----------r-2 + P2
2 1 иг , 2- ; Pi+ p2 = l,a1;a2,x > 0
ar (2 — e ai) a2 (2 — e “2)
-У- У
2e “3 2e a*
g(y) = P3—---------—2 + P4—--------7-2 ; Рз + Р4 = 1.
a3i 2 - e aA a4i 2 — e aA
As X and Y are independent then from (2), Reliability function R2 is
oo X
r2 = //ip,
0 о
_X_ X
2e ai 2e a2
--------2+P2----------
■+p4-
У
2e “4
a4 (2 — e “1) a2 (2 — e “2) у у a3 (2 — e “3) a4 (2 — e “4)
4 2 00 x 00 00 m.
j=i+2 i=l о 0 T7ij = l rrij = l J
4 2 00 00 /00 N
j=i+2 i=l о mj=l \ rrij = l
42 00 / 00 \ / 00
:6 aJ \dx
j=i+2 i=l
Rz = 1-ZZPiPi(^
i —i-L9 i —1 ' 1 J
j=i+2i=l
For three components, v = 3
__x_ x_ x_
2e 2e a2 2e аз
f(*) = Pl----:-------777 + P2-----------772 + Рз
a4 (2 — e “i) a2 (2 — e “2)
-У- _2L
2e a4 2e 2e аь
g(y) = P4----;------772+ P5—------------772+ Рб
/ ; Pi + P2 + Рз - 1
«з (2 - e “з)
a4
(2 — e as^2 — e a^ a6^2 — e a^
; P4 + Ps + Рб = t
As X and Y are independent then from (2), Reliability function R2 is
_x_
2e “3
__x_
2e “i
■ + p2 ■
X
2e a2
' + Рз'
_y_
2e «4
P4-
аг ^2 — e a^j a2(^2 — e ^ a3(2 — e a^j у у a4 ^2 — e
__x_ \
2e “6
+ P5‘
_y_
2e “s
+ p6-
a5 (2-e “s) a6 (2 - e “б)
6 3 00 X oo
dxdy
*-ХХ/ЕХ^»Хг%-^
j=i+3 i=l о 0 T7ij = l rrij = l J
6 3 00 00 /00 m.
^-ZZbZ^f-Z^
j=i+3 i=l о mj=l \ rrij = l
63 00 / OO \ / OO ™
e dx
j=i+3 i=l
6 3
Rs = 1“ZZPiP|(t+7
j=i+3 i=l 4 7
In general for v-components.
95
dxdy
(3.12)
(3.13)
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
2v
Now,
dlogL
da2
f(x) = Pifi(x) + p2f2(x) + ••• + pvfv(x); ^ Pi = l
i=l
g(y) = Pv+lfv+l(y) + Pv+2^v+2(y) + - + P2vf2v(y) J ^ Pi = 1
i=v+l
oo X
Rv = I I ^Plfl('X') + Pzf2^ + + Pvfv(x)) (p v+l^v+l (y) + Pv+2^v+2 (y) + - + P2vf2v(y))dxdy
0 о
2v V 00 X oo OO m
j=i+v i=l о 0 mi = 1 mj = 1 J
2v v 00 oo / oo m.
j=i+v i=l о mj = 1 \ rrij=l
2v v 00 / oo \ / oo
j=i+2v i=l '---------------- 7 '----
2v v
j=i+v i=l 4 y
4. Maximum Likelihood Estimation
dx
mj
:e aix I dx
Rv
11
Ua^Pi/y) = J~|
i=i
2e “i
Pi-
■ + p2-
2e “2
X- 2 x- 2
^2 - e a^j a2 ( 2 — e
n1 xi n2
2nn! (Vi\n± /Р2\П21—Г e i—г e a2
© п
гПу
/ 2nn! \
logLCa^azPi/y) = log ——- + n1logp1 - n^og^ + n2log (1 - px) - n2loga2
\П^! П2!/
ni n2
+Z[^_2to9(2_e-t)]+X[-|-2,os(2-e-t)
i=1 |^2 - e “ij i=1 ^
2 — e “2)
dlogL
dai
= 0
n 1
.m+y
ai 4—»
Xj
2xte ai
П1
«,=iy
щ L-k
i=1
Xi +
a*‘ «,2(2-^)
_4
2xte ai
= 0
(2-e “1)
= 0
n2
a2
-^+V
a 2 L-l
n2
-f
n? L-k
Xj
2xte a2
a2 2
a22
^2 - e a2^j
= 0
2xte a2
(2 - e“z)
(3.14)
(4.1)
(4.2)
96
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
di°gL _ q
dpi
Pi 1 - Pi
nl
Pi = — ; nx + n2 + 1
(4.3)
Generalizing the above results for к-components we get
(4.4)
It is oblivious that first equation is nonlinear in ccj and is not easily solvable, therefore for obtaining
their estimates we propose to use numerical iteration method. Therefore aj can be obtained by
using the nonlinear equation
since dt is a fixed point solution of h(at) = at, therefore it can be obtained by using simple iteration
procedure as h^a^) = a^k+1^ where atQ^ and а^к+1^ are kth and (k+l)th iterations of at. When the
difference between a^ and ai(^k+1) is very small we stop the iteration and once we obtain the
estimates at and the M.L.E of reliabilities for different cases are given as
1. M.L.E of reliability function when stress follows exponential distribution with known
parameter A and strength follows finite mixture of М-Transformed Exponential distribution
with parameter at is
2. M.L.E of reliability function when stress follows Lindley distribution with known
parameters A and 0 and strength follows finite mixture of М-Transformed Exponential
distributions with parameter at is
3. M.L.E of reliability function when stress follows finite mixture of Lindley distributions with
known parameters Ay and 0y and strength follows finite mixture of M-Transformed
Exponential distributions with parameter at is
h(at) = at
(4.5)
where
(4.6)
V oo
V oo
4. M.L.E of reliability function when stress and strength both follows finite mixture of M-
Transformed Exponential distributions with parameters at and aj is
97
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
5. Numerical Evaluation
RT&A, No 2 (57)
Volume 15, June 2020
In this section/ we have evaluated reliability of a system with two components for specific values
of a parameters involved in the reliability expressions in section 3 using graphical approach. Here
we will discuss how reliability of two component system behaves when stress, strength or
probability parameters are changed.
Case I: Stress follows exponential distribution and strength follows finite mixture of M~
Transformed Exponential distribution
Figure 1 Figure 2
Figure 3 Figure 4
98
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
Case II: Stress follows Lindley distribution and strength follows finite mixture of M-Transformed
Exponential distribution
Figure 5
Figure 7 Figure 8
99
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
Case III: Stress follows finite mixture of Lindley distributions and strength follows finite mixture
of М-Transformed Exponential distributions
Figure 9 Figure 10
Figure 11 Figure 12
100
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
Case IV: Stress and strength both follows finite mixture of М-Transformed Exponential
distributions
Figure 15
Figure 16
6. Real Data Analysis
In this sub section we analyze 33 leukaemia patients with two causes of death AG positive
(presence of Auer rods and/or significant granulature of the leukaemic cells) or AG negative (both
Auer rods and granulature are absent). The survival times in weeks are given
AG positive patients: 65, 156/ 100/134/16/108/121/ 4/ 39,143/ 56, 26/ 22/1, 1,5, 65
AG negative patients: 56/ 65,17, 7,16/ 22/ 3,4, 2, 3, 3, 4, 3, 30/ 4,43
101
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
We use the iterative procedure to obtain MLE of the parameters ax and a2 using equation (4.5) and
the final value of estimates are аг = 90.65061, a2 = 25.44067 and =0.5151.Based on these
estimates the MLE of the reliability are given below
Stress and strength both follows finite mixture of М-Transformed Exponential distributions R2 =
0.68
Table 1: Stress follows exponential distribution and strength follows finite mixture of M-
Transformed Exponential distribution
R 0.9161 0.9547 0.9689 0.9763 0.9809 0.9840 0.9862 0.9880 0.9892 0.9903
X 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Table 2: Stress follows Lindley distribution and strength follows finite mixture of M-Transformed
Exponential distribution
\A 0\ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5 0.9317 0.9264 0.9238 0.9223 0.9212 0.9205 0.9200 0.9195 0.9191 0.9189
1 0.9409 0.9340 0.9298 0.9270 0.9250 0.9235 0.9224 0.9215 0.9207 0.9200
1.5 0.9450 0.9375 0.9327 0.9292 0.9265 0.9245 0.9228 0.9215 0.9204 0.9195
2 0.9473 0.9399 0.9346 0.9306 0.9275 0.9250 0.9230 0.9214 0.9200 0.9187
2.5 0.9490 0.9415 0.9360 0.9317 0.9283 0.9255 0.9232 0.9212 0.9195 0.9181
3 0.9500 0.9427 0.9371 0.9326 0.9289 0.9258 0.9233 0.9211 0.9191 0.9174
3.5 0.9507 0.9436 0.9380 0.9333 0.9294 0.9262 0.9234 0.9209 0.9188 0.9169
4 0.9514 0.9444 0.9387 0.9340 0.9299 0.9264 0.9234 0.9208 0.9185 0.9164
4.5 0.9520 0.9450 0.9393 0.9344 0.9303 0.9266 0.9235 0.9207 0.9182 0.9160
5 0.9523 0.9455 0.9398 0.9348 0.9305 0.9268 0.9235 0.9206 0.9180 0.9156
Table 3: Stress follows finite mixture of Lindley distributions and strength follows finite mixture of
M-Transformed Exponential distributions, p3 = 0.35Д2 = 6.5,02 = 5.5
1 2 3 4 5 6 7 8 9 10
1 0.9680 0.9656 0.9644 0.9637 0.9632 0.9629 0.9626 0.9624 0.9622 0.9621
2 0.9784 0.9762 0.9748 0.9739 0.9733 0.9728 0.9725 0.9722 0.9719 0.9717
3 0.9818 0.9794 0.9780 0.9768 0.9760 0.9754 0.9749 0.9744 0.9741 0.9738
4 0.9833 0.9810 0.9792 0.9779 0.9770 0.9761 0.9754 0.9750 0.9744 0.9740
5 0.9842 0.9817 0.9798 0.9784 0.9771 0.9762 0.9754 0.9747 0.9741 0.9736
6 0.9848 0.9822 0.9801 0.9785 0.9772 0.9761 0.9751 0.9743 0.9736 0.9730
7 0.9852 0.9825 0.9803 0.9786 0.9771 0.9758 0.9747 0.9738 0.9730 0.9723
8 0.9855 0.9827 0.9804 0.9786 0.9769 0.9755 0.9743 0.9733 0.9723 0.9715
9 0.9858 0.9829 0.9805 0.9785 0.9768 0.9753 0.9740 0.9728 0.9717 0.9708
10 0.9860 0.9830 0.9806 0.9784 0.9766 0.9750 0.9735 0.9723 0.9711 0.9701
102
Adil H. Khan, T .R. Jan
ESTIMATION OF STRESS-STRENGTH RELIABILITY MODEL USING
FINITE MIXTURE OF M-TRANSFORMED EXPONENTIAL DISTRIB.
RT&A, No 2 (57)
Volume 15, June 2020
7. Conclusion
In this paper we have obtained stress strength reliability of a system with v-components using
different distributions for stress variables viz. exponential distribution, M-Transformed
exponential distribution and for strength variable we have used M-Transformed exponential
distribution. In the last section we showed that reliability of 2-component system can be
monotonically increasing and monotonically decreasing for specific values of parameter. Thus by
proper choice of parameters leads to high reliability. The maximum likelihood estimates of
parameters involved in the reliability function of v-components system are also obtained using
iteration method.
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