Regression equations for Marshall-Olkin trivariate exponential distribution and reliability measures of related three-unit systems Текст научной статьи по специальности «Математика»

REGRESSION EQUATIONS FOR MARSHALL-OLKIN TRIVARIATE EXPONENTIAL DISTRIBUTION AND RELIABILITY MEASURES OF RELATED THREE-UNIT SYSTEMS

B. Chandrasekar*, S. Amala Revathy

Loyola College, Chennai 600034, India

e-mail: bchandrasekar2003@yahoo.co.in, sarevathy@gmail. com

ABSTRACT

Marshall and Olkin(1967) proposed a multivariate exponential distribution and derived some properties including the moment generating function. Proschan and Sullo(1976) provide the probability density function which involves some tedious notations. In this paper, we provide an explicit expression for the probability density function in the trivariate case and derive the conditional distributions, regression equations and the moment generating function. By considering four three unit systems with trivariate exponential failure time distribution, we derive the reliability measures of the systems.

Key words: Conditional distribution, moment generating function, multiple regression, performance measure, trivariate exponential.

1 Introduction

Marshall-Olkin(1967) proposed a multivariate exponential distribution as a model arising out of Poisson shocks. The distribution is not absolutely continuous and so the distribution received considerable attention among the researchers. Bemis et al(1972) provide a probability distribution which is not absolutely continuous with respect to the Lebesgue measure in R2. Inference for bivariate exponential distribution was discussed by Arnold(1968), Bemis et al(1972), Bhattacharya and Johnson(1973) among others. For the trivariate case with equal marginals, Samanta(1983) discussed the problem of testing independence. Proschan and Sullo(1976) discussed parameter estimation for multivariate case and proposed the probability density function(pdf) involving tedious notations.

The aim of this paper is to provide an explicit expression for the pdf in the trivariate case and derive the conditional distributions, regression equations and the moment generating function. Further, by considering four three unit systems with trivariate exponential failure time distribution, we derive the reliability measures of the systems. Section 2 proposes the pdf whose bivariate marginal is the one given by Bemis et al(1972). Section 3 derives the conditional distributions, both univariate and bivariate. Further the multiple regression equations are obtained and shown that they are not linear. The moment generating function is obtained in Section 4. Finally in Section 5, three component standby, parallel, series and relay systems with trivariate exponential failure times are discussed and the performance measures are obtained.

2 The probability density function

The survival function of the Marshall-Olkin trivariate exponential distribution (MOTVE) is of the form

*Also a Guest faculty at Department of Statistics, University of Madras, Chennai-600 005, India.

F ( X1 , ^ X3 ) = eXP 1 "Z A,X, "H^ii ( X V Xj ) " ^123 ( X1 V X2 V X3 ) f , ^ ^ X3 > 0 .

[ i =1 i< j J

Here X1, A > 0, \2,\3,À23,A23 >0and x1 v x2 v .x3 = max(x13x2,.x3).

Let A, = A + ^ + ^13 + ^123 , A2 = A2 + A2 + A3 + ^123 , ^ = A + A3 + A3 + ^123 , Aji = Aj , i < j

and A — A + A2 + A3 + A2 + A3 + A3 + A23.

We propose the following pdf for the MOTVE distribution

X < xj < Xk

Xi < Xj = Xk

Xj = Xk < Xi

X1 = X2 = X3

f (xl3 x2, X3) = Xi (A;. + Xj )X*k exp {-Ai-Xi - (Xj + Xj) xj - X*k xk},

\ (Xjk + X123 ) eXP {-Xxi - (X - X ) Xj } ,

exp {-(X + X+Xjk) x- X* x-},

X23eXP (-Xx1 ),

— (21)

Remark 2.1 It can be verified that the total integral is one. While integrating the pdf, there are 6 triple integrals, 6 double integrals and one single integral. It may be noted that we have the same number of cases as in Samanta(1983) and (2.1) reduces to the one given in the paper for equal marginals situation.

2.1 Bivariate marginals

It is known that the bivariate marginals are bivariate exponential. We use the pdf approach to derive the bivariate marginals.

Theorem 2.1 The pdf of (X1 , X2) at (x1 ,x2) is given by

f12 (^ x2 )=(X1 + X13 ) X2*e XP {-(X1 + X13 ) x1 - X2*x2 } , (X2 + X23 ) X1*eXp { (X2 + X23 ) x2 - X1*x1} , (X12 + X123 ) eXp {-(X-X3 ) x1} ,

Proof The pdf of (X1 , X2) at (x1 , x2) is f12 (x1, x2) = J f (x1, x2, x3) dx

0

Three cases arise according

as x1 < x2, x1 > x2 and x = x .

Case 1: x1 < x2 In this case

f (^X2,x3) = X3 (X1 + X13)^exp{-X3x3 -(X1 + X13)x1 -x2}, X1 (X3 + X13 ) ^^ {-X1 x1 -(X3 + X13 ) x3 - A2'X2 } , X1 (X2 + X12 ) X3*eXp {-X1X1 (X2 + X12 ) X2 - XX3 } , X1 (X23 + X123 ) eXp {-X1X1 -(X + X1 ) X2 } , ^^ {-(X1 + X3 + X13 ) X1 - X2* X2 } ,

X1 < X2

X1 > X2

X1 = X2 '

X3 X1 X2

X3 X2

X1 X2 << X3

X1 X2 — X3

X2 X<1 — X3

Thus

f12 (xi , X2 ) = J f (xi, X2, X3 )dx3 + J f (xi, X2, X3 )dx3 +J f (xi, X2, X3 )dx3 + f (xi, X2, Xi ) + f (xi, X2, X2 )

0 Xi X2

Xi

= JЯ3 (* + Л3 )Я*2 exp{- (* + )xi - Я2x2 - Я3x3 fdx3 +

0

x2

JЯ (Я3 + Л3)ä*2 exp{- Äixi - Я2x2 - (Я3 + )x3 dx3 +

xi

œ

JЯ(Я2 + Л2)Я* exp{-\xi - (Я2 + Я12)x2 - Я3x3 fdx3 +

x2

Я13Я*2 exp{- (Я + Я3 + Я13 )xi - ЯЯ2 x2 f+ Я (Я23 + Я123 ) eXP{- Я1 Xi - (Я - Я1 )x2 f = (Я1 + ^13 ) ^eXP {-(Я1 + Я13 ) X1 - ЯX2 f+(1 - eXP {-Я3Xif)

ЛЛ^р {-Я1X1 - ЯX2 f (eXP {-(^3 + ^13 ) X1 -(Я1 + ^13 ) X2 f)

+ ^ (Я2 + ^2 ) eXP {-Я1X1 -(Я2 + Я2 ) X2 f eXP {-Я3* X2 f +

Я13Я2* eXP {-(Я1 + Я + Я13 ) X1 - *2 X2 f + Я1 (Я23 + Я123 ) eXP {-Я1X1 -(Я-Я1 ) X2 f

= (Я1 + Я13 )Я*2 exp{- (Я1 + Я13 )x1 - Я*2X2f Case 2: x1 > x2

As in Case 1, we can show that

f12 ( ^ X2 )=(Я2 + Я23 ) *TexP {-(Я2 + Я23 ) X2 - Я X1 f .

Case 3: x1 = x2 In this case

f ( X1, X2 , X3 ) = Я (*12 + Я23 ) eXP {-Я3 X3 -(Я-Я3 ) X1 f , Л2Я eXP {- (Л + Я2 + Я12 ) X1 - ЯX3 f ,

Я23 e XP (-*X1 ) ,

X3 xi — X2

X3 X<1-X2

X1 = X2 = X3 •

Thus f12 (x1, x2 ) = JЯ3 (*2 + Л23 )exp{- Я3x3 - (Я - Я3 )x1 fdx3

0

œ

+ JЯ2Я33 exp{- (Я + Я2 + Я2)x1 - Я3x3 f+ Л23 exp(- Àx1 )

= (*12 + *123 ) exP {-(Я-) Xif

Hence the theorem.

Remark 2.2 Thus (X1,X2) ~ BVE(* + Я3,Я2 + Я23,+ *23), in view of Bemis et al(1972). Similarly, it can be shown that the other two bivariate distributions are also bivariate exponential.

3 Conditional distributions and regressions

In this section, we determine the univariate and bivariate conditional distributions.

X

X

x

3.1 Univariate conditional distributions

The conditional pdf of X1 given (X2, X3) = (x2, x3) is

f 1.23 (X1X 2 , X3 )= f(Xy X2,X3)) , 0 < X1 <° . J 23 lX2 , X3 '

Note that f (x1, x2, x3) is given in (2.1) and

(X2 + X12 )X3 eXp{- (X2 + X12 )x2 - X3 X3 } X2 < X (X3 + X13 )X2 exp {- X2 x 2 - (X3 + X13 )x3 } X2 > X (X23 + X123 )eXp{-(X- X1 )x2 },

f 23 (x2 , X3 )=^

X2 = X3.

Three cases arise according as X2 < X3, X2 > X3, X2 — X3.

Case 1: x2 < x3

f1.23 (X1 |X2 , X3) = X1 eXp(- X1 X1 ),

X(2 X ^X1)) exp{- (X1 + X12 )x1 + X12 X2 }

\X2 + X12 /

X (X; + X23 )X1 exp{- x1 + X12 X2 + (X13 + X123 )x3 j,

\X2 + X12 JX3

X12

p{- X1X2 }

(X2 + X12 )

\ (X13+ ^) exp{-(X1 + X12 )x3 + A, X2},

\X2 + X12 JX3

Case 2: X2 > X3

f1.23 (X1 |X2 , X3) = X eXp{- A1X1 },

X( ^ + X1)) exp{- (X1 + X13 )x1 + X13 x3 } lX3 + X13 )

X (X2 + X23 "Xp{- XX1 + ^3 X3 + (X12 + ^23 )x2 },

(X3 + ^3 )X2

xtxD exp(- X x*

(K +t23) exp{- (A + A13 X + A13 r,

lA3 + A; /X2

Case 3: X2 — x,

f1.23 (X1 |X2 , X3) = A eXp(- A X1 ),

X 2

X2 << x<1 X3

X 3

X1 = X2

x-1 X3

X 3 X2

X2 << x<1

X1 = X 2 .

X2

X23 X1

(X23 + X123 )

exp

{- X*X1 + (X12 + ^3 + X123 )x2 }

X123

(X23 + X123 )

exi

p (- X1 X2 ),

X2 << x<1

X1 = X2 .

Remark 3.1 Similarly, one can find the other two univariate conditional distributions.

Theorem 3.1 The regression equation of X1 on ( X2, X3 ) is

=

1

— +

A>

A ' l(A + A3)(A + Ai3) 1 \eXp( A Хз)

+ <

A3

A3 (A + A23 )___

(A + A3К A2 (A1 + A13 )(A + A3 ),

eX

P{A13X3 -(A + A13 )X2 },

1

-+

A

A1 l(A + A12 XA2 + A12 ) A \ p( A 2 )

+

A

A2 (A + A23 )

(A2 + A2 )A* A3 (A1 + A2 )(A + A2 \ - -1 \ eXP(- AX2 \

eX

P(A12X2 - (A1 + A2 )X3 I

1

-+

A.

23

A l(A 23 '423 )A A1

X 2 > X 3

X 2 < X 3

X 2 - X 3.

Proof For x2 > x3.

E[X1 |(X2,X3) — (x2,x3)]- Jx1A1 exp{- A1 x1 }dx1 + Jx 3(A—^^exp{A3x3 -(A + A13)x1 }d;

0 L 1 (A + A3)

3 x3 - (A + A13 )x1 }dx1 +

Jx1 3( 2—a ^ exp{- Ax1 + (A2 + A23)x2 + A3x3 }dx1

(A3 + A3 ) A-

+x

A^exp(-Ax3)+ x2 A(+ ^23)exp{-(A + A3)x2}

( A + A3)

(A3 + A3 )A2

A,

'w V -~ \ exp(- A1 x3 ) +

A l(A + A13 )(A + A13) A1J

A3 (A2 + A23 )

A

I (A + As )A2 A (A + As )(A + As),

Similarly one can derive the regression equation for x2 < x3.

ex

P{-(A + A3 )x 2 + A13 x3 }

For x2 — x3.

x3

e[x 1 |(X2, X3 ) — (x2, x3 )] — J x1 A exp{- A x1 }dx1 +

A123 4exP{- A1 x2 } f xw, A23 A 4exp{- A* x1 +(A12 + A13 + A123 )x 2 }dx1

lA23 + A123> x, lA23 + A123 /

A1 l(A23 + A123 )A* A1

* - ~ \ exP(- A1 x2 )

Hence the theorem.

Remark 3.2 The other two regression equations can be derived in a similar manner. Thus the multiple regression equations are non-linear.

3.2 Bivariate Conditional Distributions

Let us find the conditional distribution of (X13 X2 ) given X3 = x3.

x

2

1

Note that f3 (x3) = A, exp (- A, x3), x3 > 0 The conditional pdf of (X1, X2) given X3 = x3 is f ( | )= f(x1, x2, x 3) 0

J 12.3 VX1, X2\X3j = f (x ) ' X1, X2 > 0 .

Here three cases arise. For x1 < x2

f12.3 (x1, X2 |X3 ) = +* exp{- (X1 + X3 )x1 - X2X2 + ( X12 + X23 + X123 )X3 },

X*

X(X +*X13 )X2 expj_ ^ X1 _ ¿2 x2 + ( ^ + Я123) X 3 }

Л-,

X3 Xi

Xi X3 X2

¿13 X2

X*

exp{_ X2X2 _ (X1 _ X23 _ X123)X3 }

Xi{X\ + Л23) exp{_ X X1 _ (X + ¿12) X 3},

X3

For Xj > X2

■/12.3 (x1, X2 |X3 )= +* eXP{_(X2 + X23 )x2 _ X1X1 + (X12 + X13 + X123)X3 }

X*

X3 ^ X2

X2 X3 X1

X2(X3 +* X23 )X1 exp{_ X1 _ X2X2 + ( X3 + X123)X3 }

3

X2 (X1 + X12 )exP{_ (X1 + X12 )X1 _ X2X2 } X X1

XTX-exP{_ Л"X1 _ ( X2 _ ^3 _ X123 )X3 } 3

^) exp{_ X2x2 _ ( X + X2)x3},

3

For Xj = x2

/12.3 (x1, X 2 IX3 )= ^^ + X123 ) exP{_ (X _ X3 )x1 + ( X2 + X23 + Л23 ) X 3 1

3

X12 exP { (X1 + X2 + X12 )X1 }

exP{_ ( X + X2 + X2) x 3}, x1 = x3

3

Remark 3.3 It can be verified with routine but tedious integration, that /12 3 (x1, x2 |x3 ) is a Pdf. Similarly one can derive the other two bivariate conditional distributions.

4 Moment generating function

Theorem 4.1 The mgf of (X1 , X2 , X3) at (t1 , t2 , t3) is given by

M (t t t )= = X1 X2X3_+ X1 (X23 + X123 )t2t3_

1 2 3 (X2 _ 12 XX3 _ 13 XX _ t1 _ 12 _ t3 ) (X2 _ 12 XX3 _ 13 XX _ t1 _ 12 _ t3 )(X _ X _ 12 _ t3 )

X 2 Л X3_+ X 2 (X13 + X123 )t1t3_

(X _ t1 JX _ t3 \X _ t1 _ 12 _ t3 ) (X _ t1 XX3 _ t3 \X _ t1 _ 12 _ 13 )(X _ X2 _ t1 _ t3 )

+

X2 - X3 .

Я з А* А* А з (A12 + Aj

23 ГГ2

(A* - t1 XA2 - ,2 )(A - t1 - ,2 - t3 ) (A* - t1 XA2 - 12 XA " t1 " , 2 " t3 )(A - A3 - t1 - , 2 )

+__A 12 t3 _ +__A13 12 _ +__A 23 ^_

(A - t1 - 12 - t3 )(A3 - t3 ) (A - t1 - 12 - 13 )(A2 - 12 ) (A - t1 - 12 - t3 )(A* - t1 )

+ A 12 + A13 +A 23 + A123

(A - t1 - 12 - 13 )

Proof M (t1,12,13 )=£{exp(t 1X1 +t 2X2 +t 3 X3)}

= A1 (A2 + A12 )A3_+ A (A3 + A3 )A2_

(A3 - t3 )(A - A - 12 - t3 )(A - t1 - 12 - t3 ) (A2 - 12 )(A - A - 12 - t3 )(A - t1 - 12 - t3 )

+ A2 (A1 + A12 )A3_+ A2 (A3 + A23 )A*_

(A3 - 13 )(A - A2 - 12 - t3 )(A - t1 - 12 - t3 ) (A - t1 )(A - A2 - 12 - 13 )(A - t1 - 12 - t3 )

A3 (A1 + A13 )A2_+ A3 (A2 + A23 )A*_

(A2 - 12 )(A - A3 - t1 - t3 )(A - t1 - 12 - t3 ) (A - t1 )(A - A3 - 12 - 13 )(A - t1 - 12 - 13 )

+ ^-t7-—-^-- - ) +-

t1 12 t3J V1! МЛ'1* ' ^3 12 13/4'^ M 12 13,

+_A (A23 + ^23 )_+ _ A2 (A13 + ^23 )

(A -A1 -t 2 -t3 )(A -t1 -t 2 -t 3 ) (A A2 -t1 -t3 )(A -t1 -t 2 -t 3 ) _A3 (A12 + A123 )_+_Л2 A3 +

(a-A3-t1 -t 2 )(a-t1 -t 2 -t 3) (a -t1 -t 2 -t3 )(a3 -t3 )

'2/V" 4 2 '3/ V-"' M *2 l3)\b3 l3, +__A13 A2 _ +__A23 A1 _ + A123

(A -t1 -t 2 -t3 )(A2 -t 2 ) (A -t1 -t 2 -t3 )(A* -t1 ) (A-t1 -t 2 -t3 )

A1 (A2 + Л2 )A3_+ A1 (A3 + A13 )A2_

I3 - t3 )(A - A - 12 - t3 )(A - t1 - 12 - t3 ) (A2 - 12 )(A - A - 12 - t3 )(A - t1 - 12 - t3 )

_A1 (A23 + A23 )_J +j A2 (A1 + A12 )A3_

(A-A1 -t 2 -t3 )(A-t1 -t 2 -t3 )J [(A3 -t3 )(A A2 -t1 -t3 )(A -t1 -t 2 -t 3 )

+ ( A - А )( A -i -i -i U II A ~ -i К A _ A -i _, „ A -t -t -t

l3№ l1 2 l3

+ A2 (A3 + A23 )A*_+_A2 (A3 + A123 )_I

(A* -t1 )(A - A2 -t1 -t 3 )(A -t1 -t 2 -t3 ) (A A2 -t1 -t3 )(A-t1 -t 2 -t3 )J

A3 (A2 + A23 )A*_+ A3 (A + A13 )A2_

_t1 )(A -A3 -t1 -t 2 )(A -t1 -t 2 -t 3 ) (A2 -t 2 )(A -A3 -t1 -t 2 )(A -t1 -t 2 -t 3 )

A (A12 + A 23 )_J + _ A2 A3__+ _ A3 A2_

(A - A3 - t1 - 12 )(A - t1 - 12 - t3 )J (A3 -t3 XA -t1 -t2 -t3 ) (A2 -t2 )(A -t1 -t2 -t3 )

A A * A

+ _ 23 1__1__123_

(A* -t1 XA -t1 -t 2 -t3 ) (A-t1 -t 2 -t3 )

__A1 A2 A3_+ __A1 (A23 + A23 )t 2t3_

(A2 -t 2 )(A3 -t 3 )(A -t1 -t 2 -t3 ) (A2 -t 2 )(A3 -t 3 )(A - t1 -t 2 - t3 )(A -A -t 2 - t3 )

+ A 2 A1 A3_+ A 2 (A13 + A23 )t1t3_

(a - ,1 )(A3 - ,3 )(A - ,1 - 12 - ,3 ) (A - ,1 )(A3 - ,3 )(A - ,1 - 12 - 13 )(A - A2 - ,1 - ,3 )

A 3 A* A2 a 3(A2 + A 23 1 2

(AГ^-tl)(A2™™-t2)(A—t1^-t7—t3^ (Ai -,1 )(A2 -t2 )(A -,1 -t2 -,3 )(A - A3 -,1 -t2 )

+__A 12 t3 _ +__A13 t2 _ +__A 23 _

(a-t1 -t2-t,)(a;-t,) (a-t1 -t2-t3)(a;-t2) At1-t2-tjx^")

+ A 12 + A13 +A 23 + X123

(A-t1 -t 2 -t 3 )

Remark 4.1 The mgf of (X1, X2) at (t1, t2) is M(t1, t2, 0) and reduces to the mgf of BVE (X + A13 , X2 + X23 , X2 + X123 ) at (t1, t2 ) in view of Barlow and Proschan(1975).

5 Reliability measures of systems with MOTVE components

In this section we derive the performance measures associated with four three component systems, assuming that the component failure times have a joint MOTVE distribution.

5.1 Standby system

Consider a three unit standby system with component failure times X1, X2, X3 respectively.

3

Then the system failure time is T =Z Xi. Assume that the component failure times are

i=1

identically distributed. As in Samantha(1983), take A1 = X2 = A3 =A1, X12 = A13 = X23 =A2, X123 =A3. Let us first find the mgf of T.

Define a1 = A + 2A2 + A3,a2 = 2A + 3A + A anda3 = 3A + 3A + A. From Theorem

4.1,

M (t1, 12, t; )= A (P1 + P2 )«1

I_1_+_1_I

[(«1 -t1 )(«2 -t1 12 )(«3 -t1 12 -t3 ) («1 -t1 )(«2 -t1 -t3 )(«3 -t1 12 - t3 )J

|_1_ 1 1

+ -w-w-V + ■

(«1 12 )(«2 t1 12 )(«3 t1 12 t3 ) («1 12 )(«2 12 t3 )(«3 t1 12 13 )J

J 1 1

+ U-w-\7-7 +

1 13 2 11 13 )(«« 3 t1 12 t, ) 1 13 2 12 13 )(« 3 t1 12 13 )

A (A + A; )|7-1 t )( 1 t

U«2 -t1 -t2 )(«3 -t1

+ A (A + A K ^-w-7 + -7-w-1-\ +

t1 12 )(«3 t1 12 13 ) («2 t1 t3 )(«3 t1 12 t3 )

1 1+ A;

(«2 12 t3 )(«3 t1 12 t3 )J («3 t1 12 t3 )

Thus the mgf of X1 + X2 + X3 at t is,

M.(t)= 6A1 (A + A )a + 3A (a + A,) + 3A2«1 + A

(a1 -1)(a2 - 2t)a3 - 3t) (a2 - 2t)(a3 - 3t) (a1 -1)(a3 - 3t) (a3 - 3t)

I a a I

Note that the mgf exists for t < min| a1,—, — >

1 1 2 3 J

Resolving the first three terms into partial fractions and simplifying we get

M * (t)= 6A A + A2 )a 24A (A + A2 «1

(A + A; )(3 A2 + 2A; )(«1 - t) (A + A; )(3 A2 + A; )(«2 - 2t) + 54A1 (A1 + A2 K + -6 A (A2 + A; ) +_ 9A1 (A2 + A;)

(3A2 + 2A; )(3A2 + A; )(a, - 3t) (3A2 + A; )(«2 - 2t) (3A2 + A; )(a, - 3t)

+ -3A2a1 + 9A2a1 + A3

(3A2 + 2A3 )(a2 - 2t) (3A2 + 2A3 )(a3 - 3t) (a3 - 3t) Let us express the mgf as the weighted average of three exponential mgfs.

( tX1 f o.v1 r ^v1

t 2t Define M1 (t )= 1--, M 2 (t )= 1--and M 3 (t ) =

v a1 y

V a2 y

1-* a

3 y

Then M * (t) = wM 1 (t) + w2M 2 (t) + w3M 3 (t), where

w = 6A1 (A1 + A 2) - 3A2

1 (A2 + A; )(3A + 2A; ) (3A2 + 2 A; )

w =- 24 A1 (A1 + A2 + -6A1 (A2 + A; ) and

2 (A2 + A; X3A2 + A; K (3A2 + A; K

54A (A + A2 )a + 9A (A2 + A;) + 9Aa1 + A; (3 A2 + 2A3 )(3 A2 + A3 )a3 (3 A2 + A3 )a3 (3 A2 + 2A3 )a3 a3

It can be verified that w1 + w2 + w3 = 1.

3 3

Therefore, the reliability function R(t) = ^ wi exp{- (ai /i)t}, t > 0, and the MTBF = ^iwi /ai

i=1 i =1

5.2 Parallel system

Consider a three unit parallel system with component failure times X1, X2, X3 respectively. Then the system failure time is T = Max Xi.

1<i<3

The distribution function of T at x is

G(x) = 1 - F (x,0,0) - F (0, x,0) - F (0,0, x) + F (x, x,0) + F (x,0, x) + F (0, x, x) - F (x, x, x) Therefore the reliability function of the system is,

33

R(t) = ^exp(- A*t)- ^ exp{- (A - At )t}+ exp(- At), t > 0.

i =1 i =1 The MTBF is

3 13 1 1

MTBF = Z -r- + .

i=1 Ai i=1 (A-Ai) A 5.3 Series system

Consider a three unit series system with component failure times X1, X2, X3 respectively. Then the system failure time is T = Min Xi.

1<i<3

The reliability function is R(t) = F (t, t, t)

= exp{- At}, t > 0.

MTBF = —. At

5.4 Relay system

A three component relay system operates if and only if component 1 and at least one of the remaining two components operate. The system failure time is T = X1 A (X2 V X3). (Barlow and Proschan, 1975). It seems natural to assume that X2 and X3 are identically distributed. Thus we assume that

F (x1, x2, x3) = exp{- A x1 - X2 (x2 + x3)- X12 (x1 v x2 + x1 v x3)- X23 (x2 v x3)- X123 (x1 v x2 v x3)}

Note that (X1, X2) and (X1, X3) are identically distributed. The reliability function is

R(t) = F (t, t,0) + F (t ,0, t) - F (t, t, t)

= 2 exp{- (A1 + A2 + 2A12 + A23 + A123 )t}- exp{- (A + 2A2 + 2A12 + A23 + A123 )t}, t > 0. The MTBF is

MTBF = -{-2-r-7-1-^

(A1 + X2 + 2A12 + X23 + X123 ) (A1 + 2A2 + 2A12 + A; + A23 )

(A1 + 3X2 + 2A12 + X23 + A1 23 )

(A1 + X2 + 2A12 + X23 + A123 )(A1 + 2A2 + 2A12 + X23 + X123 )

REFERENCES

Arnold, B.C., (1968). Parameter estimation for a multivariate exponential distribution. J. Amer. Stat. Assn. 63, 848-852.

Barlow, R. E. Proschan, F. (1975), Statistical Theory of Reliability and Life Testing/Probability models, Holt, Rinehart and Winston Inc., New York.

Bemis, B.M., Bain, L.J. and Higgins, J.J., (1972). Estimation and hypothesis testing for the parameters of a bivariate exponential distribution. J.Amer.Stat.Assn. 67, 927-929. Bhattacharya, G.K. and Johnson, R.A. (1973). On a test of independence in a bivariate exponential distribution. J.Amer.Stat.Assn. 68, 704-706.

Marshall, A.W. and Olkin, I. (1967). A multivariate exponential distribution. J. Amer.Stat. Assn. 62, 30-44.

Proschan, F and Sullo, P. (1976), Estimating the parameters of a multivariate exponential distribution. J. Amer.Stat. Assn., 71,465-472.

Samanta. M. (1983). On tests of independence in a trivariate exponential distribution. Statistics & Probability Letters, 1, 279-284.