Reliability performance measures of systems with location-scale Generalized Absolutely continuous multivariate exponential failuretime distribution Текст научной статьи по специальности «Математика»

RELIABILITY PERFORMANCE MEASURES OF SYSTEMS WITH LOCATION-SCALE GENERALIZED ABSOLUTELY CONTINUOUS MULTIVARIATE EXPONENTIAL FAILURETIME DISTRIBUTION

S. Amala Revathy, B. Chandrasekar

Loyola College, Chennai 600034, India e-mail: [email protected] [email protected],

Abstract

This paper deals with the equal marginal location-scale Generalized Absolutely Continuous Multivariate Exponential model. The distributional properties and applications of the location-scale model arising out of the k-parameter Generalized Absolutely Continuous Multivariate Exponential distribution are studied. Standby, parallel, series and relay systems of order k with location-scale Generalized Absolutely Continuous Multivariate Exponential failuretimes are discussed and their performance measures are obtained. The optimal estimators of the meantime before failure times are also derived.

Keywords: Equivariant estimation, location-scale, multivariate exponential, performance measures

1. Introduction

Though, there is an extensive literature on the reliability aspects of systems with independent failure times, not much work seems to have been carried out on systems with dependent component failure times. Rau (1970) discusses reliability analysis of systems with independent components. Chandrasekar and Paul Rajamanickam (1996), Paul Rajamanickam and Chandrasekar (1997, 1998a, 1998b), Paul Rajamanickam (1999) discuss repairable systems with dependent structures mainly assuming Marshall - Olkin type of joint distributions for the system component failure and repair times. Recently Chandrasekar and Sajesh (2013) and Chandrasekar and Amala Revathy (2016) discussed reliability applications of location-scale equal marginal absolutely continuous bivariate and multivariate exponential distributions respectively.

By considering location-scale Generalized Absolutely Continuous Multivariate Exponential (GACMVE) failuretime distribution, for k unit systems, we derive the reliability performance measures and obtain optimal estimators. In Section 2, we propose the probability density function for the location- scale GACMVE model. In Section 3, we derive some important distributional results required for further discussion. In Section 4, we consider a k unit standby system and obtain the mean time before failure (MTBF) and

48

the reliability function of the system. Further the minimum risk equivariant estimator (MREE) and the uniformly minimum variance unbiased estimator (UMVUE) of the MTBF are derived. Similar results for parallel, series and relay systems are presented in Sections 5, 6 and 7 respectively.

2. Generalized Absolutely Continuous Multivariate Exponential location scale

model

The joint pdf of GACMVE is

\ ' fi ^ . ^j+iexp k ! l=0 i=l j=0 V j

k k k

4 Zxt Z z

i=1 i = 1 i < j = 1

x. V x . " i j.

. ( X1 V X2 V .... V xk )_

X > 0 Vi; 4 >0, 4 > 0, i = 2,3....k .„.(2.1) Here x v x2 V.....V x^ = max {x, x2,......x^ } .

Let X be a random variable (vector) with the distribution function F^ r(.),4 eR,z>0 .

Let ;4 e R,T > 0} be a location-scale family, s° that F 4>T(x)= F for some distribution function F.

V * J

The location-scale GACMVE has the pdf

k-1 k-1 i f

Y k-I k-i l I j

* k ! l=0 i=l j=0 V j J

4

j

exp — — *

k k k 4Z xt +42 Z Z

i=1

i = 1 i < j = 1

x. V x . 1 j

. + 4k (X V x2 V ....V xk)"Z

k k ^

p=1

P J

x >ÇVi, ÇeR, *>0, 4>0, 4>0

. (2.2)

For fixed (4,4,-■■-,4),the distribution of ^

X2" Xk -Ç

does not depend on

(4,^) . Therefore the above family is a location - scale family with the location - scale parameter

(4,T) • Let us refer to the distribution as location-scale GACMVE. When t = 1, the resulting

family is the location GACMVE family. When 4 = 0, the resulting family is the scale GACMVE family. Since we are interested in the location-scale parameter, it is assumed that the parameters 4,4 ,••••, 4 are known.

3. Distributional properties

Theorem 3.1

Let {yXx,X2,...Xk}~ GACMVE distribution given in (2.1), and Yl,Y2...,Yk denote the order

statistics based on X1, X2 ..., Xk. Define Wl = Yl , W2=Y2-Y1, ...... Wk=Yk~Yk_ v Then

B0 W, B W,..., B-\ Wk are independent and identical standard exponential random variables,

k-1

where Bl = Z A, l = 0, 1,2,...., k - 1 and A = Z

i=l

j=0

V j)

¿1+1

i = 0,1,2,..., k -1.

Proof

The joint pdf of (X^, X2 ..., Xk ) i

is

Y k-I k-i ' j \

f{xl,x2...,xk)=-Y\YT\ . U,+i

k ! l=0 i=l j=0 V j)

exp

k k k f \ -¿1 Z Xi Z Z [*• V - •••• -K (X1 V X2 V •••• V xk ) '=1 i=1 i < j = 1V ' j )

x >0 Vi; ¿> 0,1, >0,i=2,3....k

The pdf of is

k-1

№ exp

. l=0

i - \

¿1 Z > ¿2 (>2 + 2.>3 + 3 .>4 + -l^k )

-¿3

[ [3^ ^ [k^

y3 + 2

V v 2)

>4 +

V 2 )

>5 + ••••

>k

• -¿k

V^ ) ) >1 <>2 <.....>; A >0,4 >0,i = 2,3,...k.

Consider the pdf of ••■■>Yk}

B0 B1... Bk-1eXP {-4.>1 -(A+¿2 )>2 }eXP j- [2 ^ >

4 + v 1 ) ¿2 +¿3

l V )

exp <

A +

[3^ [3 ^

A

1

¿2 +

V 2 )

¿3 +¿4

>4 f .eXP <

[ [k -^ 4 +

V1 )

>3

¿2 + ••• + ¿k

V V)

= B0 B1 ••• Bk-1 eXP {- A0 >1 - A1 >2 ••• - Ak-1 >k } In order to find the distribution of (WY, W2 ..Wn ), consider the transformation W = > ->i-x, i = 1,2,...k, with>0 = 0. Then > . = w + W + •••• + wj, j = 1,2,3,...,k. Note that the Jacobian of the transformation is 1.

>k

Amala Revathy S., Chandrasekar B. RT&A, No2 (41) RELIABILITY PERFORMANCE MEASURES OF SYSTEMS_Volume n, June 2016

The joint pdf of (WVW2 ...,Wk) is

w2 >•••'wk )=Bo Bi ••• Bk-i exP j" Ao wi - Ai (wi + w2 ) - ••••" Ak-i Z wi |

- Bo Bi ••• Bk-i exp {- Bo wi - Bi w2 - ••••- Bk-i wk } Hence B0W, BW .....B-Wk are independent and identical E(0,1) random variables.

Sufficient statistic

Let =( X, p ,X2p...,Xkp) = 1,2,.. ..n be a random sample of size n from (2.2). The joint pdf of {xlp,X2p...,Xkp} ;j = l,2,....n is

k-i k-i i f j\ In

I 1 k-i k-i i f i

p(".MnZZI ..

exp

J k! i=o i=i j=o vj y

k

A

j+i

i

p=i

k k r \

A Z "p +A2 Z Z I V xp .....+ Ak ("ip V "2p V .... V "kp )

i = i i < j = iv F JF y

^ f k ^ -Z At

m=i V m y

Let Up = (Xlp a a ... a Xkp ); and U(l) = min Up .

Y k-i k-i i f i \

pteM=nzzi .

I J k' i=o i=i j=o V j y

A

'j+i

exp 1—z

p=i

k k

A Z "p +A2 Z Z I xip V xjp +.....+ Ak ("ip V "2 p V ....V xkp )

i=i i = i i < j = i v f y

m=1

fk ^ k f k ^ / \

4U(! )+Z Am (U(l)-^)

V m y m=1 V m y \ V / /

= g^Oi, ^2 ) h ( " )

where , Tl = and

T2*=Z

p=i

k k k r \

A Z"p +A2 Z Z I "iv V"p .....+A ("pV"2pV....V"p)

i=i i = i i < j = i v ^ y

By factorization theorem, T * =(T *,T *) is a sufficient statistic

n

Theorem 3.2

(i) T~ E

nB

(ii) T G ( nk -1,t) and

(iii) T1* and T2* are independent.

Proof

(i) Let Xp — (Xlp,X2p ...,X^ ) ; p = 1,2,....n be a random sample of size n from (2.2). The joint pdf of (Xlp,X2p...,Xkp) ;j = 1,2.....n is

k-l k-l i ij\ 1 n

11 k-l k-l i 1 i

p(x,4,T)=\ n z z | .

t k! 1=q i=i j=0 V j y

A

j+l

exp

t z

t p=l

AZXp + A ^ z Pip Vj i=l i = l i < j = l V ^ ^

■A (

Xlp V X2p V •••• V Xp )-

\ ^ (k ^

v m y

!.in ( X

min\xlp /\X2p l>

Let

UP = (xiP A ^ A• • •Axkp);and u(D =min UP

Then U(1)>4 and (u(i)-4)~E(0,l)

Therefore U^ ~ E

t

i \ V nBo y

(ii) Let Y1j,Y2j ••■,Ykj denote the order statistics based on i^X^j ,X2j .. j = 1,2,.... n. Note

that Y1j = Uj, j = 1,2,____n,. Define Wrj = Yrj - Y(r-1)j , r = 1,2,3.....k; j = 1,2,3,.. ..n.

Yoj = 0 for all j. Consider

T2*=z

p=l

i = l i < j = l

AzX.+A .S z \ Xip VXjp J.....(Xlp VX2p V•••• VXp)

n k

^ (k ^

z \ Al z ^p + A z (i -1) Ym A Z (i - 2) Ym +• ••• + AkYm - z | Am U(l

= p=l I i=l

m=l V m y

(l)

k

k

i=2

i=3

Z {a (kWi p +(k - i)W p +.....W)

p=i

^ k-i

k-i

k-1

+ A

Wi p Z i+W2p Z i + W3p Z i.....+ (k - i)W

kp

V i=i

^ k-2

i=i

k-2

i=i

k-2

k-2

k-2

+ A

A3 Wip Zi +W2p Zi +W3p Zi- + W4p Zi+W5p Zi- +.... + (k -2)Wk

kp

+ •

V i=i

i=i

i=i

i=2

i=3

kk f k ^

+ A ZWm -Z Am U (i)

i= m= V m J

f

k-i

:Zl Wi p Ak + A2 Z i +A3 Z i + A4 Z i + "" + Ak

p=i | V i=i i=i

k-2 k-3

4

i=i

+

W

2p

f

V f

k- k-2

k-3

A(k - i )+ A2 Z i + A3 Z i + A4 Z i + "" + Ak

i=i i=i

k- k-2

i=i

k-3

W

3p

A (k - 2)+a Z i+ A Z i+ A Z i+••••+ A

i=

i=

i=

+

+......+

W^ (Ak + A(k-i) + A3 (k-2) +A4(k-3) +.... + Ak)+Z A,

K f k \

U,

V m y

(i )

k-i i ii\

k- i i fi\

p=i I i=o j=o Vj J

n [ k-i i f j \

k- i i

Z1Wip ZZI, A+i+Wp ZZI, aj++......wp ZZI, A+i+Z Am „„ U(i )

f k \

m= Vmy

j=o V j y i=k- j=o V j y

n I k - i i I j . , k-i i f i A k-i i f j\

ZjZZZj ,|A>i(Wim -U(i,)+W2,ZZj j......+Wp ZZ1,1

p=i I i=o j=o V J y i=i j=o VJ y i=k i j=o VJ y

A

j+i

n I k-i i f i ^

k-i i f i ^

Z1ZZ ^j+i(U(m)-U(i))+W2pZZ Aj+i+......+w^ ZZ

(m) U (i)

p= 1 j=o j=o v j j i=i J=o V j J

k- i

f i ^

A+i

v j j

Since U^ jU^) ••• are order statistics from £

A A

£ —

V Bo y

i=k- j=o

, it follows that the first term on the

right hand side follows G ( n -i, —) .

By Theorem 3.1, each of the other (k-1) terms on the right hand side follows G ( n, —) . Since wy, w2j ..., wkj are independent for each j, the k random variables on the right hand side are independent.

Hence T* ~ G( nk -1, —).

(iii) For fixed x, the joint distribution of , x2j ..., xkj j, j = 1,2,.. ..n, belongs to a location family

with the location parameter £,. The statistic T2* is ancillary and T1* is complete sufficient. Hence T1* and T2* are independent (Basu, 1955).

The following theorem will help us in obtaining the reliability performance measures of standby

n

m=i

k

n

and parallel syste Theorem 3.3

Let (TvT2,...,Tk) follow GA CMVE (A,, A2 ,...Al; ;g,r) with pdf given in equation (2.2). Then

k

(i) Z Tt — k^ d Vx + V2 +... + Vk , where Vi, V2,.... Vk are independent and

{k-(1 - 1)t} ^

i=l i

V1 ~ E

0,-

B-l

, for all l = 1, 2,.., k.

...(3.1)

(ii) (T V T V ...Tk)-k 4 d V* + V* +.....V* , where V*1, V*2,.. V*k are independent and

V* ~ E

Proof:

f k-l ^ A

vk -1/

ki >

k -1

A

, for all j = 1, 2,.., k.

.(3.2)

V !=l Vv V y

The MGF of

k k

zT, z zT VTjT vT2 v•••Tk at (Ul, U^^U) i

V i=l i<j=l j=l y

is

2 k-l k-l ^ (i

M(ul,n2...,uk)=\\..] -^nzz 7j

44 4 t k ! I=0 i=i j=0 Vj y

A

j+l

exp

k k k ul z ti + u2 z. z \ti V *j 1 +.....+ Uk (tl V t2 V •••• V ^ )>

i=l

i = l i < j = l

exp

Azt + A z z I ^v| + • ,■=l i = l i < j = lv' j

.....+ Ak (tl V t2 V •••• V ^ )

kk

I k l

V p y

p=l

Ap4

dtxdt2.......dtk

"j "j "j ^ k-1 k-1 ' (i~\

M(ui,u2...,uk) = \\...\ —-nZZ • A+1exp -44 4 T k ! 1=0 i=i j=o V J y

-TUX )z t +

i=1

k k f

(A T U2 )z z I ti V tj | +.....+ (Ak - TUk )(t1 V t2 V •••• V tk )

i = l i < j = lV J -

^ (k 1 -z( p JA4

d^ dt2.......dtk

k -l

=n

1=0

r k (k 1 \

exp -z up4

V p=l V p y y

l -

k-l i

Tzzz

i=1 j =0

(i 1 \

uj+l

V j y

Bi

Br

••z t - k 4d zv-

i=l i=l ,

where Vi's are independent and (k -1)t

V ~ E

0,-

Bl

I = l,2v„k\

(ii) M ( 0,0,_, Uk )=n

1 exP (- Uk 4)

r=0

( ¿4 (i

1 -

i=1

k - 1

B,

.-.(T VT2 V•••Tk)-k4 dVl* + v; +.....V

* t+.....v;

where Vi*'s are independent and V ~ E

k-U i \

tz

i=l

Vk - ^

B

i = 0,1,••••k -1

k

The following Lemma helps us in finding the reliability function.

55

Lemma 3.1

Let M (u ) —

k

n(l-a7-u)

j

; u <-V j .

a;

Then M (u )—^ j , ;.i (1-a;u)

a;

where w- — —:-

j k

k -1

n(

r —1 * j

a; -ar

—, and ^ w — 1. ) j—1

k

Proof

i

M (u )—

(1-a u X1-a2 u).....(1-a u)

Resolving into partial fractions,

M (u) = + +......+

(1-au) (1-a2 u) (1-a u)

w (1-a2 u).....(1-a u)+ w (1-a u )(1-a u) •••• (1-a u)+••••+w (1-a u )•••• (1-aiu)

k

n(1-a,M)

j—1

k-1

w1 n(1-aju)+w2 n(1-aju)+••••+wk n(1-aju)

j—2 j—1 j—1

#2

k

-a u)

j—1

k -1

n (1-aJu)

aj

Thus, for j = 1,2,3,.. ,,k, we get w ■ — —

n(a-ar)

r =i

* j

Corollary 3.1

The survival function corresponding to M(u) is

, u > 0.

k ( * \

G (u )—^ wj exp

j—1

1

--u

a

V j J

4 Standby system

Consider a k unit standby system with component failure times T1,T2,...,Tk having location-scale GACMVE distribution.

Then the system failure time is T = Z T .

The MTBF of the system is

wk

MTBF = E (T)

= EV + V2 +.... + V, )+kt

k (k-l)T = £-'— + k £, in view of (3.1).

i=1 Bi-1

Following the arguments of Chandrasekar and Amala Revathy (2016), the MREE of n = a 4 +p t, a, p e R, is given by

5* =a5 +

1 kn

P-

a nBn

Define,

¿01= min{^AX2iA...AXj and

1< p < n

n

502 = Z

P=1

k k

i = 1 i < j = 1

4 £ Xip + 4 Z J X.p v Xjp J +.....+ 4 (Xp V X2 p v.... v X* )

By taking a = k and P( k 1), the MREE of the MTBF is given by

/=1 Bi-1

k 501 + -1 kn

£ (k-1)+ a

l =1 B/-1 n B0

5-

Reliability function of the standby system is R(t) = P(T > t)

k

\

= PI £ T, - k t> t

V i=1 y

k \

, t > 0

. i=1 y

A 1 A

, t > 0

=P (Xv > t

k

= Zpi exP

i=1

--1

V l y in view of Lemma 3.1.

( k -1 )t

Here a =±-'— VI = 1,2,..., k, and P =

Bi-1

a

k-1

— , and £Pj =1.

na-ar ) j=!

r=1

Therefore,

* (t )=z

f{k - / ) ^

V B/-1 y

k-1

/ = ^I(k - l)r (k - r)r ^

Bl-1 Br-1 r y

exp

r=1

(k -1 )t

V B/-1 y

k

1

5 Parallel system

Consider a k unit parallel system with component failure times Tl,T2,...,Tk having the GACMVE distribution. Then the system failure time is T=Max Tt.

l<i<k

MTBF = E (T)

(* * * \ Vl + V2 + „„ + Vk)+ k4

= t

k-1

z

1=0

>i ^

zi k 1,

i=1 V k ly

B

+ k 4, in view of (3.2)

1

When n = a 4 +p t, a, p e R, the MREE of n is given by 5 = a 501 +--

kn

P-

a nBn

k-1

By taking a = k and p= z

1=0

> i

z

i=1

v k - ly

B1

the MREE of the MTBF is given by

1

5 = k --

kn

Reliability function

k-1

z

1=0

k-u i \

zi

k -1

B,

nBn

^02

R (t) = P(T > t)

=p{zV* >t

k \ *

i=1

t > 0

1=1

=z W exP

k-1 z

y

--1

V a1 y

' 1 A

k-U i \

Here a =

_ i=1 Vk ly

B1 -1

V1 = 0,1,,•••, k-1, and w =

a

k-1

1 k

V1 = k -1

n(a -ar)

r=1

6 Series system

Consider a k unit series system with component failure times TY, T2,..., Tk having the GACMVE distribution.

Then the system failure time is T = Min T.

l<i <k

From Theorem 3.2, Min T ~ E

1<l<k '

Thus, MTBF = — + 4 Bn

4 —

4 B

1

When n = a 4 +p t, a, p e R, the MREE of n is given by S = a S01 +--

kn

P-

a nBn

02

1 » 1 By taking a = 1 and P =— , the MREE of the MTBF is given by S = S01 + —

B kn

11

B nBo

02

Reliability function

R(t) = P(T > t)

= exp

Bn

(t

t >4

7 Relay system

Consider a k unit relay system with component failure times T,T,•••• ,T having the GACMVE distribution. A relay system of order k operates if the first component and anyone of the remaining (k-1) components operate. Therefore, the failure time of the system is T=T A(T vT v .... VT ) .

The reliability function of the system is

R (t )=P (T > t )

k f k -= E(- 1)r -1 Fr (t,t,...,t,0,....,0),

r = 2 V 1 J

using distributive law and routine arguments.

Here Fr (t,t,...,t,0,....,0) represents P(X >t,X2 >t,...,Xr >t,Xr+1 >0,...,Xk > 0)

Let us discuss in detail the case when k = 3. Here

R (t )=P (T > t )

=!(-1)

r = 2

= 2 exp \ -

f k -1

r -1

Fr (t,t,0)

M±i]((-f)L exp LM±2A±A)(t-4)

The MTBF is given by

MTBF =

2 —

(24+4 ) (34 + 3^2 + 4 )

' (44+ 54 + 24) ' (24 + 4 )(34+ 34+4)

+4

3

When n = a 4 +p t, a, p e R, the MREE of n is given by S* — a S01 +

kn

(-

a

nBn

02 •

By taking a = 1 and ( —

( 44 + 54 + 24) (24+4 )(34+34+4)

, in the above equation, we get the MREE

of the MTBF. Therefore, MREE of the MTBF is

S — Soi +7-kn

(44+ 54+ 24) ' (24+42 )(34Li + 34 +4)

nBn

Remark 7.1

From Theorem 3.2, we can obtain the UMVUE's of 4 and t, and hence obtain the UMVUE of a 4 +p t:

o** o

S — aS01 +

1

kn -1

(-

a

nBn

02

Hence one can obtain the UMVUE's of the MTBF in each of the four systems discussed in this chapter.

References

[1] Basu D, 1955, On statistics independent of a complete sufficient statistics. Sankhya 15, 377

-380.

[2] Chandrasekar, B. and Amala Revathy, S. (2016), Simultaneous equivariant estimation for location-scale Absolutely continuous multivariate exponential model. Communications in Statistics: Theory and Methods.

[3] Chandrasekar, B. and Paul Rajamanickam, S. (1996), A property of counting process in multivariate renewal theory, Microelectronics and Reliability, 36, 111-113.

[4] Chandrasekar, B. and Sajesh, T. A. (2013). Reliability measures of systems with ACBVE components. Reliability: Theory and Applications. 8, 7-15.

[5] Paul Rajamanickam, S. and Chandrasekar, B. (1997), Reliability measures for two-unit systems with a dependent structure for failure and repair times. Microelectronics and Reliability, 37, No 5, 829-833.

[6] Paul Rajamanickam, S. and Chandrasekar, B. (1998a), Reliability performance measures for two unit systems with a dependent structure, Proceedings of the International Conference on Stochastic Process and their Applications, Anna University, Chennai (India), Narosa Publishing House, New Delhi, 140-146.

[7] Paul Rajamanickam, S. and Chandrasekar, B. (1998b), Confidence limits for steady state availability of a system with dependent structure for failure and repair times, Journal of Applied Statistical Science, 8, 17-27.

[8] Paul Rajamanickam, S. (1999), Contributions to Reliability Analysis of Repairable Systems with Dependent Structures, Ph. D. Thesis, University of Madras, Chennai- 600 005, India.

[9] Rau, J. G. (1970), Optimization and probability in Systems Engineering, Van Nostrand Reinhold, New York.