Shock models and testing for the renewal mean remaining life classes

Microelectron.Reliab., Vol. 33, No. 5, pp. 729-740, 1993.

0026-2714/9356.00+.00 © 1993PergamonPressLtd

Printedin GreatBritain.

SHOCK MODELS AND TESTING FOR THE RENEWAL MEAN REMAINING LIFE CLASSES A. M. ABOUAMMOH, A. N. AHMED and A. M. BARRY Department of Statistics and Operations Research, College of Science, King Sand Universi~, Riyadh 11451, Saudia Arabia

(Re~ived for pubgcanon 20 December 1991) ABSTRACT

The namely

paper

considers

two

classes of life

distributions;

the new better than renewal used in expectation

harmonic

new

better

corresponding

dual

than

renewal

used

in

classes are also studied.

and

the

expectation.

The

It is known

that

ageing properties of these classes are preserved for the survival function

when

occurrence

a

device is subjected to discrete

shocks

follows the homogeneous Poisson process.

whose

A model

of

accumulated additively damages that cause the failure of a device if

exceeds a random threshold is

test

statistics

properties

are

also

investigated.

Empirical

for testing exponentiality versus these established.

Simulated value of the

ageing

tests

for

small sample sizes are calculated.

I. I N T R O D U C T I O N Different ageing criteria are used to describe the deterioration life

(position ageing) in

the

or the improvement

engineering or biological

(negative ageing)

systems.

For

detailed

accounts of these criteria we refer to Bryson and Siddiqui Rolski Singh

(1975),

Barlow and Proschan

(1986) and Abouammoh The

mean

distribution

life (MRL)

unique,

function

see Swartz

the

MRL

(1981), Gupta and Gupta ageing

engineering

and

Kocher and

properties

determines

(1973,

Researchers such as Hollander and Proschan, Muth Wellner

(1969),

(1988).

remaining

function

(1981), Desphande,

of

Theorem

the 2).

(1977), Hall and

(1983) have refered to the use of

in

biometry,

reliability analysis.

actuarial

science,

Further the renewal

concept arizes in model building for maintenance and

MRL

replacement

pokicies. Abouammoh et al.

(1991) have

classes as follows

729

introduced

two

renewal

MRL

730

A . M . ABOUAMMOH et al.

D e f i n i t i o n I.I.

A life d i s t r i b u t i o n F on (0,-),

w i t h F(0-) =

is c a l l e d new b e t t e r than renewal used in e x p e c t a t i o n i

I t

D

(NBRUE)

0 if

Q

I x

F ( u ) d u dx

~ ~ I F(u)du, t

t

~ 0

(i.i)

m

where

~ =

I F(u)du < 0

Relation

(1.1)

E(Tw-t

m . is e q u i v a l e n t to stating that

i T w > t)

(1.2)

~ ET

where

T w is the renewal random v a r i a b l e w i t h d i s t r i b u t i o n W(t) = t ~-I I F ( t ) d u and d e n s i t y function w(t) = ~-1 F(t), t 2 0 (is 0 a s s u m e d to exist). D e f i n i t i o n 1.2. is

A life d i s t r i b u t i o n F on

(0, ®),

with F(O-)

c a l l e d h a r m o n i c new b e t t e r than renewal used

(HNBRUE)

in

=

0

expectation

if D

I t

@

~

I x

F ( u ) d u dx

m

~ I ; 0 x

F ( u ) d u dx e - t / ~ ,

t

~ 0

(1.3)

This r e l a t i o n can be w r i t t e n in the form t ; [uw(u)] -I du 0

t -I The

===> NBRUE

t

2 0.

(1.4)

c o r r e s p o n d i n g dual classes NWRUE and HNWRUE,

stands for worse, (i.i) and

~ ~-i ,

where

W

can be d e f i n e d by r e v e r s i n g the i n e q u a l i t i e s in

(1.3), respectively. (NWRUE) = => H N B R U E

It has been shown that NBUE

(NW-UE)

(HNWRUE).

In section 2 the renewal M R L ageing p r o p e r t i e s are shown to be p r e s e r v e d for the s u r v i v a l to

function w h e n a device is s u b j e c t e d

d i s c r e t e shocks that o c c u r a c c o r d i n g to a h o m o g e n e o u s Poisson

processes.

In section 3

the p r e s e r v a t i o n s

of

the

cunumulative

d a m a g e m o d e l s for the renewal M R L p r o p e r t i e s are discussed. statistics

for

and the H N B R U E simulated

t e s t i n g e x p o n e n t i a l i t y v e r s u s the NBRUE

Test

(NWRUE)

(HNWRUE) p r o p e r t i e s are e s t a b l i s h e d in s e c t i o n and

v a l u e s for the p e r c e n t i l e s of the test are calculated.

Some c o m m e n t s and d i s c u s s i o n are p r e s e n t e d in s e c t i o n 5.

2. ~ H O M O G R N R O U 8

In

this

section

d i s t r i b u t i o n H, i.e. H(t) sequence according

of to

shock

POI88ON

8HOCK MODEL

we study the shock model

for

the

life

= 0 for t<0; of a device s u b j e c t e d to a

o c c u r r i n g randomly in

time.

a h o m o g e n e o u s Poisson process with

Shocks constant

arrive mean

731

Sh~k models ~ d ~stmg

value

~.

In particular suppose Pk be the probabilities

surviving the first k shocks, survival

probability

of

not

k = 0,I,... where Pk = 1 - Pk" The

H(t)

= l-H(t) that

the

device

survives

untill time t can be given by

H(t) =

~ e- ~t ( ~t)k - - Pk" k=O k!

(2.1)

Model (2.1) has been considered by Eeary, (1973)

Marshall and

Proschan

where they proved that H(t) inherits the discrete

property of Pk if Pk has IFR, Klefsjo

(1981)

similar

IFRA,

ageing

NBU and NBUE properties.

result has been proved

for

the

In

HNBUE

property. Now

we

consider the shock model (2.1) when the amount

shocks Pk' k=0,1,..,

of

follow the discrete NBRUE, NWRUE, HNBRUE and

HNWRUE properties.

Definition 2.~. A discrete distribution {Pk ) or its survival probability Pk=i=~+iPi , k = 0,1,... where Pi is the probability mass function and T 0 = 1, is said to have NBRUE (NWRUE) property if i

i

m

E E P I ~ (2) ~ E P~, j=k i=j j=k ~

k=0,1 . . . . .

(2.2)

g

where

~=

E~.

j--0 ]"

Deflnlt1~n 2.~.

A discrete

distribution (Pk}

or

its

survival

k--0,1 ....

(2.3)

probability Pk is said to have HNBRUE (HNWRUE) if

PI

~ (2) (1-1/

j=k i Note

PI, i

that the life distribution Pk has both NBRUE (HNBRUE)

and NWRUE (HNWRUE) properties iff Pk = 1-qk, for k = 0,1,... 0 ~ q

and

~ 1, i.e., is geometric. Now

we can prove that the survival function H(t) in

model

(2.1) inherits the discrete NBRUE property of Pk, k=O,1 . . . .

Theorem ~:},~ The survival function H(t) in (2.1) is k-0,1,..,

is discrete NBRUE.

NBRUE if Pk,

732

A . M . ABOUAMMOH et al.

Pzoofz

U s i n g model

(2.1) we have

i

wH =

I H(t)dt 0 e EP

=

1 --

k

k=0 =

° ;

(~t) k

e -~t

k! 0

1

m

m

-

E

Pk

k=0

I

where

a is the mean of the discrete We n e e d

is NBRUE,

to

show that

if

Pk satisfies

relation

(2.2),

then

H

i.e. m

m

m

2 ; H(u)dudx

~ wH

tx

The L.H.S.

d i s t r i b u t i o n Pk"

/ H(u)du,

t

~ 0.

(2.5)

t

of -

(2.5)

is

-_

°

*

®

(~U) k

- k!

I 2 H ( U ) d U d X = I E Pk I t X t k=0 X

e -~/ e -~/ du dx

(by using model

(2.1)),

4

1

- *

k (~)J - - e

=o

=

(by using

-~

j,

=o

dx

(5.4) of Barlow and Proschan, 1981, p.74),

=

1

-

-

~

- (~x) j ;

e -~x

j=0 t 1 -

®

j

E

E

1

.

--

E

(

j!

.

e

E

EP

.

j

~2 j=E0 i=0 .

*

-

~

o -'~u

lj=o

~ e

t

dx

EP_.

( ~x) J k

- - e

-M:

)

j!

( ~)i

)

-~

i! ( ~)J

--Pj j!

m

= - I H(u)du, ~t (by model 2.1), =

~I

2 H(u)du. t

This p r o v e s the theorem,

J

j=k

• _ ( ~x)J E ( , E Pi - - e - ~ t i=0 i=j j!

,

=

® e -~x

%2 i=0 j=i k=j 1 7

~P-. j=k J

( ~x)J

~2 j=O i=0

=

dx

j!

o

For the dual class NWRUE we state the following.

Shock models and testing

T h e o r e m 2.4. The survival if Pk, k=0,1,..,

function H(t)

T h e o r e m 2.5. The survival HNBRUE

in

model

(2.1)

is

NWRUE

has the discrete NWRUE.

Next we c o n s i d e r model

the

733

property

given by relation

(2.1)

for the HNBRUE property.

function H(t) given

if Pk'

k=0,1,..,

in mode

(2.1)

has the discrete

has

HNBRUE

(2.3).

P~OO$: We have found that = -

is

the mean of the life distribution H,

~ = £ - -Pk k=0

where

is

the

mean of the d i s t r i b u t i o n Pk" Now we need to show that ; • ; "H ( u ) d u d x t x

~ e-~t/~

(2.6)

(u)du dx.

;';H

0 x

C o n s i d e r the L. H. S. of

(2.6)

, • • . ,f~u~ k ; I H ( u ) d u dx = E P k 2 I - - e t x k =0 t x k! %

G

~

du dx,

(by using model • E

=

k=0 =

1 - Pk

"k I

~

1

-

-

E

1

-~x

- (%x) j ;

- - e

•

-~x

dx

j! j

(%X) i

E

- - e

--

%2 j=E0 i=0

i!

-%x

EP k k=j

~

E Pk

k=j

1 • -_ ( ~)J = -Z E EP k - - e %2 i=0 j=i k=j j! 1

dx

t j=0 j !

j=0 t =

(~x) J E - - e

(2.1)),

-~t

• ® ® ( %x) J E (i-i/~) i E __EjPI -j!- e i=0 j=0 k

t

(by the HNBURE property of Pk ), Now tion

by changing the order of the summation and using rela-

(3.4)

of

Barlow and Proschan

(2.6) and hence the theorem T h e o r e m ~.7. The survival if the discrete

survival

(1981, p. 74) we get relation

is proved.

function H(t) function Pk,

D in model k=0,1,..,

(2.1)

is

HNWRUE

has the discrete

HNWRUE property. The

Proofg

proof

of

proof can be carried out in parallel

Theorem

2.6.

o

steps

to

the

734

A . M . ABOUAMMOrl et al.

3.

Let that

DI~AGB

CUMULATZ~

MODZL

Pk be the p r o b a b i l i t y of s u r v i v i n g k

shocks,

shocks cause damage and the damages a c c u m u l a t e

suppose

additively.

If the a c c u m u l a t e d damage exceeds a critical t h r e s h o l d the device or the system, withstand random

fails.

shocks

D i f f e r e n t units d i f f e r in t h e i r a b i l i t y to

and

variable.

hence the t h r e s h o l d is a s s u m e d

to

a

Let the t h r e s h o l d has the d i s t r i b u t i o n F w h e r e

F(x) = 0 for x<0.

A s s u m e the damages X1,

X2,...

from s u c c e s s i v e

shocks are i n d e p e n d e n t w i t h life d i s t r i b u t i o n s F1, F2,... also

be

independent

of the t h r e s h o l d X.

Now

the

and the

probability

of

s u r v i v i n g k shocks is P k : P(

k E Xi i=l

~ X),

k = 1,2 ....

(3.1)

T h i s m e a n s that m

Pk = 0/ F I * ' ' ' * F k ( X ) d F ( x ) "

(3.2)

F u r t h e r assume that Xj follows a gamma d i s t r i b u t i o n w i t h d e n s i t y i i-i b a xa fi(x) -

e -bx,

x>0, a i > 0,

(3.3)

F(a i) for i = 1,2,...,

then

(3.1) has the form Ak-i

m

Pk = b

I F(x) 0

e -bx

(bx) F(Ak)

dx,

(3.4)

k w h e r e A k =i~lai , k=l,2 .... Esary et al. k=O,l,.., Fi

(1973)

have

studied

model

and

is d i s c r e t e NBU for all choices of Ei,

is d e c r e a s i n g in i for all x

k 0,

and P r o s c h a n

A k = k and F is DMRL (1981)

have

They,

(NBUE), proved

i=i,2,.., w h e r e Results

(3.4) for the IFR,

in particular, then Pk,

Pk,

in

also c o n s i d e r e d A - H a m e e d

(1973) have c o n s i d e r e d model

and the NBU properties.

showed

iff F is NBU.

this c o n t e x t for IFR and IFRA have been

Klefsjo

(3.1)

IFRA

have p r o v e d that if

k=0,1,..,

the latter result

is D M R L for

(NBUE).

the

HNBUE

property. Next

we

prove

the

same

result

for

the

renewal

mean

r e m a i n i n g classes.

T h e o r e m 3.i. Let the survival p r o b a b i l i t y Pk be g i v e n for k=l,2,..,

and P0=I.

NBRUE

if A k = k a n d

(HNBRUE)

The sequence Pk' F is NBRUE

k=0,1,..,

(HNBRUE).

by

(3.4)

is d i s c r e t e

Shock models and testing

Proof:

We prove

the N B U R E

the t h e o r e m

735

for the w i d e r

c a s e can be p r o v e d

in a s i m i l a r

class

HNBURE

whereas

procedure.

N o w we n e e d

k=0,1,...

(3.5)

to s h o w ®

-

i

r ~ Pi j=k i=j

k

•

E

~ (i - ,) ~ P-., ~ j=O i=j J

Note that m

~=

EP k k=l i

= 1 +

2"~ (x) e_bX ? {b 0 k=l

(bx) k-I -

-

dx},

(k-l) !

(by using model

3.4).

(bx) k-I = 1 + b

I F(x) ~ e -bx 0 k=l

-

dx

-

(k-l) !

Q

= 1 +b

I F(x) dx 0

= 1 + b

uF .

N o w the L.H.S.

of

(3.5)

__~jPl = 9

i =

is

E E b ; ' F ( x ) e -bx 9=k i=9 0 E.

~. e_bX

I . bF(X)

j=k 0

=

(bx) - - i-i (i-l) !

(bxli-i -

i=9

-

dx

dx

(i-l) !

x b J - l u j-2 ~ I bF(x)[/ e-bUdu]dx j=k 0 0 (9-2)! by using

(3.4)

of B a r l o w

,

and P r o s c h a n

(1981,p.74),

= b2

. . bJ-2uJ-2 ; E 0 9=k (9-2)!

--b 2

- u b k - 2 y k-3 ; ; e-bYdy 0 0 (k-2) ! ®

=

b 2

®

I 0

b 2

I

I =

[ ;®/'F(x)dx yu

e -by

[e-Y/PF

k-3 -by (l+i/b ~ - e (k-3) !

(by)

1 = b 2 (i +

-k )

b ~F (l

-

_)k

• I 0

. Z k-I

1 b 2

dy]du

(by) k-3

0

=

[ 2"F(x)dx]dy u

e -by

0 b2

[ I F(x)dx]du u

k-3

(by)

- -

I

e -bu

l®2~(x)dx 0 u • ® I ; F(x)dx 0 u

z k-I

Q

e -z (k-l) ! •

du dy

@

I I F(x)dx 0 u m_

- e -z [ I I F ( x ) d x 0 (k-l) ! 0 u by u s i n g (3.6). I

du]dy

dy]

dy

736

A. M. ABOUAMMOH et al,

Consider

the quantity ®

® ® ® bJ-2uJ-2 E Pk = b2 I E i=j 0 j=0 (j-2) !

j=0

(in p a r a l l e l = b2 Now

by using

satisfied.

to the first four s t e p s of

(3.7)),

I I F ( x ) d x du 0 u

(3.7)

and

(3.8)

(3.8)

the r e q u i r e d

result

(3.5)

is

can

be

[]

Now

we

obtained

® I F ( x ) d x du. u

e -bu

state

the

by r e v e r s i n g

following

all

result whose

inequalities

proof

in the p r o o f

of

Theorem

3.1. 3.2.

Theorem

is d i s c r e t e 4. T E S T I N G

The discrete NWRUE

section

Ho:

F(t)

VERSUS

NBRUE AND HNBRUE

we s t u d y the p r o b l e m

has

=

t

(3.4)

(HNWRUE) .

CLI%SSES

of t e s t i n g I 0,

I > 0

been

or H N B R U E

considered

and Doksum

Hollander

and

(1969)

Proschan

a = NBUE and

distributions ~ t(2 ) Let consider

a

(1985),

random

F with

F(t)

sample

an N B R U E

Therefore

of

~ t <

various

and Pyke

this p r o b l e m

for

this p r o b l e m

I [ I t t

size

observations

life

of d e v i a t i o n

=

for

sample

n

from

(1967)

A = IFR. for

a

=

(1983)

life t n. Let

from F w i t h t(0 ) = 0.

distribution,

then

from e x p o n e n t i a l i t y

®

the

tl, t2,...,

one

of the

may

form

m

I F(u)dudx x

-

, $ F(u)du] t

F(t)dt,

®. N o t e t h a t u n d e r the null h y p o t h e s i s

u n d e r HI: ~ ( F )

This

respectively.

independent

be

a measure

authors

Proschan

considered

~... ~ t(n ) be the o r d e r e d

~(F)

wherease

considered

(1981)

®

0

by d i f f e r e n t

respectively.

& = H N B U E have b e e n s t u d i e d by K l e f s j o

and B a s u a n d E b r a h i m i Consider

properties,

of life d i s t r i b u t i o n s .

and B i c k e l

DMRL.

4,

NBRUE

ageing properties

where

g i v e n by

if A k = k and F is N W R U E

= l - e x p ( - kx),

F(t)

a denotes

problem

t(l )

k=0,1,..,

the alternative HI:

where

(HNWRUE)

EXPONENTIALITY

In t h i s

versus

s u r v i v a l Pk,

H0:

(4.1) ~(Fn)=0

< 0, s i n c e F is a s s u m e d to be c o n t i n u o u s .

one may consider

t(l ) , . . . , t ( n ) to be d i s t i n c t .

®

Since

~(t) =

I F(u)du t

+ F(t),

then relation

(4.1)

can h a v e

the f o r m ®

=

®

I [ I ~(x)F(x)dx 0 t

, F(t) ~(t) ]F(t)dt.

(4.2)

Shock models and testing

TO

find the empirical value

of

737

~(F)

we replace F(t),

~

and

by their sample values

~(t)

Fn(t) = n-l(n-j+l),

t(j_l)

n n = n -I ? t = n -I r t-. j:l (j) j=l j

t

<

t(j),

'

and n ~(t) : (n-j+l) -I. E.(t(i)-t i 1=3 respectively.

), for t(j_l ) < t i ~ t(j)

Hence the empirical value of (4.2) n

n

n

% ( F n ) = n-2{ k=l r (n-k+l)[ j=k E (tfi)-t "J (j-l) ) i__Ej(t(i) -tz)][ t(k) -t(k_l)] n (4.3)

-tj:E1 (n-j+l) 2 (t (j) -t (j_l)) )

where t(j_l ) ~ t i < t(j). Similarly deviation

if

F

is

life

HNRUE

distribution,

then

the

from exponentiality o

m

(F) : I F(t) I I tx t

m

o

F(u)du dx - ~ t / ~ I I F(u)du dx]dt 0x (4.4)

This is equivalent to .

~(F)

=

®

IF(t) 0

®

I ,(x)F(x)dx d t t

®

I ~(x)F(x)dx IF(t) 0 0

e-t/~dt, (4.5)

The empirical values of h(Fn)

= n-21

~(F)

in (4.5) continue the form

n n n E (n-k+l) (t(k)-t(k_l))_.=Ek(t(j)-t(j_l))i_E_.(t(i )-t~j~ k=l 9 =9 n

n

- [j=El(n-j+l ) (t(j)-t(j_l)

i Ej (t(i)-tz)

]

n [ ~ (n-k+l)(t(k)-t(k_l)) k=l

exp(-t(k ) It),

n where t(j_l ) < t i ~ t(j) and t = E ti/n. i=l To

make the test statistics

~

and

~

are scale

invariant

we consider the versions Mi(n )

The

~(Fn)/t

,

i = 1,2.

believed

to

have asymptotic normality under some suitable transformation.

In

order

to

new

=

test statistics Ml(n ) and M2(n ) are

support this belief we present Tables 1 and 2 for

lower and upper percentile points

= = 0.01,

and M2(n ) for n = 2, 5(5), 30(10),

60. Each

the

0.05, 0.i0, of Ml(n ) reported

percentile

738

A . M . ABOUAMMOHet al.

is

the mean

of f o u r r e p e t i t i o n s

the percentiles tends

to

be

for the distribution symmetric

percentiles

are negative

lues

sample

of t h e

for

large values

the

percentiles

that

moves results

size

Table

zero where

Therefore

fact

statistic

Ml(n )

lower

transformation form.

statistics

and

to

Table

M2(n ) .

sample

symmetric.

this trend

Other

size n ~ 8

normality 2

(not r e p o r t e d

for e v e n as l a r g e

presents

It

the

indicates for

small

distribution

here)

sample

upper

for all va-

o f M 2 ( n ) is s k e w e d t o t h e r i g h t

For larger

size

simulated as

I00.

I. C r i t i c a l v a l u e s for N B R U E s t a t i s t i c Percentile

n

the

In

respectively

b e of l i n e a r

for t h e H N B R U E

to be more

of t h e N B R U E

and positive,

of n m i g h t

n<7.

confirm

around

size.

the distribution

sample

for i000 o b s e r v a t i o n s .

0.01

0.05

0.01

Points 0.i0

0.05

0.01

2

-0.8450

-0.1778

-0 0739

0.1141

0.2754

0.9340

5

-0.5664

-0.2666

-0

1397

0 4185

0.7794

1.8076

i0

-0.5471

-0.2261

-0

1400

0 4149

0.6925

1.3230

15

-0.3912

-0.2126

-0 1270

0 3833

0.6194

1.3708

20

-0.3173

-0.1749

-0 1 1 0 6

0 3564

0.5460

1.1901

25

-0.3241

-0.1529

-0

1052

0 3352

0.4857

1.0162

30

-0.3390

-0.1452

-0 0958

0 3026

0.4270

0.8089

40

-0.2453

-0.1391

-0 0 8 6 1

0.2508

0.3552

0.6494

50

-0.1984

-0.1259

-0 0 8 5 1

0.2165

0.3187

0.5493

60

-0.1906

-0.1093

-0 0754

0.1937

0.2810

0.5205

T a b l e 2. C r i t i c a l v a l u e s

I n

I

for H N B R U E s t a t i s t i c

Percentile

Points

0.01

0.05

0.01

0.i0

0.05

0.01

2

0.0000

0.0000

0.0001

0.i109

1.2341

0.7891

5

-0.0050

0 0003

0.0015

0.1296

0.2106

0.5039

I0

-0.0445

-0 0134

-0.0059

0.0792

0.1155

0.2528

15

-0.0595

0 0268

-0.0159

0.0654

0.1018

0.2020

20

-0.0714

-0 0358

-0.0245

0.0554

0.0841

0.1387

25

-0.0933

-0 0469

-0.0290

0.0554

0.0849

0.1311

30

-0.0838

-0 0493

-0.0331

0.0493

0.0755

0.1402

40

-0.0672

-0 0442

-0.0322

0.0497

0.0726

0.1116

50

-0.0732

-0 0 4 6 5

-0.0342

0.0454

0.0671

0.1213

60

-0.0824

-0 0492

-0.0380

0.0379

0.0588

0.0948

739

Sh~k models ~ d ~sting

AQknowledaement:

The

authors

are

grateful

to

Professor

A.

Khalique for his constructive comments on the simulation part.

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IEEE