Cost-benefit analysis of one-server two-unit imperfect switch system with degradation

Mlcroelectron. Reliab., Vol. 25, No. 5, pp. 949-958, 1985. Printed in Great Britain.

0026-2714J8553.00 +.00 © 1985 Pergamon Press Ltd.

COST-BENEFIT ANALYSIS OF ONE-SERVER TWO-U~IT IMPERFECT SWITCH SYSTEM WITH DEGRADATION • M,N. GOPALAN and S.S. WAGHMARE Department of Mathematics, Indian Institute of Technology, Powai, Bombay 400076, India

(Received for Publication 16 April |985)

A BS TRACT This paper deals %~ith the cost-benefit l-server

2-unit system with imperfect

(units/switch)

are not

The successive distributed

after the repair.

repair times of each item are arbitrarily

w]tile the successive

ing some salvage value.

failure rates are-constants,

repairable

any salvage value.

incurr-

The switch

and is scrapped only when both the units

The expected revenue,

due to different

it is discarded

After a fixed number of failures

the unit is scrapped without

are scrapped.

of

switch where the items

'as good as new'

If repair of a unit is not possible

is alWayS

analysis

expected

busy period

types of repairs of units/swltch,

expected

salvage value and expected number of repair completions t h e switch, cos t-benefit I.

all in [ O , t ~ ,

are obtained

to carry out the

analysis.

INTRODUCTION Gopalan

and Waghmare

[ 1,2,~ 3 have recently discussed

some exten.~ions of l-server switch. also,

The assumptions

are as follows (a)

f~al

of

2-unit system with imperfect

made, qulte common in literature

•

A failed unit is repairable

i.e. there are no

failures. (b)

After repair of an item is over,

as new' thereby implying s ~ e

constant

it is 'as .good

repair r a t e and

same repair time density throughout. Situations

may arise where the above assumptions

are

M

950 not Justiflable-

N GOPALAN and S S W A G H M A ~

The purpose of this paper is to carry out

the cost-beneflt analysis in such cases. The ~ d e l

is as

fol lows : Upon failure of a unit an instantaneous decides possibility of the repair. and the number of the repairs

i, a fixed number,

inspection

If the repair is possible

completed so far, is less than

it is sent for the repair.

If the unit

is not repairable it is d i s c a r d e d incurring some s a l v a g e value.

After

(l+l)-th failure it is scrapped.

The switch

is always repairable and it is scrapped only when both the units are scrapped. 2.

NOTATIONS

M = U S ~,i

if the item under o o n s i d e r a t i o n is a unit if the item under eonsideration is the switch failure rate of the item after

(i-])th r e p a i r

comp i etio n gM, i(o),

GM, I(.)

pdf

and

sf

of the r e p a i r time of the

item after i-th failure C i (u)

revenue due to a unit with failure rate in an uninterrupted interv~l of length

Pl

AU, i u

probability that a unit is repairable after i-th failure

(.,.,o)

status of a unit,

another unit and that of the

switch respectively W

'working'

UR

condition of an item

'under repair'

WR

condition of an item

'waits for repair' condition of an item 'scrapped' condition of an item

S

'standby' condition of a u ~ t

SUi # ~S I

salvage value of the unit and the switch after i-th failure indicator function of a set A

IA

The S t a t e Space O

( W, S ,

W )

1

CuR,

. )

,

w,

951

Reliability analysis 2

(--,

W,

W )

S

{w,

S, U R }

4

(UR, Wa.

W )

5

(UR, --.

W )

6

(UR,

7

(--,--,-

8

(--,

w, UR )

9

(WR,

S, UR )

10

(--,

S, UR )

II

(UR, WR, W R )

12

(UR, --, WR )

13

(-- , WR, U R )

W, WR ) )

expected revenue in [ O,u ] when the

Rm (i, J,k,u)

initial state is m

and i,Jok are the

number of the repairs completed of the items described by the state m order,

in that

m = I,Z,°..,15, e.g. Ro(i,J,k,u)

denotes the expected revenue in [ O,u 3 when the initial state is

0

and the

number of repairs completed of the operating unit, standby unit and the switch are i, j,k respectively S m (i. J ,k,u)

expected salvage value incurred in ~ O , u 3

BM (i, j,k,U) mer

expected busy period in [ O , t ] of the server due to r-th type repair of the item

~.

E~ECTED ~ N U m With

M

I~

ql - - 1 - P i "

with the initial state

m.

[O,t3 A = [i +I_<13

~:

and

ZJ+I_~13

we have No(i,J,k,t)

-

Ci+ l (t)e-(~u'l÷l+ ~N,k+1 )t

t + 1%,i~a O

e

-(~u,i~+

~S'k+l)u [

÷ IA P i ÷ l ~ (i,J,k,t-u) ÷ ~i+IIA + IAC) R ~(i,j,k,t-a)

du

ci+ 1

(u)

952

M.N. t

GOPALANand

S. S. WAGHMARE

- (Xs " + 1 +

)u

÷ R~ (i,J,k~t-u) ]du =

cj+, (t) %.,+I (t) e- ~ ' * e t t ~ . j . + kSek.)U[c + f g u , i ~ (u)

J+1 (u)

o

+ RO (Je£*Z.k,t-u)]du t

+/o gu, i ~ (u) (I -

•~%S,k+lu)•-~u*j÷lu [Cj+1 (u)

+ R3(J,i+l,k.t-~1)] du + $tgu.i+l (u)e-XS'k+ZU f u XU.J+1e-~U"j+lw [Cj+~w; O O + Is P j d ~

(J,i+l,k,t=tl)

+(qjdIB + IBC) a2(J,i+l,k,t-u)]dwdu

÷['o

°su

-~U,j +1v

÷ ~O %,I+1 (u) O Sek÷l

-

e_~S,k÷lv)Cj+I

SO %,j+le

Cj+1 (w)dw

3

R3(l,J,k,t) - ci+ i (t) QS,k÷I (t) e- ~ u , i + I t u

+ IA Pi+1 R1 (i,J.k~l,t-u) +(qi+1 IA+ IAc)R2 (i,J.k+l,t-u) ]dvdu t -~. i+lU[C + Io gS, k+l (u) e i÷i (u) + % (i.J.k+1,t-u)]du R2Ci,J,k,t) = Cj+I (t)e (~U'J+I+ ~kS,k+l)t

t

Rg(l'J'k't) " S gS,k+l (u)"I%1(i,J,k+l,t-u)du O

Reliability

analysis

953

t

RlO(laJakat) m f gSak~l (u) RR(i#Jak~lat-U)~ 0

(laJ#k#t) = Cj~ (t)~S#k~1 (t) •-~'j+It t #. .-%.j+l" + I gs.k+l (u) ~ "~.j+1 [cj~ (v~

0

f IB PJ+I R6(J*i#k+l*t-u)]dv du t j~S, k+l u = I gU#i+l (u) RE(J#i+l#k#t-u)dU O

R5(i,j#k#t)

÷ /t gu,i÷l (u) (i-e-~s" k+l u) ~ o ¢ J , l + l # k , t - u ) d u 0 This system of equations can be solved by Laplace Transform Technique (LTT) for 4.

Ro(O#O,O,t)-

EXPECTED SJ~LVAGE VAI]JE IN

[-0# t 3

t - (kU,i+l ÷ XS, k+l) u[i So(i#J#k,t) = 1 % , i + I e A PI+ISI (i,J,k,t-~1) 0

+(IAqi+l÷ IAC) (SUI+I+ S E (i#J,k,t-~))~du t - (~'k+l + kUoi+l) uS 3 (i#J#k#t-u) du + I AS,k+le O S 1 (i,J,k#t)

= ~

0

t gu#i+l (U) e- (~U, J+l + XS#k+l )S O (J,i+l ,k#t-u) du

+ f t gu,i+l (u) (l--e--(~" k+lU) e-~U#J+lUs~ (J#i÷l#k,t-u) du O t + 1%.i+i 0

(u) (I~-

~ # j+l u) e-~.~#k+l u[i

X SI (J,i+l,ktt--u)+(IBqj+l+ IBC ) (SUj+1 + SE(J,i÷l#k , t-u) 3du u (u) O~ ~S,k+l "~S*k+IV(l- e-~U#J+IV)dv t + 1%,i+i 0

u (ul 1%,j+I 0

"

--~U, j+iv

(i -

jkS,k~IV

X EIB PJ+I Sg'(J'i+1#k#t'~) ÷ (IB qJ+l + IBC)

ldv

B Pj+I

954

M. N. GOPALAN and S. S. W A G H M A R E

X (SUj+1+ SIo(J, i+l, k, t-u))~dU S2 (l.J,ket)

= ~ t J k U t j + l u ( l - e "AS'k+lu) S 8 ( i , J e k . 0

t - u ) du

+ft -~U, J+l u e-~Sek+l u 0 kU~J+le [IB PJ+I X Ss(i,j,k , t-u) + (IB qj÷l+ IBC) X (SUj+1 + SS k ) ~du t

+ ~ gS, k+1 0

(u)

e-~u" I+I u

S O (i,J,k+1,t-u)du

t %,i÷i u S~(i,J,k,t) = I gS#k+l (u) J SO (1,J,k+l,t-u)du 0 t -~U,i ÷1u ) +f (u) (l-e 0 gse k+l

BPj, 1 S 1 (i,J,k+l,t-u)

+(I B qj+l + I c) (SUj+I+ SE(i,J,k+l,t-u))Jdu B t

S 8 (i,J,k,t)

=

I gStk+1 (u%.(l-e-~'j+lu) O

B

Pj÷iSs(J,l,k+1, t'u~.

+(IB qj+l + IBC ) (SSj÷1 ÷ SSk) ~du t -~U, J+l u) S2(J,i÷l,k,t-u)du Ss(i,J,k,t) = f gu,i+1 (u) ( l - e O t -~S'k+lU) (j,i+1,k, t-u) du +I gu,i+l (u) (l-e $10 O t S9(i,J,k,t) = IO gsS k+l (u) S 1 (i,J,k÷l,t-u)du t SIo(i'J'k't)= ~ gS,k+1 (u) S2(i,J,k+l,t-u)du O This system of eouations can be solved by L~f for So(O#O, Oot). 5.

EXPECTED BUSY PERIOD DUE TO r-TH TYPE OF REPAIR OF AN ITEM

t AVo~4er(l*Jek't) = 0~ ~U. l+le (~U'i+I+ ~S'k+l)U E IA PI+I

X AV~r(i,J,k,t-u) + (qi+l IA + I c ) A

X AV~r(i,J,k,t-u)~du

955

Reliability analysis

÷~tAs,k+le- 0~U*i÷I+ kS"k÷l )UAvBM2er(i.J,k,t-u)du O

AVer ~, J ,k,t) " ~ , i+l (t)x [M-u.

=i+i3

+~tgu,i+1 (u)j (~U,j+l +kS'k+l)%l BM (J,i+1,k,t-~)dU O AVo •r

t

+I gu,i+l (u) (l-e-~u'

j+lu)e~SOk+lu[pj+I xB

O

X AVl~4r(j,i+l,k,t-u) ÷ (qJ÷l IB + IBc) X Av~j.i+l.k~t--u) ] du +It ~,i+l (u) (1-e-~S,R+lu) e-~U,i+lUAv~ J,l+l,k,t-u'du O t u -~S,k+lv (l-e-~U" J+lv)dv +[O~ gu,i+l (u)~ ~,k~le t u -~U. +O~ gu,i+l (U)o~ ~,J+l e J+IV (I e-~S'k+IV)dv~

X~pj+llB Av~(Joi+l~kot-u)+(IBqj+l+ IBc)

BM

X AVlo.~, i+l, k, t-u) du m

AVer (i,j,k,t) - I[M= S, r = k+l 3 GS'k+I (t) t j~U •i+lU BM +~O gS,k+l (u) AVo, r (i,j,k,t-u)du t +~ O gs,k+l (u) (I- e-~'U'i+lu)[PJ+I IA BM (i,j,k+l,t-u) + (qi+lIA+ IAC) X AV l,r X AVer (i,J~ k+l e t-u)~du BM r (i,J,k,t) Ave,

du O m

AV r(i'J'k't)= I~M = S. r = k+l~ GS'k+l(t)

956

M . N . CyOPALANand S. S. WAGHMARE

t Av~oMr (i, J ,k+l ,t-U) ~u + I gsok+l (u) O BM (i,J,k,t)-

I

A~8. r

[M=s.

~=k+i]

GS°k+I (t)

t +f gS,k+l (u) (l-e-)'u°j +lu) Pj ÷I IBAV~ M (j,i ,k+l, t--u5du O

AV~r(i'J'k't)=

I[M = U. r = i÷1 ~ %,I+I (t) t jAS, k+l u BM (j,i+l ,k,t-u) dU +~ gu,i*l (u) AV2, r O t (l_e-~ S °k+l u) +f guoi÷l (u) AviE~Oo~J,i+l°k.t-u)du 0

AV

'r(i'J'k't)= I E M

-- S, r = k+l

t +I gS,k÷l (uSAv~M-(i'J" k+i, t-u)du O

The LTT

~0~

can be applied to obtain

t ~: (i,j,k,t) = f Av ~'~ (u) du ~mor O m°r

6.

EXPECTED NUMBER OF REPAIR COMPLETION~ IN [ O,t OF 'DHE SWITCH With

A = Ei + 1 < 13

and

B = [ j + 1 < I']

t =

[ IA

e

0

+ IAc (~(i,j,k,t-~/) ]du t - (~S,k+l +~U,I+I )u S + I ~S,k~l e ~ (i, ~,k,t-u%,du 0

t

- ¢%, j+i +~, k÷1 ~u ~

~ (i, J,k#t) = I gu,l÷l (u) e O

(j,i+l,k,t-u} du

t +I gu#i+l (u%'e-~S#k+lU (I- e -Au'j+lu) 0

Reliability analysis

957

X[IB(PJ+I

~ (..J,i+1,k,t-U)+ qj+1 "~ (J,i+1,k,t-u))

+ IBC ~

(J,i+l,k, t-u) ]du

-~S'k+lU)e-~U#~+IU~(J,i+1,k,t-u)du t + O~gU•i+i (U) (I--e t 0

" " rUA --~' k+lV (I-e-~U" j+Iv ..,e ) 0 S#kv±

+ I gu,i+l ~ujj

XEIB(Pj+1 ~(j.i+l,k,t-u)+ qj+l ~So(j'i+l'k't-u)) + IBc ~ 0 (j' i+l. E, t-u) ]du t u -~U, j+l v --XS,k+iV + I gU,i+l (u) I ~,j+l e (l-e )

0

0

S X[IB(Pj+I ~%(j,i+1,k,t-u)+qj+l~10(J#i+l,k,t-~) S ÷ IBc ~i0 (j,i+l, k# t-u)|du

~(i.J,k,t) = It gS,k+l (u) (i- e -XU'i +lu ) [ IA( "I 0 S

+ PJ+I |L~(i,J.k+l,t-u)+ qj+l~2(i,J,k+l.t-u)) + IAc(l + ~.~(i,J, k*l, t-u.%)]du + It gS.k+l (u)e-~U' i+lUEl + ~0S(isj,k+let_u)]d u 0 ~(i#J,k#t)

[t e--~U'j+lu(1 e-~'&'k+lU) q(i,~#k#t-u)du --"0 ~8 t --hU.j ÷iu --~&#k+l u +f ~, j+le e IB (qj+l 0 ,s

+ PJ+I ~(i,J. k+l# t-uJdu ~(i,j,k,t) = I gS,k÷l (u) 0

÷ ~ (l,j,k+l,t-u) du

t 0

=/t 0 gu, j~4

(u) e- s'k*lu ~ (J,i+1,k, t-u)du

958

M.N. GOPALA.~ and S. S. WAGHMARE t ,. -~S,k+l u. S +f gu,i~4 (U) ~ 1 - - e )~lo(J,i+l, k,t-~)du 0 ~OS (O,O,O,t) •

The LTT can be applied to obtain

7°

COST--BENEFIT ANALYSIS Expected net gain 1

= R0(O,O,O,t) + So(O,C.O.t) -

where

m Z

~o~ (t)

Csm I d

Cuel, CSw i~

-1

= Z % , i ~0~ (t)

I cost per.~ unit

i-th t},pe repair time

of unit, and switch respectively S m = ~0 (O,O,O,t). REFEREN CES M.N. GOPALAN

and

S.S. WAGH~L%RE, Cost-beneflt Analysis of

l-server 2-unit Imperfect Switch System with Multistage Repairs (Accepted for Publication in Microelectronics and Rellab.) • M.N. GOPALAN

and

S.S. WAGHMARE,

Cost-benefit Annlysis of

1-server P.-unit Imperfect Switch System wi~h Delayed Repair (Accepted for Publication in Microelectronics and Reliab.). M.N° GOPALAN

and

5.S. WAGHMARE, Cost-benefit Analysis of

l-server b-unit Replacement System with Imperfect Switch. (Accepted for Publication in M/croelectronics and Reliab° ),