Reliability and maintainability of a multicomponent series-parallel system under several repair disciplines

Microelectro,. Reliah., Vol. 22. No. 6. pp. 1135-1153. 1982.

0026-2714/82/061135-191;03.00/0 Pergamon PressLtd.

Printed in Great Britain.

RELIABILITY

AND MAINTAINABILITY SYSTEM

OF A MULTICOMPONENT

UNDER SEVERAL

REPAIR

SERIES-PARALLEL

DISCIPLINES

MASANOR I KODAMA Faculty

of E c o n o m i c s ,

Kyushu

University,

FUKUOKA, JAPAN ISAO

SAWA

F a c u l t y of Engineering, Kansai

University, OSAKA, JAPAN

( R e c e i v e d for publication Ist M a r c h

198Z)

ABSTRACT This paper considers

a system consisting

series with a single repair facility. consisting connected

of two subsystems

One subsystem is K-out-of-N:G

of N identical units, while the other consists in series.

having simultaneous distributed

The life-tlmes

connected in system

of M different units

of the active units depend on each other in

failure of all the operating units and repair times are

quite generally.

Initially

all the units are operating.

The

system breaks down if more than (N-K+1) units in the parallel group are simultaneously group.

in the failed states or if any failure occures in the series

The availability

repair disciplines to deduce

and reliability

are obtained simultaneously.

the reliability

for the steady-state

of the steady-state

We use a suitable

from the availability

availability

availability

function.

transformation

Explict expressions

of the system and the mean time to system

failure under several repair disciplines

I.

function of the system under several

are obtained.

Finally some properties

for each repair discipline

are given.

INTRODUCTION In this paper we consider a multicomponent

S O and S 1 in series.

Subsystem

S o is K-out-of

system composed of two subsystem -N:G system

(more than N-K+1 must

fail for S O to fail), while subsystem S 1 consists of M different units connected in series

(the failure of any one cause S 1 to fail ).

units depend on each other in having simultaneous and repair times are distributed

quite generally. 1135

The llfe-times

of the active

failure of all the operating units Kulshrestha

[1,2] has studied

1136

MASANORI KODAMAand ISAO SAWA

the model for K=I under the assumption only when the system stops operating mutually independent.

Nakamichi

for K=I with exponential

that the repair of any unit is possible

and the life-times

et al. [3] and Kodama

failure distribution

For system presented here we derive Laplace

(see Section 2).

for each repair discipline

2.

DEFINITION

[4] have studied

availability

Finally some properties

availability,

under several repair

of steady-state

availability

are given.

OF MODELS AND NOTATIONS

Independent Poisson processes and Z(t;%*) govern the occurence with a parameter

%).

Zl(t;%o),..-,ZN(t;%o),Zl(t;ll),-..,ZM(t;%

of shock.

Zi(t;lj)

(I~j~M), and events in the process Z(t;%*)

are shocks

o

to unit i only in

to unit j only in S 1

are shocks to all operating units.

Each unit results in failure after receiving only one "shock". down if all the units in S

M)

(By Z(t;%), we meana Poisson process

Events in the process Zi(t;% o) are shocks

S o (l~i~N), and events in the process

in S 1 occurs.

the model

under the different repair policies.

transform of pointwise

mean time to system failure and steady-state disciplines

of the active units are

are simultaneously

The system breaks

in a failed state or if any failure

The repair of any unit is possible at a single service channel only

when the system stops operating,

and then we consider

the following

three repair

policies

and call the repair policies model i, model 2 and model 3, respectively:

Model 1

When the system stops operating by the stop of S , (N-K+1) units in S are o o

repaired in the batch, and when the system stops operating

by the failure of the

jth unit in S I, only the jth unit is repaired. Model 2

This model differs from model 1 in fact that only one unit is repaired

when the system stops operating by the stop of S immediately

o

and the system begin to operate

after its repair and the repair of residual failed units in S

o

is

interrupted until the next system failure by the stop of S . o Model 3

This model differs from model 1 in fact that the failed unit in S 1 and

all the units under failure in S

o

are repaired in the batch when the system stops

operating by the stop of S I. Through these models,

(N-K+I) units in S

o

and all the units in SI, one unit in S o

and all the units in S I and N units in S o and all the units in S I are repaired in model 1,model 2 and model 3, respectively when the system stops operating by the simultaneous

failure.

The repaired unit is assumed

and repair times are distributed

quite generally.

the residual operating units do not fail.

to behave like new after repair When the system stops operating,

The system is always in one of the

Reliability repair analysis

following states

I 137

: EM--- all the units in S o and S I are operative; E.-- i units in 1

S o and a l l

the units

in S1 a r e o p e r a t i v e ,

l~i~N-1;

FK_ 1 -

(N-K+I) u n i t s

i n S o have

failed and the system is undergoing repair; F..-- i units in S are operative, ij o the system s t o p s

operating

by t h e f a i l u r e

of the jth

repair,K~i~N,l~j~M; . . . .

Fo--all

undergoing repair.

We n o t e t h a t E. means t h a t

that the system is failed. Through t h i s ~=

unit

and

i n S 1 and i s u n d e r g o i n g

the units in So and S I have failed and the system is

Initially,

the system is operating

and F

means

all the units are operating.

p a p e r we u s e t h e f o l l o w i n g

notations;

M ~X. j=l 3

f(t)

is

the density

function

of repair

t f (t)=p(t)exp[-f0~(x)dx] fj(t)

is

the density

is

the density

when t h e s y s t e m s t o p s

i n So,

;

function

of repair

t fj(t)=~j(t)exp[-f0~j(x)dx], gi+M(t)

t i m e o f (N-K+I) u n i t s

time of jth

unit

i n S1,

M K~j~M, f * ( t ) = j ~ l X j f j ( t l / X . =

function operating

of repair

time o f i u n i t s

by t h e s i m u l t a n e o u s

failure

; i n S o and M u n i t s

in S1

,

t

gi+M(t)=~i+M(t)exp [-f0Ui+M(X)dx] , K
t fij(t)=Dij(t)exp[-f0~ij(x)dx]

K ,Kj,Li+ M ,K°

and Kij are the first moment

gi+M(t),fo(t) Pi(t)=P{

and fij(t) respectively

(mean repair time ) of f(t),fj(t),

;

the system is in state E i at time t }, K~i~N;

PK_I(t)=P{

the system is in state FK_ I at time t };

PK_l(t,x)dx=P{

the system is in state FK_ I at time t.and elapsed repair time lies

between x and x+dx }7

P..(t)=P{ z3

, 0~i~N-K, ljj~M ;

t PK_l(t)=f0PK_l(t,x)dx;

the system is in state F.. at time t }; 10

P..(t,x)dx=P{ the system iN in state F.. at time t and elapsed repair time lies 13 13 between x and x+dx },

Qo(t)=P{

the system is in state F ° at time t and elapsed repair time lies

between x and x+dx }, F

K~i~N, I~j~M ;

the system is in state F ° at time t };

Qo(t,x)dx=P{

MR 22:6.

t Pij(t)=f0Pij(t,x)dx,

, Qo(t)=* f0tQ°(t'x)dx;

1138

MASANORI KODAMA and ISAO SAWA

M(s) is the Laplace transform of M(t), M(s)=f0 e-StM(t)dt; Pi = lim Pi(t) ; t'-~ Qo = lira Q o ( t ) t_~-,

;

P (t)

is point-wise

PA j

is steady-state

RJ(t)

is system reliability

i

availability availability

(l__
*

of m o d e l j (l_
of m o d e l j (l=
NTSF j i s mean t i m e t o t h e f i r s t

3.

of m o d e l j

system failure

of model j (l~j_<3 j = 2 * ) ;

Model i The analysis crucially depends on the method of supplementary variables,and

the supplementary variable x denotes the elapsed time that a unit has been undergoing repair.

Viewing the nature of this model, we obtain the following set of

differential-difference equations: ,

t

M

t

t

,

,

[d/dt+N%o+~+~ ]PN (t)= /0~(X)PK-I (t'x)dx + j=l[/0~j(x)PNj(t'x)dx +]0~N+M(X)Qo(t,x)dx, (3.1) ,

M t [d/dt+i~o+~+~ ]Pi(t)=(i+l)~oPi+l(t) + ~ f U.(x)Pij(t,x)dx, K
(3.2)

[~/~t+~/~x+~(x)]PK_l(t,x)= 0,

(3.3)

[8/~t+~/~x+uj(x)]Pij(t,x)= 0,

K~i~N, I~j~M,

(3.4)

[~/~t+~/~X+~N+M(X)]Qo(t,x)= 0. Equations

(3.1)-(3.5)

are

to s o l v e d

(3.5) subject

to the following

b o u n d a r y and i n i t i a l

conditions: PK_I(t,0)=KXoPK(t),

(3.6)

Pij (t,0)=%jP i(t), , , N Qo(t,0)=X [ ei(t), i=K

(3.7)

PN(0)= i.

(3.8) (3.9)

Taking the Laplace transform of equations (3.1)-(3.8) under the initial condition (3.9), we obtain after some manipulation and simplification: x PK_I (s ,x) =KloPK(S) exp [-sx-f 0~ (t) dt ] ,

(3.10)

x , Pij (s ,x) =%jPi (s) exp [-sx-f 0~N+M (t) dt],

(3.11)

_. , N X , Qo(S,X) =l i~KPi (s) exp [-sx-J0 DN+M(t) d t] ,

(3.12)

Reliability repair analysis

1139

, , N [N+r(s)]loPN(S)=Klog(S)PK(S) + X gN+M(S)i~KPi(s ) + i,

(3.13)

[i+r (s) ]loP i (s)= (i+l) loPi+ I (s),

(3.14)

where r(s)=[s+l(l-f*(s))+l*]/I °

(3.15)

From (3.14) we have

Pi(s)= [(i+l) / (i+r (s)) ] [(i+2) / (i+l+r (s)) ] ''' [N/(N-l+r(s)) ]PN(S) = [N/(ih(s ;i,N-1)) ]PN(S)= [N/h(s ;K,N-1) ] [h(s ;K,i-1)/i]PN(S),

(3.16)

where

J h(s;i,j)= H [m+r(s)]/m, K
(3.17)

Next ,we shall use Lemma 1 to obtain the Laplace transform pl(s) of pointwise availability PAl(t). 4 l+r(s) ~ h(s;i,m-1)/m m=i The proof i s given by i n d u c t i o n on the number j .

Lemma 1

h(s;i,j)=

(3.18)

By Lemma 1 and (3.15), we have loh(0;i'J)=Xo+l* i h(O;i,m-l)/m m=i

,

M

* J

(3.19)

~oh(0;i,J)=(l+m~llmK)m~ih(0;i,m-l)/m+I*

i

h

'

(0;i,m-l)/m ,

m=i where h (0;i,j)=[

(3.20)

(s;i,J)]s=0 .

Using (3.10)-(3.13),(3.16) and Lemma i, we have PN(S) = [r(s)h(s;K,N)/(N+r(s))]{lor(s)[h(s;K,N)-f(s)]-I gN+M(S)[N(s;K,N)-I]}-I (3.21) FK_I(S)=/0FK_I(S,x)dx=KXo(I-[(s))/s =NXo[(I-f(s))/S]PN(S)/h(s;K,N-I) ,

(3.22)

Pij(s)=/0Pij(s,x)dx=ljPi(s)(l-fj(s))/s =NIj[(I-fj(s))/s][h(s;K,i-I)/i]PN(S)/N(s;K,N-I),

(3.23)

_, ~_, , N _ _, Qo(s)=f0Qo(S,x)dx=l m~KPm(S)(l-gN+M(S))/s , _, N =NI [(l-gN+M(S)/S][ ~ h(s;K,m-l)/m]PN(S)/h(s;K,N ) m=K =NI [(l-gN+M(S)/s][h(s;K,N)-l)/r(s)]PN(S)/h(s;K,N), gN+M(S)[h(s;K,N)-l]} -i . A$1(S)=m~KPi(s)=[h (s;K,N)-l]{lor(s)[h(s;K,N)-f(s)]-I *-*

(3.24) (3.25)

Using Tauberian theorem, L'Hospital's rule, Lemma 1,(3.19)-(3.24) and (3.25), we have PN =

lim SPN(S)= lim [h(s;K,N)/(N+r(s))]{S[Xo(h(s;K,N)-f(s)) s÷O s-TO

1140

MASANOR[ KODAMA and ISAO SAWA

,_, N -I gN+M(S) [ h(s;K,m-ll/m] -I) m=K , , , M , , , =I h(0;K,N)(Io(N+r(0))[K I +(l+j=l [ I.K.j 3 +I L N+M )(h(0;K,N)-I)]} -I, PK-I= s+01imSPK_I(S)=[NIoK

/h(0;K,N-I)]PN,

(3.27)

Pij = lira sPij (s)=[NIjKj/h(0;K,N-1)] [h(O;K,i-l)/i]p N , s-+0 N

M

,

M

(3,26)

(3.28)

,

~ P..=N(I /I ) ~ X.K.[h(O;K,N)-l)/h(0;K,N-1)]p N o j=l j j i=K j=l lj

(3.29)

Qo = lim SQo(S)=NIo~+M[h(0;K,N)-I)/h(0;K)N-I)]PN, s-+0 1 -I N * PA = lim SPA(S)= ~ pm=lo(N+r(0)){[h(0;K,N)-l]/[l h(0;K,N)]}pN s÷0 m=K

(3.30)

=[h(0;K,N)-I)][K

M* * * * * -i X +(i+ ~ X.K.+I ~.+.)(h(0;K,N)-I)] . i=l j 3 m m N

It is easily seen that

(3.31)

M

[ ~IPiJ+Q; +PK_I+PI = 1. i=K j

The Laplace transform of reliability function Rl(s) can be obtain from P~(s) by _, -* making suitable transformations. Putting f (s)=0)f(s)=O and gN+M(S)=0 in (3.25) yields Rl(s) since the substitution is equivalent

to the assertion that the

probability of the system moving from down state to up state is zero. if we set s=0 in Rl(s), we obtain MTSF I.

Hence we have

N N , N N RI(s)= [ (i/iX o) n.[mlo/(S+mXo+X+A )1= [ [ [Ajm/(S+Jlo+l+l i=K m=l m=K j--m N

,

N

N N ~ ~ AjmeXp[-(jlo+X+X m=K j--m N

Ajm =

)] ,

(3.32)

,

MTSF I= [ [i/(iko+X+X ) H.[mXo/(mlo+X+l i=K m=1 Rl(t) =

Horeover,

(3.33)

)] ,

(3.34)

)t],

N

H r/ H (r-j). r--m+l r--m+l

Remark i.

Noting that lim, [h(0;K,j)-l]/l* =[d---,h(0;K,j)]l*_0 X ÷0 dl

= i [i/i~ o] ) we

i=K

obtain the following results when I =0. * Pi=MN { [i+

M * * * -i _~lljKj+~lo K ] } , K_
(3.35)

, , M P K _ I = ~ K IO[i+ j=l I I.K*. M~I* *] -i j j +--l°K N

M

M =

i=K j

j

(3.36)

M [i+ ~ I.K.+M..I K ]

j=l 3 J m

M 1 * * * -i PA = [l+j__llXjKj+MNXoK ] '

o

,

(3.37)

(3.38)

Reliability repair analysis

4.

1141

Qo = O,

(3.39)

N where 1 / ~ = [ i/i. i=K

(3.40)

Model 2 Viewing the nature of this model, we obtain the following set of differential-

difference equations: M t [d/dt+N~ +~+~ ]P_(t)= ~ f ~.(x)e_.(t,x)dx, o m j=l 0 3 m3 M t [d/dt+i~ +~+X*]Pi(t)=(i+l)~oPi+l(t) + ~ f ~.(x)Po.(t,x)dx, K+I
(4.1) (4.2)

, t [d/dt+KXo+~+~ ]PK(t)=(K+I)XoPK+I(t) + f0~o(X)PK_l(t,x)dx M

t

,t

,

,

+ j~i/0~j(X)PK_l(t,x)dx + ]0~K+M(X)Qo(t,x)dx,

(4.4)

[3/~t+~/~X+Uo(X)]PK_l(t,x) = 0, [~/~t+~/~x+~j(x)]Pij(t,x) = 0,

(4.3)

K~i~N , I~j~M,

(4.5)

[3/~t+~/~X+~K+M(X)]Qo(t,x) = 0,

(4.6)

PK_I(t,0) = KloPK(t),

(4.7)

Pij(t'0) = AjPi(t), Qo(t,0) = ~

K
(4.8)

N ~ Pi(t), i=K

(4.9)

PN(0) = i.

(4.10)

Taking the Laplace transform of equations (4.1)-(4.9) under the initial condition (4.10) and solvinE them, we obtain from (3.15) x PK_l (s ,x)=K~oPK (s) exp [-sx-/0~o (t) at ] ,

(4.11)

x Pij (s,x)=ljP i(s)exp [-sx-f0~ j (z)dt] ,

(4.12)

_, , N x, Qo(S,X)=% [ Pi (s) exp [-sx-/0~K+M(t) dt], i=K

(4.13)

~N(S ) = i/[N+r(S)lo] '

(4.14)

[i+r(s)]lo~i(s ) = (i+l)loPi+l(S), _ [K+r(s)]~oPK(S)

(4.15)

K+I
_

,_,

N

_

= (K+l)~oeK+l(S)+K~ofo(S)PK(S)+~ gK+M (s) ~ Pi(s) • i=K

(4.16)

From (4.15) we have Pi(s) = NPN(S)/[ih(s;i,N-l)]=[N/h(s;K,N-l)][h(s;K,i-l)/i]PN (s) =h(s;K,i-l)/[i%oh(S;K,N)] ,

K+I~i~N-I,

(4.17)

1142

MASANORIKODAMAand ISAOSAWA

PK(s)={r(s)[r~(s)+K%~f~(s)]+K%*{~+M(s)[h(s;K~N)-~]}{r(s)[K%~h(s;K,N)r~(s)]}-~(4.~8) where rl(s)

=

s+K%o[l-fo(S)]+%[l-f*(s)]+%*

[ l - g-* K+M(S)],

(4.19)

Using (4.11)-(4.13),(4.17) and (4.18), we have =o

PK-I(S) =f0 P(s,x)dx = K%oPK(S)(I-fo(S))/s ,

(4.20)

Pij(s) = %jPi(s)(l-fj(s))/s = [%j(l-fj(s))/s]h(s;K,i-l)[iloh(S;K,N)],

(4.21)

-* k* N _, Qo(S) = [ Pi(sl(l-gN+M(S))/s i=K ,

_,

N

= ~ [(I-gN+M(S))/s]{PK(S)+

P~(s) =

[ {h(s;K,i-l)/[iloh(S;K,N)]}} i=K+l

N N [ Pi(s) = PK(S)+ ~ {h(s;K,i-ll/[iloh(S;K,N)l} i=K i=K+l

,

(4.22)

•

(4.23)

Using Tauberian theorem , L'Hospital'rule,(3.19) and (3.20), we obtain the steadystate probabilities N [K~o+ KI* [ h(0;K,i-l)/i]/[Kloh(0;K,N)] i=K , M * * * -i =[l+K~ K + ~ %.K.+I e_._] , o o j=l ] 3 ~

eK = lim SPK(S)= s+0

--

lim[s/rl(s)] s~0 (424)

*

PK-I = s÷01imSPK_I(S)= K~oKoPK,

(4.25)

Pi = lim sPi(s) = 0 , K+I
Qo = lim SQo(S) = I ~ + ~ K s-~0 It is e a s i l y seen that

(4.26)

K+I
(X)Pmj(t,x)dx,

n
(4.33)

K+l~n
(4.34)

,

,t , , [d/dt+n%o+A+E ]Pn(t)= (n+l)loPn+l(t)+ ]0~n+M(X)Qo(t,x) dx M t +j!if0~ j (X)Pnj (t,x)dx, ,

M

[d/dt+i%o+%+% ]Pi(t) = (i+l)%oPi+l(t) +

t

I / ~.(x)P..(t,x)dx, i~m,n, K
[~/3t+3/~x+u(m)(x)]PK_l(t,x) = 0,

(4.36)

[8/~t+~/~x+Uj(x)]Pij(t,x) = 0,

(4.37)

[~/3t+~/~X+~n+M(X)]Qo(t'x) = 0,

(4.38)

PK_I(t,0) = K%oPK(t),

(4.39)

Pij(t'0) = EjPi(t),

(4.40)

, , N Q°(t'0) = ~ i=IKPi(t)'

(4.41)

PN(0) = i.

(4.42)

Taking the Laplace transform of equations (4.32)-(4.41) under the initial condition (4.42) and solving them, we have PK-I(S'x) = K %oPK(S)exp [-sx-/ O u(m) (y)dy],

(4.43)

x Pij(s,x) = ~jffi(s)exp[-sx-/'0~j(y)dy],

(4.44)

_, , Qo(S,X) =

(4.45)

N

x

,

~ Pi(s)exp[-sx-f0~n+M(Y)dY], i=l

~N(S ) = [(N+r(S)%o]-l,

(4.46)

1144

MASANORI KODAMAand ISAO SAWA

[m+r(s)]loL(S ) = (m+l)loPm+l(S) + Klof(m)(S)PK(S) ,

(4.47)

N ,_, [n+r(s)]loPn(S ) = (n+l)loPn+l(S) + I gn+M(S)i=[KPi(s), K+l
(4.48)

[i+r(s)]Pi(s) = (i+l)Pi+l(S),

(4.49)

i~m,n; K~i~N-I,

From (4,49) we have Pi(s) = [n/h(s;K,n-l)][h(s;K,i-l)/i]Pn(S)

,K~i~n

= [m/h(s;n+l,m-l)][h(s;n+l,i-l)/i]L(s )

,n+l~i~m

[N/h(s;m+l,N-l)][h(s;m+l,i-l)/i]PN(S)

,m+l~i~N

=

(4.50)

Substituting (4.46) and (4.50) into (4.47) and (4.48), and rearranging with respect to Pm(S) and Pn(S) yields [m+r(s)]L(s) = i/[loh(s;m+l,N)] + [nf(m)(s)/h(s;K,n-l)]Pn(S),

(4.51)

n

[n+r(s)_n(l*/lo){*n+M(S)i__[K ~"]~nts)h(s;K'i-l) ) .r..,

m

_~

= [m/h (s ;n+l,m-l) ] [i+ (I /lo)gn+M(S ) [ h(s;n+l,i-l)/i]Pm(S ) i=n+l (I /lo)gn+M(S ) h(s;m+l,N)

+ Io

N [ h(s;m+l,i-l)/i . i--m+l

(4.52)

Using (3.17) and (3.18), we have the following Lermmas n Lemma 2. h(s;K,m)-i = r(s)[h(s;n+l,m) ~ h(s;m+l,i-l)/i i=K m + ~ h(s;n+l,i-l)/i], re>n,

(4.53)

i=n+l N

Lemma 3.

h(s;n+l,N)-i

= r(s)[h(s;n+l,m)

~

h(s;m+l,i-1)/i

i=m+l m

+

[ h(s,n+l,i-l)/i], re>n,

(4.54)

i=n+l

Lemma 4.

h(s;K,n)-i = r(s)[h(s;K,m)

n ~ h(s;m+l,i-l)/i i=m+l m

+ ~ h(s;K,i-l)/i], i=K

m
(4.55)

Hence we have from (4.51)-(4.54) en(S) = h(s;K,n-1)H(s;m,n)/[nloh(s;m+l,N)],

(4.56)

Pm(S) = [l+f (m)(s)H(s;m,n)]/[mloh(s;m,n)],

(4.57)

where H(s;m,n) ={lor(S)+l gn+M(S)[h(s;n+l,N)-l]}{[h(s;K,m)-f(m)(s)][lor(S )- I gn+M(S)] +I gn+M(S)[l-f(m)(s)]h(s;n+l,m)}-l~ Using (4.43)-(4.46),(4.50),(4.56) and (4.57), we have

(4.58)

Reliability repair analysis

1145

N

Pi(s) = {H(s;m,n)[h(s;K,n)+f (m)(s)-l]+h(s;m+l,N)}/[%or(s)h(s;m+l,N)] i=K -[l+f (m) (s)H(s ;m,n) ]/ [%or (s)h (s ;n+l,N) ] ,

(4.59)

PK_I(S) = K%oPK(S)(l-f (m)(s))/s,

(4.60)

Pij(s) = %jPi(s)(l-fj(s))/s = ~j[(l-fj(s))/s][n/h(s;K,n-l)][h(s;K,i-l)/i]Pn(S)

, K__
= ~.[(l-fj(s))/s][m/h(s;n+l,m-l)][h(s;n+l,i-l)/i]Pm(S ) J

, n+l
= %j[(l-fj(s))/s][N/h(s;m+l,N-l)][h(s;m+l,i-l)/i]PN(S)

, m+l_
, _, N Qo (s) = % [(l-gn+M (s))'/s]~ Pi (s) " i=K

(4.62)

From (3.15) [ [h (s ;K,m)-f (m) (s)] [%or (s)-%*gn*+M(S)]+%*g*+M (s)h (s ;n+l,m)(l-f (m) (s)) ]s=0=0,

(4.63)

and [ds [ (h(s ;k,m)-f-(m) (s ))(%or(S)-% *-* gn+M(S))+% *-* gn+M(S)h(s;n+l,m)(l-f(m)(s))]s=0 M

=(i+ j=l ~ %j K.+% j Ln+M) [h(0;K,m)-l]+%*K*(m)h(0;n+l,m). Using Tauberian theorem,L'Hospital's rule, (4.63),(4.64),(4.55)-(4.61)

(4.64) and (4.68),

we have

lim sH(s;m,n) = S÷0

[Xor(O)+X*-* gn+M(O)[h(0;n+l,N)-l]]lim s[[h(s;K,m)-f-(m) (s)] S~0 *-* *-* -(m) -i "[~or(S)-~ gn+M(S)]+~ gn+M(S)(l-f (s))h(s;n+l,m)]

, M , , , =~ h(0;n+l,N)[(l+ ~ ~.K.+~ Ln+M)[h(0;m,n)-l]+~*K*(m)h(0;n+l,m)] -I, j=l J J

(4.65)

P n = lim SPn(S) = h(0;K,n-l)H(m,n)/[n~oh(0;m+l,n)] $40

(4.66)

P m = lim SPm(S ) = H(m,n)/[mloh(0;m,N)] s+0

(4.67)

PN = lim SPN(S) = 0, s-+0

(4.68)

*

PA2

N

M

-i [ ~.K.+~ =i~KPi =[i+ j=l J J Ln+M+~*K*(m)h(0;n+l,m)/(h(0;K,m)-l)]

PK_I = K

*(m)

* *(m) 2 H(m,n)/h(0;m+l,N)=~ K h(0;n+l,m)P A /[h(0;K,m)-l]

Pij = ~jKj[h(0;K'i-l)/i]H(m'n)/[~oh(0;m+l'N)]

(4.7O)

' K--
= ~j.K.j [h(0;n+l,i-l)/i] H(m,n) / [~oh(0;n+l,N) ]

' n+l
=

, m+lfijN,

0

(4.69)

(4.71)

1146

MASANORI K O D A M A and ISAOSAWA

N M M i=K~ j=l~ Pij = J=~I~jK;[h(0;K'm)-I]H( m ,n)/[~ * h(0;n+l,N)] =

-* * * 2 Qo = lim SQo(S ) = ~ L n + ~ A s->0

M * 7. X.K.P. * 2 j'--'l3 J A '

,

(4.72)

(4.73)

where H(m,n) = lim sH(s;m,n) is given by (4.65) s÷0 N M * It is easily seen that ~ ~iPiJ+Q * + 2 = i. + i= K j PK-1 PA The reliability function of this model R 2 (t), its Laplace transform ~2 (s) * and the mean time to the first system failure MTSF 2 coincide with the results of the model i, respectively. Remark 3.

When

=0 , we obtain the following results using the same method as in

Remark i. M " "] -i ' H(m,n) = "~mM*lo[I+ ~~ E . K .*+ M * A K *(m~ j=13 j mo

(4. 74)

Let H(m,n)=H(m) . Pn = H(m)/n%o'

(4.75)

Pm = H(m)/mX o,

(4.76)

2 = H(m)/MmXo, PA

(4.77)

PK_I = K (m)H(m),

(4.78)

N

M M = j~iPiJ j =~l~jKj H (m)/Mm~o ,

i=K

(4.79)

Qo = 0,

(4.80)

where = (ii)

m [ i/i. i=K

(4.si)

K+l
Two new equations are added

t , , [d/dt+nXo+~+~l*]Pn(t)= (n+l)XoPn+l(t) + J ~n+M(X)Qo(t,x)dx M

+

t

[ f ~.(x)P .(t,x)dx, j=l 0 3 n3

[d/dt+ml °+l+X ]Pm(t) = (m+l)XoPm+l(t)+ f t (m) (X)PK_ l(t ,x)dx o M t + ~ / . ~ (X)Pmj (t,x)dx, j=l u J

m
(4.33)'

K+l=
(4.34)'

Taking the Laplace transform of equations of this case and using the same method as in (i), we obtain

Reliability repair analysis

Pi(s)

1147

= [m/h(s;K,m-l)] [h(s;K,i-l)/i]P (s) m

K
= [n/h(s;m+l,n-l)][h(s;m+l,i-l)/i]Pn(S)

, m+l~i~n,

= h(s;n+l,i-l)/[i,loh(s;n+l,N) ]

, n+l=
' (4.50) '

n(S) = [h(s;K,m)-f (m)(s)]H2(s;m,n)/[nEoh(s;n,N)],

(4.56)'

Pm(S) = h(s;K,m-l)H 2(s;m,n)/[mEoh(s;m+l,N)],

(4.57)'

Hg_(s;m,n) = [%or(S)+~ gn+M(S)[h(s;n+l,N)-l]] [h(s;K,m)-f (m)(s)] [%o r(s)-%*-*gn+M(s)] +7*-*gn+M(S)(1-f(m)(s))/h(s;m+l,n)] -I ,

(4.58)'

N

2 PA

i!KPi(s) = H 2(s;m,n)[h(s;m+l,n) [h(s;K,n)-~ (m)(s)]+~(m)(s)-l][%or(s)h(s;m+l,N)] -i +[h(s ;n+l,N) ] [%or(s)h(s ;n+l,N) ]-i ,

(4.59) '

PK_I(S) = KloPK(S)(l-f (m)(s))/s,

(4,60) '

Pij(s) = %j[(l-fj(s))/s] [m/h(s;K,m-])] [h(s;K,i-l)/i]Pm(S )

, K~i~m,

%j [ (l-fj (s))/s] [n/h(s ;m+l,n-1) ] [h (s ;m+l,i-l)/i]Pn(S)

, m+l~i~n,

),j[ (l-fj (s))/s]h(s ;n+l,i-l)/[i%oh(S ;n+l ,N) ]

, n+l~i~N,

_, , _, N Qo(S) = ~ [(l-gn+M(S))/s] [ Pi(s), i=K -

(4.62) '

,

H2(m,n) = lim s H 2 ( s ; m , n ) = % s-+0

(4.61) '

h(0;n+l,N)[(l+

M , , , [ %.K.+% L n + M ) ( h ( 0 ; K , m ) - i )

j=l 3 3 +k*K* (m)/h (0 ;m+l, n) ]-i ,

(4.65)'

Pn = lim SPn(S)= [h(0;K,m)-l]H2(m,n)/[n%oh(0;n,N)] , s~0

(4.66) '

P m = lira sPm (s)= h(0;K,m-l)H2(m,n)/[m%oh(0;m+l,N)] s-~0

(4.67)'

*

N

M

PA2 =i~Kei = [l+jo= . + X *K* (m) / [h(0;m+l,n)(h(0;K,m)-l)]] -I, [K%j K~+%*Ln*+M

(4.69)'

PK-I = K*(m)H 2(m,n)/h(0;m+l,N),

(4.70) '

Pij~ = %'K'[h(0;K'i-l)/i]H2(m'n)/[%oh(0;m+l'N)]3 3

, K
= ~jKj [h (0 ;m+l,i-l)/i] H2 (m,n) [h(0 ;K,m)-l] / [%oh (0 ;m+l,N) ] , m+l__
M

, M

M

[ . = [ %.K.H~(m,n)[h(0;K,m)-l]/[% *h(0,n+l,N) ]=j_~I~jK~ P2 i=K j ~I P i3 j=l 3 3* z _ •

Qo =

N * = ~* * 2 Ln+ M i=~KPi Ln+~A ,

X* *

N

It is easily seen that

M

(4.71) '

n+l__
(4.72)' ' (4.73) '

*

~ ~lPi j + Qo* + PK-I + PA2 = I. The reliability function i=K j of this model, its Laplace transform and the mean time to the first system failure

1148

MASANOR[ KODAMAand ISAO SAWA

coincide with the results of the model l,respectively. Remark 4.

When %* = 0, the steady-state probabilities coincide with the case of

Remark 3. (iii)

K+]
(4.32),(4.35)-(4.41) and (4.42) rema~t~ th,~ ~.~.r~e,one new equation is added *

[d/dt+n%o+%+% ]Pn(t)=(n+l)%oPn+l(t)+f0~

(n)

(X)PK_l(t,x)dx + f0~*+M(X)Q~(t,x)dx

M

+j=[if0~j (X)Pnj (t ,x)dx,

K+I
(4.33)"

Taking the Laplace transform of equations of this case and using the same method as in (i), we obtain (4.46)' '

PN(S) = [(N+r(S)%o]-i , Pi(s) = [n/h(s;K,n-l)][h(s;K,i-l)/i]Pn(S)

, K
= h (s ;n+l,i-l) / [i%oh (s ;n+l ,N) ]

(4.50)"

, n+l
n (s) = h(s;K,n-l)H3(s;n,n)/[n%oh(s;n+l,N)]

(4.56)"

H3(s;n,n ) = [~or(S)+% gn+M(S)[h(s;n+l,N)]][%or(s)[h(s;K,n)-f(n)(s)] +% gn+M(S)(l-h(s;K,n))]-i ,

(4.58)"

Pi(s) = [(h(s;K,n)-l)H 3(s;m,n)+(h(s;n+l,N)-l)][xor(s)h(s;n+l,N)]-I

(4.59)"

N i=K PK_I(S) = [(l-f(n)(s)/s]H3(s;n,n)/h(;n+l,N )

(4.60)"

,

Pij(s) = %j[(l-fj(s))/s][h(s;K,i-l)/i]H3(s;m,n)/[%oh(s;n+l,N)]

, K~i~n

= %j[(l-fj(s))/s][h(s;n+l,i-l)/i][%oh(s;n+l,N)]

, n+l
N _, , -* Qo(S) = % [(l-gn+M(S))/s] ~ Pi(s) , i=K H3(m,n) = lim S÷0

sH3(s;n,n)=%

*

h(0;n+l,N)[%

(4.62)" M ~ * * * ~ %.K.+% L

* *In) K " +(i+

j=l 3 J

Pn = lim SPn(S)=h(0;K,n-l)H3(n,n)/[nloh(0;n+l,N)], s÷0 *

(4.61)"

._)(h(O;m,n)-l)] - I- ' - -"

nl-m

' (4.65)" (4.66)"

N

PA2 = i!KPi =[h(0;K'n)-l]H3(n'n)/[~*h(0;n+l'N)]'

(4.69)"

PK-1 = K*(n)H3(n,n )/ h ( O ; n + l , N ) ,

(4.70)"

N

M

M

j~iPiJ = j~IXjKj[h(0;K,n)-l]H3(n,n)/[~*h(0;n+l,N)],

(4.72)"

i=K Qo =~ Ln+M[h(0;K'n)-I]H3 (n'n)/[% h(0;n+l,N)], It is easily seen that

N M * ~ Qo + PK-I + PA ~iPiJ+ * 2 = i. i=K j-

(4.73)" The reliability function

Reliability repair analysis

of this model R 2 (t) (n=m) , its Laplace transform first system failure MTSF 2 Remark 5.

When ~

1149

~2 (s) and the mean time to the

coincide with the results of the model l,respectively.

= 0 , the steady-state probabilities can be obtain from the results

of Remark 3 by putting n=m. 5.

Model 3 Viewing the nature of this model, we obtain the following set of differential-

difference equations: ,

t

t ,

,

[d/dt+N~o+X+X ]PN(t) = f0 ~(x)PK_l(t,x)dx+f0uN+M(x)Qo(t,x)dx N K

M

t

+i=0 ~ 311 f0~ij'= (X)PN-ij(t'x)dx , [d/dt+i% o +~+~ ]Pi (t) = (i+l)~ oPi+l (t) , [8/~t+~/~x+~(x)]PK_l(t,x)

K__
(5.2)

: 0,

[~/~t+3/~X+~N+M(X)]Qo(t,x) [~/~t+~/~x+uij(x)]PN_ij(t,x)

(5.3)

= 0, : 0,

(5.1)

(5.4) 0
(5.5) (5.6)

PK_I(t,0) = K~oPK(t ), PN-ij(t'0) : %jPN_i(t),

0
(5.7)

, , N Qo(t, 0) = ~ [ Pi(t), i=K

(5.8)

PN(0) = 0.

(5.9)

Taking the Laplace transform of equations (5.1)-(5.8) under the initial condition (5.9) and solving them , we obtain after some manipulation and simplification: Pi(s) : Nh°(s;K,i-I)PN(S)/[ih°(s;K,N-I),

K~i~N

(5.10)

where g°(s)

(5.11)

(s+~+~*)/~o, i h°(s;r,i) = ~ [j+g°(s)]/j, K~r~i~N ; = i, i=r-i j=r N h°(s,K,N) : l+g°(s) [ [h°(s;K,i-l)/i] i=K =

(5.12) (5.13)

N PN(S) = [h°(s;K,N-l)/N]{[h°(s;K,N)_~(s)]lo_l*~+M(S)i~KhO(s;K,i_l)/i N-K

M [ X.f..(s)h°(s;K,N-i-ll/(N-i)]} -I, -i~0 j=l ] 13 PK_I(S)

(5.14)

= N~o(I-f(s))PN(S)/[sh°(s;K,N_I)],

PN_ij(s) = NAj[(l-fij(s))/s][h°(s;K,N-i-l)/(N_i)]~N(S)/h(s;K,N_l)

(5.15) '

(5.16)

I I 50

MASANORI KODAMA a n d ISAO SAWA

_, , _, N Qo(S) =~ [(l-gN+M(S))/s ][n/h°(s;K,N-l)] [ ~ h°(s;K,i-l)/i]PN(S), i=K >3A(S) N N = ~ Pi(s)=[N/h°(s;K,N-l)] ~ [h°(s;K,i-l)/i]PN(S), i=K i=K Pi = lim sPi(s) = [N/h°(0;K,N-l)][h°(0;K,i-l)/i] s÷O

(5.17) (5.1S)

lim SPN(S)=[h°(0;K,i-I)/i]P*(N,K), s÷O (5.19)

PK-I = lim SPK_l(S) =XoK P (N,K), s-+0 --

*

(5.20)

0

*

PN-ij = lim SPN_ij(s) = XjKij[h (0;K,N-i-I)/(N-i)]P (N,K), s+0 N-K M N-K M o ~ PN-ij = [ [ [ X.K..h (0;K,N-i-I)/(N-i)]P*(N,K) , i=0 j=l i=O j=l 3 13 Qo = lim SQo(S) = [x Xo/(X +X)]~+MP s÷0

(5.21) (5.22)

(N,K)[h°(0;K,N)-I],

(5.23)

3 = [Xo/(~*+X)][hO(0;K,N)_I]P*(N,K), PA

(5.24) N-K

P (N,K) = {EoK +Xo(I+% L N+M)[h(0;K,N)-I]/(Xo+%

) +

M

~ ~ X.K..h°(0;K,N-i-I)/(N-i)} -I i=0 j=l j 13 (5.25)

N-K M * 3 It is easily seen that i=0~ j=l[PN-ij + Qo + PK-I + PA = i.

And also

N N N R3(S)= ~ h°(s;K,i-l)/[ikoh°(s;K,N)l = ~ [I/(s+i% o+x+~ )] ~ [m~o/(S+m~o+~+~ )] i=K i=K m=i+l (5.26) N , N , MTSF 3 = ~[i/(i~o+X+~ )] ~.[mXo/(mXo+~+~ )], i=K m=l Remark 6. 6.

(5.27)

If we set K=N, this model coincide with the model i and model 2 (K=N).

Properties of pl p2 and PA2 A' A

when there is no simultaneous failure

i 2 2 In this section we study the properties of PA,PA and PA From immediate c o m p a r i s o n KK o~ > ~ K *

Theorem i. Theorem 2.

between

<

models

l & 2 , we h a v e

the

* when % = 0 is true.

following

theorem;

> PAi <> PA2

(6.1)

An optimum integer m maximizing the steady-state availability of model 2

is determined by the integer m

(K
(l/i). i=K

Proof.

Example

It

is

clear

from

K* (m) = mK* o

.

K(m)=mK * m e a n s o ,

that

and

(4.77).

Uniqueness

time

of each

unit

is

m

g(K+2)>g(K+l)>g(K),

K=I

g(k)>g(K+2)>g(K+l),

K=2

g(K)>g(K+l)>g(K+2),

K~3

of m

is

not

K_=_+2. N

the mean repair

Let g(m)=mKo/i=[K(i/i),

Thus we have

,

(4.74)

we have from immediate calculation

same value.

assured.

Reliability repair analysis

1151

m =K, K=I; m =K+l, K=2; m = K+2, K$3 . We s h a l l

study

2 and PA 3 are PA

the behavior

the function

(6.2)

of steady-state

o f N and K.

availability

For convenience

i (i=1,2,3) throughout this section. o f PA

in p l a c e

1 PA '

for N and K since

we u s e t h e n o t a t i o n

P (N,M)

When the system stops operating ,

by s t o p

o f So, i f

t h e mean r e p a i r

time of each unit

is

t h e same v a l u e

(i.e.,K

=

(N-K+I)Ko) , then we have the following lemma and theorem. N

Lemma 5.

Let

F(N,K)=(N-K+l)Ko/l~K(i/i).=

(I~K~N) , then

(i)

F(N,K) is increasing in K for any fixed N.

(ii)

F(N,K) is increasing in N. for_any fixed K.

Proof. Since

N i/i < (N-K2+I)/[KI+ j ), NsK2>KI$1, K2-KI-ISj$O

i=K 2 it follows that N

K2-Kl-i

K2-1

(K2-K I) [ (i/i)< [ i=K 2 j=0

(N-K2+I)/(KI+J) = (N-K2+l) [ (i/i) i=K 1

Hence we have

K2_I

N N N F (N, K 2 )-F (N, K I) =K~ [(N-K2+1) [ (i/i) - (K2-K I) [ (i/i)]/[ [ (i/i) [ (i/i)] i=K 2 i=K I i=K 2 i=K I K2-1

>Ko[(N-K2 +1)

K 2-I N N (i/i) - (N-K2+1) [ (i/i)][ [ (i/i) [ (i/i)] = 0 i=K 1 i=K 1 i=K I i=K 2

which proves (i). (ii).

Proof is similar to that of (i).

Theorem 3. (ii)

(i)

Each P~(N,K) (i=l,2) is decreasing in K (I~K=
P~(N,K) and P~(N,K) are a decreasing function of N and a constant function of

N for fixed K~I (N~K),respectively.

(iii)

P~(N,K)÷0 as N-~= for any fixed K~I. ,

Proof.

N

,

,

(i) and (ii) are clear from (3.38), i / ~ = i~K(1/i) = , K =(N-K+I)K ° ,(4.24)

and Lemma 5. (iii)

Since N

N

(N-K+I)/ [ (I/i)>(N-K+I)/ f (i/x)dx i=K K = (N-K+I) log (N/K)-~

(6.3)

(N-~)

it follows that (iii) is true. M

,

Let g(1)=j_l'iAjKij , f(m)=h°(0;K,m-1)/m

(K_
and theorem. Lemma 6.

(i)

m for I/lo
f(m) is increasing in m for A/lo>l.

(ii)

f(m) is decreasing in

1152

MASANORI KODAMA and ISAO SAWA

Proof.

We consider n 2 and n I such that NSn2>nl~K .

The Proof is given by induction

on n 2 and we omit the details. Theorem 4.

(i)

if g(m) is decreasing in m for fixed M and I/Io<1 , then P~(N,K) is

decreasing in K (I~K~N) for fixed N$1.

(ii)

if g(m) is increasing in m for any

fixed M and I/Io>i, then P~(N,K) is increasing in K for any fixed N>I.= Proof.

From (5.24) we have

N p3(N,K) = P*(N,K) ~ h°(0;K,m-l)/m m=K N

N

M

={l+(N-K+l)K°~°/[m=K ~ h°(0;K'm-1)/m] + [m--[K(j=[llj~-mj)h°(0;K'm-l)/m] N /[ ~ hO(0;K,m_l)/m] }-i m=K N N N =[l+(N-K+l)Ko*1o/ ~ f(m) + ~ f(m)g(N-m)/ ~ f(m)] -I, m=K m=K m=K N N [ f(m)g(N-m)]/ ~ f(m) ,then we have m=K m=K K2-1 N N N G(N,K2)-G(N,KI)=K~Xo[(N-K2+I ) ~ f(m)-(K2-Kl) ~ f(m)]/[ ~ f(m) ~ f(m)] m=K I m=K 2 m=K I m=K 2 K 2-I N N N + ~ f(m) ~ f(n)[g(N-n)-g(N-m)]/[ ~ f(m) ~ f(m)],N>K2>Kl>l, m=K I n=K 2 m=K I m=K 2

(6.4)

Let G(N,K) = [(N-K+I)Kolo+

(i)

If %/%

o

(6.5)

, then using Lemma 6, we have

N

f (m) < (N-K2+I) h° (0 ;K, El+J-l) / (Kl+J)= (N-K2+I) f (KI+J), m=K 2 Hence we have N (K2-KI) ~ f(m)<(N-K2+l m=K 2 Let numerator of (6.5)

0~J ~K2-KI-I-

K2-KI-I K2-1 ) ~ f(Kl+J) = (N-K2+I) ~ f(m) j=0 m=K I be G (N,K 2) - G (N,KI). Noting that g(m) is decreasing in m

for any fixed M, we have K2-1 N g (N,K2)-G (W,Kl)> ~ f(m) ~ f(n)[g(N-n)-g(N-m)] m=K I n=K 2

$ 0

Hence P~(N,K) is decreasing in K for any fixed N$1 by(6.4) (ii)

Proof is similar to that of (i). REFERENCES

[1]

D.K.Kulshrestha,

" Reliability of a Parallel Redundant Complex System," Opns.

Res. 16,No.i,1968. [2]

D.K.Kulshrestha,

" Reliability of a Repairable Multicomponent System with

Redundancy in Parallel," IEEE.Trans.on Reliability.19,No.2,1970.

Reliability repair analysis

[3]

N.Nakamichi,J.Fukuta,S.Takamatsu,

and M°Kodama, " Reliability Consideration on

a Repairable Muliticomponent System with Redundancy in Parallel," 3.0pns. Res.Soc. of Japan.17,No.l,1974. [4]

M.Kodama, " Probabilistic Analysis of a Multicomponent Series-Parallel System under Preemptive Repeat Repair Discipline," Opns. Res.24,No.3,1976.

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