A repairable system with N failure modes and one standby unit

Mlcroelectro~ ReSab., Vol. 20, pp. 831435 O Pergamon Press Ltd. 1980. Printed in Great Britain

0026-2714,/80/1201--0931g)Z00/0

A REPAIRABLE SYSTEM WITH N FAILURE MODES A N D ONE STANDBY U N I T Mrrsuo Y ~ m R o Department of Planning Technology, Ashikaga Institute of Technolo~, 268-1, Oomae-cho, Ashikaga-shi, Tochigi-ken, Japan 326

(Receivedfor publication 29 September 1980)

ABSTRACT In this paper, we deal with a repairable and one stanby unit.

Laplace t r a n s f o r m s o f

such a system are obtained by u s i n g t h e A particular

system with N failure modes state probability

supplementary

for

variable method.

case is considered.

1.INTRODUCTION Markov model for a repairable

system with two failure modes and

one stanby unit has been analyzed by Elsayed and Dhillon[l]. paper extends repairable

the

model to "a more genera! model.

ststem with N fail~re modes(:[2i-[5]),

unit and g e n e r a l r e p a i r state probability supplementary

time distributions.

We deal w i t h a

one identicai

Laplace transforms

stanby of

for the system are obtained by using the

variable method.

and corresponds

This

A particular

case is also considered

to the model of Elsayed et a1.[1].

2.NOTATION The

following notations are used to analyzed the model in next

section; N

number o f

failure modes,

hi

failure (hazard)

rate of failure mode i,

N i=1 831

MITSUOYAMASI/IRO

832

time since repair began,

X

gi (t) ,~i(x) pdf and repair(hazard) rate of failure mode i at repair time x, replacement rate of failure mode i, probability that 1st unit is under operational state at

Po(t)

time t,

Pl(t)

probability that stanby unit is under operational state at time t,

P0,i(t, x)

probability density(with respect to x) that is state(0,i)

ist

unit

under repair at time t and its elapsed

repair time is x, Pl,i(t, X)

probability density(with respect to x) that stanby unit is state(l,i) under repair at time t and its elapsed repair time is x,

Pj,i (t)

Pj,i(t)=

i(t,xldx, for j=0,1,

,

implies Laplace transform with respect to t,

for i=l,2,...,N.

3.ANALYSIS Fig.l shows the state transition diagram for a repairable system with N failure modes and one stanby unit.

"" ~ . ~ N ~ / ~

_~.,~.~~

~ N ( X ) ~

~N(X)~

O

Operati°nal

NFailed

state

state

Fig.l State transition diagram

The following assumptions are applied to analyze the model. l.The states of the ststem are represented by its state number and pair(.,.).

State 0 and 1 imply that,lst and atanby unit are under

operational state.

States (0,i) and (1,i) imply that ist and

stanby units are under failed state for failure mode i. 2.0perational and etanby units are identical. 3.Failure rate of failure mode i is constant, ~i"

833

RepmrableSymenmFmlum Modm 4.Rate that the failed unit for failure ~ d e

i

isrep~aced

the

stanby unit is constant, ~i' 5. The times to repairare

generally distributed with repair rate

i(x) from the failed state(0,i) or (l,i) at repair time x. 6. The system is state 0 at t=0. By using the supplementary variable method, the following sets of integro-differential dP0(t) dt

equations ca.be derived; N i~ ~" ~ i(x) pO'i (t'x)dx' i=l 0

= - APo (t) +

P0ri(t,x)

(t,x) + ~P0ri ~x

~t

(i)

= ~[~i(x) +~i]P0,i(t,x),

P0,i(t,0) = ~iP0(t), = -~Pl(t)

dt

~Pl,i (t'x)

0

~i(XlPl,i(t,xldx + "

(2,a)

x=0

(2,b)

iP0,i(t),

(3)

~)Plti(t' x) +

~t

+

x~0

= - L~i(X)Pl, i(t'x) '

~x

AiPl(t),

Pl,i(t,O) =

x>0

(4,a)

x=O

(5,b)

for i=l,2,...,N. The initial conditions are as follows; (6)

Po(O) = 1,

Pl(0)

(6)

= 0,

(7)

Pj,i(0,x) = Pj,i(0) =0 for j=0,1 and i=1,2,...,N.

Taking Laplace transforms of (1)-(4) with initaial c0ndition s (5)-(7), we have sP~(s)

- 1 = -lP~(s)

(8)

~i(x)P~,i(s, x}dx,

+

i=1

0

~p*. ~ (s,x)

sP~,ils,x) +

= -[ ~i(x) +4i]P~,i(S,X),

(9,a)

~X (9,b)

p~,i(s,O) = AiP~(s), N sP~(s) = - A P ~ ( s )

+

i=l sp~,i(s,x)

+ ,~P~,i (s'x)

0 _- _ ~i(~)p~,i(s,x),

i=l

(ii ;a)

~x

p,ics,o) M ~ ~IOIO - E

--

(ll,b)

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MhTsoO YAMAsmao

The solutions for (6)=(8) are ; N

P~ = [s + k P~'i(s'x)

~

~ig[(s+~i

)]-I

(12)

= )'iP~(s)exp[-Is+~iIx-

P~,i(s)

=

0

0

~iluIdul'

(13)

p~,i(s,x)dx [1 - g . * ( s + ~ i ) ]

=

4iP~(s)

,. ,

J-,,,.,, ,..

k ill[l* gi(s+~i)l i=l (" + ~i)

]

"

q (s)=

rs

....

.

p~ _.(s,x) - - ~ i P ~ ( s ) e x p [ - s x ,~ i ,

q , i (s) =

I o P *l(, Si , X )

(14)

,~

( s+~i~

-

I

....P ~ ( s ) ,

(15)

?i(u)du],

(16)

x 0

dx,

= kiP~ (s) [i

g~(s) ]. ,

(17)

s

4.A PARTICULAR CASE ~ As a particular case of the system in the previous section, the system(Model I) of Elsayed et al. [1] is considered as follows~ we put N=2, ~I = ~0' 42 = ~i'

~l(X)= ~0'

72 (x}=~l' a n d ~ i = ~

When into

(12) and (15), P~(s) and P~(s) are

P~(s)=

(s+~ + ~o ) (s+~ ~- ~.l )

,.,

(18)

[ (s+ 40 + 4l ) (s+ ~ + ~o ) (s+ ~ + ~ )

- ~o ~o (s+~+ ~l ) - )'1/'~t (s+'j,+ ~o )] and P~(S) =

~0)] ¢s+~+ ~)¢s-W,+ p.1 ) Its+ 40 + A])¢,+/,'o ) (s+p. l) ~ ( s + ~o ) (s+ F I ) [ ~o(S+oL + Fz)+ A l ( S + ~ +

-

~0 ~0 (s+/~l )

- ~ 1 PI

,P~ (s), (19)

(s+/~o ) ]

The inverse transforms of (18) and (19) coincide with (7) and (8) in [I].

For the explicit expressions of P0(t) and Pl(t), refer tO [i].

ACEROWLEDGEMENT I thank

Dr. Balbir S. Dhillon for providing the reference[5]

and Dr. Yasunobu Yuasa for his helpful advice.

Rcpairable Systems Failure Modes REFERENCES

[I]E.A. Elsayed and Babir S. Dhillon,"Repairable Systems with One Standby Unit", Microelectron., Relia., Vol.19,pp.243-245,(1979). [2]Shunji Osaki,"Reliability Analysis of an Intermittently Used System with N Types of Failures", The Trans. of IECE of JAPAN, Voi.54-C, No.3,pp.279-280,(1971). [3]E.A. Elsayed and A. Zebib,"A Reparable Multistate Device", IEEE Trans. on Reliability, Vol.R-28, N~.I, pp.81-82(1979). [4]Mitsuo Yamashiro,"A Repairable Multistate Device with General Repair Time", IEEE Trans. on Reliability, VoI.R-29,(1980). [5]Balbir S. Dhillon,"Non-Repairable Multi-Failure Mode Systems", Proc. of the 11th Annual Modeling & Simulation Conference, Pittsburgh, USA(1980).

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