Availability analysis of a repairable system with common-cause failure and one standby unit

Microelectro,. Reliab., VoL 27, No. 4, pp. 741-754,1987. Printed in Great Britain.

0026-2714/8753.00+ .00 Pergamon Journals Ltd.

AVAILABILITY ANALYSIS OF A REPAIRABLE SYSTEM WITH COMMON-CAUSE FAILURE AND ONE STANDBY UNIT M. MAHMOUD, M.A. MOKHLES and E.H. SALEH

Department of Mathematics, Faculty of Science, Ain ShahS University, Cairo, Egypt (Received for publication 23 June 1987)

Abstract This paper investigates the mathematical model of a system consisting of two non-identical parallel redundant active units, with common-cause unit.

failure,

and a cold standby

The failed units are repaired one at a time or are

repaired together,

if they fail due to common cause failure.

All repair time distributions

are arbitrary and different.

The analysis is carried out under the assumption of having a single service facility for repair and replacement. Applying the supplementary variable technique, transforms of the various state probabilities Explicit expressions

Laplace

are developed.

for the steady state probabilities

and

the steady state availability are derived. Some well known results are obtained as special cases. A numerical example is given to illustrate the effect of the repair policy on the steady state

probabilities

and the

availability of the system.

i.

Introduction In classical reliability models, it is assumed that the

redundant units fail independently. redundant

systems,

(see Dhillon

in complex

the occurrence of common-cause

failure

[3]) violates the assumption of independent

redundant units failure. common-cause

However,

This is due to the fact that the

failure is multiple 741

failures which occur

M. MM-IMOUDetal.

742

because of a single initiating cause occurs, constitute

all the other failures

a complete

Several models, developed

factor or cause.

are triggered to

system failure. for example Dhillon

for redundant

failure Dhillon

failure and repair rates.

failure,

Dhillon

parallel

In his models,

the units,

analysis

of a

active redundant having constant

[4] obtained the Laplace

transform of the state probabilities

unit.

common-cause

[2] studied the availability

subject to common-cause

non-identical

[2-4], have been

systems including

system consisting of two non-identical units,

When this

of a system with two

active units and one cold standby the system is only repaired when all

including the standby unit,

fail with general

repair time distribution. It may be thought that the availability

and reliability

of the system will be increased by repairing unit immediately,

instead of waiting

each failed

for all the units

(three)

of the system to fail. In this paper, we consider the model given by Dhillon but with a main difference have a single service that whenever

facility

a unit fails,

of the various

certain conditions, Moreover,

We assume to

for repair and replacement,

it is repaired.

state probabilities

by using the supplementary

obtained.

in the repair policy.

[4],

variable

and

Laplace transforms

of the system are developed technique

[1].

the results given by Dhillon the Laplace transforms

Under

[4] are

of the availabi-

lity, and the mean up and down times of the system during (0,t]

are obtained.

availability

by Dhillon

probabilities

and

of the system are derived explicitly.

A numerical availability

Also, the steady-state

comparison

of the system,

of the state probabilities

and

under the repair policy adopted

[4] and the policy adopted in this paper is

presented.

2.

Assumptions i.

Initially,

the system is in a good state; both the

743

Av~lability analysis non-identical

units, unit i and

active parallel redundant

unit 2, say, are operating.

The standby unit, s say, is

kept in cold standby state. 2.

All failure rates are constant.

3.

Common-cause

failure can only occur when there are

more than one unit operating. 4.

Repairs,

common-cause and other failures,

are

statistically independent. 5.

Whenever an active unit fails,

it is sent for repair

at once, where it is repaired on "first come-first basis.

served"

A repaired unit is like new and returns into operation

immediately. 6.

When both active units

to common-cause

failure,

(unit I and unit 2) fail due

they are repaired and sent back into

operation together. 7.

When both active units

to common-cause

(unit 1 and unit 2) fail due

failure, or otherwise,

rupted and the standby unit replaces ing the replacement,

the repair is inter-

any one of them.

the repair process

Follow-

restarts from the

beginning. 8.

The system is in the failed state

including the standby unit, 9.

if all the units,

are non-operating.

When the standby unit fails, the repair is inter-

rupted and the system is repaired as a whole with a general repair rate. i0. All the repair time distributions

are arbitrary and

different. 11. After repairing any one of the active units, or both (if failed due to common-cause sent into operation

failure),

it

(they) is (are)

immediately and the operating standby

unit returns into the cold standby state.

3.

MR 27:4-J

Notation

Ak

Constant

failure rate of unit k,

(k = 1,2).

A3

Constant com~aon-cause failure rate.

744

M. MAI~MOtn)et al.

As

Constant

failure rate of the standby unit°

Constant replacement rate to replace any one of the failed active units by the standby unit. i

State of the system!

i = 0

(both active units; are operating) ,

i = 1

(unit 1 under repair,

and unit 2 operating),

i = 2

(unit 2 under repair,

and unit 1 operating),

i = 3

(both active units, units 1 and 2, are failed due to c o m m o n - c a u s e failure),

i = 4

(unit

i = 5

(unit 1 fails, w h i l e unit 2 under repair),

i = 6

(standby unit operating, while both active units, units 1 and 2, are under repair due to commoncause failure) ,

i = 7

(standby unit operating, while unit 1 under repair and unit 2 waiting for repair),

i = 8

(standby unit operating, while unit 2 under repair and unit 1 w a i t i n g for repair),

i = 9

(the s y s t e m is in the failed

Pi(t)

Probability 0 < i < 9.

Pi(x,t)

P r o b a b i l i t y density, (with respect to repair time x) that the system is in state i at time t, and the unit under repair has an elapsed repair time x; i < i < 9.

Pk(X) ,fk(x)

units

1 and 2, of the system

2 fails, while unit 1 under repair),

state and under repair).

that the system is in state i at time t,

Repair rate and p r o b a b i l i t y density function (p.d.f.) of repair time of a failed unit K (K = 1,2) and has an e l a p s e d repair time of x. X

fk(x)=~k(X) exp[-

f ~k(U)du],

k = 1,2

0 ~o(X),fo(X)

Repair rate and p.d.f, of repair time of the active units if failed due to c o m m o n - c a u s e failure and has e l a p s e d repair time of x. X

fo(X)

= Po(X)exp[-

n(x),h(x)

0

f ~o(U)du].

Repair rate and p.d.f, of the system in state 9, and has an e l a p s e d repair time of x. X

h(x)

=

n(x)exp[-

/ n(u)du]. 0

t I Pi(x,t)dx,

Pi(t)=

1 <_ i <_ 9.

0 Mean repair time of the s y s t e m in state v

=

f xh (x) dx. 0

P*(s)

=

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

P*(S)

=

f exp [-st] P (t)dt° 0

P(t).

9.

Availability analysis 4.

The State

Probabilities

The transition dered is shown

The following

of the S~stem

diagram

in Figure

of the redundant

d (~

set of i n t e g r o - d i f { e r e n ~ i a l e q u a t i o n s the method

for

of supplementary

[I].

3 Z li)Po(t) i=l

+

system consi-

i.

the model may be set up by using variables

745

Z

=

t lui(x)Pi(x,t)dx

Z i=l

t I Uo(t)P6(x,t)dx

+

0

0

t £ n (x)P9 (x,t)dx

+

(i)

0

(~ d (~

+ ~-~ + U i + A3_i)Pi(x,t)

+ a)P3+i(t)

= 0,

= A3_iPi(t),

(~'X + ~-6 + "i (x) + As)P6+i(x't) (~-~ + ~

+ n(x))P9(x,t)

= ~iPo(t)

t I

+

(2)

i = 0,1,2

(3)

i = 0,1,2

(4)

= 0

We also have the following

Pi(0,t)

= 0,

i = 1,2

(5) boundary

conditions:

~3_i(x)P9_i(x,t)dx,

i = 1,2

(6)

i = 0,1,2

(7)

0 P6+i(0,t)

= aP3+i(t) ,

2 P9 (0't) = ~s i~ 0 P6+i (t)

(8)

~o( x )

Figure

(i) : Transition

diagram of thesvstem;

the down states; the system.

state

stars

9 is the failed

represent state of

+

746

M. MAHMOUDet al.

Since we are assuming

that the system is initially

good state at time t = 0, then Po(0) probabilities

in a

= 1, and all other initial

equal to zero.

The set of the differential using the Laplace t r a n s f o r m

technique.

forms of the set of equations

P[(s) = I

equations

P~(x,s)dx,

(1-8)

(1-5)

Taking

are solved by

Laplace

trans-

and using the definition

1 <_ i I 9

(9)

0

we get: 4~

I

(s+

Ii)P_"(S)-I~ =

Z $ i=1 0

i=l

+

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

/Po(X)P6(x,s)dx

+ $ 0

0

(s+ ~-~ + ~i(x)

+ X3_i)Pi(x,s)

P3+i(s) *

= (S + a) -I A3_iPi(s), *

(s + ~

+ ~i(x)

+ Xs)P6+i(x,s)

(S + ~-~ + , ( x ) ) P 9 ( x . s )

Pi(O,s)

= 0

= ~iPo(S)

=

0

,

= 0,

+

,(x)P

9(x,s)dx

(I0)

i = 1,2

(11)

i = 0,1,2

(12)

i = 0,1,2

(13)

.

(14)

+ I ~i(x)P6+i(x,s)dx,

i = 1,2

(15)

i = 0,1,2

(16)

0 p*

*

6+i(0,s)

= aP3+i(s)

,

P9(O's)

2

= As i~O

,

.

(17)

P6+i(s)

Equations

(ii),

(13), and

equations

of first order,

(14)

are homogeneous

their

solutions

differential

are respectively

as follows: .

Pi(x,s)

~

x

= P (0,s)exp[-(s+A3_i)x-

$ ~i(u}du],

i = 1,2

(18)

0 x

P6+i(x,s)

£ ~i(u)du],

= P6+i(0,s)exp[-(S+As}X-

i=0,i,2

(19)

0 *

P9(x,s)

x

= P;(O,s)exp[-sx

-

I .(u)du]

(20)

0 Integrating

the set of equations

(18}-(20)

with respect

to x

Av~labilityanalysis

from 0-~® , and using

747

(9), we obtain:

* * [l-f*(s+Xi 3_i ) ] Pi(s) = (s+A3_ i) -i Pi(0,s)

, i

* * [l-f~(s@A s)] , P6+i(s) ffi (S+A s) -i P6+i(0"s)

=

1,2

(21)

i = 0,1,2

(22)

and * -i * P9(S) = S P9(O,s) [l-h*(s)] Substituting

P6+i(s)

•

(23)

for i = 1,2 into (15), and using

(16) and

(17), we obtain: P:(0,S)

= XiP (s) + (s+a)

Using equations

-i

* uAiP3-i(s) f3-i* (S+Xs) , i = 1,2

(24)

(24) and (21), we get explicit expressions

for Pl(S) and P2(s) in terms of Po(S) as follows: Pi (s) = AiAi(s)P:(s)[(s+A3_i)B(s)]-I,

i = 1,2

(25)

where Ai(s) = [l-fl(s+X s) ] [l+uX3_i((s+s) (s+Xi))-If*3_ i(s+As) (l-f3_i(s+Xi)) ], i = 1,2

(26)

and 2 B(s) -- 1- . (~ qf[Cs+~ s) [1-f~ (s+ A3_ i) ] [ (s+Xi) (s+a) ]-I} i=1 Substituting

(27)

from (25) into (12) and (16), we see that

Pi(s), i = 3,...,9 may be written in terms of Po* (s) as * follows : *

P3 (s) = X3(s+a)

-I

*

Po(S)

(28)

P3+i (s) = AIA2Ai(S)Po(S) [(s+s) (s+~3_i)B(s)]-i , i = 1,2

(29)

P6(s) -- ~3[(S+~s) (s+a)I-1[1-fo(s+x s))Po(s),

(30)

P6+i(s) =a~iA 2(l-f i(s+Xs))Ai(s)P o(s) [(S+a) (S+Xs) (s+13_i)]-l, i = 1,2

(31)

and P;(S) = aA s[s(s+A s) (S+u)]-Ic(s) (1-h*(s))P:(s),

(327

748

M. MAHMOUDet al.

where .

2

C(s) = ~3[l-fo(S+As)]

+ ~IA2B-I(s)

,

E (s+~3_i)-IAi(s) (l-fi(S+~s)) i=l (33)

Having

E Pi(t) = l, i.e., i

Z P*(s)~ = s -1, i

we get:

*(s) 1 A3 aC(s) A PO = ~ {1 + S--~ + (s+a) (S+As) [i+ -~(1-h*(s))]

1 2 + B(S-----~i~ 1

where Ai(s);

Ai(s) (Ai(s+a) (s+A3_i)(s+a)

i = 1,2, B(s)

and

(33) respectively.

5.

Special Case Now,

+ AIA2)

+

}-I (34)

and C(s)

are given by

let us consider the special

(26),

(27),

case where the system

will be repaired only when all its units,

including

the

standby unit fail. The states

4 and 5 will be combined

into one state,

state "f" say, where both active units 1 and 2 fail, one after the other.

States

6, 7, and 8 will form one state,

state "r" say, where unit 1 or 2 is replaced by the standby unit.

This model

is similar to that of Dhillon

[4].

The

state diagram is shown in Figure 2.

(x)

AI A2

E

Figure

(2): Transition

diagram of the system in the case

where repair is performed of the system fail; of the system; system.

only when all units

stars represent

the down states

state 9 is the failed state of the

Availability analysis Let ~o(X)

= Vl(X) = ,2(x) = 0.

AI(S ) = A2(s) C(s) Hence,

equations

Pi(s) *

(25)

and

= li(s+13_i)-IP

P3(s) (we see that P3(s) Equations

Consequently,

= B(S) = i, 2 ~ i=l

= A 3 + 1112

749 we have:

and

(s+A3_ i)

-i

(28) will be: o

(s),

(35)

i = 1,2

(36)

= 13(s+s)-iPo(S ) did not change).

(29) for i = 1,2 will combine to form 2

P~(s) = ~112(s+~)-IPoCS)

~ (s+13_i)-l

(37)

i=l Similarly,

equations

(30) and

Pr• (s) = u[(s+~) (S+As) ] Equation ,

(31) will combine

-I * Po(S) [A3+AIA 2

-i]

2

~ (s+A3_i) i=l

(38)

(32) will take the form ,

P9(S)

to give

= ~As[S(S+As) (s+u)]-l(l-h*ls))Po(S)

[A3+~IA 2

2

~ (s+~3_ i) i=l

_11 ,,

(39) Finally, * Po(S)

equation

(34), will be

= s - i {I+A3(s+u)-I+a(A3+AIA 2

2 Z (s+A3_ i)-l) ((s+u) (s+A s))-I i=l

2 x [I+A s-l(l-h*(s))] S

+

[Ai(s+~)+AIA2] [ (s+u) (s+A3_i) ]-i} -I i=l

= ((s+u) (S+As))-l[s+

3 , 2 Z A -uA h (s) (A3+AIA Z (s+A3_i)-l] -I i=l i s 2 i=l (4O)

We find that the set of equations by Dhillon

6.

~

(35-40)

are those obtained

[4].

Steady

State Probabilities

of the S~stem

Using the well known relation Pi = lira Pi(t) t+o.

= limsP i (s) s+0

0
750

M.M.Jd-I~Otn9 et al. we get the steady state probabilities using equations

(25-34),

Pi = liAi(A3-iB) - 1 P o

of the system, by

as follows:

'

(41)

i = 1,2

(42).

P3 = a - l A 3 P o

P3+i = III2AiPo(UA3-i B)-I' -i

*

= As 1 3 ( 1 - f o ( A s ) ) P

P6

(43)

i = 1,2

(44)

o

* P6+i = AIA2Ai(I-fi(As))Po(AsA3-iB)

-I

i

'

=

(45)

1,2

(46)

P9 = CVPo and 2

Po = [1+ a-1/~3 + ~slC(I+/~s~)

+ B-1

Ai( i+ -l l 2 )fh47)

r. i=l

where A i = lim s~0

Ai(s)

= [l-fi(ls)] [l+AilA3_if3_i(A s) (l-f 3_i(Ai))],

i=1,2

(48)

B = lim B(s) s~0 2 l-

f~ (l$)(l-fi (A3_i))

,

(49)

i=l C = lim s÷0

C(s) ,

= A3(I-f°(As))+

2

AIA2B-I

_

.

i=iF A31 i Ai(l-f i(A s )),

(50)

and oo

= -h*'(0)

=

f xh(x) dx

(51)

0 = mean repair time of the system

7.

The Steady-State

Availability

The system availability

of the System

is given by

Av (t) = Po (t) + P1 (t) + P2 (t) + P6(t) Thus,

the steady-state

availability

+ P7(t)

+ P8(t)"

is given by

Availability analysis

751

Av = Po + P1 + P2 + P6 + P7 + ~8 Substituting

with

(41),

(44),

(45), and

(47), we get:

AV = LI/L 2 where

L1

1 ÷

÷ B -1

S

iAi

i=l

and

L 2 = 1 + u-iA3 + AslC(I+AsV) with Ai;

8.

+ B -I

2 Z A31iAi(li+u-illA2)_ i=l

i = 1,2, B, C and v as given by

The Case of Similar

Units

The case of similar

units,

above results,

may be evaluated

from the

by taking

l I = 12 = A s = A, and fl(x)

(48-51).

= f2(x)

Under these

~l(X)

= ~2(x)

= ~(x),

= f(x). conditions,

we get the following

transformations: (i) States

1 and 2 in Figure

fails while (2) States active (3) States

the other

fail),

The transition

state

diagram

state

(one unit

"I" say°

(I) form one state

state

7 and 8 in Figure

unit operates),

states

is operating),

4 and 5 in Figure units

(I), form one state

(both

"2" say. (i) form one state

(the standby

"4" say.

showing

in the case of similar

the flow between units

is given

different

in Figure

Consequently, A 1 = A 2 = [l-f

(l) ] [i + f (l) (l-f

B = 1 - f*2(l)(l-f

(l)) ],

*(I)) 2 ,

and C = 13(l-f~(l)) Hence,

the steady

availability

+ 2l(1-f*(l))2(1-f*(l) state probabilities

are as follows:

(1-f*(l)))-I and steady

state

3.

752

M. MAHMOUDet al.

FJ(x) A

E

Figure

(3}: Transition diagram of the system in the case of identical units; stars represent the down states; state 9 is the failed state of the system.

*

P1 = 2(l-f

(A))Po[I-f

*

(A) (l-f * (A))

*

P2 = 2All-f

*

(A))Po[Ull-f

]-i

,

*

(A) (l-f

--l

(A)))]

P3 = A3Po/~'

P4 = 2(l-f

*

(A} ) 2 P o [l-f * (A)(l-f

*

(A))] -i ,

P6 = A-IA3Po (l-f: (A)) '

* P9 = VPo[A3(I-fo(A))

+ 2(l-f

*

(A))

2

[l-f * (A) (l-f * (~))]-i],

and Po = {l+2(A+a) (l-f*lA)) (u(l-f*(A) (l-f*(A))))-l+u-iA3

+ ~-Ic(I+A~) }-i . Finally, Av = Po + P1 + P4 + P6"

9.

Numerical Assuming

Erlanglan

Example that the repair

form with different

time distributions rates,

(Uix) (Pixr)r-le -rpix fi (x) ..... (r-l) l and h(x)

r-I erqX = .[nx) (nxr) (r-l)!

have an

i.e., ,

i = 0,1,2

+

753

Availabilityanalysis Taking

r =

i,

exponential In steady

Table

A1 =

A2 =

n

10000

200,

(I),

the

400

h(x)

will

have

the

probabilities

and

the

and

The

A3 =

0.05,

values

are

~ =

calculated

2000,

of

~,

where

r,

the

steady

for

steady

state

the

in Table

steady-state

availability values

system

values

calculated

(i) :

the

~o = ~i

=

for

i000, P2

=

~ = 0,

500°

different

are

PO Pl P2 P3 P4

and

state

of

different

and

probabilities

~ ' 1

0,1,2,

steady

A = 0.i,

and

Taking

Table

i =

availability

As =

300,

system

for

forms°

state

=

fi(x)

of

availability

of

the

(2).

probabilities

of

state

the

system

and

(r =

the

1 and

steady-state different

u)

0

200

300

400

500

0.18181322 0.36362645 0.00001818 0.00000455

0.9989256! 0.00099892 0.00000005 0.00002497 0.00000050

0.99925863 0.00066617 0.00000003 0.00002498 0.00000022

0.99942519 0.00049970 0.00000002 0.00002499 0.00000012

0.99952513 0.00039982 0.00000002 0.00002499 0.00000008

I 0.00004994 I 0.504E-09 I I 0.99997497

0.00004996 0.501E-09

0.00004997 0.500E-09

0.00004998 0.500E-09

0.99997498

0.99997499

0.99997500

0.45453306 el P6 P9

0.00000455

Rv

0.99997273

. . . . . . . . . . . ....... . I-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... * according

to

Dhillon's

4 and

6 combine

aEn

axl0 n

=

Table

(2):

The

assumptions

into

one

values V

.

.

_

;

IPr

PO P! P2 P3 P4 P6 P9 Av

_

_

_

.

.

.

. . . . .

. .

of

. . . . . .

u =

0 and

probabilities

of

the

system

and

(u =

the

500

.

.

.

.

.

.

.

.

.

_ . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. . . . . . .

3

0.99952513 0.00039982 0.00000002 0.00002499 O.O000000B 0.00004998 0.500E-09

0.99952510 0.00039983 0.00000002 0.00002499 0.00000008 0.00004997 0.500E-09

0.99952500 0.00039983 0.00000002 0.00002499 0.00000008 0.00004997 0.500E-09

0.99952495 0.00049984 0.00000003 0.00002499 0.00000008 0.00004997 0.500E-09

0.99997498

0.99997488

0.99997484

I .

different

r)

2

0.99997500

0 states

steady-state

and

!

. . . . . . . . . . .

8 =

state.

steady-state

availability

when

.

. . . . .

4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

. . . . . .

5 0.999524B1 0.00039984 0.00000003 0.00002499 0.00000008 0.00004997 0.500E-09 0.99997480 . . . . . .

..

. . . . .

..

M. MAHMOUD et al

754 COb~4ENTS

From Table (i)

(1), we see that:

Po increases as

~(the repair rate of the units)

increases. (li)

The steady state availability increases as ~ increases. From Table

availability

(2), we see that Po and the steady state

(Av) decrease as the repair time distributions

deviate from the exponential form.

REFERENCES [i]

D.R. COX: The Analysis of Non-Markovlan Stochastic Processes by Supplementary Variables; Phil. SOCo 51, 433

[2]

(1955).

B.S. Dhillon: A Common-Cause Failure Availability Model; Microelectronic

[3]

BoS. Dhillon:

and Reliab.17,

583

(1977) o

On Common-Cause Failure-Bibliography;

Microelectronlc and Reliab. [4]

Proc. Camb.

18, 533

(1979).

B.S. Dhillon: State Device Redundant System with Common-Cause Failure, electronic and Reliab.

and One Standby Unit; Micro20, 411

(1980).