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The reliability function and the availability of a duplication redundant system with a single service facility for preventive maintenance and repair
Microelectron. Reliab., Vol. 29, No. 5, pp. 751 756, 1989.
0026 2714/8953.00+ .00 © 1989 Pergamon Press pie
Printed in Great Britain.
THE RELIABILITY F U N C T I O N A N D THE AVAILABILITY OF A DUPLICATION R E D U N D A N T SYSTEM WITH A SINGLE SERVICE FACILITY FOR PREVENTIVE MAINTENANCE A N D REPAIR G. S. MOKADDIS and S. S. ELIAS Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt (Received for publication 14 November 1988) Abstract--This paper concerns a two-unit system with a cold standby and a single service facility for the performance of preventive maintenance and repair. Explicit expressions for the Laplace transforms of the availability of the system, the reliability, the mean down time during (0, t) and for the mean time to system failure have been obtained under the assumption that the failure times, the inspection times, the repair times and the preventive maintenance times of the two units are governed by distinct arbitrary general distributions. The results obtained by Srinivasan and Gopalan and by Gopalan and d'Souza are derived from the present results as special cases.
1. INTRODUCTION
operative and inoperative periods, known as the up and the down periods. In this paper the system is characterized by the probability of its being in the up state or the down state. In order to find the reliability and the availability of this system, integral equations for these probabilities corresponding to different initial conditions are set up by identifying suitable regenerative stochastic processes.
Many authors have considered the reliability analysis of systems with standbys and several independent repair facilities. However, it may not be possible to have more than one repair facility in many practical situations. Srinivasan and G o p a l a n [1] have carried out the probabilistic analysis of a two-unit system with a cold standby and a single repair facility. Recently, G o p a l a n and d'Souza I'2] have considered the same situation where the system is subject to preventive maintenance under the assumption that the repair time and the preventive maintenance time of a unit are exponentially distributed. The purpose of this paper is to analyse the availability and the reliability of a two unit system with a cold standby and a single service facility for the performance of preventive maintenance and repair under the assumption that the two units are not similar and that the failure times, the inspection times, the repair times and the preventive maintenance times of the two units are governed by distinct arbitrary general distributions. Initially, the first unit is switched on while the second unit is kept as a cold standby. The failure rate of a cold standby is assumed to be zero. The standby is switched on instantaneously when the operating unit is sent for preventive maintenance or for repair. If the service facility completes the preventive maintenance or the repair for a unit while the system is still functioning, the unit is placed in standby. If, however, the unit that has been operating fails or is sent for preventive maintenance before the service facility becomes free, the unit waits as if in a queue and is taken up immediately for maintenance or for repair when the service facility becomes free. In such a case, the system ceases to operate until the unit already undergoing either preventive maintenance or repair is made available for operation. The system thus passes through cycles of NR 29:5-F
The following notation is adopted (note that i = 1, 2). f~(.) probability density function of the failure time of the ith unit with distribution function Fi(.) h~(.) probability density function of the inspection time of the ith unit with distribution function Hi(.) #i(-) probability density function of the repair time of the ith unit with distribution function Gi(.) v~(.) probability density function of the preventive maintenance time of the ith unit with distribution function Vii(.) Z(t) a discontinuous random variable assuming the value 1 if the system is found in the down state at instant t and the value 0 otherwise For simplicity, we write q~(.) = 1 - qS(.) for any function ~b(.). ai(u) = hi(u)Fi(u),
bi(u) = fi(u)Hi(u).
n~i)(t) PrI-Z(t) = l I Z(0) = 0, the ith unit enters the service facility at t = 0 for preventive maintenance] n~(t) Pr[Z(t) = l I Z(0) = 0, the ith unit enters the service facility at t = 0 for repair] wt~'~(t) Pr[Z(t) = 1 ]Z(0) = 1, the ith unit is under preventive maintenance and the other waits for preventive maintenance] w~i~(t) Pr[Z(t) = l l Z ( 0 ) = 1, the ith unit is under preventive maintenance and the other waits for repair] wt~(t) Pr[Z(t) = I l Z ( 0 ) = 1, the ith unit is under repair and the other waits for preventive maintenance] w~)2(t) Pr[Z(t) = 1 r Z(0) = 1, the ith unit is under repair and the other waits for repair] e~(t) P r [ Z ( t ) = l , Z(u) v~ l V-O <. u < tt Z(O) = O, the ith unit enters the service facility at t = 0 for preventive maintenance] 751
752
G . S . MOKADDISand S. S. ELIAS
P~)(t)
Pr[Z(t)---l, Z(u)# l VO~
and
W(j)2(t)
Pr[Z(t) = 11 initially the ith unit was switched on and the other unit was kept as cold standby].
In order that the system is found to be in the down state at instant t, it is necessary that the ith unit that was switched on at t = 0 either goes for preventive maintenance or fails somewhere between 0 and t, say in (u,u+du; u < t), the instant u is then a renewal point (see [3, 4]). We have the relation
C~ gi(u)n(2J)(t-- u)du,
Gi(t) +
Jo
wherei, j = l , 2 and i + j = 3 . Taking the Laplace transforms of equations (2.1), (2.2), (2.3) and (2.4) we obtain
pi(s) = ai(s ) ~z~°(s)+ bl(s) g~)(s), ~,"(s) = C,gs)w~,'Vs) + C*(s)~?(s)d,As)w~(s) +d*(s)~'(s),
7Z~)(S) = eij(s)w(~)l (s ) -4-e*(s)n~J'(s) + uq(s)w~2(s) +u*(s)~'(s),
Ii ai(u)rc~°(t - u)du + _fl bi(u)~)(t - u)du'
Pi(t)
(2.4)
w~{ (t) = Gi(t) + fl gi(u) x~jl(t - u)du
We introduce the function
Pi(t) =
vi(u)~J)(t-u)du,
w([~(t) = ffi(t)+ f£ vi(u)n~)(t-u)du,
f(s). 2. THE AVAILABILITY ANALYSIS OF A DUPLICATION REDUNDANT SYSTEM WITH PREVENTIVE MAINTENANCE AND REPAIR
fo
w(li{(t) = ffi(t)+
w~3(s) = i = 1,2.
(2.1)
In order to write an equation for ~]0(.), we consider the following possibilities. (1) The operating unit goes for preventive maintenance before the ith unit comes back from the service facility after preventive maintenance. (2) The service facility completes the preventive maintenance work on the ith unit before the operating unit goes for preventive maintenance. (3) The operating unit fails before the ith unit comes back from the service facility after preventive maintenance. (4) The service facility completes the preventive maintenance work on the ith unit before the operating unit fails. Therefore, we have the relation
w~;(s) = fli(s) + gi(s) x(~)(s),
(2.5)
wi'~(s) = ~i(s) + v~(s)~ ( s ) and
w~(s) = Ms) + g,(s)~(s), where
Cij(s) = f / e Staj(t) ffi(t)dt, C*(s) = f /~ e-St aj(t) Vi(t)dt, dij(s) = fo~ e ~bj(t) V/(t)dt, d*(s) = f / e-~tbj(t)Vi(t) dt,
l~li)(t) = Ii aj(u) [ Vi(u)w]i] (t - u) + Vi(u)~ o/)(t - u)] du
+ .t~ bj(u) [~(u)w~i~(t - u) + V~(u)r@(t -
cO(s) = f/ e Staj(t)Gi(t)dt,
u)] du, (2.2)
wherei, j = 1,2 and Similarly, we obtain rc~)(t)
i+j=3.
f] a~(u) [(~i(u) w~ (t -- u) + Gi(u ) rt]J)(t+
o
e*(s) = f o e-~taj(t)Gi(t)dt'
u)] du
= fie
Stbj(t)Gi(t)dt,
;/
e-~tbj(t) Gi(t)dt,
u*(s) =
bj(u) [(7,i(u)w~)2(t -- u) oh(s) =
+ Gi(u) n~)(t -- u)] du,
e -" ~(t)dt,
(2.3) wherei, j =
1,2
We obtain further
and
fii(s) = f / e-~tGi(t)dt,
i+j=3. where
i,j = 1, 2
and
i + j = 3.
(2.6)
Duplication redundant system It is obvious that
Co(s ) + C*(s) = eu(s) + e*(s) = aj(s)
(2.7)
do(s ) + d*(s) = uq(s) + u~(s) = bj(s).
(2.8)
753
by the use of the standard expression for a bilinear form
and
X'A-IY=
Oy X'
for
IAI#0,
1.41 '
The set of transformed equations (2.5) can be solved
where X' is a row vector and Y is a column vector. Thus pl(s) is given by the relation
pi(s) = -X'A-iY, where
X=[-a,(s) Y ' = [0
0
0
0
-bl(S ) 01:0
01 ~l(s)
a2(s)
0
0
0
0
O~s(S) ill(s)
~l(S)
0
0
0],
fls(s)]
fl2(s) ill(s)
and A= 1
-C%(s)
0
-Oh(s)
1
-a%(s)
-d?s(s )
- C 12(s)
0
- dl2(s)
0 0 0
--C21(S) 0 0
0 0 0
0 0
1 0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
-v~(s)
0
0 0
0 0
o
I o
0 0 0 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 I 0 -e*s(S) 1 --u*2(s) -e'~l(s) 0 -u~,(s) 1 I
0 - v2(s) 0
- v ds) 0 0
0 0 0
0
0
0
-~(s)
-g2(s)
0
0
0
0
0
0
0
-g2(s)
-
v2(s) 0
-ods) I
o
0
I
o
0
0
0
0
-d21(s ) 0 0 0 0 --e12(s) 0 --ut2(s ) 0 0 --esl(s ) 0
By partitioning X', Y and A as before and by the use of the following property,
LL AA4_] 2
- A 4 . 1 A s K -1
A~.I+A41A2K-1A3AT*IJ
where K = A 1- A3A ~ tA s, we get pl(s) = A [ L - M N ] -
tMB,
where
A=[-at(s ) 0
B'=[a,ts)
L=
M=
-bl(S)
0],
as(s) ~l(s) as(s) /h(s) fl2(s) /h(s) fl,(s)],
1 -C~l(s )
-C*ds) 1
0 -d*t(s)
0
--e*2(s)
1
--u~'s(s)] '
-qds)
o
-u*l(s)
1_]
I --Cls(S )
0 -- C s 1(s) 0 0
-d12(s) 0 0 0
0 - d s l(s) 0 0
0 0 0
-d%(s)-]
O
[
0 0 - e 12(s) 0
0 0
0 0
0
-uMs)
-- e21 (s)
0
-u,~(s)
0 0 0
--U21(S)
754
G.S. MOKADDISand S. S. ELIAS
and
N =
0
-vds)
0
0
-v2(s) 0
0 0
0 0
0 -Ul(S )
0 0
0 -Ods)
-vds) 0
0 0
-02(s)
0
0
0
0
0
0
-eds)
0
0
-g2(s)
0
After some manipulation we obtain Laplace transforms of known functions, then the mean downtime of the system can be evaluated.
p,(s) = {al(S) [c 12(s) + d, 2(s)] I t(s) + ct2(s) [C2 ,(s) + d 21(s)] 12(s) -4- fl, (S) [e, 2(s) + u l 2(S)] 13(s) + fl2(s) [e2,(s)+u21(s)]I,(s)} J-l(s), (2.9)
3. THE RELIABILITY ANALYSIS OF A DUPLICATION REDUNDANT SYSTEM WITH PREVENTIVE MAINTENANCE AND REPAIR
where We introduce the function
11(s) = a,(s) [1 - Zds) Z 2 ( s ) - x,(s) y2(s)]
Pi(t) = Pr[Z(t) = 1,
+ bl(s) [x,(s)h2(s) + x2(s)Z,(s)],
I2(s)= al(s) [ h i ( s ) -
h,(s) Zl(s)Z2(s) + x,(s) y,(s)Zz(s)] + bt(s) [x,(s) + hl(s)x2(s)Zl(s) -- XI(S) X2(S)yl(S)],
Z(u) ¢ 1 ¥ 0 ~< u < t Iwhere initially the ith unit was switched on and the other unit was kept as cold standby] and obtain the following relation, Pi(t) =
13(S) = ai(s) [h,(s) y2(s) + y,(s)Zz(s)]
f£a,(u)P~,"(t-u)du+fl bi(u)P~)(t-u)du" (3,1)
+ bl(s) [1 -hl(s) hz(s)-xz(s)yl(s)],
I,(s) = al(s) [hl(s) y2(s)Zt(s) + y l ( x ) - xt(s) yl(s)y2(s)] + bl(S) [ Z l ( s ) - Zl(S) h,(s)h2(s ) + xt(s)yl(s)h2(s)]
We proceed to write equations for P(tO(.) and P~)(.). We obtain
P~)(t) = Ii aj(u) Vi(u)P~'(t-- u)du
and
+
J(s) = [1 - h,(s)h2(s)] [1 - Zl(s)Z2(s)] + [xl(s)xz(s)Yl(S)y2(s)-xi(s)y2(s) -- X2(S)yl(S ) - Xz(s) y2(s)hl(s)Zl(s ) - xds)y,(s)hz(s)Z2(s)],
f'
b~(u)V~(u)P~)(t-u)du
o
+ Vi(t) [at(t ) + b~(t)], (3.2)
P~)(t) = 1, as(u)Gi(u)P°l)(t-u)du
where
xi(s) -= e*(s) + gi(s)eu(s),
+ t~ bj(u) Gi(u) P~)(t-
yi(s) = d*(s) + Vi(s)dij(S),
+ Gi(t) [aj(t) + bj(t)],
hi(s) = C*(s) + vi(s) Cq(s)
u)du
for i,j= 1,2 and i + j = 3. Taking the Laplace transforms of equations (3.1) and (3.2) we obtain
and
Zi(s) = u*(s) + oi(s)%(s),
P i(s) = ai(s) P~i)(s)+ b i(s) P~)(s),
for i,j= 1,2 and i + j = 3 . The function pl(t) is then obtained after inversion numerically or otherwise. The mean downtime for the system during (0, t) is given by
E[fl Z(u)du] = fl E[Z(u)]du---f£ P,(u)du, (2.10) so that the Laplace transform of the mean downtime is p,(s)/s. Since pl(s) is known explicity in terms of the
e~)(s) = C~(s) P~J)(s)+ d*(s)P~)(s) + C,j(s) + do(s ) (3.3) and
P~)(s) = e~(s)P°/)(s) + u*(s)P~)(s) + e0(s ) + uo(s), for i,j= t,2 and i + j = 3 . Solving the set of equations (3.3) we obtain Al
nt(s)
= A~2 ,
(3.4)
Duplication redundant system
755
and
where
u*2(s) = u ' A s ) = u*(s).
A 1 = [C12(s)+d12(s)] {al(s ) [1 --u*2(s)u*t(s)
--e*z(s)d*l (s) ] + b ,(s) [u*2(s)e*l (s) + C*l(s)e*2(s)]}
Therefore (4.1) will take the form
f (s) fl(s) u(s) p(s) - 1 - u*(s)- 9(s)u(s)"
-t- [ C 2 l(S) + d 2 t(S)] {al(s) [C~2(s)
C'~2(s) u*2(s)u*l(s) + e*2(s) u~l(s)d%(s)] + b,(s) [e*l (s) u'~2(s) C*2(s) -d*2(s)e'~2(s)e*l(s)+ e*2(s)] } + [ex2(s) + ul2(s)] {al(s) [C~2(s)d~l(s)
(4.2)
-
+ d*2(s)u* ds)]
+ bl(s ) [1 - C*z(s)C'~I(s ) - d*2(s)e'~l(s)] } + [e21(s) + u2 l(s)] {al(s) [C*2(s)d'~l(s)u*2(s)
If the repair time is exponentially distributed 9(x) = / r e -~x, equation (4.2) gives the result by Srinivasan and Gopalan [1]. Similarly, for the reliability of the system, (3.4), under the assumption of no preventive ance but only repair, will take the form
such that obtained
~" "[u12(s)+u21(s)u*z(S)] PI(s) =JI~s) i_u*2(s)u,~l(S ~ "
(4.3)
+ d*2(s ) - d ~ 2 ( s ) d ~ l ( s ) e ~ 2 ( s ) ]
-b bl(S)[U~2(s)--u~2(s)C~2(s)C~l(S) + d*2(s)e*2(s) C~l (s)] }
Iffl(.) =f2(.) = f ( . ) and 91(.) = 92(.) = g(.), equation (4.3) will take the form
and
A 2 = 1 -u~2(s)u~l(s)-d~x(s)e~2(s)-
P(s)
C~2(s)C*l(s)
+ C*2(s) C'~(s)u*2(s)u*l(s) -- C~2(s) d~l(s)u~2(s)e~l(s ) -C21(s)d12(s)u21(s)e12(s)
d12(s)e21(s)
+ e*E(s)e*l(s)d*2(s)d'~l(s). The function Pl(t) is obtained by inverting the Laplace transform (3.4). The reliability function Rl(t ) is given by
so that its Laplace transform Rl(s ) is given by
C*(s) = C*(s),
cl,i(s) = d(s),
d*(s) = d*(s),
e,j(s) = e(s),
e*(s) = e*(s),
% ( s ) = u(s),
u*(s) = u*(s),
~(s) = ~(s),
ill(s) = 3(s),
~=o
(4.4)
1 -u*(s)
Gj(s) = C(s),
RI(s ) = [1 -- Pa(s)]/s. Since Pl(s) is known explicitly, R l(s) can be evaluated. The function R~(t) is obtained after inversion. The mean time to system failure is then
f(s) u(s)
If the repair time is exponentially distributed, the results obtained by Srinivasan and Gopalan [1] are directly obtained from equation (4.4). (2) If the two units are similar, so that fl(-) = fz(.) = f ( . ) , hx(. ) = h2(. ) = h(.), 91(.) = g2(-) = g(.) and vl(.) = v2(.) = v(.), then the Laplace transforms (2.6) will reduce to
g l ( t ) = f ~ Pl(u) du,
To = RI(S) ~=o = - d p l ( s )
equation mainten-
ai(s) = a(s) and bi(s) = b(s),
and hence equation (2.9) will take the form
p(s) =
a(s) D l (s) + b(s) D 2(s) , O(s)
(4.5)
where D 1(s) = 0t(s) [C(s) + d(s)] [ 1 - u* (s) - ,q(s) u(s)] + fl(s) [e(s) + u(s)] [d*(s) + v(s)d(s)],
O2(s) = ~(s) [C(s) + d(s)] [e*(s) + O(s) e(s)] + fl(s) [e(s) + u(s)] [ 1 - C* ( s ) - v(s) C(s)]
4. SOME SPECIAL CASES FOR THE AVAILABILITY AND RELIABILITY OF DIFFERENT REDUNDANT SYSTEMS
(1) If there is no preventive maintenance but only repair, i.e. v(.) = h(.) = 0, equation (2.9) will take the form
and
C(s) = [1 - C*(s) - v(s) C(s) [1 - u*(s) - O(s)u(s)] - [e*(s) + g(s)e(s)] [d*(s) + v(s) a(s)].
pl(S) =
fl(s){fll(s)u12(s) + fl2(s)u2,(s)[u*2(s) + 91(s)u12(s)] } 1 -- [u'~e(s) "4-yl(s)ul z(S)] [u*l(s) + O2(S)Uz l(S)] (4.1) F o r the case of similar units, i.e. for the case of f~(.) =fz(.) = f ( . )
and
01(.) = 02(.) = 0(.),
the Laplace transforms (2.6) will reduce to 3 d s ) = lh(s) = B(s),
u l g s ) = u2ds) = u(s)
Similarly, equation (3.4) will take the form
P(s)
a(s) A l (s) + b(s) A 2(s ) , a(s)
(4.6)
where A l(s) = [C(s) + d(s)] [ 1 - u*(s)] + d*(s) [e(s) + u(s)],
A2(s ) = e*(s) [C(s) + d(s)] + [e(s) + u(s)] [1 - C*(s)] and
756
G.S. MOKADDISand S. S. ELIAS A(s) = [1 - u*(s)] [1 - C*(s)] - e*(s)d*(s).
For exponential preventive maintenance time and repair time the results obtained by Gopalan and d'Souza [2] can be easily found by direct substitution in equations (4.5) and (4.6). REFERENCES
1. S. K. Srinivasan and M. N. Gopalan, Probabilistic
analysis of a two-unit cold standby system with a single repair facility, IEEE Trans. Reliab. R-22, 250-254 (1973). 2. M. N. Gopalan and C. A. d'Souza, Probabilistic analysis of a two-unit system with a single service facility for preventive maintenance and repair, Ops Res. Soc. Am. 23, 173-177 (1975). 3. R. Bellman and T. E. Harris, On the theory of age dependent stochastic branching processes, Proc. natn. Acad. Sci. U.S.A. 34, 601-604 (1948). 4. W. L. Smith, Regenerative stochastic processes, Proc. R. Soc. A 232, 6-13 (1955).
0026 2714/8953.00+ .00 © 1989 Pergamon Press pie
Printed in Great Britain.
THE RELIABILITY F U N C T I O N A N D THE AVAILABILITY OF A DUPLICATION R E D U N D A N T SYSTEM WITH A SINGLE SERVICE FACILITY FOR PREVENTIVE MAINTENANCE A N D REPAIR G. S. MOKADDIS and S. S. ELIAS Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt (Received for publication 14 November 1988) Abstract--This paper concerns a two-unit system with a cold standby and a single service facility for the performance of preventive maintenance and repair. Explicit expressions for the Laplace transforms of the availability of the system, the reliability, the mean down time during (0, t) and for the mean time to system failure have been obtained under the assumption that the failure times, the inspection times, the repair times and the preventive maintenance times of the two units are governed by distinct arbitrary general distributions. The results obtained by Srinivasan and Gopalan and by Gopalan and d'Souza are derived from the present results as special cases.
1. INTRODUCTION
operative and inoperative periods, known as the up and the down periods. In this paper the system is characterized by the probability of its being in the up state or the down state. In order to find the reliability and the availability of this system, integral equations for these probabilities corresponding to different initial conditions are set up by identifying suitable regenerative stochastic processes.
Many authors have considered the reliability analysis of systems with standbys and several independent repair facilities. However, it may not be possible to have more than one repair facility in many practical situations. Srinivasan and G o p a l a n [1] have carried out the probabilistic analysis of a two-unit system with a cold standby and a single repair facility. Recently, G o p a l a n and d'Souza I'2] have considered the same situation where the system is subject to preventive maintenance under the assumption that the repair time and the preventive maintenance time of a unit are exponentially distributed. The purpose of this paper is to analyse the availability and the reliability of a two unit system with a cold standby and a single service facility for the performance of preventive maintenance and repair under the assumption that the two units are not similar and that the failure times, the inspection times, the repair times and the preventive maintenance times of the two units are governed by distinct arbitrary general distributions. Initially, the first unit is switched on while the second unit is kept as a cold standby. The failure rate of a cold standby is assumed to be zero. The standby is switched on instantaneously when the operating unit is sent for preventive maintenance or for repair. If the service facility completes the preventive maintenance or the repair for a unit while the system is still functioning, the unit is placed in standby. If, however, the unit that has been operating fails or is sent for preventive maintenance before the service facility becomes free, the unit waits as if in a queue and is taken up immediately for maintenance or for repair when the service facility becomes free. In such a case, the system ceases to operate until the unit already undergoing either preventive maintenance or repair is made available for operation. The system thus passes through cycles of NR 29:5-F
The following notation is adopted (note that i = 1, 2). f~(.) probability density function of the failure time of the ith unit with distribution function Fi(.) h~(.) probability density function of the inspection time of the ith unit with distribution function Hi(.) #i(-) probability density function of the repair time of the ith unit with distribution function Gi(.) v~(.) probability density function of the preventive maintenance time of the ith unit with distribution function Vii(.) Z(t) a discontinuous random variable assuming the value 1 if the system is found in the down state at instant t and the value 0 otherwise For simplicity, we write q~(.) = 1 - qS(.) for any function ~b(.). ai(u) = hi(u)Fi(u),
bi(u) = fi(u)Hi(u).
n~i)(t) PrI-Z(t) = l I Z(0) = 0, the ith unit enters the service facility at t = 0 for preventive maintenance] n~(t) Pr[Z(t) = l I Z(0) = 0, the ith unit enters the service facility at t = 0 for repair] wt~'~(t) Pr[Z(t) = 1 ]Z(0) = 1, the ith unit is under preventive maintenance and the other waits for preventive maintenance] w~i~(t) Pr[Z(t) = l l Z ( 0 ) = 1, the ith unit is under preventive maintenance and the other waits for repair] wt~(t) Pr[Z(t) = I l Z ( 0 ) = 1, the ith unit is under repair and the other waits for preventive maintenance] w~)2(t) Pr[Z(t) = 1 r Z(0) = 1, the ith unit is under repair and the other waits for repair] e~(t) P r [ Z ( t ) = l , Z(u) v~ l V-O <. u < tt Z(O) = O, the ith unit enters the service facility at t = 0 for preventive maintenance] 751
752
G . S . MOKADDISand S. S. ELIAS
P~)(t)
Pr[Z(t)---l, Z(u)# l VO~
and
W(j)2(t)
Pr[Z(t) = 11 initially the ith unit was switched on and the other unit was kept as cold standby].
In order that the system is found to be in the down state at instant t, it is necessary that the ith unit that was switched on at t = 0 either goes for preventive maintenance or fails somewhere between 0 and t, say in (u,u+du; u < t), the instant u is then a renewal point (see [3, 4]). We have the relation
C~ gi(u)n(2J)(t-- u)du,
Gi(t) +
Jo
wherei, j = l , 2 and i + j = 3 . Taking the Laplace transforms of equations (2.1), (2.2), (2.3) and (2.4) we obtain
pi(s) = ai(s ) ~z~°(s)+ bl(s) g~)(s), ~,"(s) = C,gs)w~,'Vs) + C*(s)~?(s)d,As)w~(s) +d*(s)~'(s),
7Z~)(S) = eij(s)w(~)l (s ) -4-e*(s)n~J'(s) + uq(s)w~2(s) +u*(s)~'(s),
Ii ai(u)rc~°(t - u)du + _fl bi(u)~)(t - u)du'
Pi(t)
(2.4)
w~{ (t) = Gi(t) + fl gi(u) x~jl(t - u)du
We introduce the function
Pi(t) =
vi(u)~J)(t-u)du,
w([~(t) = ffi(t)+ f£ vi(u)n~)(t-u)du,
f(s). 2. THE AVAILABILITY ANALYSIS OF A DUPLICATION REDUNDANT SYSTEM WITH PREVENTIVE MAINTENANCE AND REPAIR
fo
w(li{(t) = ffi(t)+
w~3(s) = i = 1,2.
(2.1)
In order to write an equation for ~]0(.), we consider the following possibilities. (1) The operating unit goes for preventive maintenance before the ith unit comes back from the service facility after preventive maintenance. (2) The service facility completes the preventive maintenance work on the ith unit before the operating unit goes for preventive maintenance. (3) The operating unit fails before the ith unit comes back from the service facility after preventive maintenance. (4) The service facility completes the preventive maintenance work on the ith unit before the operating unit fails. Therefore, we have the relation
w~;(s) = fli(s) + gi(s) x(~)(s),
(2.5)
wi'~(s) = ~i(s) + v~(s)~ ( s ) and
w~(s) = Ms) + g,(s)~(s), where
Cij(s) = f / e Staj(t) ffi(t)dt, C*(s) = f /~ e-St aj(t) Vi(t)dt, dij(s) = fo~ e ~bj(t) V/(t)dt, d*(s) = f / e-~tbj(t)Vi(t) dt,
l~li)(t) = Ii aj(u) [ Vi(u)w]i] (t - u) + Vi(u)~ o/)(t - u)] du
+ .t~ bj(u) [~(u)w~i~(t - u) + V~(u)r@(t -
cO(s) = f/ e Staj(t)Gi(t)dt,
u)] du, (2.2)
wherei, j = 1,2 and Similarly, we obtain rc~)(t)
i+j=3.
f] a~(u) [(~i(u) w~ (t -- u) + Gi(u ) rt]J)(t+
o
e*(s) = f o e-~taj(t)Gi(t)dt'
u)] du
= fie
Stbj(t)Gi(t)dt,
;/
e-~tbj(t) Gi(t)dt,
u*(s) =
bj(u) [(7,i(u)w~)2(t -- u) oh(s) =
+ Gi(u) n~)(t -- u)] du,
e -" ~(t)dt,
(2.3) wherei, j =
1,2
We obtain further
and
fii(s) = f / e-~tGi(t)dt,
i+j=3. where
i,j = 1, 2
and
i + j = 3.
(2.6)
Duplication redundant system It is obvious that
Co(s ) + C*(s) = eu(s) + e*(s) = aj(s)
(2.7)
do(s ) + d*(s) = uq(s) + u~(s) = bj(s).
(2.8)
753
by the use of the standard expression for a bilinear form
and
X'A-IY=
Oy X'
for
IAI#0,
1.41 '
The set of transformed equations (2.5) can be solved
where X' is a row vector and Y is a column vector. Thus pl(s) is given by the relation
pi(s) = -X'A-iY, where
X=[-a,(s) Y ' = [0
0
0
0
-bl(S ) 01:0
01 ~l(s)
a2(s)
0
0
0
0
O~s(S) ill(s)
~l(S)
0
0
0],
fls(s)]
fl2(s) ill(s)
and A= 1
-C%(s)
0
-Oh(s)
1
-a%(s)
-d?s(s )
- C 12(s)
0
- dl2(s)
0 0 0
--C21(S) 0 0
0 0 0
0 0
1 0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
-v~(s)
0
0 0
0 0
o
I o
0 0 0 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 I 0 -e*s(S) 1 --u*2(s) -e'~l(s) 0 -u~,(s) 1 I
0 - v2(s) 0
- v ds) 0 0
0 0 0
0
0
0
-~(s)
-g2(s)
0
0
0
0
0
0
0
-g2(s)
-
v2(s) 0
-ods) I
o
0
I
o
0
0
0
0
-d21(s ) 0 0 0 0 --e12(s) 0 --ut2(s ) 0 0 --esl(s ) 0
By partitioning X', Y and A as before and by the use of the following property,
LL AA4_] 2
- A 4 . 1 A s K -1
A~.I+A41A2K-1A3AT*IJ
where K = A 1- A3A ~ tA s, we get pl(s) = A [ L - M N ] -
tMB,
where
A=[-at(s ) 0
B'=[a,ts)
L=
M=
-bl(S)
0],
as(s) ~l(s) as(s) /h(s) fl2(s) /h(s) fl,(s)],
1 -C~l(s )
-C*ds) 1
0 -d*t(s)
0
--e*2(s)
1
--u~'s(s)] '
-qds)
o
-u*l(s)
1_]
I --Cls(S )
0 -- C s 1(s) 0 0
-d12(s) 0 0 0
0 - d s l(s) 0 0
0 0 0
-d%(s)-]
O
[
0 0 - e 12(s) 0
0 0
0 0
0
-uMs)
-- e21 (s)
0
-u,~(s)
0 0 0
--U21(S)
754
G.S. MOKADDISand S. S. ELIAS
and
N =
0
-vds)
0
0
-v2(s) 0
0 0
0 0
0 -Ul(S )
0 0
0 -Ods)
-vds) 0
0 0
-02(s)
0
0
0
0
0
0
-eds)
0
0
-g2(s)
0
After some manipulation we obtain Laplace transforms of known functions, then the mean downtime of the system can be evaluated.
p,(s) = {al(S) [c 12(s) + d, 2(s)] I t(s) + ct2(s) [C2 ,(s) + d 21(s)] 12(s) -4- fl, (S) [e, 2(s) + u l 2(S)] 13(s) + fl2(s) [e2,(s)+u21(s)]I,(s)} J-l(s), (2.9)
3. THE RELIABILITY ANALYSIS OF A DUPLICATION REDUNDANT SYSTEM WITH PREVENTIVE MAINTENANCE AND REPAIR
where We introduce the function
11(s) = a,(s) [1 - Zds) Z 2 ( s ) - x,(s) y2(s)]
Pi(t) = Pr[Z(t) = 1,
+ bl(s) [x,(s)h2(s) + x2(s)Z,(s)],
I2(s)= al(s) [ h i ( s ) -
h,(s) Zl(s)Z2(s) + x,(s) y,(s)Zz(s)] + bt(s) [x,(s) + hl(s)x2(s)Zl(s) -- XI(S) X2(S)yl(S)],
Z(u) ¢ 1 ¥ 0 ~< u < t Iwhere initially the ith unit was switched on and the other unit was kept as cold standby] and obtain the following relation, Pi(t) =
13(S) = ai(s) [h,(s) y2(s) + y,(s)Zz(s)]
f£a,(u)P~,"(t-u)du+fl bi(u)P~)(t-u)du" (3,1)
+ bl(s) [1 -hl(s) hz(s)-xz(s)yl(s)],
I,(s) = al(s) [hl(s) y2(s)Zt(s) + y l ( x ) - xt(s) yl(s)y2(s)] + bl(S) [ Z l ( s ) - Zl(S) h,(s)h2(s ) + xt(s)yl(s)h2(s)]
We proceed to write equations for P(tO(.) and P~)(.). We obtain
P~)(t) = Ii aj(u) Vi(u)P~'(t-- u)du
and
+
J(s) = [1 - h,(s)h2(s)] [1 - Zl(s)Z2(s)] + [xl(s)xz(s)Yl(S)y2(s)-xi(s)y2(s) -- X2(S)yl(S ) - Xz(s) y2(s)hl(s)Zl(s ) - xds)y,(s)hz(s)Z2(s)],
f'
b~(u)V~(u)P~)(t-u)du
o
+ Vi(t) [at(t ) + b~(t)], (3.2)
P~)(t) = 1, as(u)Gi(u)P°l)(t-u)du
where
xi(s) -= e*(s) + gi(s)eu(s),
+ t~ bj(u) Gi(u) P~)(t-
yi(s) = d*(s) + Vi(s)dij(S),
+ Gi(t) [aj(t) + bj(t)],
hi(s) = C*(s) + vi(s) Cq(s)
u)du
for i,j= 1,2 and i + j = 3. Taking the Laplace transforms of equations (3.1) and (3.2) we obtain
and
Zi(s) = u*(s) + oi(s)%(s),
P i(s) = ai(s) P~i)(s)+ b i(s) P~)(s),
for i,j= 1,2 and i + j = 3 . The function pl(t) is then obtained after inversion numerically or otherwise. The mean downtime for the system during (0, t) is given by
E[fl Z(u)du] = fl E[Z(u)]du---f£ P,(u)du, (2.10) so that the Laplace transform of the mean downtime is p,(s)/s. Since pl(s) is known explicity in terms of the
e~)(s) = C~(s) P~J)(s)+ d*(s)P~)(s) + C,j(s) + do(s ) (3.3) and
P~)(s) = e~(s)P°/)(s) + u*(s)P~)(s) + e0(s ) + uo(s), for i,j= t,2 and i + j = 3 . Solving the set of equations (3.3) we obtain Al
nt(s)
= A~2 ,
(3.4)
Duplication redundant system
755
and
where
u*2(s) = u ' A s ) = u*(s).
A 1 = [C12(s)+d12(s)] {al(s ) [1 --u*2(s)u*t(s)
--e*z(s)d*l (s) ] + b ,(s) [u*2(s)e*l (s) + C*l(s)e*2(s)]}
Therefore (4.1) will take the form
f (s) fl(s) u(s) p(s) - 1 - u*(s)- 9(s)u(s)"
-t- [ C 2 l(S) + d 2 t(S)] {al(s) [C~2(s)
C'~2(s) u*2(s)u*l(s) + e*2(s) u~l(s)d%(s)] + b,(s) [e*l (s) u'~2(s) C*2(s) -d*2(s)e'~2(s)e*l(s)+ e*2(s)] } + [ex2(s) + ul2(s)] {al(s) [C~2(s)d~l(s)
(4.2)
-
+ d*2(s)u* ds)]
+ bl(s ) [1 - C*z(s)C'~I(s ) - d*2(s)e'~l(s)] } + [e21(s) + u2 l(s)] {al(s) [C*2(s)d'~l(s)u*2(s)
If the repair time is exponentially distributed 9(x) = / r e -~x, equation (4.2) gives the result by Srinivasan and Gopalan [1]. Similarly, for the reliability of the system, (3.4), under the assumption of no preventive ance but only repair, will take the form
such that obtained
~" "[u12(s)+u21(s)u*z(S)] PI(s) =JI~s) i_u*2(s)u,~l(S ~ "
(4.3)
+ d*2(s ) - d ~ 2 ( s ) d ~ l ( s ) e ~ 2 ( s ) ]
-b bl(S)[U~2(s)--u~2(s)C~2(s)C~l(S) + d*2(s)e*2(s) C~l (s)] }
Iffl(.) =f2(.) = f ( . ) and 91(.) = 92(.) = g(.), equation (4.3) will take the form
and
A 2 = 1 -u~2(s)u~l(s)-d~x(s)e~2(s)-
P(s)
C~2(s)C*l(s)
+ C*2(s) C'~(s)u*2(s)u*l(s) -- C~2(s) d~l(s)u~2(s)e~l(s ) -C21(s)d12(s)u21(s)e12(s)
d12(s)e21(s)
+ e*E(s)e*l(s)d*2(s)d'~l(s). The function Pl(t) is obtained by inverting the Laplace transform (3.4). The reliability function Rl(t ) is given by
so that its Laplace transform Rl(s ) is given by
C*(s) = C*(s),
cl,i(s) = d(s),
d*(s) = d*(s),
e,j(s) = e(s),
e*(s) = e*(s),
% ( s ) = u(s),
u*(s) = u*(s),
~(s) = ~(s),
ill(s) = 3(s),
~=o
(4.4)
1 -u*(s)
Gj(s) = C(s),
RI(s ) = [1 -- Pa(s)]/s. Since Pl(s) is known explicitly, R l(s) can be evaluated. The function R~(t) is obtained after inversion. The mean time to system failure is then
f(s) u(s)
If the repair time is exponentially distributed, the results obtained by Srinivasan and Gopalan [1] are directly obtained from equation (4.4). (2) If the two units are similar, so that fl(-) = fz(.) = f ( . ) , hx(. ) = h2(. ) = h(.), 91(.) = g2(-) = g(.) and vl(.) = v2(.) = v(.), then the Laplace transforms (2.6) will reduce to
g l ( t ) = f ~ Pl(u) du,
To = RI(S) ~=o = - d p l ( s )
equation mainten-
ai(s) = a(s) and bi(s) = b(s),
and hence equation (2.9) will take the form
p(s) =
a(s) D l (s) + b(s) D 2(s) , O(s)
(4.5)
where D 1(s) = 0t(s) [C(s) + d(s)] [ 1 - u* (s) - ,q(s) u(s)] + fl(s) [e(s) + u(s)] [d*(s) + v(s)d(s)],
O2(s) = ~(s) [C(s) + d(s)] [e*(s) + O(s) e(s)] + fl(s) [e(s) + u(s)] [ 1 - C* ( s ) - v(s) C(s)]
4. SOME SPECIAL CASES FOR THE AVAILABILITY AND RELIABILITY OF DIFFERENT REDUNDANT SYSTEMS
(1) If there is no preventive maintenance but only repair, i.e. v(.) = h(.) = 0, equation (2.9) will take the form
and
C(s) = [1 - C*(s) - v(s) C(s) [1 - u*(s) - O(s)u(s)] - [e*(s) + g(s)e(s)] [d*(s) + v(s) a(s)].
pl(S) =
fl(s){fll(s)u12(s) + fl2(s)u2,(s)[u*2(s) + 91(s)u12(s)] } 1 -- [u'~e(s) "4-yl(s)ul z(S)] [u*l(s) + O2(S)Uz l(S)] (4.1) F o r the case of similar units, i.e. for the case of f~(.) =fz(.) = f ( . )
and
01(.) = 02(.) = 0(.),
the Laplace transforms (2.6) will reduce to 3 d s ) = lh(s) = B(s),
u l g s ) = u2ds) = u(s)
Similarly, equation (3.4) will take the form
P(s)
a(s) A l (s) + b(s) A 2(s ) , a(s)
(4.6)
where A l(s) = [C(s) + d(s)] [ 1 - u*(s)] + d*(s) [e(s) + u(s)],
A2(s ) = e*(s) [C(s) + d(s)] + [e(s) + u(s)] [1 - C*(s)] and
756
G.S. MOKADDISand S. S. ELIAS A(s) = [1 - u*(s)] [1 - C*(s)] - e*(s)d*(s).
For exponential preventive maintenance time and repair time the results obtained by Gopalan and d'Souza [2] can be easily found by direct substitution in equations (4.5) and (4.6). REFERENCES
1. S. K. Srinivasan and M. N. Gopalan, Probabilistic
analysis of a two-unit cold standby system with a single repair facility, IEEE Trans. Reliab. R-22, 250-254 (1973). 2. M. N. Gopalan and C. A. d'Souza, Probabilistic analysis of a two-unit system with a single service facility for preventive maintenance and repair, Ops Res. Soc. Am. 23, 173-177 (1975). 3. R. Bellman and T. E. Harris, On the theory of age dependent stochastic branching processes, Proc. natn. Acad. Sci. U.S.A. 34, 601-604 (1948). 4. W. L. Smith, Regenerative stochastic processes, Proc. R. Soc. A 232, 6-13 (1955).