- Email: [email protected]
Optimum preventive maintenance policies for a 2-unit redundant system with repair and post-repair
Mtcroelectron. Reliab., Vol.20, pp. 887-890.
0026--2714/80/1201-0887502.0010
PergamonPressLtd 1980.Printed in Great Britain
OPTIMUM I~REVENTIVE MAINTENANCE POLICIES FOR A 2-UNIT R E D U N D A N T SYSTEM WITH REPAIR A ~ POST-REPAIR i
,
S. M. SINHA Department of Operational Research, University of Delhi, India and D. V. S. KAPm Department of Mathematics, Madhav Institute of Technology and Science, Gwalior-474005, India
(Receivedfor publication 12 April 1979) Abstract--Stochastic behaviour of a 2-unit cold-standby redundant system with repair and post-repair and subject to slide preventive maintenance is discussed under the most generalized assumptions. We employ (MRP) Marker Renewal processes technique to obtain various reliability measure~ Optimum preventive policies which maximize the availability are derived under suitable conditions. A numerical example is presented.
INTRODUCTION
That system reliability can be vastly improved by introducing redundancy is well known. Reliability of the s y s ~ is also improved by repair and/or pre~re~tive maintenance. Many contributions to such systems have been made. Optimum PM (preventive maintenance) policies by maximizing or minimizing some criterion have been derived for simple as well as redundant systems; see [L3,4 ]. Recently authors [2] discussed the stochastic behaviour of 2-unit redundant system with repair an d post-repair under the most general assumptions. However, it is of interest to take account of a preventive maintenance for such a system. With this objective in view, we first discuss the stochastic behaviour of a 2-unit redundant system with repair and post-repair and subject to slide PM policy. Such a PM policy is reasonable and practical in application; see [4, 6]. Further, we discuss the optimum PM policies maximizing the steady state availability. Finally, a numerical example is presented. A model with repair, post-repair and PM includes various earlier models as special cases. ASSUMPTIONS
I. The system has two identical units. A unit in standby never deteriorates or fails. 2. Each switchover is perfect and instantaneous. 3. When a unit goes down ("fails" or "its PM time arrives") it goes for service ("repair, post-repair" or PM). 4. A unit on repair goes for post-repair. After postrepair a unit is like new. 5. When a unit operates for a specified length of time without failure, it goes to PM. After PM a unit is like new.~ 6. PM of an operative unit is made only if another unit is in standby. 7. An operative unit, which forfeited PM because
8.
9.
10. 11. 12. 13.
14.
15.
of assumption 6, undergoes PM upon service completion of the down unit. If the operative unit goes down and another is in standby, the downunit undergoes service immediately and the standby unit takes over the operation. If the operative unit goes down while another is under service, it awaits service until the service completion of the other unit. There is one server which is always available. Each operative unit has increasing failure rate so as to make PM effective. The failure time"distribution of each unit is F(t) with mean 1/L The repair and post-repair time distributions are Gi(t) and G2(t) with means 1//al and 1//a2, respectively. The time to begin PM is a random variable with distribution A(t) and time to accomplish PM has distribution G3(t) with mean 1//a3. All the random variables are mutually independent. ANALYSIS
We define the foUowjng time instants at which the system makes a transition into states: - 1 A unit begins to operate and another begins to be in standby. 0 Post-repair or PM of a unit is completed and the other is operating. t A unit begins to operate and repair of the failed unit begins. 2 A unit is operating and post=repair of another unit begins after repair completion. 3 A unit begins to Operate and PM of other begins, 4 An operating unit fails while other is under repair. 5 An operating unit fails while other is under postrepair or PM. 887
888
S.M. S~NnAand D. V. S. KAPIL
Note that the time instants at which the system enters states 0, 2, 4 and '5 are non-regeneration points and time instants for states i and 3 are regeneration points. By similar argutnen~ as used in [2~, 5], ttie following-mass~anctions of the MRPs are derived:
Q- 11(t) = fl A(x)dF(x),
(1)
Q- la(t)
= fl F(x)dA (x),
(2)
Q12(t) =
foF(X)dG1 (x),
(3)
Q~,(O= fo' gdx)dF(x~
(4)
Q~5~(t)=
fl F(x)A(x) d [Gl(x) * G2(x)],
(5)
Q~2J(t)=
fl F(x)A(x) d [Gt(x) * G2(x)],
(6)
(18)
whore Q~(t) is Cdf of MRP, i~e., the' probability that after transi "t~g into state i, the process noxt enters into state j, in an amount of time less than or equal to t, QlT)(t)is Cdf of MRP from state i intoj via state m, and Q~7")(t) is Cdf of M R P from state i into j via states m and n. From the Laplace-Stieltjes (LS) transforms of (1)(18), we can derive the mean recurrence time for a certain state, the limiting probabilities that the system is in a certain state in the steady state, and the expected number of visits to a certain state per unit time in the steady state. Let I1dl33) be the mean mourrence time for state 1(3). Then 111 =
Q~2c~(t)=
foF(x) dGa (x),
[ 1/#1 + I/#2 + fo°F(t)A(t){Gl(t)*G2(t)} dt1
+IoP(t)A(t)G3't)dt}]/[1-;:A(t'G3(t)dF(t)], (19)
Q~2~(t) = f l [Gl(X) * (~2(x)] (iF(x),
Qi'~(t)= f~ F(x)dG, (x),
(8)
~,ooA(t)[Gl(t), G2(t)] dF(t).
(20)
30
Q~'°~(t) = fl X(x)[Gl(x), 6~(x)] dr(x), Q[4"s)(t)= [fl F(x)dG(x )
* G2(t),
(9)
(lO)
Let Pj (j = 0-5) denote the limiting probabilities that the process is in statej in the steady state. Then
Po =
F(t)X(t)[Gl(t)*. G 2 (t)] dt/111 +
Q~21s)(t)= f~f~[F(x)-F(u)]dGl(u)dG2(x-u), (11) Q~2~°)(t)= f~ [Gdx) * G2(x)]F(x) dA(x), (12)
Q~o(t)= f~F(x)A(~)dG~(x~
(13)
Pi
P2 =
Q~(t)--f~e,~(x)dF(x),
= I- l~(t)Gq(t)dt/111, Jo
fo
F(t)[G~(0, g2(t)]
dt/111,
(21) (22)
(23)
fooF(t)G3(t)dt/133,
(24)
= [; F(t)Gl(t)dt/ll 1,
(25)
Pa = P~
f- F(t)A(t)G3(t)dt/la~,
jo
do
(14)
P~ = l; F(t)EGl(t), ~2(t)] dt/tl , Q3a(t) = f l
F(x)A(x) dGa (x),
Q~°:~(t)-- fo Gs(x)F(x) dA (x),~
(15) (16)
+
Nt)G3(t)dt/133.
(26)
Note that Po + Pt + P2 + P3 is the steady state availability and P4 + P~ is the steady state,u~vailability. 5
Q~°i(t)= fo A(x)G3(x)dF (x),
(17)
It is evident that ~ Pj -- 1. 1=o Let Mj ( / = 0-5) denote the expected numbers of
Optimum preventive maintenance policies for a 2-unit redundant system
889
M 1 = 1/ljj, j = 1,3,
(28)
M2. = qlz(O)/ltl,
(29)
L(0) is.the availability that tmoperating unit undergoes P M upon post-repair or P M completion (i.e. to ffi 0), and L(oo) is the availability that no PM is mtide. Our interest is to obtain the optimum scheduled time tff maximizing tim availability in (33). Therofore, putting U(to) = O, we have
M+ = q1+(0)/111,
(30)
r(to)[P/m - P3(x/m
visits to state j per unit time in the steady state. Then
Mo =
q~12d(O)llll + q3o(0)/133,
M s = q~*~(0)/111 + q35(0)/133.
(27)
(31)
+ 1/#2) - Ps3(o) + P3Jl~(0)]
+ fit3(0) -- fl3112(0) -- fl Io ° 83 (t)dS(t) + #3
OPTIMUM PM POLICIES
We have discussed a random PM in general in the prig sections. Now we consider an age PM policy which could be adopted in practical applications, we assume {~
A(t) =
for t < to, for t __>to.
x
/.(to) ffi
0' = J o F(0G~(t)dt
13(oo)]
[1/#1 + t/#2 - J12(0)] [1 - i3(oo)] + [ 1 / m - S3(0)]lt2(~) 03)
~12 ~
(44)
P = #1
(45)
fo
+
#12,
and r(t) = f(O/F(t) denotes the failure rate. It is assumed that F(t) has a dcnsityf(t). Further introduce the notation
fl f o G3(t)dF(t) + [J3 f ; [Gl(t)*G2(t)]dF(t) film -
(i -- 1, 3),
(43)
. Nt)[Gt(t)iG2(t)]dt,
m~
where OF(t) G~(t) dt
(i ffi 1, 3),
(32)
+ ['i'3 - J_,,(0)]I12(oo)
?~ =
(42)
where
Steady state availability L(to) is,obtained from (21)-(24) as [71 + ?r, - S l i ( O ) ] [ 1 -
;oo[ G l ( t ) * G2(t)] d F (t),
P3(l/gx + 1/#2) (46)
(34) M =/J~lOl
-
P,).
(47)
712 = I o F(t)[ Gt(t) * G2(t)] dr,
(35)
Then we have
I3(t)fftlG3(t)dF(t),
(36)
112(t)=f,i[dl(t)*G2(t)]dF(t),
(37)
Theorem 1. Suppose [Gl(t)*G2(t)]
J3(t)f f+i F(t)G3(t)dt,
(38)
J12(t) =
F(t)[Gl(t)* G2(t)] dr.
(39)
t~ = oo, i.e. no scheduling.
o
Consider the following special cases of the PM policies. From (33~ we have
['}11 "l- ~111]
o
[l//~x + 1//~2
if
+ ?3 [ ~ [GI(0* tT,(t)] de(t) ao G-s(t)dF(t) + l/m
(iii) If fl/#3 > fl3(1/#1 + 1/#2) and r(0) ~_ m, then
the optimum scheduled time is t~ = O, i.e. PM is made just upon post-repair or PM completion. Theorem 1 is easily proved in a similar manner to Nakagawa and Osaki [4]. We omit the proof.
G3(t)dF(t)
L(0) =
(i) / f r(oo) > M, P/#3 > P3(1/#1 + 1/#2) and r(0) < m, or r(oo) > M and ill#3 ~ fla(l/pt + 1/#2), then there exists a finite and unique optimum scheduled time t~ satisfyin# (42). (ii) Ifr(ov) _< M, then the optimum scheduled time is
[610)* G , ( t ) ] ~ ( t ) (4O)
1/2 L(oo) = 1/#1 + 1/#2 + 1/2 - Yl - ~12"
(41)
NUMERICAL EXAMPLE
We have discussed the optimum PM policies by adopting as the criterion of optimality the steady state availability. Now we consider a numerical example. We assume that G~t) = 1 - exp ( - pit) (iffi 1, 2, 3) and dF(t) = ~(~) exp (-art) (a > 0). Failure time distribution is a Gamma distribution with shape parameter 2 and has increasing failure rate.
890
S, M. Sn~ilx and D. V. S. KApU+ Table 1
1/Mean PM time Optimum sc+heduledtime ' Availability 2.0 3.0 4.0 5.0
0.5330 0.2886 0.2042 0.1613 0.1344 0.1157 0.1018 0.0910 0.0825
6.0
7.0 8.0 9.0 10.0
0.5882 0.6175 0.6403 0.6591 0.6764 0.6898 0.70:~7 0.7145 0.7252
Guessing t h e injti~! valt~ of to from r0) < M+ we have obtained unique solution t~ by using successive approximation in (48). For Optimum scheduled time the availability is derived from (33). Table 1 shows the dependence of the mean PM time in the optimum scheduled time t~ maximizing the availability. Setting 2 / ~ - 1, 1/#t = 1, 1//z2 -- 1/2, the availability with no PM L(oo) is 0,5669. CONCLUSIONS
Then we have r(t) = ~2t/(1 -t- ~t). Failure rate is continuous and monotonely hlfreasing with r(0) = 0 and From Theorem 1, if u < M, then no PM is made. We adopt a finite and unique sclleduled time t~ as the optimum policy, if t8 satisfies (42) and > #2/(# - #A i.e. # > 2#3, i.e.
We derived the reliability measures by+using the mass ~ n s of MI~Ps. Our results i n c l u ~ + ~ a l cases, those for the 2-unit r e d u n ~ t S y s t ~ ' + W t t h ; ~ and PM see [4]; 2'unit redundant system with ~ i r and post-repair see [2] and other models also. F i J ~ e r we discussed the optimum PM policies by maximizing the steady state availability. The above numerical results on comparing with those obtained in [4] enable us ~o determine the effect of introducing post-repair on optimum scheduled time and availability. R~n~NC~
(].12 " /21)
I((Z~ ]~1)~-
]A2(~ "1- ] ~ p J
P3(0[ "~ p3) 2"
For given distribution (42) is simplified as 0(2~ e - (~t+ ~3)to
P - (P - p3)e-"° +
(~ + # 3 ) 2 ~2j~3 J-p1e-('+~')io
to=
+
L (.
+
f12e-(,+ Pt)lo1
-J
- 2P3)
(48)
1. R. E. Barlow and F. Prosehan, Mathematical Theory of Reliability. Wiley, New York (1965), 2, P. K: Kaput and K. R. Kapoor, 1EEE Trans. Reliab. R-27, 382-385 (1978). 3. P. M. Morse, Queues,Inventories and Maintenance. Wiley, New York (1958). 4. T. Nakagawa and S. Osaki, Z. Ops Res. 20, 171-187 (1976)i 5. T. Nakagawa and S. Osaki, Microelectron. Reliab. 15, 633-636 (1976). 6. D. V. Rozhdestvenskiy and G. N. Fanarzhi, Enono Cybernetics. 475--479 (1970).
0026--2714/80/1201-0887502.0010
PergamonPressLtd 1980.Printed in Great Britain
OPTIMUM I~REVENTIVE MAINTENANCE POLICIES FOR A 2-UNIT R E D U N D A N T SYSTEM WITH REPAIR A ~ POST-REPAIR i
,
S. M. SINHA Department of Operational Research, University of Delhi, India and D. V. S. KAPm Department of Mathematics, Madhav Institute of Technology and Science, Gwalior-474005, India
(Receivedfor publication 12 April 1979) Abstract--Stochastic behaviour of a 2-unit cold-standby redundant system with repair and post-repair and subject to slide preventive maintenance is discussed under the most generalized assumptions. We employ (MRP) Marker Renewal processes technique to obtain various reliability measure~ Optimum preventive policies which maximize the availability are derived under suitable conditions. A numerical example is presented.
INTRODUCTION
That system reliability can be vastly improved by introducing redundancy is well known. Reliability of the s y s ~ is also improved by repair and/or pre~re~tive maintenance. Many contributions to such systems have been made. Optimum PM (preventive maintenance) policies by maximizing or minimizing some criterion have been derived for simple as well as redundant systems; see [L3,4 ]. Recently authors [2] discussed the stochastic behaviour of 2-unit redundant system with repair an d post-repair under the most general assumptions. However, it is of interest to take account of a preventive maintenance for such a system. With this objective in view, we first discuss the stochastic behaviour of a 2-unit redundant system with repair and post-repair and subject to slide PM policy. Such a PM policy is reasonable and practical in application; see [4, 6]. Further, we discuss the optimum PM policies maximizing the steady state availability. Finally, a numerical example is presented. A model with repair, post-repair and PM includes various earlier models as special cases. ASSUMPTIONS
I. The system has two identical units. A unit in standby never deteriorates or fails. 2. Each switchover is perfect and instantaneous. 3. When a unit goes down ("fails" or "its PM time arrives") it goes for service ("repair, post-repair" or PM). 4. A unit on repair goes for post-repair. After postrepair a unit is like new. 5. When a unit operates for a specified length of time without failure, it goes to PM. After PM a unit is like new.~ 6. PM of an operative unit is made only if another unit is in standby. 7. An operative unit, which forfeited PM because
8.
9.
10. 11. 12. 13.
14.
15.
of assumption 6, undergoes PM upon service completion of the down unit. If the operative unit goes down and another is in standby, the downunit undergoes service immediately and the standby unit takes over the operation. If the operative unit goes down while another is under service, it awaits service until the service completion of the other unit. There is one server which is always available. Each operative unit has increasing failure rate so as to make PM effective. The failure time"distribution of each unit is F(t) with mean 1/L The repair and post-repair time distributions are Gi(t) and G2(t) with means 1//al and 1//a2, respectively. The time to begin PM is a random variable with distribution A(t) and time to accomplish PM has distribution G3(t) with mean 1//a3. All the random variables are mutually independent. ANALYSIS
We define the foUowjng time instants at which the system makes a transition into states: - 1 A unit begins to operate and another begins to be in standby. 0 Post-repair or PM of a unit is completed and the other is operating. t A unit begins to operate and repair of the failed unit begins. 2 A unit is operating and post=repair of another unit begins after repair completion. 3 A unit begins to Operate and PM of other begins, 4 An operating unit fails while other is under repair. 5 An operating unit fails while other is under postrepair or PM. 887
888
S.M. S~NnAand D. V. S. KAPIL
Note that the time instants at which the system enters states 0, 2, 4 and '5 are non-regeneration points and time instants for states i and 3 are regeneration points. By similar argutnen~ as used in [2~, 5], ttie following-mass~anctions of the MRPs are derived:
Q- 11(t) = fl A(x)dF(x),
(1)
Q- la(t)
= fl F(x)dA (x),
(2)
Q12(t) =
foF(X)dG1 (x),
(3)
Q~,(O= fo' gdx)dF(x~
(4)
Q~5~(t)=
fl F(x)A(x) d [Gl(x) * G2(x)],
(5)
Q~2J(t)=
fl F(x)A(x) d [Gt(x) * G2(x)],
(6)
(18)
whore Q~(t) is Cdf of MRP, i~e., the' probability that after transi "t~g into state i, the process noxt enters into state j, in an amount of time less than or equal to t, QlT)(t)is Cdf of MRP from state i intoj via state m, and Q~7")(t) is Cdf of M R P from state i into j via states m and n. From the Laplace-Stieltjes (LS) transforms of (1)(18), we can derive the mean recurrence time for a certain state, the limiting probabilities that the system is in a certain state in the steady state, and the expected number of visits to a certain state per unit time in the steady state. Let I1dl33) be the mean mourrence time for state 1(3). Then 111 =
Q~2c~(t)=
foF(x) dGa (x),
[ 1/#1 + I/#2 + fo°F(t)A(t){Gl(t)*G2(t)} dt1
+IoP(t)A(t)G3't)dt}]/[1-;:A(t'G3(t)dF(t)], (19)
Q~2~(t) = f l [Gl(X) * (~2(x)] (iF(x),
Qi'~(t)= f~ F(x)dG, (x),
(8)
~,ooA(t)[Gl(t), G2(t)] dF(t).
(20)
30
Q~'°~(t) = fl X(x)[Gl(x), 6~(x)] dr(x), Q[4"s)(t)= [fl F(x)dG(x )
* G2(t),
(9)
(lO)
Let Pj (j = 0-5) denote the limiting probabilities that the process is in statej in the steady state. Then
Po =
F(t)X(t)[Gl(t)*. G 2 (t)] dt/111 +
Q~21s)(t)= f~f~[F(x)-F(u)]dGl(u)dG2(x-u), (11) Q~2~°)(t)= f~ [Gdx) * G2(x)]F(x) dA(x), (12)
Q~o(t)= f~F(x)A(~)dG~(x~
(13)
Pi
P2 =
Q~(t)--f~e,~(x)dF(x),
= I- l~(t)Gq(t)dt/111, Jo
fo
F(t)[G~(0, g2(t)]
dt/111,
(21) (22)
(23)
fooF(t)G3(t)dt/133,
(24)
= [; F(t)Gl(t)dt/ll 1,
(25)
Pa = P~
f- F(t)A(t)G3(t)dt/la~,
jo
do
(14)
P~ = l; F(t)EGl(t), ~2(t)] dt/tl , Q3a(t) = f l
F(x)A(x) dGa (x),
Q~°:~(t)-- fo Gs(x)F(x) dA (x),~
(15) (16)
+
Nt)G3(t)dt/133.
(26)
Note that Po + Pt + P2 + P3 is the steady state availability and P4 + P~ is the steady state,u~vailability. 5
Q~°i(t)= fo A(x)G3(x)dF (x),
(17)
It is evident that ~ Pj -- 1. 1=o Let Mj ( / = 0-5) denote the expected numbers of
Optimum preventive maintenance policies for a 2-unit redundant system
889
M 1 = 1/ljj, j = 1,3,
(28)
M2. = qlz(O)/ltl,
(29)
L(0) is.the availability that tmoperating unit undergoes P M upon post-repair or P M completion (i.e. to ffi 0), and L(oo) is the availability that no PM is mtide. Our interest is to obtain the optimum scheduled time tff maximizing tim availability in (33). Therofore, putting U(to) = O, we have
M+ = q1+(0)/111,
(30)
r(to)[P/m - P3(x/m
visits to state j per unit time in the steady state. Then
Mo =
q~12d(O)llll + q3o(0)/133,
M s = q~*~(0)/111 + q35(0)/133.
(27)
(31)
+ 1/#2) - Ps3(o) + P3Jl~(0)]
+ fit3(0) -- fl3112(0) -- fl Io ° 83 (t)dS(t) + #3
OPTIMUM PM POLICIES
We have discussed a random PM in general in the prig sections. Now we consider an age PM policy which could be adopted in practical applications, we assume {~
A(t) =
for t < to, for t __>to.
x
/.(to) ffi
0' = J o F(0G~(t)dt
13(oo)]
[1/#1 + t/#2 - J12(0)] [1 - i3(oo)] + [ 1 / m - S3(0)]lt2(~) 03)
~12 ~
(44)
P = #1
(45)
fo
+
#12,
and r(t) = f(O/F(t) denotes the failure rate. It is assumed that F(t) has a dcnsityf(t). Further introduce the notation
fl f o G3(t)dF(t) + [J3 f ; [Gl(t)*G2(t)]dF(t) film -
(i -- 1, 3),
(43)
. Nt)[Gt(t)iG2(t)]dt,
m~
where OF(t) G~(t) dt
(i ffi 1, 3),
(32)
+ ['i'3 - J_,,(0)]I12(oo)
?~ =
(42)
where
Steady state availability L(to) is,obtained from (21)-(24) as [71 + ?r, - S l i ( O ) ] [ 1 -
;oo[ G l ( t ) * G2(t)] d F (t),
P3(l/gx + 1/#2) (46)
(34) M =/J~lOl
-
P,).
(47)
712 = I o F(t)[ Gt(t) * G2(t)] dr,
(35)
Then we have
I3(t)fftlG3(t)dF(t),
(36)
112(t)=f,i[dl(t)*G2(t)]dF(t),
(37)
Theorem 1. Suppose [Gl(t)*G2(t)]
J3(t)f f+i F(t)G3(t)dt,
(38)
J12(t) =
F(t)[Gl(t)* G2(t)] dr.
(39)
t~ = oo, i.e. no scheduling.
o
Consider the following special cases of the PM policies. From (33~ we have
['}11 "l- ~111]
o
[l//~x + 1//~2
if
+ ?3 [ ~ [GI(0* tT,(t)] de(t) ao G-s(t)dF(t) + l/m
(iii) If fl/#3 > fl3(1/#1 + 1/#2) and r(0) ~_ m, then
the optimum scheduled time is t~ = O, i.e. PM is made just upon post-repair or PM completion. Theorem 1 is easily proved in a similar manner to Nakagawa and Osaki [4]. We omit the proof.
G3(t)dF(t)
L(0) =
(i) / f r(oo) > M, P/#3 > P3(1/#1 + 1/#2) and r(0) < m, or r(oo) > M and ill#3 ~ fla(l/pt + 1/#2), then there exists a finite and unique optimum scheduled time t~ satisfyin# (42). (ii) Ifr(ov) _< M, then the optimum scheduled time is
[610)* G , ( t ) ] ~ ( t ) (4O)
1/2 L(oo) = 1/#1 + 1/#2 + 1/2 - Yl - ~12"
(41)
NUMERICAL EXAMPLE
We have discussed the optimum PM policies by adopting as the criterion of optimality the steady state availability. Now we consider a numerical example. We assume that G~t) = 1 - exp ( - pit) (iffi 1, 2, 3) and dF(t) = ~(~) exp (-art) (a > 0). Failure time distribution is a Gamma distribution with shape parameter 2 and has increasing failure rate.
890
S, M. Sn~ilx and D. V. S. KApU+ Table 1
1/Mean PM time Optimum sc+heduledtime ' Availability 2.0 3.0 4.0 5.0
0.5330 0.2886 0.2042 0.1613 0.1344 0.1157 0.1018 0.0910 0.0825
6.0
7.0 8.0 9.0 10.0
0.5882 0.6175 0.6403 0.6591 0.6764 0.6898 0.70:~7 0.7145 0.7252
Guessing t h e injti~! valt~ of to from r0) < M+ we have obtained unique solution t~ by using successive approximation in (48). For Optimum scheduled time the availability is derived from (33). Table 1 shows the dependence of the mean PM time in the optimum scheduled time t~ maximizing the availability. Setting 2 / ~ - 1, 1/#t = 1, 1//z2 -- 1/2, the availability with no PM L(oo) is 0,5669. CONCLUSIONS
Then we have r(t) = ~2t/(1 -t- ~t). Failure rate is continuous and monotonely hlfreasing with r(0) = 0 and From Theorem 1, if u < M, then no PM is made. We adopt a finite and unique sclleduled time t~ as the optimum policy, if t8 satisfies (42) and > #2/(# - #A i.e. # > 2#3, i.e.
We derived the reliability measures by+using the mass ~ n s of MI~Ps. Our results i n c l u ~ + ~ a l cases, those for the 2-unit r e d u n ~ t S y s t ~ ' + W t t h ; ~ and PM see [4]; 2'unit redundant system with ~ i r and post-repair see [2] and other models also. F i J ~ e r we discussed the optimum PM policies by maximizing the steady state availability. The above numerical results on comparing with those obtained in [4] enable us ~o determine the effect of introducing post-repair on optimum scheduled time and availability. R~n~NC~
(].12 " /21)
I((Z~ ]~1)~-
]A2(~ "1- ] ~ p J
P3(0[ "~ p3) 2"
For given distribution (42) is simplified as 0(2~ e - (~t+ ~3)to
P - (P - p3)e-"° +
(~ + # 3 ) 2 ~2j~3 J-p1e-('+~')io
to=
+
L (.
+
f12e-(,+ Pt)lo1
-J
- 2P3)
(48)
1. R. E. Barlow and F. Prosehan, Mathematical Theory of Reliability. Wiley, New York (1965), 2, P. K: Kaput and K. R. Kapoor, 1EEE Trans. Reliab. R-27, 382-385 (1978). 3. P. M. Morse, Queues,Inventories and Maintenance. Wiley, New York (1958). 4. T. Nakagawa and S. Osaki, Z. Ops Res. 20, 171-187 (1976)i 5. T. Nakagawa and S. Osaki, Microelectron. Reliab. 15, 633-636 (1976). 6. D. V. Rozhdestvenskiy and G. N. Fanarzhi, Enono Cybernetics. 475--479 (1970).