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Reliability analysis of a two-unit redundant system with a replaceable repair facility
M:croelectron Rehab, Vol 32, No 9, pp 1237-1240, 1992 Pnnted m Great Britain
0026-2714/925500 + 00 © 1992PergamonPress Ltd
RELIABILITY ANALYSIS OF A TWO-UNIT R E D U N D A N T SYSTEM WITH A REPLACEABLE REPAIR FACILITY TONGDE GUO Department of Mathematics and Mechamcs, Zhengzhou Insutute of Technology, Zhengzhou, People's Repubhc of China and JINHUA CAO Institute of Apphed Mathematics, Acadenua Simca, Belting, People's Repubhc of China (Received for pubhcatzon 8 March 199 I)
Abstract--This paper considers a two-unit redundant system where the repmr facility is subject to fadure and can be replaced by a new one when it fads. By using Markov renewal theory we obtain some rehabdlty quantities of the system and the repair facdity, respectively
1. INTRODUCTION In reliability analysis of a reparable system, it is usually assumed that the repair facility neither fads nor deteriorates. In fact, this need not be so There are many instances wherein this assumption ~s not true. Recently, Cao and W u [1-3] have studied some reparable systems where the repair facthty ,s subject to fadure and can be replaced by a new one after it fails. This is one kind of useful reparable system. The usual reparable systems are special cases of the systems stuched Keeping this factor m wew, we study a two-unit redundant system where the repair facihty is subject to failure and can be replaced by a new one. Employing the Markov renewal process, we obtain the following measures of rehabihty: (a) the distribution of the time to the first system failure and its mean; (b) the system's point availability and steady-state avadabihty; (c) the expected n u m b e r of system fadures d u n n g the tame interval (0, t] and the steady-state system failure frequency; (d) the probabthty that the repair facthty is being replaced at time t and its steady-state probabihty; (e) the expected number of repair facility fadures in (0, t] and its steady-state fadure frequency, (f) the distribution of the tame to the first repair facihty failure and its mean.
(2) The probabihty that either of the two units fails d u n n g the time interval (t, t + At], given that the two units are operating at time t, is a0 At + o(At) and the probability that one unit fails during the interval (t, t + At], given that one umt is operating at time t, is a 1 At + o(At). The above model is the system discussed m refs [4-6] and implies several interesting models as specaal cases. In this paper we further assumed that the repair facility has an exponential failure distribution and can be replaced by a new one after it fails. (3) The hfetame U of the repair facility has an exponential dlstnbuUon U ( t ) = P { U <~ t} = 1 - e -c', c, t/> 0. It is assumed that the repair facility neither fails nor deteriorates m its idle periods (4) When the repair facility fails, it ,s immediately replaced by a new one and the failed umt must wait for repair. The replacement time V of the repair facility has a general dlstnbutaon V(t) with mean d (0 < d < + o o) When the replacement of the faded repair facility is completed, the new repair facility continues to repair the faded unit. The repair t~me for the failed umt is cumulative, i.e. the repatr of the faded unit is completed when the total cumulated repair t,me for the failed unit exceeds Y (5) All the random variables X, Y, U and V are mutually independent (6) There is a single repair facility and the repair diclphne for failed untts is "first come first served". (7) After repair, the faded unit is as good as new.
2. MODEL AND ASSUMPTIONS (1) Consider a two-s,mllar-umt redundant system with one repair facihty. For each umt, the lifetame X has an exponential distribution with mean l/aj and the repa,r time Y of the faded umt has a general distribution with mean b MR~2/9--D
3. MATHEMATICAL DISTRIBUTION OF THE MODEL Let g(s) be the Laplace-Stieljes transform of G(t). In general, the small letter function denotes the LS transform of the corresponding capital letter throughout this paper.
1237
1238
TONGDEGuo and JINHUACAO
Let 17denote the time since the unit fails and begins to be repaired until this repair IS completed, where 17 contains some replacement period for the repair facility which may fail during the repair of the unit Let Girl(t) = P { t " <~ t, the number of the repair facility failures during 17 is exactly equal to r I , r =0,1,2, (1) and
Q Up-state i ~ Down-state Fig 1 Transition diagram
Let To = 0, Tj, 7"2,, . be the successive time instants of change in the state of the system We define Z.=X(T.+O),
C J ( t ) = P { ~ " <~t}
Then {Z., T.} is a homogeneous Markov renewal process with state space E = {0, 1} Let
L e m m a 1, g(s) = g(s + c - cv(s)),
n = 0 , 1,2,
~ = E17 = b(1 + cd)_
Q,:(t)=P{Z,=j, Tl<~tlZo=,},
P r o o f Obviously, we have G(t) = P{17 <~ t} = ~ Girl(t)
(2)
i=0,1,
j=0,1,2,
In a m a n n e r similar to ref [5]. we have
r=0
Qol(t) = 1 - e .... ,
From (1) we have Gtrl(t)=P
Y+
V,,~t, m=l
=
U,,~ Y< m~l
Q~o(t) =
e . . . . dC(u),
Qi~)(t) =
(1 - e . . . . ) d C ( u ) ,
Um m=l
V(rl(t -- u ) [ U i ~ ) ( u ) -- U C r + ' ( u ) l dG(u)
(3) Q~2(¢)=
where UI, U2, .. and V~, V2, denote the successive operating lifetimes and replacement time lengths of the repair facility, respectively, A (r)(t) iS the r-fold convolution of A ( t ) with itself, A(°)(t)= 1 for t f> 0 Throughout this paper we promise I~ = i Xk = 0. From (2) we get G(t) = r=
I
V~r)(t - u) e
r!
"~
~+ ~
Jo
(ct)" __..
= g (s + c + cv ( s ) ) .
Further, we have dE(S)ds , = 0 = b ( l + c d ) which completes the proof To analyze the above system, we introduce the following states of the system 0: no unit fails, i. one unit is operating and the other is being repaired, 2: one unit is being repaired and the other is waiting for repair Possible states and transitions are shown in Fig 1, Obviously, state 0 and state 1 are regenerative states, and state 2 IS a non-regenerative state
-
qol (s) = ao/(ao + s), ql0(S) = g ( a 0 + S)
uutu;
e-(~+~)' r! ao(t)
a~e . . . . G ( . ) d u ,
where ~(t) = I - G(t). Taking the LS transforms on both sides of the above equations, we get
qlz(s) = al [1 - g(s + al)]/(aL + s),
Taking the LS transform of the above equation we have g(s) = r~=O[D(S)]r
I0
q]~)(s) = ~(s) -- ~(s + al ) By means of the above mathematical description of the model, quoting from the results of ref [5] we can get the following rehablhty quantities of the system (l) the distribution of the time to the first system failure and its mean, (2) the system's point availabihty and steady-state availability, (3) the expected number of system failures during the time interval (0, t] and the steady-state system failure frequency. We omit the above quantities for brevity. 4. THE PROBABILITY THAT THE REPAIR
FACILITY IS BEING REPLACED AT TIME t Let B~(t) = P{the repair facility is being replaced at time t lZo = z} i = 0 , 1,
t>~0
Rellabdtty analym of a two-unit redundant system
Theorem 1. The L transforms of Bj(t), 1 = 0, 1, are
b~ (s) = aob ~ (s)/(ao + s)
1239
Theorem 2. The LS transforms of Mj(t), i = O, 1, are too(S) = qOl(s)ml (S)
b ~ (s) = co(s)~(s)/[l - ~(s)
+s~(aj + s)/(ao + s)]s(s + cO(s))
m, (s) = c~(s)/I(s + co(s))(~(s)
(4)
+sg(s + a l)/(a o + s))]
(10)
and and
B = lira B,(t) = lira sb,*(s) l~oO
M = hm M,(t)/t = l i m s m , ( s ) = B / d .
s~0
t~ao
= aobcd/Iaob(l + cd) + g ( a 0 ] .
(5)
Proof. For convenience, we introduce the random event: e(t) = {the repair facflRy is being replaced at time t} By probabfllstic arguments we have
(11)
s~O
Proof. (1) Derive the Markov renewal equations. For convenience, we introduce the following random events' e(r, t) = {the number of repair faclhty failures during (0, t] is exactly equal to r}, r = 0, 1, 2,.. Let
Bo(t) = Qo, (t) • B l ( t )
Q~'~(t) = P{Z, = O, T 1 <~t, e(r,
B~(t) = Q~o(t) * Bo(t) + Ql~)(t) * Bl(t )
+ P { T l > t, e(t)lZo = 1}
(6)
rl)lZo =
I}
Ql~)['1(t)=P{Z~=l, T l < ~ t , e ( r , T . ) l Z o = l } Ot'l(t) = P{TI > t, e(r, t)[Zo = 1}
where
Mo(t) = Qol (t) * Mj (t)
(7)
MI (t) = Qjo(t) * Mo(t)Q~2)(t) * M, ( t ) + D ( t ) + C(t)
Obvmusly, we have
=
P
(13)
We have the following Markov renewal equations:
P{T~ > t, e(t)lZo = 1} = e { ? > t, e(t)lZo = 1}.
P{?> t , e ( t ) l Z o =
(12)
(14) 1} where
Sn+Un+l<~t
n--0
Um
D(t) = ~, rDl'](t),
mffil
yml
•~ V~")(t)
17(0
~r(u) dU C"+l)(u)
(8)
n~0
C(t) = ~ r(Q~'~(t) + Q~21)t'l(t)). ?=1
where
S.= ~ (U.+V..),
The first equation of (14) is easily proved Applying the total probability rule we have
n=0,1,2,
m~J
Taking the L transforms on both sides of (8), we get
f;e-"P{T,
e l ( t ) = ~ Q~'~(t) • [mo(t) + rl + r~0
r=0
× Ql~)[rl(t) • [ g I (t) + r] + ~ rD['l(t)
> t, e(t)lZo = 1} dt
r=O
=cO(s)~(s)/ts(s + c6(s))].
(9)
Taking the L transform on (6) and using (9), we obtain b~ (s) = q0l (s)b'{ (s)
b'~ (s) = qlo (s)b~ (s) + q~])(s)b ~ (s) +co(s)~s)/[s(s +
c~(s))].
On solving the above equatmn we get (4). 5. THE EXIPF_,C-'TEDNUMBER OF REPAIR FACILITY FAILURES DURING THE TIME INTERVAL (0, tl
Let N(t) denote the number of repair facility fmlures during (0, t] and let M~(t)= E [ N ( t ) I Z o = i], i --0, 1
=Qlo(t) • Mo(t)+Ql~)(t) * M l ( t ) +D(t)+C(t) which is the second equation of (14) (2) Fred Q['~(t) + Q121)['](t)and Dt'l(t). By defimtion (12),
Q['~(t) + Q121)['1= P{Z, = O, T l <. t, e(r, r,)l Zo = 1} + P { Z , = 1, T, <~ t, e(,', T,)IZo = 1}
=P{X>
f', f < t,e(r, T,)IZo = 1}
+P{f ~
1}
- - P { g ~< t, e(r, TI)IZo = 1} = al'](t)
(15)
1240
TONGDEGUO and JINI-IUACAO 6. THE DISTRIBUTION OF THE TIME TO THE FIRST REPAIR FACILITY FAILURE
where Gt'l(t) is given by (1)
D1°l(t) = P { Y > t, U > t} = G(t) e "
(16)
olrl(t) = P { T I > t, e(r, t)lZ0 = 1]
Denote by w the time to the first repair facdlty failure let
H,(t)=P{w<~tlZo=t},
l=0,1,
t~>0
= P { Y > t,e(r, t ) J Z 0 = 1} and H , = E ( w I Z = t) Theorem 3 The LS transforms of H,(t), l = 0, 1, are
ho(s) = (ao + s)hl (s)/ao h~ (s) = cg(s + c)/[(s + c)(~(s + c) =
+ sgis + c + a I )/(ao + s))]
fo ;o
( v ~- ~) * f i t - u))~(u) dU"'(u)
+
(19)
and
(Ut'~* ff(t - u ) ) G ( t - u ) dVl"(u)
(17)
11o = l/ao + H~ H l = 1/c + g ( c + a I )/~,(c)ao
(3) Find c ( s ) + d(s), It is evident that
Proof_ By regenerative theory we have
Q~'~(t) = Ql2)t'l(t) + Dtrl(t) = Gi'l(t) + Ot'](t), r=0,1,_
HI(t) = Q o l ( t ) . H i ( t ) H l (t) = Q[°c}(t) • no(t) + Q/~)[°l(t) • Hj (t) +e{u<~t,
Taking the LS transforms on both sides of the above equations and forming the generating function we get
b(s, w) = ~, w'[q[r~(S) + q~])t'](S) + ,=0
(20)
dt'lis)]
Y>u}
Taking the LS transforms of the above equation we get
ho(s) = qol (s) * h I (s) hL (s ) = q~°rJ(s) * ho(s ) + q~t°l(s) * hi (s)
= ~ wr[gt'l(S) +
dt,l(s)]
+cg(s + c)/(s + c)
r=0
Substituting the LS transforms of (3), (16) and (17) into the above equation we get
Solving the above equations we get (19). Substituting (19) into the following.
n ( t ) = - ~dh, s is) s=0
b (s, w) = g(s + c - cv(s )w ) + (s + cw - cv(s )w ) x [1 - g(s + c - cv(s)w)/(s + c - cv(s)w)] Therefore the LS transform of C ( t ) + D ( t ) is
c(s) + d(s) = -~w db (s, w) ,. = l = c~(s)/(s + cO(s)) (18) (4) P r o o f the theorem. Taking the LS transforms of (14) and using (18) we get
too(S) = qol (s )ml (s ) ml(s) = qto(s)mo(s) + q]])(s)ml (s) + c~(s)/(s + cf(s)). Solving the above equattons we get (10), using the T a u b e n a n theorem we have hm M : ( t ) / t = h m s m , ( s ) ,
t =0, 1
SubstRuting (10) into the right sides of the above equaUons we get (l l) This completes the p r o o f
(20) is obtained
Acknowledgement--This work was supported by the National Natural Science Fund of China REFERENCES
I J Cao, Analysis of a machine service model with a repairable service equipment J Math Res Exposzt 5, 93-100 (1985) 2 J Cao and Y Wu, Reliability analysis of a multistate repairable system with a replaceab}e repair facility Acta Math Appl Sin (English Series)4, 113-121 (1988) 3 J Cao and Y Wu, Rehabtlity analysis of a two-umt cold standby system with a replacement repair facility Microelectron Rehab 29, 145-150 (1989) 4, D P Gaver, Time to failure and avallabihty of paralleled system with repair IEEE Trans Rehab R-12, 30--38 (1963). 5 T Nakagawa and S Osaki, Stochastic behavlour of two-unit paralleled redundant systems with repair maintenance_ Mwroelectron Rel, ab. 14,457-461 0975) 6 J Cao and K Cheng, lntrodueuon to Rehability Mathemancs Science Press, Beijxng (1986)
0026-2714/925500 + 00 © 1992PergamonPress Ltd
RELIABILITY ANALYSIS OF A TWO-UNIT R E D U N D A N T SYSTEM WITH A REPLACEABLE REPAIR FACILITY TONGDE GUO Department of Mathematics and Mechamcs, Zhengzhou Insutute of Technology, Zhengzhou, People's Repubhc of China and JINHUA CAO Institute of Apphed Mathematics, Acadenua Simca, Belting, People's Repubhc of China (Received for pubhcatzon 8 March 199 I)
Abstract--This paper considers a two-unit redundant system where the repmr facility is subject to fadure and can be replaced by a new one when it fads. By using Markov renewal theory we obtain some rehabdlty quantities of the system and the repair facdity, respectively
1. INTRODUCTION In reliability analysis of a reparable system, it is usually assumed that the repair facility neither fads nor deteriorates. In fact, this need not be so There are many instances wherein this assumption ~s not true. Recently, Cao and W u [1-3] have studied some reparable systems where the repair facthty ,s subject to fadure and can be replaced by a new one after it fails. This is one kind of useful reparable system. The usual reparable systems are special cases of the systems stuched Keeping this factor m wew, we study a two-unit redundant system where the repair facihty is subject to failure and can be replaced by a new one. Employing the Markov renewal process, we obtain the following measures of rehabihty: (a) the distribution of the time to the first system failure and its mean; (b) the system's point availability and steady-state avadabihty; (c) the expected n u m b e r of system fadures d u n n g the tame interval (0, t] and the steady-state system failure frequency; (d) the probabthty that the repair facthty is being replaced at time t and its steady-state probabihty; (e) the expected number of repair facility fadures in (0, t] and its steady-state fadure frequency, (f) the distribution of the tame to the first repair facihty failure and its mean.
(2) The probabihty that either of the two units fails d u n n g the time interval (t, t + At], given that the two units are operating at time t, is a0 At + o(At) and the probability that one unit fails during the interval (t, t + At], given that one umt is operating at time t, is a 1 At + o(At). The above model is the system discussed m refs [4-6] and implies several interesting models as specaal cases. In this paper we further assumed that the repair facility has an exponential failure distribution and can be replaced by a new one after it fails. (3) The hfetame U of the repair facility has an exponential dlstnbuUon U ( t ) = P { U <~ t} = 1 - e -c', c, t/> 0. It is assumed that the repair facility neither fails nor deteriorates m its idle periods (4) When the repair facility fails, it ,s immediately replaced by a new one and the failed umt must wait for repair. The replacement time V of the repair facility has a general dlstnbutaon V(t) with mean d (0 < d < + o o) When the replacement of the faded repair facility is completed, the new repair facility continues to repair the faded unit. The repair t~me for the failed umt is cumulative, i.e. the repatr of the faded unit is completed when the total cumulated repair t,me for the failed unit exceeds Y (5) All the random variables X, Y, U and V are mutually independent (6) There is a single repair facility and the repair diclphne for failed untts is "first come first served". (7) After repair, the faded unit is as good as new.
2. MODEL AND ASSUMPTIONS (1) Consider a two-s,mllar-umt redundant system with one repair facihty. For each umt, the lifetame X has an exponential distribution with mean l/aj and the repa,r time Y of the faded umt has a general distribution with mean b MR~2/9--D
3. MATHEMATICAL DISTRIBUTION OF THE MODEL Let g(s) be the Laplace-Stieljes transform of G(t). In general, the small letter function denotes the LS transform of the corresponding capital letter throughout this paper.
1237
1238
TONGDEGuo and JINHUACAO
Let 17denote the time since the unit fails and begins to be repaired until this repair IS completed, where 17 contains some replacement period for the repair facility which may fail during the repair of the unit Let Girl(t) = P { t " <~ t, the number of the repair facility failures during 17 is exactly equal to r I , r =0,1,2, (1) and
Q Up-state i ~ Down-state Fig 1 Transition diagram
Let To = 0, Tj, 7"2,, . be the successive time instants of change in the state of the system We define Z.=X(T.+O),
C J ( t ) = P { ~ " <~t}
Then {Z., T.} is a homogeneous Markov renewal process with state space E = {0, 1} Let
L e m m a 1, g(s) = g(s + c - cv(s)),
n = 0 , 1,2,
~ = E17 = b(1 + cd)_
Q,:(t)=P{Z,=j, Tl<~tlZo=,},
P r o o f Obviously, we have G(t) = P{17 <~ t} = ~ Girl(t)
(2)
i=0,1,
j=0,1,2,
In a m a n n e r similar to ref [5]. we have
r=0
Qol(t) = 1 - e .... ,
From (1) we have Gtrl(t)=P
Y+
V,,~t, m=l
=
U,,~ Y< m~l
Q~o(t) =
e . . . . dC(u),
Qi~)(t) =
(1 - e . . . . ) d C ( u ) ,
Um m=l
V(rl(t -- u ) [ U i ~ ) ( u ) -- U C r + ' ( u ) l dG(u)
(3) Q~2(¢)=
where UI, U2, .. and V~, V2, denote the successive operating lifetimes and replacement time lengths of the repair facility, respectively, A (r)(t) iS the r-fold convolution of A ( t ) with itself, A(°)(t)= 1 for t f> 0 Throughout this paper we promise I~ = i Xk = 0. From (2) we get G(t) = r=
I
V~r)(t - u) e
r!
"~
~+ ~
Jo
(ct)" __..
= g (s + c + cv ( s ) ) .
Further, we have dE(S)ds , = 0 = b ( l + c d ) which completes the proof To analyze the above system, we introduce the following states of the system 0: no unit fails, i. one unit is operating and the other is being repaired, 2: one unit is being repaired and the other is waiting for repair Possible states and transitions are shown in Fig 1, Obviously, state 0 and state 1 are regenerative states, and state 2 IS a non-regenerative state
-
qol (s) = ao/(ao + s), ql0(S) = g ( a 0 + S)
uutu;
e-(~+~)' r! ao(t)
a~e . . . . G ( . ) d u ,
where ~(t) = I - G(t). Taking the LS transforms on both sides of the above equations, we get
qlz(s) = al [1 - g(s + al)]/(aL + s),
Taking the LS transform of the above equation we have g(s) = r~=O[D(S)]r
I0
q]~)(s) = ~(s) -- ~(s + al ) By means of the above mathematical description of the model, quoting from the results of ref [5] we can get the following rehablhty quantities of the system (l) the distribution of the time to the first system failure and its mean, (2) the system's point availabihty and steady-state availability, (3) the expected number of system failures during the time interval (0, t] and the steady-state system failure frequency. We omit the above quantities for brevity. 4. THE PROBABILITY THAT THE REPAIR
FACILITY IS BEING REPLACED AT TIME t Let B~(t) = P{the repair facility is being replaced at time t lZo = z} i = 0 , 1,
t>~0
Rellabdtty analym of a two-unit redundant system
Theorem 1. The L transforms of Bj(t), 1 = 0, 1, are
b~ (s) = aob ~ (s)/(ao + s)
1239
Theorem 2. The LS transforms of Mj(t), i = O, 1, are too(S) = qOl(s)ml (S)
b ~ (s) = co(s)~(s)/[l - ~(s)
+s~(aj + s)/(ao + s)]s(s + cO(s))
m, (s) = c~(s)/I(s + co(s))(~(s)
(4)
+sg(s + a l)/(a o + s))]
(10)
and and
B = lira B,(t) = lira sb,*(s) l~oO
M = hm M,(t)/t = l i m s m , ( s ) = B / d .
s~0
t~ao
= aobcd/Iaob(l + cd) + g ( a 0 ] .
(5)
Proof. For convenience, we introduce the random event: e(t) = {the repair facflRy is being replaced at time t} By probabfllstic arguments we have
(11)
s~O
Proof. (1) Derive the Markov renewal equations. For convenience, we introduce the following random events' e(r, t) = {the number of repair faclhty failures during (0, t] is exactly equal to r}, r = 0, 1, 2,.. Let
Bo(t) = Qo, (t) • B l ( t )
Q~'~(t) = P{Z, = O, T 1 <~t, e(r,
B~(t) = Q~o(t) * Bo(t) + Ql~)(t) * Bl(t )
+ P { T l > t, e(t)lZo = 1}
(6)
rl)lZo =
I}
Ql~)['1(t)=P{Z~=l, T l < ~ t , e ( r , T . ) l Z o = l } Ot'l(t) = P{TI > t, e(r, t)[Zo = 1}
where
Mo(t) = Qol (t) * Mj (t)
(7)
MI (t) = Qjo(t) * Mo(t)Q~2)(t) * M, ( t ) + D ( t ) + C(t)
Obvmusly, we have
=
P
(13)
We have the following Markov renewal equations:
P{T~ > t, e(t)lZo = 1} = e { ? > t, e(t)lZo = 1}.
P{?> t , e ( t ) l Z o =
(12)
(14) 1} where
Sn+Un+l<~t
n--0
Um
D(t) = ~, rDl'](t),
mffil
yml
•~ V~")(t)
17(0
~r(u) dU C"+l)(u)
(8)
n~0
C(t) = ~ r(Q~'~(t) + Q~21)t'l(t)). ?=1
where
S.= ~ (U.+V..),
The first equation of (14) is easily proved Applying the total probability rule we have
n=0,1,2,
m~J
Taking the L transforms on both sides of (8), we get
f;e-"P{T,
e l ( t ) = ~ Q~'~(t) • [mo(t) + rl + r~0
r=0
× Ql~)[rl(t) • [ g I (t) + r] + ~ rD['l(t)
> t, e(t)lZo = 1} dt
r=O
=cO(s)~(s)/ts(s + c6(s))].
(9)
Taking the L transform on (6) and using (9), we obtain b~ (s) = q0l (s)b'{ (s)
b'~ (s) = qlo (s)b~ (s) + q~])(s)b ~ (s) +co(s)~s)/[s(s +
c~(s))].
On solving the above equatmn we get (4). 5. THE EXIPF_,C-'TEDNUMBER OF REPAIR FACILITY FAILURES DURING THE TIME INTERVAL (0, tl
Let N(t) denote the number of repair facility fmlures during (0, t] and let M~(t)= E [ N ( t ) I Z o = i], i --0, 1
=Qlo(t) • Mo(t)+Ql~)(t) * M l ( t ) +D(t)+C(t) which is the second equation of (14) (2) Fred Q['~(t) + Q121)['](t)and Dt'l(t). By defimtion (12),
Q['~(t) + Q121)['1= P{Z, = O, T l <. t, e(r, r,)l Zo = 1} + P { Z , = 1, T, <~ t, e(,', T,)IZo = 1}
=P{X>
f', f < t,e(r, T,)IZo = 1}
+P{f ~
1}
- - P { g ~< t, e(r, TI)IZo = 1} = al'](t)
(15)
1240
TONGDEGUO and JINI-IUACAO 6. THE DISTRIBUTION OF THE TIME TO THE FIRST REPAIR FACILITY FAILURE
where Gt'l(t) is given by (1)
D1°l(t) = P { Y > t, U > t} = G(t) e "
(16)
olrl(t) = P { T I > t, e(r, t)lZ0 = 1]
Denote by w the time to the first repair facdlty failure let
H,(t)=P{w<~tlZo=t},
l=0,1,
t~>0
= P { Y > t,e(r, t ) J Z 0 = 1} and H , = E ( w I Z = t) Theorem 3 The LS transforms of H,(t), l = 0, 1, are
ho(s) = (ao + s)hl (s)/ao h~ (s) = cg(s + c)/[(s + c)(~(s + c) =
+ sgis + c + a I )/(ao + s))]
fo ;o
( v ~- ~) * f i t - u))~(u) dU"'(u)
+
(19)
and
(Ut'~* ff(t - u ) ) G ( t - u ) dVl"(u)
(17)
11o = l/ao + H~ H l = 1/c + g ( c + a I )/~,(c)ao
(3) Find c ( s ) + d(s), It is evident that
Proof_ By regenerative theory we have
Q~'~(t) = Ql2)t'l(t) + Dtrl(t) = Gi'l(t) + Ot'](t), r=0,1,_
HI(t) = Q o l ( t ) . H i ( t ) H l (t) = Q[°c}(t) • no(t) + Q/~)[°l(t) • Hj (t) +e{u<~t,
Taking the LS transforms on both sides of the above equations and forming the generating function we get
b(s, w) = ~, w'[q[r~(S) + q~])t'](S) + ,=0
(20)
dt'lis)]
Y>u}
Taking the LS transforms of the above equation we get
ho(s) = qol (s) * h I (s) hL (s ) = q~°rJ(s) * ho(s ) + q~t°l(s) * hi (s)
= ~ wr[gt'l(S) +
dt,l(s)]
+cg(s + c)/(s + c)
r=0
Substituting the LS transforms of (3), (16) and (17) into the above equation we get
Solving the above equations we get (19). Substituting (19) into the following.
n ( t ) = - ~dh, s is) s=0
b (s, w) = g(s + c - cv(s )w ) + (s + cw - cv(s )w ) x [1 - g(s + c - cv(s)w)/(s + c - cv(s)w)] Therefore the LS transform of C ( t ) + D ( t ) is
c(s) + d(s) = -~w db (s, w) ,. = l = c~(s)/(s + cO(s)) (18) (4) P r o o f the theorem. Taking the LS transforms of (14) and using (18) we get
too(S) = qol (s )ml (s ) ml(s) = qto(s)mo(s) + q]])(s)ml (s) + c~(s)/(s + cf(s)). Solving the above equattons we get (10), using the T a u b e n a n theorem we have hm M : ( t ) / t = h m s m , ( s ) ,
t =0, 1
SubstRuting (10) into the right sides of the above equaUons we get (l l) This completes the p r o o f
(20) is obtained
Acknowledgement--This work was supported by the National Natural Science Fund of China REFERENCES
I J Cao, Analysis of a machine service model with a repairable service equipment J Math Res Exposzt 5, 93-100 (1985) 2 J Cao and Y Wu, Reliability analysis of a multistate repairable system with a replaceab}e repair facility Acta Math Appl Sin (English Series)4, 113-121 (1988) 3 J Cao and Y Wu, Rehabtlity analysis of a two-umt cold standby system with a replacement repair facility Microelectron Rehab 29, 145-150 (1989) 4, D P Gaver, Time to failure and avallabihty of paralleled system with repair IEEE Trans Rehab R-12, 30--38 (1963). 5 T Nakagawa and S Osaki, Stochastic behavlour of two-unit paralleled redundant systems with repair maintenance_ Mwroelectron Rel, ab. 14,457-461 0975) 6 J Cao and K Cheng, lntrodueuon to Rehability Mathemancs Science Press, Beijxng (1986)