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Analysis of a two unit standby system with partial failure and two types of repairs
Microelectron. Reliab., Vol. 24, No. 5, pp. 873-876, 1984. Printed in Great Britain.
0026-2714/8453.00 + .00 @ 1984 Pergamon Press Ltd.
ANALYSIS OF A TWO UNIT STANDBY SYSTEM WITH PARTIAL FAILURE AND TWO TYPES OF REPAIRS L. R. GOEL Department of Statistics, Institute of Advanced Studies, Meerut University, Meerut-250005, India ASHOKKUMAR D.S.C., Metcalfe House, Delhi, India and A. K. RASTOGI Department of Statistics, Meerut College, Meerut, India (Receivedfor publication 2 May 1984) Abstract--A two dissimiliar unit standby system is analysed. The priority unit can either be in normal or partial operative mode. When the unit fails from the partial mode, it undergoes minor repair and the unit becomes operative with different failure rate. If this unit fails again, it goes to major repair after which it works as good as new. The standby unit while in use is either operative or failed. This non priority unit fails without passing through the partial failure mode and undergoes only one type of repair with differentrepair time distribution. Failure and repair time distributions are negative exponential and general respectively. Regenerative technique in MRP is applied to obtain several reliability characteristics of interest to system designers.
1. I N T R O D U C T I O N
2. S Y S T E M D E S C R I P T I O N
The concept of dissimilar units serves a good purpose (1) Two dissimilar units operate in cold standby in modern costly system, because it is profitable at configuration. Priority unit has the two modes of times to keep a basic system and supplement it by a operation--normal (O) and partial (P). Non-priority system which is less expensive but may serve the unit has a single operation mode--normal (O). The purpose for short durations, when the basic system is priority unit in normal operative mode cannot fail not available. In fact the concept of dissimilarity in without going through P-mode. units is reflected mathematically by taking different (2) When the priority unit fails from P-mode, it failure/repair time distributions. Further the concepts goes to minor repair and becomes operative as O1 of different types of repairs (major/minor) is also with different failure rate. Upon failure this O1 unit prevalent in the context of industrial systems. A minor goes under major repair and after repair it works like repair may bring the system to operational conditions new and assumes O-mode, Non-priority unit keeps but with the changed failure features (may be increased the system operative but with reduced efficiency. This failure rate or a different failure time distribution) and unit fails completely without passing through P-mode a major repair may completely restore the unit to new and after repair it works as new. like condition. Nakagawa and Osaki [6] have studied (3) Failure time distributions are negative expothe behaviour of a two unit priority standby system nential and repair time distributions are arbitrary and Osaki et al. I-5] have studied a two unit redun- with different cdf's. dant system with two phase repairs. (4) Switching is perfect and instantaneous. In the present paper we incorporate the two con(5) Single repair facility is available for all repairs cepts of dissimilar units and different failure modes and preference is given to priority unit. The service of simultaneously in a redundant system with two modes the non-priority unit is pre-emptive repeat type. of operative unit, i.e. normal (O) and partial (P). The (6) If the unit in P-mode fails completely, its service underlying failure time distributions are taken as starts afresh and the time already spent in P-mode negative exponential and repair time distributions are goes to waste. taken as general. The technique of regenerative pro3. N O T A T I O N S cess is applied to obtain distributions of time to first failure, MTFF, the pointwise and steady state Priority unit can be in one of the following modes: O availabilities. operative; P partial failure; rl,r 2 under minor or major 873
L.R. GoH. et al.
874
repair; R 1, R2 continued minor or major repair; O~ operative after minor repair. Non-priority unit can be in one of the following modes:
Laplace Stieltjes transform Laplace-convolution Laplace-Stieltjes convolution rnij mean (conditional) sojourn time in state S~ given that transition is to state S~. ~q mean (unconditional) sojourn time in state Q
0 operative; S standby; r under repair; w waiting for repair. The system may be in one of the following states at any instant where the first letter denotes the mode of priority unit and the second corresponds to non-priority unit. Up states So(O,S)
S4(r2,0)
SI(P,S)
S2(rl, O)
$3(01,S)
Ss(.Ol,r)
S6(O,r)
S7(P,w).
S i = ~ mij J ui time to first failure when the system starts from state Si. Vi P[u i <~ t] ml, m 2 mean repair time for minor, major repair M~(t) probability that the system starting from Si is up at time "t" without transiting into any other regenerative state.
Down states
Ss(rl,w)
S9(r2, w) Slo(Rl,w)
Sll(R2, w)
ct constant failure rate of priority unit in Omode. /3 constant failure rate of priority unit in Pmode. 0 constant failure rate of priority unit in O~mode. e constant-failure rate of non-priority unit in Omode. g(t), G(t) pdf and cdf of time to repair of a P unit. 01(0, Gl(t) pdfand cdfoftime for minor repair. g2(t), G2(t ) pdf and cdf of time for major repair. h(t), H(t) pdf and cdf of time to repair a non-priority unit. qu(t),Qij(t) pdf and cdf of time for transition from state S~ to S~. * Laplace transform
ff;',unlessstatedotherwise. T h e t r a n s i t i o n d i a g r a m a l o n g w i t h the m o r e effective rate of t r a n s i t i o n is given in Fig. 1. Simple p r o b a b i l i s t i c c o n s i d e r a t i o n yield the following expressions for d i s t r i b u t i o n f u n c t i o n s of t r a n s i t i o n times
Qol(t) = fi ~e-"' dt = l _ e - ~tt Qlo(t) =
e -~t d G ( t ) ; Q12(t) =
f
fie - ¢ ' G ( t ) d t
0
h(t)
$6
gz(~)
g (t )
~ S,o
-5"5
Up stores
Down stores ( ~
Fig. 1.
Non-recjenerotive states
$9
Reliability analysis
875
where
Q23(t)= f l e-St dG1(t); Q2,,o(t)= f i ~ e - a Gl(t)dt NI(S
Q~2½°)(t)= G, (t) - Q2 3(t) Q3,,(t) =
Q4o(t) =
) =
QolQ12Q2,1o.+O0101202303404,11
Dl(S ) ~
(3)
1-Qo1010-Qox012023034~40.
(4)
Hence we obtain
fo
0 e -°t dt = 1 - e - ° '
d ~ M T F F = E(uo) = - d s V°(s)ls=°
e -~t dGE(t); Q4A1 m =
~ e - a (~2(t)dt
= (/20 +/11)+PIa{/12 +P23(/13 +/14)}
(s)
P12( 1 -P23P4o) Q~'(t)
= a2(t)-
Q59(t) = Q67(t) =
Q40(t); Qs3(t) --
e -°' dH(t)
0 e - ° t / 4 ( t ) dt; Q6o(t) =
1 =
e -~' dH(t)
@1
1
q~2
1
(6)
6(1 -(b~4'.,)
5. POINTWISE AND STEADY STATE AVAILABILITIES
f:
e e - ~ ' B ( t ) d t ; Q76(t)
e -f*dG(t) F r o m the theory of regenerative process the pointwise availabilities Ai(t ) of system starting from a given regenerative point are seen to satisfy the following recursion relations
Q7s(t) = ; B e -~' G(t)dt; Qss(t) = Grit) Q96(t) = G2(t); Q~s(t) = 0, otherwise.
Ao(t) = Mo(t) + qol (t) (~) A l(t) Al(t ) = Ml(t)+q10(t)(~ Ao(t)+q12(t)@ A2(t ) A2(t ) = M 2 (t) + q23(0 (g) A3(t)+ q(2½°)(t)© A5 (t) A 3(t) = M3(t) + q3,(t) @ A4(t) A4(t) = M4(t) + q,o(t) @ Ao(t) + q{4161)(t) (~) A6(t) As(t) = Ms(t) + q53(t) (g) A3(t) + q59(t) @ A9(t) A6(t) = M6( t ) + q60(t) (~) Ao(t) + q67 (t) @ A 7(t) AT(t) = MT(t)+ q76(t) (~) A6(t)+ qTs(t) @ As(t) As(t) = qss(t) (~) As(t) A9(t ) = q96(t) @ A6(t)
Letting t ---, oo a n d using Q~s(oo) = Po, we get pol=p3,=pss=p96=
1
p~o = d(fl) = 4'; p ~ = ~ ; p ~ = d # ) PZ,IO = P(210)= ~1, PgO -- d2(t;) = 4'2;
= 4',
P.,,, = pkl61) = 62; Ps3 = / ~ ( 0 ) = O; P59 = P60 = ]~(~) = re; P67 = if; P76 = 4'; P78 = ~" M e a n sojourn times are #0 = 1/c~; 41 = j ; - ~ ' G(t)dt = 6/fl
(7)
where as easy probability consideration shows .2
=
fe -'t Ga(t)dt
=
~l/e;
/13 =
Mo(t ) = e -~t, Ml(t ) = e -fit G(t), M2(t ) = e -~t Gl(t)
1/0
l~. = fe-~' d2(t)dt = ,2/~; .s = fe-°tR(t)dt = ~/O #6= fe-~'H(t)dt=rVc*;
/18=fdl(t)dt=m1;
Ma(t ) = e -°t, M4(t ) = e -~t G2(t), M s ( t ) = e -°t/-I(t) M6(t ) = e -st l~(t), My(t) = e -fit G(t). T a k i n g Laplace transform and solving for A*(s) one has
~7= fe-~rG(t)dt=d~/fl
A*(s)
/19=fd,2(t)dt=m2.
*
*
*
*
*
*
*
*
.{-r~* r~* ~ * ( 1 I } ~ * / , * r** ] * * tff23~/34t/46 t/67~78~/853 [Ms +q53(M3 , , , , ,(10) , , ,
+ q34M4)] +qolq12[q25 (1)
+q59q96}+q23q34q46][ *
*
*
*
*
*
*
*
*
*
*
(qs3q34q46
M*
6 + q 6*7 M T*]
*
*
(9)
*
D2(s ) = [(1--q67q76)--q67q78q85(q53q34q46 * * * * * * * * * + qs9q96)] [1 - - qo 1qlo - - qo1 ql 2q23q34q40
T a k i n g Laplace-Stieltjes transform and solving we o b t a i n the Laplace-Stieltjes transform of distribution function of failure time. If the system is initially in So then (with a r g u m e n t s omitted for brevity)
~'o(S) = Na(s)/D~(s)
(8)
+qs9q96)][Mo + q o l M l +qolql2M2 * * * * * * * * * +qolq12q23M3 +qolqt2q23qa4M4] , , ,(10) +qolq12[q25 (1--q67q76)* *
Considering renewal process a r g u m e n t s one has
MR 24: 5-D
N2(s)/D2(s)
N 2 ( s ) = [ ( 1 - ~ t,/67 * t,/76 "* !~ - ~i f*6 7 ff~*7 8 t,/85 ~* ~v/53 t ' * t-/34t/46 0" 0,~11)
4. TIME TO FIRST FAILURE
Vo(t) = Qoi(t) ~) Vt(t) Vl(t ) = Qo~(t) (~) Vo(t)+ Q~2(t) ~) v2(t) Vz(t ) = Qz3(t)(~) v3(t)+Q2,10(t ) Vs(t) = Os4(t) ® VAt) V~(t) = Qao(t) (~) Vo(t)+Q4,1~(t).
=
where (omitting for brevity the a r g u m e n t Is) from all transform)
(2)
* * * * * --qolqlzq23q34q46q60] * * *
*
--"/01t/X2t/2511* ~* .*(10}
*
× [qs3q34{q40( 1 --q67q76) "/46 *
*
"/60 ) t ~/59ff96"/60A * * *(lll * *
--qolq12q23q3,*q46
*
*
*
*
q67q78q85qs3q3a-q40" (10)
L, R. GOELet al.
876 Hence the steady state availability A o = lim Ao(t ) = limSA*(s) = a/(a+b) t-+oc
(11)
s~O
where a = [(1 --P67P76) --P67PTs(P53P46 ~(11} +P59)][//0+/21
as a particular case of our model with no partial failure mode. This can be affected by taking I/fl = 0 and 1/c~ = )~o. Also 0 and c will be )-1 and H(t) = G2It). The service time distribution G(t) will be absent. Hence ~ = 1, qJ = 02, rt = 03- Substituting these values in (6) we get
+ p~2{m~+ P23(u3+ m2)}] + Pl 2[p~½°)(1 - P67P76)
M T F F = vo -
+Pa3P46P67PTs] [#5 + Ps3(#3 + mE)] (10) + PI2[P2s (PsaP46PS9)+ P23P46] [#6 + P6797] 02) b = Plz[P46 + PzsPs9P4o]P67Pvsml + Pl 2[( 1 -- P23P40}P67Pvs + P25P60]P59ra2 (13)
or
v'o = 2av o -
1 +p(1 + 01q52)
2o(1-2,02) 1 +p(1 + 0 1 0 2 )
pO -0102)
which is in agreement with (lO) of [4].
Substituting different values of Pij and ~i we get REFERENCES
+f~{m, +g& @ + m 2 ) } l + ( ~ [ ~ , ( 1 - ~ O )
+ ~[q$, O,bq~2 + t~, + 0 t q~2] [ ~ + r~~ ]
+ ~[(1- 4,~0~)e4+ ~,~]~m2. 6. P A R T I C U L A R
(14)
05)
CASE
The model of S. M. Sinha et al. [4] may be treated
1. A. Kumar, Stochastic behaviour of a two-unit standby redundant system with two switching failure modes, Math. Operationsforsch. Statist., Ser. Optimization 8, 5715800977). 2. R. E. Barlow and F. Proschan, Mathematical Theory o! Reliability. John Wiley, New York (1965). 3. S. M. Gupta, N. K. Jaiswal and L. R Goel. Analysis of two-unit standby redundant system under partial failure and pre-emptive repair priority, Int. J. Systems Sci. 13, 675 (1982). 4. S. M. Sinha, A. Kumar and M. Aggarwal, Analysis of a two-unit priority standby system with different operative states, J. Mathl Sci. 12-13, 65-79 (1978). 5. S. Osaki and K. Okumato, Repair limit policies for a twounit standby redundant system with two phase repairs, Microelectron. Reliab. 16, 41 (1977). 6. T. Nakagawa and S. Osaki, Stochastic behaviour of a two-unit priority standby redundant system with repair, Mieroelectron. Reliab. 14, 309 (1975).
0026-2714/8453.00 + .00 @ 1984 Pergamon Press Ltd.
ANALYSIS OF A TWO UNIT STANDBY SYSTEM WITH PARTIAL FAILURE AND TWO TYPES OF REPAIRS L. R. GOEL Department of Statistics, Institute of Advanced Studies, Meerut University, Meerut-250005, India ASHOKKUMAR D.S.C., Metcalfe House, Delhi, India and A. K. RASTOGI Department of Statistics, Meerut College, Meerut, India (Receivedfor publication 2 May 1984) Abstract--A two dissimiliar unit standby system is analysed. The priority unit can either be in normal or partial operative mode. When the unit fails from the partial mode, it undergoes minor repair and the unit becomes operative with different failure rate. If this unit fails again, it goes to major repair after which it works as good as new. The standby unit while in use is either operative or failed. This non priority unit fails without passing through the partial failure mode and undergoes only one type of repair with differentrepair time distribution. Failure and repair time distributions are negative exponential and general respectively. Regenerative technique in MRP is applied to obtain several reliability characteristics of interest to system designers.
1. I N T R O D U C T I O N
2. S Y S T E M D E S C R I P T I O N
The concept of dissimilar units serves a good purpose (1) Two dissimilar units operate in cold standby in modern costly system, because it is profitable at configuration. Priority unit has the two modes of times to keep a basic system and supplement it by a operation--normal (O) and partial (P). Non-priority system which is less expensive but may serve the unit has a single operation mode--normal (O). The purpose for short durations, when the basic system is priority unit in normal operative mode cannot fail not available. In fact the concept of dissimilarity in without going through P-mode. units is reflected mathematically by taking different (2) When the priority unit fails from P-mode, it failure/repair time distributions. Further the concepts goes to minor repair and becomes operative as O1 of different types of repairs (major/minor) is also with different failure rate. Upon failure this O1 unit prevalent in the context of industrial systems. A minor goes under major repair and after repair it works like repair may bring the system to operational conditions new and assumes O-mode, Non-priority unit keeps but with the changed failure features (may be increased the system operative but with reduced efficiency. This failure rate or a different failure time distribution) and unit fails completely without passing through P-mode a major repair may completely restore the unit to new and after repair it works as new. like condition. Nakagawa and Osaki [6] have studied (3) Failure time distributions are negative expothe behaviour of a two unit priority standby system nential and repair time distributions are arbitrary and Osaki et al. I-5] have studied a two unit redun- with different cdf's. dant system with two phase repairs. (4) Switching is perfect and instantaneous. In the present paper we incorporate the two con(5) Single repair facility is available for all repairs cepts of dissimilar units and different failure modes and preference is given to priority unit. The service of simultaneously in a redundant system with two modes the non-priority unit is pre-emptive repeat type. of operative unit, i.e. normal (O) and partial (P). The (6) If the unit in P-mode fails completely, its service underlying failure time distributions are taken as starts afresh and the time already spent in P-mode negative exponential and repair time distributions are goes to waste. taken as general. The technique of regenerative pro3. N O T A T I O N S cess is applied to obtain distributions of time to first failure, MTFF, the pointwise and steady state Priority unit can be in one of the following modes: O availabilities. operative; P partial failure; rl,r 2 under minor or major 873
L.R. GoH. et al.
874
repair; R 1, R2 continued minor or major repair; O~ operative after minor repair. Non-priority unit can be in one of the following modes:
Laplace Stieltjes transform Laplace-convolution Laplace-Stieltjes convolution rnij mean (conditional) sojourn time in state S~ given that transition is to state S~. ~q mean (unconditional) sojourn time in state Q
0 operative; S standby; r under repair; w waiting for repair. The system may be in one of the following states at any instant where the first letter denotes the mode of priority unit and the second corresponds to non-priority unit. Up states So(O,S)
S4(r2,0)
SI(P,S)
S2(rl, O)
$3(01,S)
Ss(.Ol,r)
S6(O,r)
S7(P,w).
S i = ~ mij J ui time to first failure when the system starts from state Si. Vi P[u i <~ t] ml, m 2 mean repair time for minor, major repair M~(t) probability that the system starting from Si is up at time "t" without transiting into any other regenerative state.
Down states
Ss(rl,w)
S9(r2, w) Slo(Rl,w)
Sll(R2, w)
ct constant failure rate of priority unit in Omode. /3 constant failure rate of priority unit in Pmode. 0 constant failure rate of priority unit in O~mode. e constant-failure rate of non-priority unit in Omode. g(t), G(t) pdf and cdf of time to repair of a P unit. 01(0, Gl(t) pdfand cdfoftime for minor repair. g2(t), G2(t ) pdf and cdf of time for major repair. h(t), H(t) pdf and cdf of time to repair a non-priority unit. qu(t),Qij(t) pdf and cdf of time for transition from state S~ to S~. * Laplace transform
ff;',unlessstatedotherwise. T h e t r a n s i t i o n d i a g r a m a l o n g w i t h the m o r e effective rate of t r a n s i t i o n is given in Fig. 1. Simple p r o b a b i l i s t i c c o n s i d e r a t i o n yield the following expressions for d i s t r i b u t i o n f u n c t i o n s of t r a n s i t i o n times
Qol(t) = fi ~e-"' dt = l _ e - ~tt Qlo(t) =
e -~t d G ( t ) ; Q12(t) =
f
fie - ¢ ' G ( t ) d t
0
h(t)
$6
gz(~)
g (t )
~ S,o
-5"5
Up stores
Down stores ( ~
Fig. 1.
Non-recjenerotive states
$9
Reliability analysis
875
where
Q23(t)= f l e-St dG1(t); Q2,,o(t)= f i ~ e - a Gl(t)dt NI(S
Q~2½°)(t)= G, (t) - Q2 3(t) Q3,,(t) =
Q4o(t) =
) =
QolQ12Q2,1o.+O0101202303404,11
Dl(S ) ~
(3)
1-Qo1010-Qox012023034~40.
(4)
Hence we obtain
fo
0 e -°t dt = 1 - e - ° '
d ~ M T F F = E(uo) = - d s V°(s)ls=°
e -~t dGE(t); Q4A1 m =
~ e - a (~2(t)dt
= (/20 +/11)+PIa{/12 +P23(/13 +/14)}
(s)
P12( 1 -P23P4o) Q~'(t)
= a2(t)-
Q59(t) = Q67(t) =
Q40(t); Qs3(t) --
e -°' dH(t)
0 e - ° t / 4 ( t ) dt; Q6o(t) =
1 =
e -~' dH(t)
@1
1
q~2
1
(6)
6(1 -(b~4'.,)
5. POINTWISE AND STEADY STATE AVAILABILITIES
f:
e e - ~ ' B ( t ) d t ; Q76(t)
e -f*dG(t) F r o m the theory of regenerative process the pointwise availabilities Ai(t ) of system starting from a given regenerative point are seen to satisfy the following recursion relations
Q7s(t) = ; B e -~' G(t)dt; Qss(t) = Grit) Q96(t) = G2(t); Q~s(t) = 0, otherwise.
Ao(t) = Mo(t) + qol (t) (~) A l(t) Al(t ) = Ml(t)+q10(t)(~ Ao(t)+q12(t)@ A2(t ) A2(t ) = M 2 (t) + q23(0 (g) A3(t)+ q(2½°)(t)© A5 (t) A 3(t) = M3(t) + q3,(t) @ A4(t) A4(t) = M4(t) + q,o(t) @ Ao(t) + q{4161)(t) (~) A6(t) As(t) = Ms(t) + q53(t) (g) A3(t) + q59(t) @ A9(t) A6(t) = M6( t ) + q60(t) (~) Ao(t) + q67 (t) @ A 7(t) AT(t) = MT(t)+ q76(t) (~) A6(t)+ qTs(t) @ As(t) As(t) = qss(t) (~) As(t) A9(t ) = q96(t) @ A6(t)
Letting t ---, oo a n d using Q~s(oo) = Po, we get pol=p3,=pss=p96=
1
p~o = d(fl) = 4'; p ~ = ~ ; p ~ = d # ) PZ,IO = P(210)= ~1, PgO -- d2(t;) = 4'2;
= 4',
P.,,, = pkl61) = 62; Ps3 = / ~ ( 0 ) = O; P59 = P60 = ]~(~) = re; P67 = if; P76 = 4'; P78 = ~" M e a n sojourn times are #0 = 1/c~; 41 = j ; - ~ ' G(t)dt = 6/fl
(7)
where as easy probability consideration shows .2
=
fe -'t Ga(t)dt
=
~l/e;
/13 =
Mo(t ) = e -~t, Ml(t ) = e -fit G(t), M2(t ) = e -~t Gl(t)
1/0
l~. = fe-~' d2(t)dt = ,2/~; .s = fe-°tR(t)dt = ~/O #6= fe-~'H(t)dt=rVc*;
/18=fdl(t)dt=m1;
Ma(t ) = e -°t, M4(t ) = e -~t G2(t), M s ( t ) = e -°t/-I(t) M6(t ) = e -st l~(t), My(t) = e -fit G(t). T a k i n g Laplace transform and solving for A*(s) one has
~7= fe-~rG(t)dt=d~/fl
A*(s)
/19=fd,2(t)dt=m2.
*
*
*
*
*
*
*
*
.{-r~* r~* ~ * ( 1 I } ~ * / , * r** ] * * tff23~/34t/46 t/67~78~/853 [Ms +q53(M3 , , , , ,(10) , , ,
+ q34M4)] +qolq12[q25 (1)
+q59q96}+q23q34q46][ *
*
*
*
*
*
*
*
*
*
*
(qs3q34q46
M*
6 + q 6*7 M T*]
*
*
(9)
*
D2(s ) = [(1--q67q76)--q67q78q85(q53q34q46 * * * * * * * * * + qs9q96)] [1 - - qo 1qlo - - qo1 ql 2q23q34q40
T a k i n g Laplace-Stieltjes transform and solving we o b t a i n the Laplace-Stieltjes transform of distribution function of failure time. If the system is initially in So then (with a r g u m e n t s omitted for brevity)
~'o(S) = Na(s)/D~(s)
(8)
+qs9q96)][Mo + q o l M l +qolql2M2 * * * * * * * * * +qolq12q23M3 +qolqt2q23qa4M4] , , ,(10) +qolq12[q25 (1--q67q76)* *
Considering renewal process a r g u m e n t s one has
MR 24: 5-D
N2(s)/D2(s)
N 2 ( s ) = [ ( 1 - ~ t,/67 * t,/76 "* !~ - ~i f*6 7 ff~*7 8 t,/85 ~* ~v/53 t ' * t-/34t/46 0" 0,~11)
4. TIME TO FIRST FAILURE
Vo(t) = Qoi(t) ~) Vt(t) Vl(t ) = Qo~(t) (~) Vo(t)+ Q~2(t) ~) v2(t) Vz(t ) = Qz3(t)(~) v3(t)+Q2,10(t ) Vs(t) = Os4(t) ® VAt) V~(t) = Qao(t) (~) Vo(t)+Q4,1~(t).
=
where (omitting for brevity the a r g u m e n t Is) from all transform)
(2)
* * * * * --qolqlzq23q34q46q60] * * *
*
--"/01t/X2t/2511* ~* .*(10}
*
× [qs3q34{q40( 1 --q67q76) "/46 *
*
"/60 ) t ~/59ff96"/60A * * *(lll * *
--qolq12q23q3,*q46
*
*
*
*
q67q78q85qs3q3a-q40" (10)
L, R. GOELet al.
876 Hence the steady state availability A o = lim Ao(t ) = limSA*(s) = a/(a+b) t-+oc
(11)
s~O
where a = [(1 --P67P76) --P67PTs(P53P46 ~(11} +P59)][//0+/21
as a particular case of our model with no partial failure mode. This can be affected by taking I/fl = 0 and 1/c~ = )~o. Also 0 and c will be )-1 and H(t) = G2It). The service time distribution G(t) will be absent. Hence ~ = 1, qJ = 02, rt = 03- Substituting these values in (6) we get
+ p~2{m~+ P23(u3+ m2)}] + Pl 2[p~½°)(1 - P67P76)
M T F F = vo -
+Pa3P46P67PTs] [#5 + Ps3(#3 + mE)] (10) + PI2[P2s (PsaP46PS9)+ P23P46] [#6 + P6797] 02) b = Plz[P46 + PzsPs9P4o]P67Pvsml + Pl 2[( 1 -- P23P40}P67Pvs + P25P60]P59ra2 (13)
or
v'o = 2av o -
1 +p(1 + 01q52)
2o(1-2,02) 1 +p(1 + 0 1 0 2 )
pO -0102)
which is in agreement with (lO) of [4].
Substituting different values of Pij and ~i we get REFERENCES
+f~{m, +g& @ + m 2 ) } l + ( ~ [ ~ , ( 1 - ~ O )
+ ~[q$, O,bq~2 + t~, + 0 t q~2] [ ~ + r~~ ]
+ ~[(1- 4,~0~)e4+ ~,~]~m2. 6. P A R T I C U L A R
(14)
05)
CASE
The model of S. M. Sinha et al. [4] may be treated
1. A. Kumar, Stochastic behaviour of a two-unit standby redundant system with two switching failure modes, Math. Operationsforsch. Statist., Ser. Optimization 8, 5715800977). 2. R. E. Barlow and F. Proschan, Mathematical Theory o! Reliability. John Wiley, New York (1965). 3. S. M. Gupta, N. K. Jaiswal and L. R Goel. Analysis of two-unit standby redundant system under partial failure and pre-emptive repair priority, Int. J. Systems Sci. 13, 675 (1982). 4. S. M. Sinha, A. Kumar and M. Aggarwal, Analysis of a two-unit priority standby system with different operative states, J. Mathl Sci. 12-13, 65-79 (1978). 5. S. Osaki and K. Okumato, Repair limit policies for a twounit standby redundant system with two phase repairs, Microelectron. Reliab. 16, 41 (1977). 6. T. Nakagawa and S. Osaki, Stochastic behaviour of a two-unit priority standby redundant system with repair, Mieroelectron. Reliab. 14, 309 (1975).