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A two-unit redundant system
Microelectron. Reliab, Vol. 19, pp. 277 to 278. © Pergamon Press Ltd. 1979. Printed in Great Britain
0026-2714/79/0601-0277S02,00/0
A TWO-UNIT R E D U N D A N T SYSTEM R . S U B R A M A N I A N and N. R A V I C H A N D R A N Department of Mathematics, Indian Institute of Technology, Madras 600 036, India
(Received for publication 12 January 1979) Abstract--A two-unit standby redundant repairable system with repair limit suspension policy is studied. Expressions for the availability and reliability of the system are provided by identifying suitable regeneration points.
INTRODUCTION
Recently Osaki [I] considered some repair limit suspension policies for a two-unit standby redundant repairable system. His study was confined to cold standby systems in which the online unit has a constant failure rate. By employing Markov renewal processes, he obtained the Laplace transform of the availability of the system and optimized the same in the stationary case. The object of this contribution is to obtain expressions for the operating characteristics for a more general model without recourse to any transform technique. This is achieved by identifying certain regeneration points [2]. It is also demonstrated how this concept considerably simplifies the analysis. ASSUMPTIONS
(1) The system consists of two units. Either unit performs the desired operation satisfactorily. (2) The failure rate of the standby is assumed to be a constant. (3) There are two repair facilities. The time needed for repairing a unit in facility i(i-- 1, 2) is a random variable having a general distribution G~(-), with mean 1/#i. We assume that 1/#2 < 1/#1. (4) Repair Limit Suspension Time (RLST) is a random variable with a general distribution. This is the maximum time up to which any particular repair is allowed to continue in facility 1. (5) On failure a unit is taken to facility I. (6) If a unit fails when the other unit is undergoing repair in facility 1, then the unit undergoing repair is switched to the facility 2, which is faster. (7) If a repair is not completed in facility 1 before RLST then it is switched to facility 2, where the repair starts afresh. (8) Switch is perfect and switchover is instantaneous. (9) Each unit is "new" after repair. (10) When facility 2 functions, facility I is closed down. NOTATION
f~(t) adt) b(t) © ~(t) C(t) g(t) O("~(t)
3. e -z* pdf of the repair time in facility i (i = 1, 2) pdf of the random variable denoting the RLST convolution symbol 1 - c(t), c(') any function ~o c(u) du gl(t)g(t) + [Gl(t)b(t)] ©g2(t) n-fold convolution of the function O(t) with itself over (0, t) ANALYSIS
First we investigate the behaviour of the standby unit during the operative period of the online unit. Let S(t) denote the state of the standby at any time t. The following symbols are used to describe the state of the standby at any time t. 0 ri
operable. under repair in facility i; i = 1,2.
Further we define:
ni3(t) = Pr{S(t) = j [ S(0) = i 4: S(-0)}
i = 0, rl, r2
j = O , rl ni,:(t, x ) d x = Pr{S(t)= r2 and the stay in this state during its last visit to this state lies in (x, x + dx) [ S(0) = i ~ s(-0)}, i = 0, r l , r 2.
Defining h(t) = fs(t) © g(t) and
?(t) = ~ h~n~(t), n=l
we get by renewal theoretic arguments [3] 7Zoo(t) = Fs(t) + 7(t)(~ ff-s(t),
(1)
no,,(t) = fs(t) (~ [G,(t)B(t)] q- [?(t)(~)fs(t)] (~) [G,(t)B(t)], nor:(t, x ) = [ f ~ ( t - x ) ( ~ ( G l ( t -
(2)
x ) b ( t - x))] (~2(x)
+ 7(t - x) (~)fs(t - x) (~) [Gl(t - x)b(t - x)] G2(x), (3)
nr, o(t) = g(t)(~)noo(t), Event E One unit just switched online and the other unit just enters facility l for repair nr,r~(t) = g(t)(~)nor,(t) + Gl(t)B(t), fo~,(t) pdf of the life time of a unit while online (in standby) 7[r,r2(t, x) = Gl(t - x)b(t - x)G2(x) 277
(4) (5)
278
R. SUBRAMANIANand N. RAVK'HANDRAN ~(t x
,(](U)7~Orz(t-- U, X) du,
-F
(6)
)
Tc,:,:(t,x)=
~,:o(t) = .qz(t) (D 7ro0(t),
(7)
Rr:r,(t) =
(8)
gz(t)
(~ ~'0r,(t),
~t
t
j',~/~o(t) dt ~ .... .
R*{O~ =
As a consequence of key renewal theorem we get from (12) the steady state avilability of the system as
G:(t)cS(t - x) +
Now it is easy to get the mean time to system failure from {13),
x
g2(u)Tro,,(t - u, x) du.
.f~ Yo(t)
(9)
0
We use the regenerative nature of the E-event to obtain ~b(t) and @t) which are, respectively, the pdfs of the time intervals between two successive E-events with or without a system break-down in between. The expression for 4>(t) is derived by considering the following mutually exclusive and exhaustive possibilities in which the standby can be found, when the online unit fails : (1) Operable. (2) Under repair in facility 1. (3) Under repair in facility 2.
Special case
We now specialize to the case discussed in [1]. Setting ;t=0 Jo(t) = 21 e - ~'''
and
b(t) = 6 ( t -
to),
and simplifying, we get
fl =
)~-1 x
+
For the calculation of ~(t) possibilities 2 and 3 are not permissible.
--
It2
)
~"'~ (~t(t) dt
e -
O(t) = yo(t)~,o(t) + [fo(t)~,,,(t)] © g2(t) +
+
;o
fotu) du
o ~r,rz(U X)
g2(t
~- - ( xU )-~- x)" dx,
6(t) = Jo(t)~,, oct).
(10) ( 11 )
Now we are in a position to obtain the reliability
12
Zl
A
where g~()~ ) =
e ~,t ,qa(t} dr. )
R(t) and availability A(t) of the system conditioned at
the origin by an E-event. Writing
This result is in agreement with that given in [l].
REFERENCES n=l
n=l
we get by probabilistic arguments, A(t) =/To(t) + a(t)©/To(t),
(12)
R(t) = Fo(t) + fl(t)@Fo(t).
(13)
I. S. Osaki and K. Okumoto, Repair limit suspension policies for a two-unit standby redundant system with two phase repairs, Microelectron. Reliab. 16, 41 (1977). 2. S. K. Srinivasan, Stochastic Point Processes and their Application, Griffin, London (1974). 3. D. R. Cox, Renewal Theory, Methuen, London (1962).
0026-2714/79/0601-0277S02,00/0
A TWO-UNIT R E D U N D A N T SYSTEM R . S U B R A M A N I A N and N. R A V I C H A N D R A N Department of Mathematics, Indian Institute of Technology, Madras 600 036, India
(Received for publication 12 January 1979) Abstract--A two-unit standby redundant repairable system with repair limit suspension policy is studied. Expressions for the availability and reliability of the system are provided by identifying suitable regeneration points.
INTRODUCTION
Recently Osaki [I] considered some repair limit suspension policies for a two-unit standby redundant repairable system. His study was confined to cold standby systems in which the online unit has a constant failure rate. By employing Markov renewal processes, he obtained the Laplace transform of the availability of the system and optimized the same in the stationary case. The object of this contribution is to obtain expressions for the operating characteristics for a more general model without recourse to any transform technique. This is achieved by identifying certain regeneration points [2]. It is also demonstrated how this concept considerably simplifies the analysis. ASSUMPTIONS
(1) The system consists of two units. Either unit performs the desired operation satisfactorily. (2) The failure rate of the standby is assumed to be a constant. (3) There are two repair facilities. The time needed for repairing a unit in facility i(i-- 1, 2) is a random variable having a general distribution G~(-), with mean 1/#i. We assume that 1/#2 < 1/#1. (4) Repair Limit Suspension Time (RLST) is a random variable with a general distribution. This is the maximum time up to which any particular repair is allowed to continue in facility 1. (5) On failure a unit is taken to facility I. (6) If a unit fails when the other unit is undergoing repair in facility 1, then the unit undergoing repair is switched to the facility 2, which is faster. (7) If a repair is not completed in facility 1 before RLST then it is switched to facility 2, where the repair starts afresh. (8) Switch is perfect and switchover is instantaneous. (9) Each unit is "new" after repair. (10) When facility 2 functions, facility I is closed down. NOTATION
f~(t) adt) b(t) © ~(t) C(t) g(t) O("~(t)
3. e -z* pdf of the repair time in facility i (i = 1, 2) pdf of the random variable denoting the RLST convolution symbol 1 - c(t), c(') any function ~o c(u) du gl(t)g(t) + [Gl(t)b(t)] ©g2(t) n-fold convolution of the function O(t) with itself over (0, t) ANALYSIS
First we investigate the behaviour of the standby unit during the operative period of the online unit. Let S(t) denote the state of the standby at any time t. The following symbols are used to describe the state of the standby at any time t. 0 ri
operable. under repair in facility i; i = 1,2.
Further we define:
ni3(t) = Pr{S(t) = j [ S(0) = i 4: S(-0)}
i = 0, rl, r2
j = O , rl ni,:(t, x ) d x = Pr{S(t)= r2 and the stay in this state during its last visit to this state lies in (x, x + dx) [ S(0) = i ~ s(-0)}, i = 0, r l , r 2.
Defining h(t) = fs(t) © g(t) and
?(t) = ~ h~n~(t), n=l
we get by renewal theoretic arguments [3] 7Zoo(t) = Fs(t) + 7(t)(~ ff-s(t),
(1)
no,,(t) = fs(t) (~ [G,(t)B(t)] q- [?(t)(~)fs(t)] (~) [G,(t)B(t)], nor:(t, x ) = [ f ~ ( t - x ) ( ~ ( G l ( t -
(2)
x ) b ( t - x))] (~2(x)
+ 7(t - x) (~)fs(t - x) (~) [Gl(t - x)b(t - x)] G2(x), (3)
nr, o(t) = g(t)(~)noo(t), Event E One unit just switched online and the other unit just enters facility l for repair nr,r~(t) = g(t)(~)nor,(t) + Gl(t)B(t), fo~,(t) pdf of the life time of a unit while online (in standby) 7[r,r2(t, x) = Gl(t - x)b(t - x)G2(x) 277
(4) (5)
278
R. SUBRAMANIANand N. RAVK'HANDRAN ~(t x
,(](U)7~Orz(t-- U, X) du,
-F
(6)
)
Tc,:,:(t,x)=
~,:o(t) = .qz(t) (D 7ro0(t),
(7)
Rr:r,(t) =
(8)
gz(t)
(~ ~'0r,(t),
~t
t
j',~/~o(t) dt ~ .... .
R*{O~ =
As a consequence of key renewal theorem we get from (12) the steady state avilability of the system as
G:(t)cS(t - x) +
Now it is easy to get the mean time to system failure from {13),
x
g2(u)Tro,,(t - u, x) du.
.f~ Yo(t)
(9)
0
We use the regenerative nature of the E-event to obtain ~b(t) and @t) which are, respectively, the pdfs of the time intervals between two successive E-events with or without a system break-down in between. The expression for 4>(t) is derived by considering the following mutually exclusive and exhaustive possibilities in which the standby can be found, when the online unit fails : (1) Operable. (2) Under repair in facility 1. (3) Under repair in facility 2.
Special case
We now specialize to the case discussed in [1]. Setting ;t=0 Jo(t) = 21 e - ~'''
and
b(t) = 6 ( t -
to),
and simplifying, we get
fl =
)~-1 x
+
For the calculation of ~(t) possibilities 2 and 3 are not permissible.
--
It2
)
~"'~ (~t(t) dt
e -
O(t) = yo(t)~,o(t) + [fo(t)~,,,(t)] © g2(t) +
+
;o
fotu) du
o ~r,rz(U X)
g2(t
~- - ( xU )-~- x)" dx,
6(t) = Jo(t)~,, oct).
(10) ( 11 )
Now we are in a position to obtain the reliability
12
Zl
A
where g~()~ ) =
e ~,t ,qa(t} dr. )
R(t) and availability A(t) of the system conditioned at
the origin by an E-event. Writing
This result is in agreement with that given in [l].
REFERENCES n=l
n=l
we get by probabilistic arguments, A(t) =/To(t) + a(t)©/To(t),
(12)
R(t) = Fo(t) + fl(t)@Fo(t).
(13)
I. S. Osaki and K. Okumoto, Repair limit suspension policies for a two-unit standby redundant system with two phase repairs, Microelectron. Reliab. 16, 41 (1977). 2. S. K. Srinivasan, Stochastic Point Processes and their Application, Griffin, London (1974). 3. D. R. Cox, Renewal Theory, Methuen, London (1962).