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Analysis of a two-unit parallel redundancy with an imperfect switch
M~roele~u'mL Rehah., Vol 23, No. 2, pp. 309 318. 1983. Printed in Greztt Britain
ANALYSIS
OF
A
WITH
0026~2714/83/020309 10503.00 © 1983 Pergamon Press El
TWO-UNIT AN
PARALLEL
IMPERFECT
REDUNDANCY
SWITCH +
S.
K.
SR
INIVASAN
D e p a r t m e n t of Mathematics, NATIONAL
UNIVERSITY
OF SINGAPORE,
Kent Ridge 0511, SINGAPORE. (Received for publication 15th N o v e m b e r
1982)
Abstract
A parallel redundant system of two identical
units is studied
when the switchover from repair to on-line is defective.
It is assumed
that there is a single repair facility and that either unit has priority over the switchin~ device while queuing for repair.
The reliability and
aw~ilability functions are obtained explicitly when the units have a constant failure rate.
The method of extension to cover the case of dis-
similar units with non-constant failure rates is also indicated. I. Introduction. Two-unit parallel redundant system is an important one in the theory of reliability system analysis.
Gaver [l7 studied the system quite early
in the devolopment of the theory and this was followed by Liebowitz [3I who dealt with the special system in which the units have constant failure rates and repair times distributed quite arbitrarily.
This was improvized
by Linton [41. who used the supplementary variable technique and dealt with a more general system in which the failure and repair time distributions of one of the units are Erlangian and those of the other, arbitrary. Ohasbi et al [~
essentially followed the same technique and dealt with a
system of two identical units with arbitrary repair and failure rates. Subramanian and Ravichandran [81 studied the most general system dealt with so far and derived explicit expressions for the various operating characteristics.
In their analysis which is by far the most elegant one in as much
l e a v e o f a b s e n c e from Department o f M a t h e m a t i c s , I n d i a n I n s t i t u t e T e c h n o l o g y , Madras, I n d i a . 309
of
310
S . K . SRINIVASAN
as i t
c i r c u m v e n t s tile u g l y s u p p l e m e n t a r y v a r i a b l e
deal w i t h
tile system
in which t h e
b u t e d in an Erlan~!ian m a n n e r , arbitrarily.
taneously,
is r e p a i r e d ,
contribution
for
state
device
to enable the unit analyzes
'['lie consequences o f past
can be r e s t o r ( ' d
art
imperfect
o f the : ~ u b j e c t
car
We c o n f i n e o u r a t t e n t i o n
in p a r a l l e l , t h e
units
case of dissimilar uf t h e p a p e r ,
facility fails,
for the repair
tancously.
device
in
is undertaken
the switching is restored theory it
is t m d e r t a k c n
condition;
The e x t e n s i o n
repair
completed,
otherwise
t o h a v e an e n d l e s s before
the unit
will
line.
In t h e p a p e r we s h a l l
assume t h a t
device
and t h e u n i t
o r in s t a n d b y )
tically
(on l i n e
and s w i t c h i n g
distributed.
in a straight the unit
device
The r e s u l t s
device
the system
rate
are constants.
waits
to for
o f tile s w i t c h i n g The r e p a i r
times
and i d e n -
i n t h e p a p e r can be g e n e r a i i z e d
f o r w a r d m a n n e r when t h e d i s t r i b u t i ? n
and s w i t c h i n g
Of c o u r s e
i s s w i t c h e d on t o t h e main
the failure
presented
In
c o u l d be r e s t o r e d unit
of it
be r e p a i r e d .
a r e a s s u m e d t o be i n d e p e n d e n t l y
of the repair
times
of
are not identical,
The l a y o u t o f t h e p a p e r i s cribing
earlier
repair fail,
sequence of such e v e n t s . under repair
is k e p t
o f t h e switchiJ~2
in s t a n d b y does n o t
the unit
unti]
iristru/-
the unit
I f hy t h e t i m e t h e
repair
of the unit
had f a i l e d
If a unit
facility
otherwise
t h e main s y s t e m , we have a s y s t e m b r e a k d o w n and t h e o t h e r that
t o tile
is c o n n e c t e d t o ti~e s y s t e m
o f warm r e d u n d a c n y and r e p a i r
the unit
the unit
connected
in t h e c o n c l u d i n g p a r t
by t h e
the unit
is
fails
units
and t h e switchin}5 d e v i c e .
facility.
is possible unit
briefly
redun-
(assumed t o be) s u p p o r t e d bv a common re,p a i r
t o t h e main s y s t e m ;
if the other
for the ease of parallel
by t h e r e p a i r
device
and Subramanian [ 7 ] .
t o a s y s t e m o f two i d e n t i c a l
i s in o p e r a b l e
state
The
A summary o f t h e
in S r i n i v a s a n
be i n d i c a t e d
of the imit
a
to the system.
conditions,
On c o m p l e t i o n o f t h e r e p a i r ,
standby
then the device
s w i t c h i n t t had been a n a l y z e d i n the
he f ound
of the units
if the switching device in
will
The system i s
the repair
is d e f e c t i v e ,
b e i n g s w i t c h e d on a t t i m e t - ( ) .
units
that
such a system.
Ik)wever tile n r o b l e m had n u t been a n a l y z e d dancy.
:50 f a r i s
ba(k t o t h e s v s t c i n i n s t a n -
t o be r e s t o r e d
s t a n d b y systems u n d e r d i f f e r e n t
of
is dislr'i-
bein~ distributed
cltaracteristics
the models analyzed
tlowever i f t h e s w i t c h i n g
has t o be r e p a i r e d present
in a l l it
t h e authur.~
t i m e o f one ()f t h e u n i t s
~tll o t h e r
A COlllmOn f e a t u r e
whenever a unit
life
technique,
as f o l l o w s .
(along with the notation)
The s t o c h a s t i c is
introduced
model d e s -
in section
2.
Redundancy analysis
The subsequent
section deals with the analysis
of the reliability the regenerative availability
function.
311
leading to the determination
The final section contains
the analysis of
structure of the process which in turn leads to the
function and the Kingman p-functi~m.
2. System description
and notation
The system can be described by the discrete
valued stochastic
pro-
cess {Z(t); t > O} where Z(t) for any arbitrary t denotes the number of nonoperable units at the epoch t. it
c a n n o t be p u t on l i n e
We s h a l l
We shall take a unit to be non-operable
due t o n o n - o p e r a b l e
d e n o t e t h e s y s t e m by a f i n i t e
by s l a s h e s .
The f i r s t
of the units
(assumed i d e n t i c a l )
while the repair
last
is reserved
system.
Erlangian
model)
that
identified
of the switching device
characterizes
the nature
rate,
while G
The symbol Ek can be u s e d
of order
we . s h a l l
of the
be u s e d f o r e x p o n e n t i a l
with the failure
distribution.
distribution
time distribution
k.
indicate
Thus we h a v e an M/M/G how t h e r e s u l t s
for
s y s t e m s can be o b t a i n e d .
of the stochastic the switch is
we o b s e r v e t h a t
process
{Z(t))
in o p e r a b l e
we a l s o o b t a i n
a regenerative
by t h i s
we s h a l l
state,
does not consist
the point
into the state
condition state.
events corresponding
to entries
O are regenerative.
Moreover
when t h e p r o c e s s
The s t a t e
exclusive
Z(t)
In view o f t h e s p e c i a l
u s e t h e symbol 3 t o d e n o t e
of mutually
tageous to use it. Kingman
and t h e s e c o n d ,
At t h e c o n c l u d i n ~ s e c t i o n
At t h e o u t s e t
if
is
f o r t h e most g e n e r a l
more g e n e r a l
of the life
The symbol M (Markov) w i l l
whose p a r a m e t e r
to denote s pecial
s e q u e n c e o f t h e s y m b o l s M, G s e p a r a t e d
(in the present
time d i s t r i b u t i o n .
distribution
nature of the switching device.
symbol d e n o t e s t h e n a t u r e
symbol
if
it.
we s h a l l
a regenerative
= 3, i = 0,1,2,...,n
find
O,
enjoyed {0,1,2,3} it
advan-
phenomenon o f
~2] in the sense that for 0 = t o < t 1 < t2 ..... < t n Pr {Z(ti)
status
Although the
set of points,
3 generates
i s in s t a t e
we have
}
n
= Pr {Z(to) }
[i P(ti_ti_l ) i=l
where p(.) is the Kingman - p function to our initial the process
conditions,
Pr { Z(to)} = i.
can be characterized
the following notation
characterizing We shall
the process.
later on indicate how
in terms of the p-function.
:
pdf
:
probability
cdf
:
cumulative
density ftmction distribution
(f,g,q)
Function
According
(F,G,Q)
We shall use
312
S . K . SRIN1VASAN
sf
survivor
f(n) (t)
n-fold
x)
function
(F, G, Q )
convolute of f(t)
failure
rate
of the units
w h i l e on o n l i n e
failure
rate
of the units
while
failure
rate
of the switching
pdf of the repair
a-event
entry of the process
{Z(t)}
into
state
0
b-event
entry of the process
(Z(t)}
into
state
2
c-event
entry of the process
{Z(t)}
into
state
1 from s t a t e
:
time of the unit
t h e number o f y - e v e n t s
We n e x t n o t e t h a t
the a-events
in
a r e s w i t c h e d on,
the time o r i g i n
t o s y n c h r o n i z e w i t h an a - e v e n t .
and i t s
repair
of repair
completed.
o f two such p o l i c i e s the cost
structure
equivalent at
Since
Further
it
will
initially
d e v i c e may in t u r n
units
fail.
to a later
to assuming that
manner.
the state
venient
of the units. repair
facility.
{Y(t]}(the unit
It consists
origin
in question)
Its
state
failures
we d e f e r
device
arises
it
the switching
switching Device
only
it
i s cono f one
d e v i c e and t h e
by t h e s t o c h a s t i c
with states
is
of the unit.
defined
process epoch o f t h e as f o l l o w s .
State of Y(t]
even
the comparison
i m m e d i a t e l y on f a i l u r e
unit,
0,1,2
has failed
is inspected
f o r t h e p r o c e s s b e i n g c h o s e n as t h e f a i l u r e values
~n
the completion
l e a d t o an a - e v e n t ,
can be d e s c r i b e d
which t a k e s
need.
Thus f o r o u r p u r p o s e s ,
of switching
of the failed
the failure
w h e r e i n we p r o p o s e to d i s c u s s
the course of events that
to study the sub-system that
since
ause further
t i m e s o f commencement o r c o m p l e t i o n o f t h e r e p a i r To a n a l y z e
loss of generality,
one o f t h e u n i t s
In v i e w o f t h i s
contribution
in a d e t a i l e d
b o t h tile u n i t s
be a s s u m e d t h a t
until
2
renewal process while
only at the time of its
n o t be r e p a i r e d
device
(y=a,b,c)
Such a p o l i c y may be o p t i m a l
of the switching
any o f t h e o n l i n e
Ctl,t2]
can be a s s u m e d , w i t h o u t
o f ti~e s w i t c h i n g mect~anism g e t s d e t e c t e d words t h e s w i t c h w i l l
or switching
form an o r d i n a r y
t h e c-('vcn~:s form a delglyed r e n e w a l p r o c e s s .
before
device
g(.)
N (tl,t2)
other
in s t a n d b y
Unit
0
operable
1
under repair
under repair operable
2
non o p e r a b l e
under repair
3
operable
operable
Redundancy analysis
313
Normally on failure, the unit goes for repair so that the process Y(t) takes the initial value 0 or 2.
Then the process Y(t) makes transitions of the
type 0 + I (2 ÷I) and possibly loops of the type l +0 ~l before the state 3 is reached. (i:
Thus it is convenient to introduce the functions qi(t)
0,1,2) where qi(t) =
lim Pr{Y(t+A)=3,Y(~)~3 ~/'~ [ 0 , t ] IY(0)=i#Y(-A')}/k ~,h'÷0
(I)
3. Reliability of the system We now proceed to express the statistical characteristics of the system in terms of qi(.), g(.) and the failure rates X,v and ~.
The system
reliability and availability characteristics of the function are determined by the usual functions R(t) and A(t) where
R(t)
= Pr { N b ( 0 , t
) }=
(2)
0
A(t) = Pr {Z(t) / 2 }
(3)
it is assumed that initially (at t =0) both the units are switched on with the switch in operable condition.
As long as the assumption of constant failure
rates holds, it is equivalent to assuming that the units and switch are operable.
Thus the initial condition can be taken as Z(0} = 3. We first obtain explicit expressions for the functions qi(. ) (i=0,I,2).
to obtain qo(t) for any t > 0, we note that the first a-event at t may or may not be intercepted by a switch failure.
If there is a switch failure, we
note that the point event corresponding to the repair completion of the unit (which must follow the switch failure) is a regenerative event.
It must
be borne in mind that the on line unit should not fail throughout such otherwise it would it would lead to a system failure and such events would not contribute to the probability that the system is neliable.
Thus we have
qo(t) = g(t)e -St + it g(u)(l _e_BU) ql(t-u)du.
(4)
O
In an exactly similar manner, we have ql(t) = g(t)e -vt + I t g(u) (l -e
l ~ U
)qo ( t - u ) d u
(S)
O
q2(t) =
g(u)ql (t-u)du
(6)
O
Equations
(4) - (6) are amenable to Laplace Transform technique and denoting
the Laplace Transform of q i ( t )
by q i ( s ) we f i n d
314
S . K . SRINIVASAN
qo(S) = {g*(l~+s)+g*(,~+s)(g*[s)-~*!~3,+s}!/ f 1 -(.,.,*(s)-g*!~.s))(£*(s)-~(v+s)) An e x p l i c i t
form f o r q ~ ( s )
and "v w h i l e q ~ ( s )
(7)
}
can be n b t a i r l e d from (7} by an i n t e r c h a n g e
o f !-'
is !;iron hy
q~(s/ ; q~(s)S(.~) To o b t a i n (0,t)
the reliability'
function
may o r may n o t be i n t e r c e p t e d
the a-avoiding
contribution
(sO we note t h a t
R(t),
by an a - e v e n t .
Thus we d e n o t e by
- Pr{Na(O,t]=O , Nb(O,t)=0}
We n e x t n o t e t h a t
aR(t)
can be e x p l i c i t l y
that
(O,t)
may o r may n o t he i n t e r c e p t e d
Iakin£ t h i s
into account,
aR(t) To o b t a i n
= e -2at
R(t)
in t e r m s o f q i ( t )
by n o t i n g
by an on i i n e f a i l u r e .
we f i n d
+ 2
e
> e '
.o(t-u)
+ (I-c-Su)u,(t-u)}eX(t-U)du
(10)
we n o t e
The s e c o n d t e r m i s e v a l u a t e d (0,t).
This
of renewal theory
by f i x i n g
a-events
is the renewal dengity
(ll)
on t h e l a s t
we can s t i l l
and S u b r a m a n i a n
o f such a - e v e n t s ,
>0 ~
[7],
a-event
in t h e
of a-events.
use the machinary
Chapter 2),
If bha(-)
we have
N b ( 0 , t ) = 0 1=
a
bha(")
our attention
are defective,
(see Srinivasan
Pr (N ( 0 , t )
>.l} N b ( ( I , t ] , 0 }
i s b e s t done by u s i n g t h e r e n e w a l d e n s i t y
Although the b-avoiding
To o b t a i n
(9)
obtained
R(tJ = ;jR(t) + P r { N a ( 0 , t }
interval
~(t)
so t h a t
aR(t)
the interval
the interval
bha(u}aR(t-u)du
(t2)
O
explicity,
b e t w e e n two s u c c e s s i v e
we n o t e t h a t
b-avoiding
bha(t)
=
i f bf}(.)
a-events,
is the pdf of the interval
we t h e n have
~ b@(n)(t)
(13)
1
b~h(t) = 2
e -2xu X { e - g U q o ( t - u ) O
+ (l_e-~P)q2(t_u) Thus t h e r e l i a b i l i t y
function
we u s e L a p l a c e T r a n s f o r m ,
i s g i v e n by e q u a t i o n s
an e x p l i c i t a* R * ( s ) = a R * ( s ) {1 * b h (s) }
1
2x
-
2),
2),+~+--~
expression
s+X
(qo(S+),)_q~(s+X)]_
(14)
(11) t h r o u g h
(14).
If
f o r R*(s) can be o b t a i n e d :
qo (s+~)-q~(~+~)
= {2X+---7- 2 x + g + s
I
}e-~(t-U)du
2~ + 2~+s
2),
1-q~(s+~) s+X
q~(s+X)}
} /
(15)
R e d u n d a n c y analysis
Then m e a n t i m e t o s y s t e m f a i l u r e
315
(MTS[:) c a n be i d e n t i f i e d
a s R*(O)
:
(16)
We next proceed to obtain the availability
4. A v a i l a b i l i t y So f a r function
it
function we h a v e d e a l t
is necessary
with b-avoiding
to relax
this
to take into account b and c-events. circumwmt c-event
Renewal p r o c e s s
To obtain the availability Once this is done, we have
As b-events :ire not regenerative,
Thus i t
is necessarily
is sufficient
generated by a and c - e v e n t s .
notation
~(-) h~(-)
one.
events.
constraint.
them by noting that every b-event
with probability
(further)
function of the system.
if
wc
We i n t r o d u c e
followed by a study
t h e Markov
the following
:
: pdf of the interval between events of types a,B (a,B =a,c) : the renewal density matrix of Tile bIRP generated by a and c-events
(~,8 = a,c).
Next we n o t e by t h e d e f i n i t i o n
o f a and c - e v e n t s
it
follows
~aa (t) = b ~(t) 0ac(t)
= 2 Ii
(17)
e-2~P X{e-~Pqo(t-u )
+ (I -e-BP)q2(t-u )}(I -e-~(t-u))du
(18)
~ca(t) '= qo(t) e-~t
(19)
Occ(t) = qo(t) (I -e -£t)
(20)
The renewal density functions satisfy the equations Srinivasan
we
(see for example
~6])
h~6(t) = ~a~(t) +
~ y=a,c
(u)hy6(t-u)du
(a,B = a,c)
The functions h B(. ) can be obtained by Laplace Transform technique.
(21)
We next
observe that A(t)
= Pr { Z ( t )
~ 2 )
= Pr {Z(t) J 2, Na,c(0,t ) = O} +Pr{Z(t) 12, Na,c(0,t ) > O}
The first term on the rhs of (22) can be easily identified to be aR(t).
(22)
The
second term is evaluated by observing that there can be one or more events of type a or c.
Using renewal arguments,
we obtain
316
S . K . SRINIVASAN
Pr { Z ( t )
I 2, N a , c ( 0 , t ) > 0 }
I t0
=
haa(U)aR(t-u)du
i t0
+
hac(U)e-X[t-U)Q~(t-u)du
1221
Thus we finally have
A(t)
= aR(t)
+
It h a a ( U ) a R ( t _ u ) d u
+
(u)e-~(t-u)~..o(t-u)du
0 The s t e a d y
state
availability
for example Srinivasan
where ha(~)
by a l i m i t i n g
proceedure
(sc.e
[7] C h a p t e r 2): (24)
a R(u)du ÷ hc (~) JO[°~e-~Uo°(u)du
and h (¢o) a r e t h e e q u i l i b r i u m
values of haa(-)
C
be d e t e r m i n e d
~
through the renewal equations
We n e x t n o t e t h a t h aa ( , )
can be o b t a i n e d
and S u b r a m a n i a n
A(°°) = h a ( ~ ) I ~
(25)
0 ac
and h
(.)
and can
ac
(21).
the Kingman-p-function
can be o b t a i n e d
through
: p(t)
=
jt
-2Xt
haa(U)e-2X(t-U)du
+ e
of the process
{Z(t)}
(25)
0 The o t h e r
state
probabilities
]'he two t i m e p r o b a b i l i t i e s
explicitly.
can a l s o he o b t a i n e d
of the process
of the type aij(tl,t2)
where aij(tl,t are useful
quantities
for the determination
can a g a i n he e x p r e s s e d the results
presented
v a l u e and v a r i a n c e
S. Conclusion
2) = Pr ( Z ( t 1) = i ,
section
first
consider
tril)utions repair
o f t h e c o s t o v e r any a r b i t r a r y
failure rates.
rates
device,
i
to state
0 by r e s t o r a t i o n
add a s u b s r i p t
units
of the units
of the switching
while in standby.
consists
Thus
for estimates
of expected
horizon.
will
tt~e o n - l i n e
of the failed
i to a to identify
Thus we have t o d e a l w i t h a . - e v e n t s
units that
can be relaxed,
constant:;.
l,ct ~
repair time di'
1't1(' I1(({"
be d e n o t e d by' ? , ] ( ' )
by g O ( ' ) .
Since the a-event
1
of identical
with di>:tinct
that a r e d i s t i n c t
X. and ~. t o d e n o t e t e s p e c t i v e l y rates
ht~l~(-).
plmming
Both these constant~
t h e ca:¢e o f d i s s i m i l a r
time distributions
1
These
and outlook
anti f a i l u r e
while that
functions
form t h e b a s i s
The system that we have analyzed have constant
of the cost structure.
in terms of the renewal in t h i s
Z(t 2) ~ j t
t) t"
II[
ii:l,21
Likewi:~e we use t h e symbol~ failure
rates
and t h e f a i l n r e
d e n o t e s t h e e n t r y o f t h e systen~
u n i t back to t h e s y s t e m , we can u~e
the unit that
is restored
and t h e d i f f e r e n t
to the system,
q-functions
that
ari'~(
Redundancy
can be distinguished
by the superscript
analysis
i.
317
With this modification,
equations
(4) through (6) a r e to be r e p l a c e d by the f o l l o w i n g ones : _e-gU) i gi (a) (1 ql ( t - u ) d u 0 q ii ( t ) = g o ( t ) e - ~ .1t + I t g o ( u ) ( l _ e -~iu) qi~ (t_u) d u 0 q i2(t ) = f t gi (u) qli ( t - u ) d u •
qlo(t)
=
gi(t)
"
e_B t
+
ft
(4') (5') (6')
0
Likewise (10) g e t s r e p l a c e d by aR(t) = e (Xl+X2)t + ~ "
f
t e (Xl+X2)t ~i {e-gU -qoi ( t - u ) 0
+ ( 1 - e-f~n)Q21 (t-u) }e - x 3 - i ( t - u ) d u To o b t a i n the r e l i a b i l i t y
function,
(10')
we note t h a t the a. e v e n t s form a simple 1
type o f btarkov renewal p r o c e s s and e q u a t i o n s
(12) and (13) are s t i l l
valid
w h i l e (14) is to be r e p l a c e d by
b~(t)
= ~
i t e_(Xl.X2)u t i { -Su i ( t _ u ) 0 e qo + ( 1 - e -gu) q2i (t-u) }e -~3-i (t-u) du
(14')
The results of section 4 can be modified on similar lines, Finally we indicate how the results of sections Ic~ cover the case when the life time is distributed
[!rlangian type. characterize
3 and 4 can be extended
according
to the special
Since the epochs o f a - e v e n t s a r e c o n v e n i e n t p o i n t s ,
t h e s e epochs by a n o t h e r d i s c r e t e
we
index j t h a t d e s c r i b e s the
l-r'langian phase of the on l i n e u n i t at t h a t epoch. renewal p r o c e s s g e n e r a t e d by a - e v e n t s of type j .
Thus we have a Markov The a n a l y s i s p r e s e n t e d in
.~eetions 3 and 4 can be c a r r i e d through for the new type of a - e v e n t s . results
wJIl be v e r y s i m i l a r except f o r n o t a t i o n a l
complexity.
The
Of c o u r s e
f u r t h e r r e s e a r c h is n e c e s s a r y to e x t e n d , w i t h o u t r e c o u r s e supplementary variables, distributed
the a n a l y s i s to t h e most g e n e r a l case when the l i f e times are arbitrarily.
Reference 1
Gaver, l). P.,
(1963), Time to F a i l u r e and A v a i l a b i l i t y
Systems with Repair,
IEEE Trans.
2
Kingman, J. F . ,
(1971),
3
L i e b o w i t z , B. R. (1966), redundant system with
of P a r a l l e l e d
Rel. R-12, 30-38.
R e g e n e r a t i v e Phenomena, John Wiley, London. Reliability
considerations
for a two-unit
g e n e r a l i z e d r e p a i r t i m e s , (~pns. Res. 14, 233-4i.
318
S . K . SRINIVASAN
4.
IAnton, parallel
5.
Ohashi,
B. (;.,
(1976),
redundant
Some a d v a n c e m e n t s
system,
7.
S.
Springer-Verlag,
Subramanian, 2-unit
15, 3 9 - 4 6 .
T.,
Micro.
K. and S u b r a m a n i a n R . ,
Redundant Systems,
8.
system,
o t two Lm}~
1980), Rel.
Stochastic
20,
behavi¢~,n"
,171-7~.
London.
Srinivasan,
175,
Rcl.
/~L, Huang ,J. M. and N i s h i d a ,
of a tyro-unit paralleled
griffin,
~licro.
ill t h e a n a l y s i s
Lecture
Notes
in E c o n o m i c s :rod .~l~thc.ln:tt:ical Sy>;tum~:
Berlin.
t~. and R a v i c h a n d r a n ,
Parallel
Pr~@pb)li2tj~2_A21a!.?'2js of
(198()),
RedlJndanl S y s t e m ,
N.,
(1979),
IEI!E.
Stochastic
Trans.
R¢'.I. R.
gehaviour
28,
of
419-420.
ANALYSIS
OF
A
WITH
0026~2714/83/020309 10503.00 © 1983 Pergamon Press El
TWO-UNIT AN
PARALLEL
IMPERFECT
REDUNDANCY
SWITCH +
S.
K.
SR
INIVASAN
D e p a r t m e n t of Mathematics, NATIONAL
UNIVERSITY
OF SINGAPORE,
Kent Ridge 0511, SINGAPORE. (Received for publication 15th N o v e m b e r
1982)
Abstract
A parallel redundant system of two identical
units is studied
when the switchover from repair to on-line is defective.
It is assumed
that there is a single repair facility and that either unit has priority over the switchin~ device while queuing for repair.
The reliability and
aw~ilability functions are obtained explicitly when the units have a constant failure rate.
The method of extension to cover the case of dis-
similar units with non-constant failure rates is also indicated. I. Introduction. Two-unit parallel redundant system is an important one in the theory of reliability system analysis.
Gaver [l7 studied the system quite early
in the devolopment of the theory and this was followed by Liebowitz [3I who dealt with the special system in which the units have constant failure rates and repair times distributed quite arbitrarily.
This was improvized
by Linton [41. who used the supplementary variable technique and dealt with a more general system in which the failure and repair time distributions of one of the units are Erlangian and those of the other, arbitrary. Ohasbi et al [~
essentially followed the same technique and dealt with a
system of two identical units with arbitrary repair and failure rates. Subramanian and Ravichandran [81 studied the most general system dealt with so far and derived explicit expressions for the various operating characteristics.
In their analysis which is by far the most elegant one in as much
l e a v e o f a b s e n c e from Department o f M a t h e m a t i c s , I n d i a n I n s t i t u t e T e c h n o l o g y , Madras, I n d i a . 309
of
310
S . K . SRINIVASAN
as i t
c i r c u m v e n t s tile u g l y s u p p l e m e n t a r y v a r i a b l e
deal w i t h
tile system
in which t h e
b u t e d in an Erlan~!ian m a n n e r , arbitrarily.
taneously,
is r e p a i r e d ,
contribution
for
state
device
to enable the unit analyzes
'['lie consequences o f past
can be r e s t o r ( ' d
art
imperfect
o f the : ~ u b j e c t
car
We c o n f i n e o u r a t t e n t i o n
in p a r a l l e l , t h e
units
case of dissimilar uf t h e p a p e r ,
facility fails,
for the repair
tancously.
device
in
is undertaken
the switching is restored theory it
is t m d e r t a k c n
condition;
The e x t e n s i o n
repair
completed,
otherwise
t o h a v e an e n d l e s s before
the unit
will
line.
In t h e p a p e r we s h a l l
assume t h a t
device
and t h e u n i t
o r in s t a n d b y )
tically
(on l i n e
and s w i t c h i n g
distributed.
in a straight the unit
device
The r e s u l t s
device
the system
rate
are constants.
waits
to for
o f tile s w i t c h i n g The r e p a i r
times
and i d e n -
i n t h e p a p e r can be g e n e r a i i z e d
f o r w a r d m a n n e r when t h e d i s t r i b u t i ? n
and s w i t c h i n g
Of c o u r s e
i s s w i t c h e d on t o t h e main
the failure
presented
In
c o u l d be r e s t o r e d unit
of it
be r e p a i r e d .
a r e a s s u m e d t o be i n d e p e n d e n t l y
of the repair
times
of
are not identical,
The l a y o u t o f t h e p a p e r i s cribing
earlier
repair fail,
sequence of such e v e n t s . under repair
is k e p t
o f t h e switchiJ~2
in s t a n d b y does n o t
the unit
unti]
iristru/-
the unit
I f hy t h e t i m e t h e
repair
of the unit
had f a i l e d
If a unit
facility
otherwise
t h e main s y s t e m , we have a s y s t e m b r e a k d o w n and t h e o t h e r that
t o tile
is c o n n e c t e d t o ti~e s y s t e m
o f warm r e d u n d a c n y and r e p a i r
the unit
the unit
connected
in t h e c o n c l u d i n g p a r t
by t h e
the unit
is
fails
units
and t h e switchin}5 d e v i c e .
facility.
is possible unit
briefly
redun-
(assumed t o be) s u p p o r t e d bv a common re,p a i r
t o t h e main s y s t e m ;
if the other
for the ease of parallel
by t h e r e p a i r
device
and Subramanian [ 7 ] .
t o a s y s t e m o f two i d e n t i c a l
i s in o p e r a b l e
state
The
A summary o f t h e
in S r i n i v a s a n
be i n d i c a t e d
of the imit
a
to the system.
conditions,
On c o m p l e t i o n o f t h e r e p a i r ,
standby
then the device
s w i t c h i n t t had been a n a l y z e d i n the
he f ound
of the units
if the switching device in
will
The system i s
the repair
is d e f e c t i v e ,
b e i n g s w i t c h e d on a t t i m e t - ( ) .
units
that
such a system.
Ik)wever tile n r o b l e m had n u t been a n a l y z e d dancy.
:50 f a r i s
ba(k t o t h e s v s t c i n i n s t a n -
t o be r e s t o r e d
s t a n d b y systems u n d e r d i f f e r e n t
of
is dislr'i-
bein~ distributed
cltaracteristics
the models analyzed
tlowever i f t h e s w i t c h i n g
has t o be r e p a i r e d present
in a l l it
t h e authur.~
t i m e o f one ()f t h e u n i t s
~tll o t h e r
A COlllmOn f e a t u r e
whenever a unit
life
technique,
as f o l l o w s .
(along with the notation)
The s t o c h a s t i c is
introduced
model d e s -
in section
2.
Redundancy analysis
The subsequent
section deals with the analysis
of the reliability the regenerative availability
function.
311
leading to the determination
The final section contains
the analysis of
structure of the process which in turn leads to the
function and the Kingman p-functi~m.
2. System description
and notation
The system can be described by the discrete
valued stochastic
pro-
cess {Z(t); t > O} where Z(t) for any arbitrary t denotes the number of nonoperable units at the epoch t. it
c a n n o t be p u t on l i n e
We s h a l l
We shall take a unit to be non-operable
due t o n o n - o p e r a b l e
d e n o t e t h e s y s t e m by a f i n i t e
by s l a s h e s .
The f i r s t
of the units
(assumed i d e n t i c a l )
while the repair
last
is reserved
system.
Erlangian
model)
that
identified
of the switching device
characterizes
the nature
rate,
while G
The symbol Ek can be u s e d
of order
we . s h a l l
of the
be u s e d f o r e x p o n e n t i a l
with the failure
distribution.
distribution
time distribution
k.
indicate
Thus we h a v e an M/M/G how t h e r e s u l t s
for
s y s t e m s can be o b t a i n e d .
of the stochastic the switch is
we o b s e r v e t h a t
process
{Z(t))
in o p e r a b l e
we a l s o o b t a i n
a regenerative
by t h i s
we s h a l l
state,
does not consist
the point
into the state
condition state.
events corresponding
to entries
O are regenerative.
Moreover
when t h e p r o c e s s
The s t a t e
exclusive
Z(t)
In view o f t h e s p e c i a l
u s e t h e symbol 3 t o d e n o t e
of mutually
tageous to use it. Kingman
and t h e s e c o n d ,
At t h e c o n c l u d i n ~ s e c t i o n
At t h e o u t s e t
if
is
f o r t h e most g e n e r a l
more g e n e r a l
of the life
The symbol M (Markov) w i l l
whose p a r a m e t e r
to denote s pecial
s e q u e n c e o f t h e s y m b o l s M, G s e p a r a t e d
(in the present
time d i s t r i b u t i o n .
distribution
nature of the switching device.
symbol d e n o t e s t h e n a t u r e
symbol
if
it.
we s h a l l
a regenerative
= 3, i = 0,1,2,...,n
find
O,
enjoyed {0,1,2,3} it
advan-
phenomenon o f
~2] in the sense that for 0 = t o < t 1 < t2 ..... < t n Pr {Z(ti)
status
Although the
set of points,
3 generates
i s in s t a t e
we have
}
n
= Pr {Z(to) }
[i P(ti_ti_l ) i=l
where p(.) is the Kingman - p function to our initial the process
conditions,
Pr { Z(to)} = i.
can be characterized
the following notation
characterizing We shall
the process.
later on indicate how
in terms of the p-function.
:
:
probability
cdf
:
cumulative
density ftmction distribution
(f,g,q)
Function
According
(F,G,Q)
We shall use
312
S . K . SRIN1VASAN
sf
survivor
f(n) (t)
n-fold
x)
function
(F, G, Q )
convolute of f(t)
failure
rate
of the units
w h i l e on o n l i n e
failure
rate
of the units
while
failure
rate
of the switching
pdf of the repair
a-event
entry of the process
{Z(t)}
into
state
0
b-event
entry of the process
(Z(t)}
into
state
2
c-event
entry of the process
{Z(t)}
into
state
1 from s t a t e
:
time of the unit
t h e number o f y - e v e n t s
We n e x t n o t e t h a t
the a-events
in
a r e s w i t c h e d on,
the time o r i g i n
t o s y n c h r o n i z e w i t h an a - e v e n t .
and i t s
repair
of repair
completed.
o f two such p o l i c i e s the cost
structure
equivalent at
Since
Further
it
will
initially
d e v i c e may in t u r n
units
fail.
to a later
to assuming that
manner.
the state
venient
of the units. repair
facility.
{Y(t]}(the unit
It consists
origin
in question)
Its
state
failures
we d e f e r
device
arises
it
the switching
switching Device
only
it
i s cono f one
d e v i c e and t h e
by t h e s t o c h a s t i c
with states
is
of the unit.
defined
process epoch o f t h e as f o l l o w s .
State of Y(t]
even
the comparison
i m m e d i a t e l y on f a i l u r e
unit,
0,1,2
has failed
is inspected
f o r t h e p r o c e s s b e i n g c h o s e n as t h e f a i l u r e values
~n
the completion
l e a d t o an a - e v e n t ,
can be d e s c r i b e d
which t a k e s
need.
Thus f o r o u r p u r p o s e s ,
of switching
of the failed
the failure
w h e r e i n we p r o p o s e to d i s c u s s
the course of events that
to study the sub-system that
since
ause further
t i m e s o f commencement o r c o m p l e t i o n o f t h e r e p a i r To a n a l y z e
loss of generality,
one o f t h e u n i t s
In v i e w o f t h i s
contribution
in a d e t a i l e d
b o t h tile u n i t s
be a s s u m e d t h a t
until
2
renewal process while
only at the time of its
n o t be r e p a i r e d
device
(y=a,b,c)
Such a p o l i c y may be o p t i m a l
of the switching
any o f t h e o n l i n e
Ctl,t2]
can be a s s u m e d , w i t h o u t
o f ti~e s w i t c h i n g mect~anism g e t s d e t e c t e d words t h e s w i t c h w i l l
or switching
form an o r d i n a r y
t h e c-('vcn~:s form a delglyed r e n e w a l p r o c e s s .
before
device
g(.)
N (tl,t2)
other
in s t a n d b y
Unit
0
operable
1
under repair
under repair operable
2
non o p e r a b l e
under repair
3
operable
operable
Redundancy analysis
313
Normally on failure, the unit goes for repair so that the process Y(t) takes the initial value 0 or 2.
Then the process Y(t) makes transitions of the
type 0 + I (2 ÷I) and possibly loops of the type l +0 ~l before the state 3 is reached. (i:
Thus it is convenient to introduce the functions qi(t)
0,1,2) where qi(t) =
lim Pr{Y(t+A)=3,Y(~)~3 ~/'~ [ 0 , t ] IY(0)=i#Y(-A')}/k ~,h'÷0
(I)
3. Reliability of the system We now proceed to express the statistical characteristics of the system in terms of qi(.), g(.) and the failure rates X,v and ~.
The system
reliability and availability characteristics of the function are determined by the usual functions R(t) and A(t) where
R(t)
= Pr { N b ( 0 , t
) }=
(2)
0
A(t) = Pr {Z(t) / 2 }
(3)
it is assumed that initially (at t =0) both the units are switched on with the switch in operable condition.
As long as the assumption of constant failure
rates holds, it is equivalent to assuming that the units and switch are operable.
Thus the initial condition can be taken as Z(0} = 3. We first obtain explicit expressions for the functions qi(. ) (i=0,I,2).
to obtain qo(t) for any t > 0, we note that the first a-event at t may or may not be intercepted by a switch failure.
If there is a switch failure, we
note that the point event corresponding to the repair completion of the unit (which must follow the switch failure) is a regenerative event.
It must
be borne in mind that the on line unit should not fail throughout such otherwise it would it would lead to a system failure and such events would not contribute to the probability that the system is neliable.
Thus we have
qo(t) = g(t)e -St + it g(u)(l _e_BU) ql(t-u)du.
(4)
O
In an exactly similar manner, we have ql(t) = g(t)e -vt + I t g(u) (l -e
l ~ U
)qo ( t - u ) d u
(S)
O
q2(t) =
g(u)ql (t-u)du
(6)
O
Equations
(4) - (6) are amenable to Laplace Transform technique and denoting
the Laplace Transform of q i ( t )
by q i ( s ) we f i n d
314
S . K . SRINIVASAN
qo(S) = {g*(l~+s)+g*(,~+s)(g*[s)-~*!~3,+s}!/ f 1 -(.,.,*(s)-g*!~.s))(£*(s)-~(v+s)) An e x p l i c i t
form f o r q ~ ( s )
and "v w h i l e q ~ ( s )
(7)
}
can be n b t a i r l e d from (7} by an i n t e r c h a n g e
o f !-'
is !;iron hy
q~(s/ ; q~(s)S(.~) To o b t a i n (0,t)
the reliability'
function
may o r may n o t be i n t e r c e p t e d
the a-avoiding
contribution
(sO we note t h a t
R(t),
by an a - e v e n t .
Thus we d e n o t e by
- Pr{Na(O,t]=O , Nb(O,t)=0}
We n e x t n o t e t h a t
aR(t)
can be e x p l i c i t l y
that
(O,t)
may o r may n o t he i n t e r c e p t e d
Iakin£ t h i s
into account,
aR(t) To o b t a i n
= e -2at
R(t)
in t e r m s o f q i ( t )
by n o t i n g
by an on i i n e f a i l u r e .
we f i n d
+ 2
e
> e '
.o(t-u)
+ (I-c-Su)u,(t-u)}eX(t-U)du
(10)
we n o t e
The s e c o n d t e r m i s e v a l u a t e d (0,t).
This
of renewal theory
by f i x i n g
a-events
is the renewal dengity
(ll)
on t h e l a s t
we can s t i l l
and S u b r a m a n i a n
o f such a - e v e n t s ,
>0 ~
[7],
a-event
in t h e
of a-events.
use the machinary
Chapter 2),
If bha(-)
we have
N b ( 0 , t ) = 0 1=
a
bha(")
our attention
are defective,
(see Srinivasan
Pr (N ( 0 , t )
>.l} N b ( ( I , t ] , 0 }
i s b e s t done by u s i n g t h e r e n e w a l d e n s i t y
Although the b-avoiding
To o b t a i n
(9)
obtained
R(tJ = ;jR(t) + P r { N a ( 0 , t }
interval
~(t)
so t h a t
aR(t)
the interval
the interval
bha(u}aR(t-u)du
(t2)
O
explicity,
b e t w e e n two s u c c e s s i v e
we n o t e t h a t
b-avoiding
bha(t)
=
i f bf}(.)
a-events,
is the pdf of the interval
we t h e n have
~ b@(n)(t)
(13)
1
b~h(t) = 2
e -2xu X { e - g U q o ( t - u ) O
+ (l_e-~P)q2(t_u) Thus t h e r e l i a b i l i t y
function
we u s e L a p l a c e T r a n s f o r m ,
i s g i v e n by e q u a t i o n s
an e x p l i c i t a* R * ( s ) = a R * ( s ) {1 * b h (s) }
1
2x
-
2),
2),+~+--~
expression
s+X
(qo(S+),)_q~(s+X)]_
(14)
(11) t h r o u g h
(14).
If
f o r R*(s) can be o b t a i n e d :
qo (s+~)-q~(~+~)
= {2X+---7- 2 x + g + s
I
}e-~(t-U)du
2~ + 2~+s
2),
1-q~(s+~) s+X
q~(s+X)}
} /
(15)
R e d u n d a n c y analysis
Then m e a n t i m e t o s y s t e m f a i l u r e
315
(MTS[:) c a n be i d e n t i f i e d
a s R*(O)
:
(16)
We next proceed to obtain the availability
4. A v a i l a b i l i t y So f a r function
it
function we h a v e d e a l t
is necessary
with b-avoiding
to relax
this
to take into account b and c-events. circumwmt c-event
Renewal p r o c e s s
To obtain the availability Once this is done, we have
As b-events :ire not regenerative,
Thus i t
is necessarily
is sufficient
generated by a and c - e v e n t s .
notation
~(-) h~(-)
one.
events.
constraint.
them by noting that every b-event
with probability
(further)
function of the system.
if
wc
We i n t r o d u c e
followed by a study
t h e Markov
the following
:
: pdf of the interval between events of types a,B (a,B =a,c) : the renewal density matrix of Tile bIRP generated by a and c-events
(~,8 = a,c).
Next we n o t e by t h e d e f i n i t i o n
o f a and c - e v e n t s
it
follows
~aa (t) = b ~(t) 0ac(t)
= 2 Ii
(17)
e-2~P X{e-~Pqo(t-u )
+ (I -e-BP)q2(t-u )}(I -e-~(t-u))du
(18)
~ca(t) '= qo(t) e-~t
(19)
Occ(t) = qo(t) (I -e -£t)
(20)
The renewal density functions satisfy the equations Srinivasan
we
(see for example
~6])
h~6(t) = ~a~(t) +
~ y=a,c
(u)hy6(t-u)du
(a,B = a,c)
The functions h B(. ) can be obtained by Laplace Transform technique.
(21)
We next
observe that A(t)
= Pr { Z ( t )
~ 2 )
= Pr {Z(t) J 2, Na,c(0,t ) = O} +Pr{Z(t) 12, Na,c(0,t ) > O}
The first term on the rhs of (22) can be easily identified to be aR(t).
(22)
The
second term is evaluated by observing that there can be one or more events of type a or c.
Using renewal arguments,
we obtain
316
S . K . SRINIVASAN
Pr { Z ( t )
I 2, N a , c ( 0 , t ) > 0 }
I t0
=
haa(U)aR(t-u)du
i t0
+
hac(U)e-X[t-U)Q~(t-u)du
1221
Thus we finally have
A(t)
= aR(t)
+
It h a a ( U ) a R ( t _ u ) d u
+
(u)e-~(t-u)~..o(t-u)du
0 The s t e a d y
state
availability
for example Srinivasan
where ha(~)
by a l i m i t i n g
proceedure
(sc.e
[7] C h a p t e r 2): (24)
a R(u)du ÷ hc (~) JO[°~e-~Uo°(u)du
and h (¢o) a r e t h e e q u i l i b r i u m
values of haa(-)
C
be d e t e r m i n e d
~
through the renewal equations
We n e x t n o t e t h a t h aa ( , )
can be o b t a i n e d
and S u b r a m a n i a n
A(°°) = h a ( ~ ) I ~
(25)
0 ac
and h
(.)
and can
ac
(21).
the Kingman-p-function
can be o b t a i n e d
through
: p(t)
=
jt
-2Xt
haa(U)e-2X(t-U)du
+ e
of the process
{Z(t)}
(25)
0 The o t h e r
state
probabilities
]'he two t i m e p r o b a b i l i t i e s
explicitly.
can a l s o he o b t a i n e d
of the process
of the type aij(tl,t2)
where aij(tl,t are useful
quantities
for the determination
can a g a i n he e x p r e s s e d the results
presented
v a l u e and v a r i a n c e
S. Conclusion
2) = Pr ( Z ( t 1) = i ,
section
first
consider
tril)utions repair
o f t h e c o s t o v e r any a r b i t r a r y
failure rates.
rates
device,
i
to state
0 by r e s t o r a t i o n
add a s u b s r i p t
units
of the units
of the switching
while in standby.
consists
Thus
for estimates
of expected
horizon.
will
tt~e o n - l i n e
of the failed
i to a to identify
Thus we have t o d e a l w i t h a . - e v e n t s
units that
can be relaxed,
constant:;.
l,ct ~
repair time di'
1't1(' I1(({"
be d e n o t e d by' ? , ] ( ' )
by g O ( ' ) .
Since the a-event
1
of identical
with di>:tinct
that a r e d i s t i n c t
X. and ~. t o d e n o t e t e s p e c t i v e l y rates
ht~l~(-).
plmming
Both these constant~
t h e ca:¢e o f d i s s i m i l a r
time distributions
1
These
and outlook
anti f a i l u r e
while that
functions
form t h e b a s i s
The system that we have analyzed have constant
of the cost structure.
in terms of the renewal in t h i s
Z(t 2) ~ j t
t) t"
II[
ii:l,21
Likewi:~e we use t h e symbol~ failure
rates
and t h e f a i l n r e
d e n o t e s t h e e n t r y o f t h e systen~
u n i t back to t h e s y s t e m , we can u~e
the unit that
is restored
and t h e d i f f e r e n t
to the system,
q-functions
that
ari'~(
Redundancy
can be distinguished
by the superscript
analysis
i.
317
With this modification,
equations
(4) through (6) a r e to be r e p l a c e d by the f o l l o w i n g ones : _e-gU) i gi (a) (1 ql ( t - u ) d u 0 q ii ( t ) = g o ( t ) e - ~ .1t + I t g o ( u ) ( l _ e -~iu) qi~ (t_u) d u 0 q i2(t ) = f t gi (u) qli ( t - u ) d u •
qlo(t)
=
gi(t)
"
e_B t
+
ft
(4') (5') (6')
0
Likewise (10) g e t s r e p l a c e d by aR(t) = e (Xl+X2)t + ~ "
f
t e (Xl+X2)t ~i {e-gU -qoi ( t - u ) 0
+ ( 1 - e-f~n)Q21 (t-u) }e - x 3 - i ( t - u ) d u To o b t a i n the r e l i a b i l i t y
function,
(10')
we note t h a t the a. e v e n t s form a simple 1
type o f btarkov renewal p r o c e s s and e q u a t i o n s
(12) and (13) are s t i l l
valid
w h i l e (14) is to be r e p l a c e d by
b~(t)
= ~
i t e_(Xl.X2)u t i { -Su i ( t _ u ) 0 e qo + ( 1 - e -gu) q2i (t-u) }e -~3-i (t-u) du
(14')
The results of section 4 can be modified on similar lines, Finally we indicate how the results of sections Ic~ cover the case when the life time is distributed
[!rlangian type. characterize
3 and 4 can be extended
according
to the special
Since the epochs o f a - e v e n t s a r e c o n v e n i e n t p o i n t s ,
t h e s e epochs by a n o t h e r d i s c r e t e
we
index j t h a t d e s c r i b e s the
l-r'langian phase of the on l i n e u n i t at t h a t epoch. renewal p r o c e s s g e n e r a t e d by a - e v e n t s of type j .
Thus we have a Markov The a n a l y s i s p r e s e n t e d in
.~eetions 3 and 4 can be c a r r i e d through for the new type of a - e v e n t s . results
wJIl be v e r y s i m i l a r except f o r n o t a t i o n a l
complexity.
The
Of c o u r s e
f u r t h e r r e s e a r c h is n e c e s s a r y to e x t e n d , w i t h o u t r e c o u r s e supplementary variables, distributed
the a n a l y s i s to t h e most g e n e r a l case when the l i f e times are arbitrarily.
Reference 1
Gaver, l). P.,
(1963), Time to F a i l u r e and A v a i l a b i l i t y
Systems with Repair,
IEEE Trans.
2
Kingman, J. F . ,
(1971),
3
L i e b o w i t z , B. R. (1966), redundant system with
of P a r a l l e l e d
Rel. R-12, 30-38.
R e g e n e r a t i v e Phenomena, John Wiley, London. Reliability
considerations
for a two-unit
g e n e r a l i z e d r e p a i r t i m e s , (~pns. Res. 14, 233-4i.
318
S . K . SRINIVASAN
4.
IAnton, parallel
5.
Ohashi,
B. (;.,
(1976),
redundant
Some a d v a n c e m e n t s
system,
7.
S.
Springer-Verlag,
Subramanian, 2-unit
15, 3 9 - 4 6 .
T.,
Micro.
K. and S u b r a m a n i a n R . ,
Redundant Systems,
8.
system,
o t two Lm}~
1980), Rel.
Stochastic
20,
behavi¢~,n"
,171-7~.
London.
Srinivasan,
175,
Rcl.
/~L, Huang ,J. M. and N i s h i d a ,
of a tyro-unit paralleled
griffin,
~licro.
ill t h e a n a l y s i s
Lecture
Notes
in E c o n o m i c s :rod .~l~thc.ln:tt:ical Sy>;tum~:
Berlin.
t~. and R a v i c h a n d r a n ,
Parallel
Pr~@pb)li2tj~2_A21a!.?'2js of
(198()),
RedlJndanl S y s t e m ,
N.,
(1979),
IEI!E.
Stochastic
Trans.
R¢'.I. R.
gehaviour
28,
of
419-420.