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Cost analysis of a two-dissimilar unit cold standby redundant system with administrative delay and no priority in repair
MIcroelectron. Reliab., Vol. 30, No. 6, pp. 1155-1177, 1990.
0026--2714/9053.00 + .00 © 1990 Pergamon Press pic
Printed in Great Britain.
COST ANALYSIS OF A TWO-DISSIMILAR UNIT COLD STANDBY REDUNDANT SYSTEM WITH ADMINISTRATIVE DELAY AND NO
PRIORITY IN REPAIR S.S. ELIAS and S.W. LABIB Ain Shams University, Faculty of Science, Department of Mathematics, Cairo, Egypt (Received for publication 20 February 1990)
This paper deals with the cost analysis similar unit cold standby redundant for each unit under the assumption rative delay and no priority
system with three modes that there is administ-
in repair.
repair time and administrative
The failure time,
time distributions
ral and arbitrary.
Some reliability measures
to system designers
have been obtained.
vious results are derived
of a two-dis-
are gene-
of interest
Moreover
some pre-
from the present results as spe-
cial cases.
1.
IMTRODUCTIONAND
Gupta
F O R M U L A T I O M OF T H E PROBLEM:
[2] discussed
the stochastic
similar unit cold standby redundant unit works in three modes, and total failure.
nistrative
assumption
partial
failure
the same problem of [2], but with
delay and no priority
in repair.
The admi-
was assumed to be exponential.
Elias and Labib
state availability redundant
i.e., normal,
time distribution
Mokaddis,
system in which each
and arbitrary.
[3] discussed
administrative
[i] have derived
of a two-dissimilar
the steady
unit cold standby
system where each unit has three modes under the that there is an administrative
delay in getting
• the repairman at the system location and assuming randoms
of a two
The repair and failure time distribut-
ions are exponential Gupta
behaviour
variables
having general distributions. 1155
that all
1156
S.S. ELIAS and S. W. LABIB
The purpose of the present paper is to study the cost function of a two-dissimilar unit cold standby redundant system and three different modes, with administrative delay and no priority in repair. The following assumptions and notations are adopted for our model.
(1)
Two dissimilar units operate in cold standby configuration
i.e., a unit does not fail du=ing its standby
state.
(2)
Each unit of the system has three modes
:
i. normal mode, which means the functioning of the unit with full capacity, ii. partial mode which means the functioning of the unit with reduced capacity at a specified level, iii. total failure mode which means the functioning of the unit with a capacity below the specified level. (3)
Upon total failure of an operating certain administrative actions, trative delay in getting repair, a unit comprises two parts:
unit and due to
there is an adminisso the down time of
administrative delay
and repair time. (4)
No unit of the system can attain the total failure mode without passing through the partial failure mode.
(5)
Whenever the operative unit fails completely, standby unit is switched. instantaneous
(6)
the
The switching is perfect,
and without any damage to the system.
In P-mode, a unit is working as will as under going repair. mode,
On getting repaired,
the unit enters the N-
it is also possible that the unit during repair
deteriorates
further and enters the F-mode.
The events
of a P-unit passing into N- and F-modes are mutually exclusive.
Two-dissimilar unit cold standby redundant system
(7)
In case the N-unit
fails partially,
unit is in F-mode and under repair, till the repair of the F-unit
(8)
When one unit is o p e r a t i v e led unit is under repair, unit is completed, as standby,
1157
while the other the former waits
is completed.
and the other totally
fai-
if the repair of the failed
the o p e r a t i v e
unit will be taken
and the repair unit will be taken as ope-
rative. (9)
The repair time for a P-unit and F-unit,
administra-
tive delay time and times to transit between different modes are random variables ted such that Di
: Pi
generally
distribu-
:
> FAi
Hi :
FA i
>
Fr i ,
L i : Pwi----> Fw i
Vi :
Oi
>
Pi
>
0 i.
Gli:
Pi
> Oi
and
G2i : Fr i
We refer to the d i f f e r e n t
aspects
'
of unit i; i = 1,2
by : O i , St i
operative mode,
Pi
partially repair)
standby mode,
failed
(and stipulated
,
FAi
failed unit
Fri
repair of unit
i
under a d m i n i s t r a t i v e i
delay has finished Fci
to be under
totally
Pwi
partially
Fwi
totally
, i
and repair continued
preceding
failed unit
failed
unit
i i
state
from
,
waiting
for repair
and w a i t i n g
Under the above assumptions, one of the following
,
started after a d m i n i s t r a t i v e
failed unit
the i m m e d i a t e l y
delay
,
for repair.
the system will be in
states at any instant;
So E (01, St2)
,
S1 E (Stl, 02)
'
$2 ~ (PI ' St2)
'
$3 i (Stl , P2)
,
$4 m (FAI ' 02 )
,
S5 ~ (01 , FA2)
,
1158
S.S. ELIASandS. W. LABIs
S 6 • (Frl, 0 2 )
,
S 7 ~ (01 , Fr 2)
,
S8 E (FAI , p2 ) ,
$9 m (P1 ' FA2)
'
Sl0m
(Fc1' Fw2)
'
Sll ! (Pw I, Fc 2) ,
$12 ~ (Fr I, Pw 2) ,
S13 a (Pw I, Fr 2)
,
S14 a (FA I, Fw 2) ,
$15a
(Fw I, FA 2) ,
$16a
(Frl, Fw 2)
,
S17 a (Fw I, Fr 2) ,
$18E
(Fc I, Fw 2) ,
Slga
(Fw 1, Fc 2 )
Let tively,
U
thus
and
F
denote the up and failed state respec-
U e (So, S I, .-- , SI3),
The state diagram of the system
Fig.
Q
Up states
D
Down stores
F e (s14, s15,...,s19) is depicted
in Fig. I.
i. The state diagram of the system.
2. T R A N S I T I O N P R O B A B I L I T Y AND M E A N W A I T I N G T I M E S IN S T A T E S
Let
:
Eo
state of the system at time t = 0 ,
E
set of regenerative
states
i = 0, 1, 2, ..., 9, 12, 13,
~Si} ..., 17 ,
Two-dissimilar unit cold standby redundant system
set of non-regenerative
states
1159
{S j} ;
j = 10 , 11 , 18 , 19. Pij
one-step transition probability from state state
p(U,V)
ij
Sj ;
&
(S i
Sj e
S i to
E) ,
transition probability from state
S i to state Sj
through states Su&S v, (S i & Sj e E and S u & S v e E). Denote the complementary function of any distribution F(.) by
F(.) ffi l-F(.). Therefore,
P20
ffi I O
P31
= I
P46
OI
P02 = PI3 = P14,16 = P15,17 = PI6,7=PlT,6 ml' OO Dl(t)dGll (t) P24 " S allCt~ aDlCt~ O cO ' P35 = J" Gl2lt) dD2lt) D2(t)dGl2(t) O CO ' P48 = J~ HI (t) dV2(t) ' V2(tldHl(t )
O
O
P57
GO = I Vl(tldH2 (t) O Go
P60
" I o
p(10) 63
'
P59 =
,J~CDH2lt) dVllt) o
V2 (t) dG21(t) ffi f O O l t t=o x=o CO
-
L2(t-x) dV2(x) dG21(t)
0~
I I XffiO t=x
L2(t-x) dV2(x) dG21(t), put t-x=y
O0 X=O
L2(Y) dV2(x) dG21(Y + x) ,
y=o
p(10,18)= _(i0) r6'18
67
~ =
G9
I
I
XffiO
G21(t) dV2(x) dL2(t-x)
G21(t) dV2(x) dL2(t-x)
t=x
co
-
x=o
co
S
X=O
y
t=o
CO
S
y=o
G21(Y + x) dV 2 (x) d L 2 (y)
OD P71
=
I
(11) P72
-
I
V1 (t) d G22(t)
o
CO X~O
S
y=o
Ll(Y) d VI(X) d G2~Y + x) , oo
p(11,19)= 76
p~ll) ,19
-- S
x=o
CO
S
y=o
G22(Y+X) dVl(X) d Ll(Y)
'
1160
S.S. ELIASand S. W. LABIB
GO
GO , J~ Hz(t)D2(t)dGl2 (t) P8,12 ffi D2(t)Gl2(t)dHl(t) o
I o
P84
UB
:
P8,14
I
G12(t)Hl(t)dD2 (t)'
o
P95 :
CO I Dl(t)Gll(t)dH2(t), o
P9,13
=
P12,3
ffi I o
co
£2(tldG21(t)'
H2(-t)Dl(t)dGll(t)
I
o
P9,15 :
P12,18
T GII (t) H2(t)dDl (t)' o
= p(18) = 12,7
I o
G21(t)dL2(t),
GO £i (t)d S22(t) ,
P13,2
ffi I o
and
Pij = o
P13,19 = p(19) = T - G22 (t)dLl(t), 13,6 o
for other
i
and
j .
To obtain the mean waiting times in states, let mij denotes the mean time for direct transition from state
S.1 Sj ; (i, j = 0, i, 2, ..., 19) and Pij(s) denotes
to state
the Laplace-transform of time
Bi of the system in
PO = m02
'
Pij(t).
Therefore,
the mean stay
Si(i = 0,1,2,...,19) ~].
is given by
ffi m13
CO -d * (s) + * (s)] = .PGll(t)Dl(t)dt, ~2 ffim20+ m24 = ~ [P20 P24 sffio co -d • • = ~3 = m31 + m35 " ~ [51 Is) + 5 5 (Sl]s=o ~ Gl2(tID2(tldt' co g4 " m46
+ m48 " ~ s
[P46
(s) + P*
48
(s)]
sffio
ffi ~"o ~Z(t)~2(t)dt, co
~5 ". m57 + m59" ~sd [ p
* (s)] 7 (s) + P59
s=o
o
~2(t)gZCt)dt
,
(10,18)* ~6 " m60 + m63 + m6 ,i0 = ~-6 -d [P:0 (s) + p(10 )*(s) + p 63
•
67
(s)] s =o
co
Io ~7 " m71
+ m
72
+ m
-d [p;l ( _(ii _(11,19)* 7,11 = ~ s) + P72 )*(s) + P76
OD
~J~ Ql(t) G22(t)dt, o
V2(t) G21(t) dt ,
(s)] S=O
Two-dissimilar unit cold standby redundant system
1161
P8 = m84 + m8,12 + m8,14 = ~ss [P8,t s) + p ,12 (s) + P8,14 ( s ) ] oO I H1 ( t ) o
"
Gi2 I t )
52 I t )
s o
,
at
* (s) + p; ,13 (s) + p*9, 15(S)]sffio H9 = m95 + m9,13 + m9,15 = ~-d [P95
- ~
N2(t) ~lz(t) 5 z (t)
o
dt
GD PI2 = m12,3 + m12,18
ffi ~-d [~2,3 (s)
, p12,1e(s)] = ~ ~.2(t)~n(t)dt,
+
S=O
O
Co PI3 = m13,2 + m13,19 = ~-d [P~3, 2 (s)
3.
T I M E TO SYSTEM FAILURE
+ P13,19 * ( s )s=o ] = IoL l ( t ) G 2 2 ( t ) d t .
(TSF)
Let • TTi (t)
cdf of the time to system failure
T[~(s)
Laplace - Stieltjes transform o f ~ ( t )
®
I E
symbol for ordinary convolution, t
Act) @
s(t) =
~
s (t-
o symbol for Stieltjes
®
A(t)
Let
TO ,
T
the state
~
B(t)
=
u)
i.e., A
(u)
convolution
It o
o = Si
,
an
,
i.e. ,
B (t - u) dA (u)
denote the epoch at which the system enters
i (i e E) and let
state visited at time
Tn
T O = o.
If X n denotes
then ~ X n , Tn~
the
constitutes a
J
Markov renewal process with state space E , Q
=
Qij ( t )
where Qij " P ~Xn+l " J is
a
semi
-
Markov
;
Tn+l - Tn < t / Xn ffi i}
kernal over
E.
Using the theory of regenerative the following equations
:
~ o (t)
"
Q02 (t)
~
~2
(t)
,
]71 (t)
"
QI3 (t)
Q
]-[3 (t)
,
process, we obtain
1162
S. S. EUAS and S. W. LAmB
(t) TT2 TT3(t) .
,~,. ,t,
@ IT,,,,.
T4(t) .
o,, ,t,
-- , ~ ,t,
TF5(t)
, ~ ,t,
® . Tr~ ,t,
®
TT,,,:, .. Q,, ,,:,
T T 8 (t)
,
®
TI-~,,~, + Q, ,,:, @ IF9 (t,
,
IT6(t) = Q60 (t) @
~°(t)+
TFT(t, = TT,(t, =
%~6^(I0)(t)3@
TT3(t) + ~(I0)%16,18(t),
Q~.,,:,
®
Fr,,,:, .. ^,,~,,,:, ® TL,,:, + ^ " ~ ' ,,:,.
Q84(t)
®
Tr,t, + Q~..~,,:, ®
~72
~7,19
® %,,, +
[9(t) " Q95(t}
TL,,:, + o~..,,:,.
®
~12(t) =
Q12,3(t)@
if3 (t) + Q12,18(t) ,
% 3 (t) "
Q13,2( t' Q
~2 (t) + Q13,19 (t)
+
Taking Laplace-Stieltjes transform of these relations and solving fOrgo
(s), the
L-S
transform of the distribu-
tion function of the first passing time is given by :
•
(s)
-
:
N 1 (s) / D (s)
(1}
where Nl(S) "• Q02 Q24
{(Q48 [ Q8,14 + Q8,12 Q12,18 ] + W46~6,18 ^ ^1101"111;
QI3Q31 ) (I'Q59Q95) - Q35 Q57 QTI QI.3 }
+
(Q59[Q9,15 + Q9,13QI3,19] + ~57 ~ ^ ^(ii)7,19) ( Q35 [ l
•• 10 •
Q48 Q8,12 Q12,3 + Q4 6w63
] )$
'
Dl(S) - [(I-Q02Q20)(I-Q48Q84)-Q46Q60Q02Q24][(I-QI3Q31)(I-Q59Q95) -Q57QTIQI3Q35]-Q24Q35[Q48Q8,12QI2,3
+
Q59 Q9,13 Q13,2 + Q57 Q~½1) ] where for brevity we have omitted the argument s
n(Io) ] [
Q46 ~63
Two-dissimilar unit cold standby redundant system
The mean
time
1163
to system failure for the first
time
is given by : E(T)
=
ds
S=O
(2)
D1
where D1 -
P24 P35
E[(1-P48 P84 ) - P46 P60][(I-P59 P95)'P57P71 ] +
- [P48P8,12PI2,3 N1
=
N~(o)
=
"2
+ P24
-
(10)]
+
P46P63
[P59P9,13PI3,2
(11).7
P57P72
~
'
D'(o)
[CoCl
+ C2C3]
ICoC
+
%ci
+ %%
- C~C4 - CoC~
c c3
+
+
+ %%
c2c ]
where CO(S} = P~5 (s) [(I-P59(s) q95 (s)) - P57 (s) P71(s) ]
,
(i0) ' Cl(S) = P48 (s) (P8,14 (s) + P8,12 (s) P12,18(s) ) + P46(s) P6,18 C2(s)
=
P59(s) (pg,15(s) + P9,13 (s) P13,19 ('s)) + P57(s) P7,19
C3(s ) = P35(s) (P48(s) P8,12(s) P12,3(s) + P46(s) P63 _(I0)
C4(s) = P24(s)
[(1-P48(s)
P84(s))
- P46(s) P60 (s) |
(s),
(s})
,
Cs(S) = P24(S) [P59(s) Pg, 13(s) P13,2 (s) + P57(s) P72 -(ll)(s) ] , Ci(o) - C i , and d Ci(s) cl = i ds
;
i = 0,I,...,5
s=o
Special caset
If the two units are similar
NI(8) - Q01QI2 E(Q24[Q47 + Q46Q69 ] + ^~23w35 ^(9))((I-Q01QI0)( 1-Q24Q42 )QI2Q23Q30Q01 ) + (Q24[Q47 +
+ QI2 [ Q24 Q46 Q61
Q46Q69 ] +
^(5) Q23 ~31
(5))( Q23 Q39
] )~
" Q01QI2[Q24(Q47 + Q46Q69 ) + ^~23~39 ^(5)][(I-Q01QI0)
(I -
(5) + Q24Q42} + Q12(Q23Q31 Q24Q46Q61 - Q23Q30Q01 ) ] MR 30/(~--J
1164
S.S. ELIASand S. W. LABIB
2 Dl(S) = [(I - Q01QI0)(I - Q24Q42)
- Q23Q30Q01QI2] 2
[QI2(Q24Q46Q61
-
+ Q23Q(~I))]
Therefore,
(5) Q01Q12[Q23Q39 + Q24(Q47 + Q46 Q69 ) ] = [Q ~(5) (I-Q01Q10)(I-Q24Q42)-Q12 23~31 +Q24Q46Q61+Q23Q30Q01 ]
~l(S)
and the mean time to
system
failure
for
the
first time
is given by : (5) E(T) = P12[~2 + P23 ~3 + P24(~4 + P46~6)]-(P23P31 +P24P46P61)~o + (1-P24P42)(go
+ ~l)/P12[P2 3 p~5)+ 9 P 2 4 ( 1 - P 4 2 - P46 P61 )]
which coincide with the results given in [3]. 4.
POINTWISE
AVAILABILITY
We have define
AND
Ai(t)
system is up at time
t
generative state
at t=0
Si
STEADY
STATE
AVAILABILITY
as the probability that the
given that the system enters rei.e.
Ai(t) = P(System is up, i.e., available at time Eo
=
t = o /
Si )
and Mi(t) is the probability that the system is up initially in state t
Si
and
continues in that state at time
or in a non-regenerative
state.
Therefore,
Mo(t) = Vl(t)
,
Ml(t)
M2(t) = Gll(t)Dl(t)
,
M3(t) = Gl2(t) D2(t)
,
M4(t) = V2(t) ~l(t)
,
M5(t) = Vl(t) H2(t)
,
= V2(t)
,
M6(t) = G21 (t) IV2 (t) +
t o
L2 (t-u) dV2(u)
]
,
M7(t) = G22 (t) [VI (t) +
t o
£i (t-u) dVl(U)
]
,
M8(t) = Hl(t)Gl2(t)D2(t) Ml2(t) = L2 (t) G21 (t)
, '
S9(t) = H2(t)Gll(t)
Dl(t)
Ml3(t) = L1 (t) G22 (t)
,
1165
Two-dissimilar unit cold standby redundant ~ s ~ m
By probabilistic argument we have : Ao(t) = Mo(t) + q02 (t) @
A2(t)
,
Al(t) = Ml(t) + q13 (t) @
A3(t)
,
A2(t) = M2(t) + q20 (t) @
Ao(t) + q24(t)
A4(t),
A3(t) = M3(t) + q31 (t) @
Al(t) + q35(t)
A5(t),
A4(t) = M4(t) + q46 (t) @
A6(t) + q48(t)
A8(t),
A5(t) " M5(t) + q57 (t) @
A7(t) + q59(t)
A9(t),
A6(t) " M6(t) + q60 (t) @
Ao(t ) + q63 _(i0) (t) @
A3(t)
+ q67(10'18)(t) @
A7(t) '
A7(t) = M7(t) + q71(t)
Al(t) + q72
+ q76(ll'19)(t) Q
A4(t) + qs,12(t) @
@
Al2(t)
AI4 (t) :
A9(t) = M9(t) + q95 (t) @ + q9,15(t)
A2(t)
A6(t) '
A8(t) = M8(t) + q84(t) @ + q8,14(t) @
(t)
A5(t) + q9,13(t) @
Al3(t)
Als(t)'
Al2(t) = Ml2(t) + q12,3(t) Q
A3(t ) + q12,7 (18) (t) @
A7(t)
A13(Z) = Ml3(t) + q13,2(t) @
A2(t) + q13,6(19)(t) @
A6(t),
Al4(t) = q14,16(t) @
Al6(t)
Als(t)
= q15,17(t) @
Al7(t)
Al6(t)
= q16,7 (t) @
A 7 (t)
'Al7(t)
= q17,6 (t) @
A 6 (t)
,
Taking Laplace transform of these equations and solving for A* (s) we have : o
A~(s)u
=
Na(S) / Da(S)
(3)
| 166
S.S. ELIASand S. W. LABm
where Na(S)
=
* K?
M i
i=o
Da(S) = R(s)-Jo(S)Ro(S)-Jl(S)Rl(S)+J2(s)R2(s)+J3(s)R3(s)+J4(s)R4(s ........ ,
.
Ko
(4)
(l_q(10,18) (11,19) 67 q76 )[(l-q13q31)(l-q48 q84)(l-q59 q95 ) q24 q48 q8,12 q12,3 q35 q59 q9,13 q13,2 ]
_(ll) [( )( )(q46 _(10,18) b) - q72 q24 l-q13q31 I-q59 q95 q67 - q48 (q59 q67 -(10'18)a - q57 )(q48 q8,12 q12,3 q35 )] (I0) q35[ ( )( a (I1,19)) + q63 1-q48q84 q59 q57 q76
+
q59qg,13q13,2q24 (q48 q76 -(ii'19) b - q46 ) ] (10,18) - q71q13q35 [(1-q48q84)(q57 - q59q67 a) + q59qg,13q13,2q24 (q46 q67-(10'18) - q48 b) ] _(i0) -
q24 q35 q63
_(ll)
[
q72
q
q46
-
57
a bq48 q59 ]
where a
_ = -q9,15 q15,17 q17,6
b =
K1
'
_(18) -q8,14 q14,16 q16,7 - q8,12q12,7
~ q02q24 _
(19) qg,13q13,6
l_q59q9511- - (10,181 _(11,191)( q67 q76 q48q8,12q12,3q31 )
(l_q59q95). (i0) . (11,19) _ . . (10,18) |q63 q31(q76 q48D-q46)-q711q46q67 -bq48)]
• (10,18 )q59 a)+q~10) + q71q35[q48q8,12q12,3~q57-q67 3 (q46q57_abq48q59)]~ K*
2 = q02(l-q48 q84 ) C6 ;
C6
= (l-q13q31)(l-q59q95)('~-q67(10'18)-(ll'lg)q76) + q35 [_(I0)(q63qs~ (11,19) ) (q57 q57 q76 q71 ql3
K~
-
q02 q24 (i- q59 q95 ) C7 ;
(10,18) a ) ] , q59 q67
-
)
Two-dissimilar unitcoldstandbyredundantsystem =
(I
C7
q48q8,].2q12,3 +
K4* K~
=
_(10,18)_(].1,19))+ _(10)(q ¢ (11,19)b) -q67 q76 ~63 46-|48q76
(10,18) b ) , qTl q13 ( q46 q67 - q48
q02 q24
=
1167
C6
,
q02 q24 q35 C7 '
K*
q02q24 [(l-q13q31)(l-q59q95)(q46-q48
6
(11,19) q76
b)
(11,19) + q35 [q48q8,12 q12,3 (q57 q76 - q59 a - q71 q13 (q46 q57 - q48 q59 ab) ] , K* 7
q02q24 El_ql3q31) (l-q59q95)t,q 46q67 (10,18) - q48 b) (10,18) + q35[q48q8, 12 q12,3(q57-q59 q67
a)
+
(i0) (q46 q57 - q48 q5 9 a b)]~ q63 K*
=
q02 q24 q48
K*
9
=
q02 q24 q35 q59
K*
=
K*
=
8
12 13
C6
' C7
q02 q24 q48 q8,12
' C
,
6
C
q02 q24 q35 q59 q9,13 +
(10,18)
(ii)
Jo(S) =
q60 q02
Jl(S) =
q71 q13 +
J2(s) =
(11,19) q60 q02 q76
J3 (s) =
q71 q13 q(170'18) + q(6130) '
J4(s)
=
_(10) _(11) q60 q02 q71 q13 - q63 g72
R(s)
=
(I-
q67
7
q72
'
q76(11'19) q63(10) , +
_(ii)
g72
-(10'18)-(11'19))[(1-
q67
q76
'
'
q02q20
)(
l-q13q31)(l-q48q84)(l-
q59 q95)-q24q48qs,12q12,3q35q59q9,13q13,2 ] ' Ro(S ) . q24[q46(l-q13q31)(l-q59q95)-q48q 8,12q12,3q35q59 a ] ,
1168
S.S. ELIAS and S. W. LABIB
Rl(S) = q35 [q57(l-q02q20)(l-q48q84 ) - q59q9,13q13,2q24q48 b]'
R2(s) =
q24q48[(l-q13q31)(l-q59q95 )b- q8,12q12,3q35
=
--
)a
q57 ]
--
43(s)
q35q59 [(I q02q20)(l-q48q84
q9,13q13,2q24
R 4 (s) =
q2 4 q35 [q46 q57 - q48 q59 ab ]
The steady state availability
q46 | '
is given by : Na(S)
AoC-) =
~
AoCtl =
lim
S ~O
.... D'(S)
sA*ls) ~
a
(5) s=o
where N a (0)
13 ~--O= Ti ~i
=
.
=
TI
K?
;
l
I
qij =
Pij
and t
D'a (s) I
=
R'
S=O
- J o R'o
- J'o R o - J l
R1
!
-
43 ÷J34 where R(o),
, J4(O)
stand for
and
R'(s)
Special case:
M 8*
=
K* 0
+ K* = 1
M 9*
,
+
J2
44 ÷J44
t
R'O
,6,,
,
R~
, J'O
R2' (6)
* MI3
=
,,*
,
J~
..... J~ (s) I S=O
If two units are similar,
* MI2
#
then :
and
[(i-~'9))[( l-q24q42)-q12q24q46q61]
= (I~ q33 -(5, -
R'
J , R~(s) I S=O S=O
- q24a]~ C8
R1
Jl
R , R o, "''' R4' Jo' Jl .... ' J4 stand for ooo
_ (5 q31)q12[~23
C8 ;
))[( l-~lqlo) (l-q24q42)
+ q12 q24 q46 q61 ]
q12 (q30 q01- q31)(q23 + q24 a) ,
K2* + K3* =
(5,9)) q01 (i -q33
K~ + K~ =
qol ql2 (1-q33-(5'9))c8
'
( i-q24 q42) C8 '
Two-dissimilar unit cold standby redundant system
K~ + K* =
7
K*
8
+ K* =
*
(q
q01 q12
a
23- q24
(5,9)
9
q01 q12 q24 (1-q33
* -
) C8
,
) c8
'
(1 _ ( 5 , 9 ) )
KI2 + K13- q01 q12 q24 q46
C8
-g33
1169
•
Therefore , (9) ÷ q2,~q47q78%31 1 % 1 M*~-c"(~I M~1 Nacs~ = E %2 [q 2~+ %4q46%3 +
(5,9)) (1 - q33
[ (1
+
q12 (q01 (M*2 + q24 M~4
- q24 q42 +
) ( M* 0
+
M*) 1
q01
+
q24 q46 M*6 ) -
-M: q24q46q61 )]~ E('+ (5'9))[(l-q01q10) (1-q24 q42 ) z q33 + q12q24q46q61 ] - q12(q30 q01 - q31 ) ( q23 (9) q24 q46 q63 - q24 q47 q78 q83)~
'
and Da(S) =
-
~(1-q(5'9))[(1-q q )(1-q q ) - q q q q ] 33 01 I0 24 42 12 24 46 61 %
%2'%1 ÷ %o%1 ) (%3 ÷ q24q46q63 + % , % 7 % 8 % 3 ~ (1 + _ ( 5 , 9 ) ) [
g33
(l-qOlqlo)(l-q24q42
- q12(q30qOl-q31)(q23-q24q46q63
) +
[-"
q12q24q46q61 ]
- q24 q47 q78 q83
Then (9)+
A:(S) i
12[q23 + q24q46q63 (5)M*]
q31
O
+
(i
q24 q47 q78 q83
][
qOl
M* -
3
_(5,9)) [( ) (M: + *) + -q33 1-q24 q42 qolM1 q24 4
q24q46M~ )-
M*
/
E.. (5,9) )[ (l-q01ql0 )( i-q24 q42 )-q12 q24 q46 q61 ]%1-q33 q12(q23 + q24q46q63 + q24q47q78q83)(q31
+ q30 q01)~
The steady state availiability is given by : Na(S)
Ao(~) N a ( s ) ljlo =
=
D a(s) s=ol
)
(l-q(5~91)[(l-q24q42)~l+P12(~2+P24~4+P24P46
~6 ) ]
1170
S.S. ELIASand S. W. LABm
+ P12 [i - P24 P42 - P24 P46 P61 ] P3 + [(1- P24P42)(1 - P33 _(5,9) -P31 (5)P121-P30P12P24P46P61 . ]~O' D'(s)i - Na(S) I + P P P (I-p(5'9)) ( P7 s-o s=o 12 24 46 33
+
~8 ) "
which agree with the corresponding results obtained in [3]
5. E X P B C T E D BUSY PERIOD OF SERVER FOR R E P A I R
IN (O f t~
Let BJk(t)-the probability that at time
t
the server is busy
with repair , according to the distribution Gjk (t) given that the system entered regenerative state S i at
t =o.
By probabilistic argument we have the following relations for BiJk(t). For j = I,
k = 1
B101(t'" q02 (t, Q
Bll(t)2
'
Bl11(t'" qz3 (t, Q
BII3 (t)
,
B~ l(t) = q20 (t) Q
Bll(t)0 + q24 (t) Q
B141(t)+ GI1 (t' '
B~l(t) = q 3 1 ( t ) Q
Bll(t) + (t) Q 1 q35
Bll(t) 5
Bn(t)4 : q46 (t) (~ ezzctl6 + q48(t) (~) BiZet)8 BZl(t)5 " ~57 (t) (~BlZ(t)7 + q59 (t) (~)Bzzlt}9 BZi(t) :6
q60(t) (~BlZct) l ( t+-(10)(t)(~B~ "63 ~ o q 10'lS)(t) 67
B~z(t) - qTl(t) ~
Q
Bll(t) 7
BnIt) ÷ q72 -(n)(t) 1
q(11'19)(t) 76
Q
(t) 8,14
Q
Q
Bll.t) 2 t
Q
11 B12(t)
B~ l(t)
B~l(t~ " qe4 (t~ C) Bll(t~4 q
,
÷ qs,n(t)
Bll(t) 14
,
Two-dissimilar unit cold standby redundant system
B191(t) " q95(t) G
Blsl(t) + q9,13 (t)
%,lSCt> ~
Blltt)12 = ql2,3(t> Bll(t> 13
ffi q13,2
1171
11 B13(t)
Q
+
B~t> + ~11 ct~ , 15
(~
Bll~t~3 + ql2-ClB>,7(t~ (~) ell(t~ ,
(t) ~
8llct> +_~19> ~ 2
q13,6
Bll(t)14 = q14,16 (t)
Q
Bll(t)16
'
811(t)15 = qlS,l~ (t>
~)
sll(t)17
'
Bll(t) (~t) 16 = q16,7
Q
Bll(t) 7
,
Bll(t) (t) 17 = q17,6
Q
Bll(t) 6
Taking Laplace-Stieltjes
8Zl.t~ 6 t
,
transform and simplifying,
we obtain ~-11. mo ~s), as follows : ~ll(s) o
=
Mll(s) / Da(S)
(7)
where
Mll(s) and
Da
=
Gll(S) (K 2
+
K9 )
as given in (4). In the long run, the fraction of time for which system
is under repair according to the distribution
Gll(t)
is
given by : ~11 Bo
lim t )oo Mllls) D,a(S)
=
B II 't)
lim s >o
(8)
1 smo
where Mll(s) and For
I
s=o D~(o) J - 1
=
'
T2
+
T9
as given in 16) ,
ii s B o (s)
k = 2
Bl2(t) = q02 (t) Q
B12(t)2
'
Bl2(t)l " q13 (t) Q
B12(t)3
'
1172
S.S. ELIAS and S. W. LABIB
,,;,,,:, .-q,o,t, ® ,,,,,t, +o
B121t)4 = %6~t)
,t, @
C) ~12c~)6+ %8 (t)
B12(t)5 : q57(t) @
.
~
B12(t)7 + q59 (t) @
Bn~t)8 B12(t)9
I~o)itl ® B12(t)6 = q60(t)@ Bl2(t) + q63
c~o,.~(t) Q ~ , t , , B~ 3 (t)+q67 -
BT12(t) = q71(t) ~2(t)+q7211(t)~B12(t)+p(ll'lg)(t ) 2 76
Bn(t) 8
= %4
~t) ~Bl2(t) + 4
q8,14 (t) @
(tl ~
%,n
q9,15 (t) @
nct)
B12(t)14 + G12 (t) , Bl2(t) 13
B12(t)15 '
B12(t)12 = ql2,3(t) @
B12(t)3 + q12,7-(18)(t)@
Bl2(t) = (t) @ 13 q13,2
Bl2(t)+ _(19) 2 q13,6
B12(t)14 = q14,1~t)
@
Bl2(t) = (t) @ 15 q15,17 Bl2(t) (t) 16 = q16,7
@
Bl2(t) (t) @ 17 = q17,6
B12(t)'6
Bn
(t) @ Bl2(t) + 5 q9,13
B~2(t) = q 9 5 ( t ) @
~
@
B12(t)7 ' Bl2(t), 6
B12(t)16 ' Bl2(t)
,
17
Bl2(t)
,
7
Bl2(t) , 6
Taking Laplace-Stieltjes transform and simplifying, we obtain ~12(s), as follows : o ~12(s)
=
Ml2(s) / D (s)
MI2(s)
=
GI2(S) (K~
o
(9)
a
where +
K 8*)
and Da(S) as given in (4). In the long run, the fraction of time for which the
Two-dissimilar unitcoldstandbyredundantsystem
1173
server is busy according to the distribution law Gl2(t) is given by : B 12 o
=
lim t--->~ MI2(s)
=
D'(s)
BO12( t )
lim S s-->o
=
~ 1 2 (S)
(i0)
I
s
o
where
and
Ml2(s)
[ s=o
=
D'(s)
i s=o
as in
For
T3
j = 2
+
T8
(6)
,
B21(t)0 = q02(t) Q
B21(t)2
B21(t) (t) Q 1 -- q13
B~l(t)
.
k = 1 '
B21(t) = q20(t) , Q B201(t) +
(t) Q
B21(t) 4 '
B~l(t) + q35(t) Q
B215 (t) ,
B241(t) = q 4 6 ( t ) Q
B261(t) +
B21(t) , 8
B251(t) = q57(t) Q
B~l(t) + q 5 9 ( t ) Q
B21(t) , 9
B261(t) = q60 (t) Q
B21(t)+0 q63-(10)(t)Q
B231(t) +
B21(t) = 3
q31(t) Q
q(10'lS) (t) Q 67 B21(t) = 7
q71(t) ~
(t) Q
B21(t) + - (t) 7 G21 '
B21(t) 12
21 B14(t) ,
B21(t)5 4 q9,!3 (t) Q
q9,15 (t) G
+
B21(t), 6
B21(t)4 + q8,12 (t) Q
q8,14 (t) Q B21(t)9 = q95 (t) Q
q48
B21(t)+I ~72-(ii)(t)C)B221(t)
q(ll'lg)(t) ~ 76 B21(t)8 = q84 (t) Q
q24
B21(t) + 13
B21(t)15 '
G
B21(t) = (t) Q 12 q12,3
B21(t) + -(18)(t) 3 q12,7
21 (t) Q Bl3(t) = q13,2
B21(t) + -(19)(t) Q 2 q13,6
B21(t)
7
+
B21(t) , 6
21
(t),
1174
S.S. ELIAS and S. W. LABIB
B21Ct'14= %4,1~t' G
~211t'16
B21(t)15 = q15,1~ t)
Q
B21(t)17 '
B21(t) (t) 16 = q16,7
Q
B21(t) + - (t) 7 G21 '
B21(t) = (t) 17 q17,6
@
B21(t) 6
Taking Laplace-Stieltjes transform, we obtain~21(s), as follows : ~21(s')
=
M21(s) / Da(S)
(ii)
where M21(s)
=
G21(s)
and
Da(S)
[K;
K* + q8,14 q14,16 8
+
K* ], 12
as given in (4).
In the long run, the fraction of time for which the server is busy according to the distribution law G21(t) is given by :
B~l
=
lim
s
s
"~_21,
~o
)o
=
~)
M 2~s) D ~ (s)
I
~z21
s=o
where M21(s) ~=o
and
D'a(S)
J
=
T6 +
as in
P8,14 T8 + TI2
(6)
s=o
For
j
=
2
,
k=2,
B22 (t) '
B~2(t) = q02(t) B~2(t) = ql3(t)
B 22 (
B~2(t) = q20 ct, @
Bnct'0 * %4(tI Q
4 t,
B~2(t) = q31 ~t, Q
B2~It'l* %s ct, @ ,~nIt,s '
Bnct~4 = q4~t~, (~
B=2ct~ * q48(tl (~
snctl8
'
B~2(t~ =qs~ (t~ C)B22't)~ *qs9 (t) ( ~ B22(t~9 ' '~ 2't~ =%o `tl (~)Bn(tl *o q(10';18)(t) @ 67
'~,3-(l°l(t~ C ) ' ~ 2(t~ * B 22 (t) 7 '
Two-dissimilar unit cold standby redundant system
B~ 2(t) = q71 (t) Q
B22(t) + i
q76(ll'19)(t) Q
B22 8 ~t~ = q84(t~ ~
q72(ll)(t) Q
B~2(t)
B22(t~4 + q8,1~t~ ~ n
BI4 (t) Q
+
B22(t)6 + G22-(t),
q8,14(t) B22(t) = q95 (t)
1175
Bn(t~12 +
, B22(t) 13
B22(t)5 + q 9 , 1 3 ( t ) ~
+
q9,15 (t)
Q
B22(t)15 '
(t~
~
Bn(t)
B22(t)13 = ql3,2(t)
~
B22(t)2 + q13,6-(19)(t)Q B22(t)6 + G22(t) '
B22(t~ 12
= q12,3
+
3
(18~,.,
(~
q12,7 It;
B22(t) (t) 14 = q14,16
Q
B22(t) 16
,
B22(t) (t) 15 = q15,17
Q
B22(t)
,
B22(t) (t) 16 = q16,7
~)
B22(t) 7
Q
B22(t) + - (t) 6 G22 "
B22(t) , 7
17 ,
p
B22(t) = (t) 17 q17,6
Taking Laplace-Stieltjes transform, we obtainB22(s) as follows : ~22(S) o
=
M22(S) / Da(S)
(13)
where
and
M'2(S) = G22(s)(K~7 + -q9,15 q15 ' 17 K*9 + K*13 ). Da(S) as given in (4). In the long run, the fraction of time for which the
server is busy according to the distribution law
G22(t)
is given by z B ss o
=
lim s B~'(s) S'--->O
=
M''(s)
I
D~(s)
(14)
S=O
where M2S(s) and
i
=
s=0
D~(s) as in (6)
T
+
7
P
T
9,15
+
9
T
,
13
I176
S.S. ELIASand S. W. LABIB 6.
COST
ANALYSIS:
(I) The e x p e c t e d
up-time
of the s y s t e m
in
(o,t]
is
t ~up(t)
=
~ o
Ao(U)
du
so that ~*up(S) (2) The e x p e c t e d
=
down-time
A* (s)/s o
(15)
of the s y s t e m
~dn(t)
=
• Udn(S)
=
i6
(o,t]
is
:
t - Nup(t)
so that
(3) The e x p e c t e d
s12
busy period
~ *up (s)
of the s e r v e r
(16)
for repair
in (o,t]
t ~ll(t) P
=
~ o
B 11 o
*ll(s) ~b "
= ~ll~s) o '
(u) du
so that / s
(17) '
also ~12(s)
= ~12(s)/So
• 21 ~b (s)
=
~'22(s)
= ~22(s)/S~o
'
(18)
~21(s)/s o
(19)
and
Denote which
by G(t)
is e q u a l
revenue
the e x p e c t e d
to the d i f f e r e n c e
and the e x p e c t e d
total
total
(20) gain
between
cost
incurred
in
the e x p e c t e d
(o,t] total
in the same interval.
Therefore, Glt)
= Zl~uplt)
ii, t 12(t - z 2 ~b ~ ) - Z3 Ub
) - Z4~ b21 (t) - Z5~ p2 2 ( t ) (21)
where Z1
is the r e v e n u e
Z i, i = 2,3,4,5,
per unit
are the cost
ver is b u s y a c c o r d i n g 22,
,
per unit time
to the d i s t r i b u t i o n
for w h i c h
the ser-
Gij(t) , i j = l l , 1 2 , 2 1 ,
respectively. The e x p e c t e d
given G
up-time
by
= t =
total
cost per unit
time
~n s t e a d y
state
:
lim >~
G(t) t
= s
lim >o
s G*(s)
ZIAo - Z2 BIIo - Z3 B12o - Z4 Bo21 _ Z5 B22o
(22)
is
Two-dissimilar unit cold standby redundant system
REFERENCES I.
G. S. Mokaddis, Reliab,
2.
S.M. Gupta, Reliab.
3.
(1989).
N.K. Jaiswal and L.R. Goel, Microelectron.
23, 329-331
S.M. Gupta, Reliab.
S.S. Elias and S.W. Labib, Microelectron
29, 511-515
(1983).
D.K. Pandey and Renu Gupta, Microelectron,
26, 847-850
(1986).
1177
0026--2714/9053.00 + .00 © 1990 Pergamon Press pic
Printed in Great Britain.
COST ANALYSIS OF A TWO-DISSIMILAR UNIT COLD STANDBY REDUNDANT SYSTEM WITH ADMINISTRATIVE DELAY AND NO
PRIORITY IN REPAIR S.S. ELIAS and S.W. LABIB Ain Shams University, Faculty of Science, Department of Mathematics, Cairo, Egypt (Received for publication 20 February 1990)
This paper deals with the cost analysis similar unit cold standby redundant for each unit under the assumption rative delay and no priority
system with three modes that there is administ-
in repair.
repair time and administrative
The failure time,
time distributions
ral and arbitrary.
Some reliability measures
to system designers
have been obtained.
vious results are derived
of a two-dis-
are gene-
of interest
Moreover
some pre-
from the present results as spe-
cial cases.
1.
IMTRODUCTIONAND
Gupta
F O R M U L A T I O M OF T H E PROBLEM:
[2] discussed
the stochastic
similar unit cold standby redundant unit works in three modes, and total failure.
nistrative
assumption
partial
failure
the same problem of [2], but with
delay and no priority
in repair.
The admi-
was assumed to be exponential.
Elias and Labib
state availability redundant
i.e., normal,
time distribution
Mokaddis,
system in which each
and arbitrary.
[3] discussed
administrative
[i] have derived
of a two-dissimilar
the steady
unit cold standby
system where each unit has three modes under the that there is an administrative
delay in getting
• the repairman at the system location and assuming randoms
of a two
The repair and failure time distribut-
ions are exponential Gupta
behaviour
variables
having general distributions. 1155
that all
1156
S.S. ELIAS and S. W. LABIB
The purpose of the present paper is to study the cost function of a two-dissimilar unit cold standby redundant system and three different modes, with administrative delay and no priority in repair. The following assumptions and notations are adopted for our model.
(1)
Two dissimilar units operate in cold standby configuration
i.e., a unit does not fail du=ing its standby
state.
(2)
Each unit of the system has three modes
:
i. normal mode, which means the functioning of the unit with full capacity, ii. partial mode which means the functioning of the unit with reduced capacity at a specified level, iii. total failure mode which means the functioning of the unit with a capacity below the specified level. (3)
Upon total failure of an operating certain administrative actions, trative delay in getting repair, a unit comprises two parts:
unit and due to
there is an adminisso the down time of
administrative delay
and repair time. (4)
No unit of the system can attain the total failure mode without passing through the partial failure mode.
(5)
Whenever the operative unit fails completely, standby unit is switched. instantaneous
(6)
the
The switching is perfect,
and without any damage to the system.
In P-mode, a unit is working as will as under going repair. mode,
On getting repaired,
the unit enters the N-
it is also possible that the unit during repair
deteriorates
further and enters the F-mode.
The events
of a P-unit passing into N- and F-modes are mutually exclusive.
Two-dissimilar unit cold standby redundant system
(7)
In case the N-unit
fails partially,
unit is in F-mode and under repair, till the repair of the F-unit
(8)
When one unit is o p e r a t i v e led unit is under repair, unit is completed, as standby,
1157
while the other the former waits
is completed.
and the other totally
fai-
if the repair of the failed
the o p e r a t i v e
unit will be taken
and the repair unit will be taken as ope-
rative. (9)
The repair time for a P-unit and F-unit,
administra-
tive delay time and times to transit between different modes are random variables ted such that Di
: Pi
generally
distribu-
:
> FAi
Hi :
FA i
>
Fr i ,
L i : Pwi----> Fw i
Vi :
Oi
>
Pi
>
0 i.
Gli:
Pi
> Oi
and
G2i : Fr i
We refer to the d i f f e r e n t
aspects
'
of unit i; i = 1,2
by : O i , St i
operative mode,
Pi
partially repair)
standby mode,
failed
(and stipulated
,
FAi
failed unit
Fri
repair of unit
i
under a d m i n i s t r a t i v e i
delay has finished Fci
to be under
totally
Pwi
partially
Fwi
totally
, i
and repair continued
preceding
failed unit
failed
unit
i i
state
from
,
waiting
for repair
and w a i t i n g
Under the above assumptions, one of the following
,
started after a d m i n i s t r a t i v e
failed unit
the i m m e d i a t e l y
delay
,
for repair.
the system will be in
states at any instant;
So E (01, St2)
,
S1 E (Stl, 02)
'
$2 ~ (PI ' St2)
'
$3 i (Stl , P2)
,
$4 m (FAI ' 02 )
,
S5 ~ (01 , FA2)
,
1158
S.S. ELIASandS. W. LABIs
S 6 • (Frl, 0 2 )
,
S 7 ~ (01 , Fr 2)
,
S8 E (FAI , p2 ) ,
$9 m (P1 ' FA2)
'
Sl0m
(Fc1' Fw2)
'
Sll ! (Pw I, Fc 2) ,
$12 ~ (Fr I, Pw 2) ,
S13 a (Pw I, Fr 2)
,
S14 a (FA I, Fw 2) ,
$15a
(Fw I, FA 2) ,
$16a
(Frl, Fw 2)
,
S17 a (Fw I, Fr 2) ,
$18E
(Fc I, Fw 2) ,
Slga
(Fw 1, Fc 2 )
Let tively,
U
thus
and
F
denote the up and failed state respec-
U e (So, S I, .-- , SI3),
The state diagram of the system
Fig.
Q
Up states
D
Down stores
F e (s14, s15,...,s19) is depicted
in Fig. I.
i. The state diagram of the system.
2. T R A N S I T I O N P R O B A B I L I T Y AND M E A N W A I T I N G T I M E S IN S T A T E S
Let
:
Eo
state of the system at time t = 0 ,
E
set of regenerative
states
i = 0, 1, 2, ..., 9, 12, 13,
~Si} ..., 17 ,
Two-dissimilar unit cold standby redundant system
set of non-regenerative
states
1159
{S j} ;
j = 10 , 11 , 18 , 19. Pij
one-step transition probability from state state
p(U,V)
ij
Sj ;
&
(S i
Sj e
S i to
E) ,
transition probability from state
S i to state Sj
through states Su&S v, (S i & Sj e E and S u & S v e E). Denote the complementary function of any distribution F(.) by
F(.) ffi l-F(.). Therefore,
P20
ffi I O
P31
= I
P46
OI
P02 = PI3 = P14,16 = P15,17 = PI6,7=PlT,6 ml' OO Dl(t)dGll (t) P24 " S allCt~ aDlCt~ O cO ' P35 = J" Gl2lt) dD2lt) D2(t)dGl2(t) O CO ' P48 = J~ HI (t) dV2(t) ' V2(tldHl(t )
O
O
P57
GO = I Vl(tldH2 (t) O Go
P60
" I o
p(10) 63
'
P59 =
,J~CDH2lt) dVllt) o
V2 (t) dG21(t) ffi f O O l t t=o x=o CO
-
L2(t-x) dV2(x) dG21(t)
0~
I I XffiO t=x
L2(t-x) dV2(x) dG21(t), put t-x=y
O0 X=O
L2(Y) dV2(x) dG21(Y + x) ,
y=o
p(10,18)= _(i0) r6'18
67
~ =
G9
I
I
XffiO
G21(t) dV2(x) dL2(t-x)
G21(t) dV2(x) dL2(t-x)
t=x
co
-
x=o
co
S
X=O
y
t=o
CO
S
y=o
G21(Y + x) dV 2 (x) d L 2 (y)
OD P71
=
I
(11) P72
-
I
V1 (t) d G22(t)
o
CO X~O
S
y=o
Ll(Y) d VI(X) d G2~Y + x) , oo
p(11,19)= 76
p~ll) ,19
-- S
x=o
CO
S
y=o
G22(Y+X) dVl(X) d Ll(Y)
'
1160
S.S. ELIASand S. W. LABIB
GO
GO , J~ Hz(t)D2(t)dGl2 (t) P8,12 ffi D2(t)Gl2(t)dHl(t) o
I o
P84
UB
:
P8,14
I
G12(t)Hl(t)dD2 (t)'
o
P95 :
CO I Dl(t)Gll(t)dH2(t), o
P9,13
=
P12,3
ffi I o
co
£2(tldG21(t)'
H2(-t)Dl(t)dGll(t)
I
o
P9,15 :
P12,18
T GII (t) H2(t)dDl (t)' o
= p(18) = 12,7
I o
G21(t)dL2(t),
GO £i (t)d S22(t) ,
P13,2
ffi I o
and
Pij = o
P13,19 = p(19) = T - G22 (t)dLl(t), 13,6 o
for other
i
and
j .
To obtain the mean waiting times in states, let mij denotes the mean time for direct transition from state
S.1 Sj ; (i, j = 0, i, 2, ..., 19) and Pij(s) denotes
to state
the Laplace-transform of time
Bi of the system in
PO = m02
'
Pij(t).
Therefore,
the mean stay
Si(i = 0,1,2,...,19) ~].
is given by
ffi m13
CO -d * (s) + * (s)] = .PGll(t)Dl(t)dt, ~2 ffim20+ m24 = ~ [P20 P24 sffio co -d • • = ~3 = m31 + m35 " ~ [51 Is) + 5 5 (Sl]s=o ~ Gl2(tID2(tldt' co g4 " m46
+ m48 " ~ s
[P46
(s) + P*
48
(s)]
sffio
ffi ~"o ~Z(t)~2(t)dt, co
~5 ". m57 + m59" ~sd [ p
* (s)] 7 (s) + P59
s=o
o
~2(t)gZCt)dt
,
(10,18)* ~6 " m60 + m63 + m6 ,i0 = ~-6 -d [P:0 (s) + p(10 )*(s) + p 63
•
67
(s)] s =o
co
Io ~7 " m71
+ m
72
+ m
-d [p;l ( _(ii _(11,19)* 7,11 = ~ s) + P72 )*(s) + P76
OD
~J~ Ql(t) G22(t)dt, o
V2(t) G21(t) dt ,
(s)] S=O
Two-dissimilar unit cold standby redundant system
1161
P8 = m84 + m8,12 + m8,14 = ~ss [P8,t s) + p ,12 (s) + P8,14 ( s ) ] oO I H1 ( t ) o
"
Gi2 I t )
52 I t )
s o
,
at
* (s) + p; ,13 (s) + p*9, 15(S)]sffio H9 = m95 + m9,13 + m9,15 = ~-d [P95
- ~
N2(t) ~lz(t) 5 z (t)
o
dt
GD PI2 = m12,3 + m12,18
ffi ~-d [~2,3 (s)
, p12,1e(s)] = ~ ~.2(t)~n(t)dt,
+
S=O
O
Co PI3 = m13,2 + m13,19 = ~-d [P~3, 2 (s)
3.
T I M E TO SYSTEM FAILURE
+ P13,19 * ( s )s=o ] = IoL l ( t ) G 2 2 ( t ) d t .
(TSF)
Let • TTi (t)
cdf of the time to system failure
T[~(s)
Laplace - Stieltjes transform o f ~ ( t )
®
I E
symbol for ordinary convolution, t
Act) @
s(t) =
~
s (t-
o symbol for Stieltjes
®
A(t)
Let
TO ,
T
the state
~
B(t)
=
u)
i.e., A
(u)
convolution
It o
o = Si
,
an
,
i.e. ,
B (t - u) dA (u)
denote the epoch at which the system enters
i (i e E) and let
state visited at time
Tn
T O = o.
If X n denotes
then ~ X n , Tn~
the
constitutes a
J
Markov renewal process with state space E , Q
=
Qij ( t )
where Qij " P ~Xn+l " J is
a
semi
-
Markov
;
Tn+l - Tn < t / Xn ffi i}
kernal over
E.
Using the theory of regenerative the following equations
:
~ o (t)
"
Q02 (t)
~
~2
(t)
,
]71 (t)
"
QI3 (t)
Q
]-[3 (t)
,
process, we obtain
1162
S. S. EUAS and S. W. LAmB
(t) TT2 TT3(t) .
,~,. ,t,
@ IT,,,,.
T4(t) .
o,, ,t,
-- , ~ ,t,
TF5(t)
, ~ ,t,
® . Tr~ ,t,
®
TT,,,:, .. Q,, ,,:,
T T 8 (t)
,
®
TI-~,,~, + Q, ,,:, @ IF9 (t,
,
IT6(t) = Q60 (t) @
~°(t)+
TFT(t, = TT,(t, =
%~6^(I0)(t)3@
TT3(t) + ~(I0)%16,18(t),
Q~.,,:,
®
Fr,,,:, .. ^,,~,,,:, ® TL,,:, + ^ " ~ ' ,,:,.
Q84(t)
®
Tr,t, + Q~..~,,:, ®
~72
~7,19
® %,,, +
[9(t) " Q95(t}
TL,,:, + o~..,,:,.
®
~12(t) =
Q12,3(t)@
if3 (t) + Q12,18(t) ,
% 3 (t) "
Q13,2( t' Q
~2 (t) + Q13,19 (t)
+
Taking Laplace-Stieltjes transform of these relations and solving fOrgo
(s), the
L-S
transform of the distribu-
tion function of the first passing time is given by :
•
(s)
-
:
N 1 (s) / D (s)
(1}
where Nl(S) "• Q02 Q24
{(Q48 [ Q8,14 + Q8,12 Q12,18 ] + W46~6,18 ^ ^1101"111;
QI3Q31 ) (I'Q59Q95) - Q35 Q57 QTI QI.3 }
+
(Q59[Q9,15 + Q9,13QI3,19] + ~57 ~ ^ ^(ii)7,19) ( Q35 [ l
•• 10 •
Q48 Q8,12 Q12,3 + Q4 6w63
] )$
'
Dl(S) - [(I-Q02Q20)(I-Q48Q84)-Q46Q60Q02Q24][(I-QI3Q31)(I-Q59Q95) -Q57QTIQI3Q35]-Q24Q35[Q48Q8,12QI2,3
+
Q59 Q9,13 Q13,2 + Q57 Q~½1) ] where for brevity we have omitted the argument s
n(Io) ] [
Q46 ~63
Two-dissimilar unit cold standby redundant system
The mean
time
1163
to system failure for the first
time
is given by : E(T)
=
ds
S=O
(2)
D1
where D1 -
P24 P35
E[(1-P48 P84 ) - P46 P60][(I-P59 P95)'P57P71 ] +
- [P48P8,12PI2,3 N1
=
N~(o)
=
"2
+ P24
-
(10)]
+
P46P63
[P59P9,13PI3,2
(11).7
P57P72
~
'
D'(o)
[CoCl
+ C2C3]
ICoC
+
%ci
+ %%
- C~C4 - CoC~
c c3
+
+
+ %%
c2c ]
where CO(S} = P~5 (s) [(I-P59(s) q95 (s)) - P57 (s) P71(s) ]
,
(i0) ' Cl(S) = P48 (s) (P8,14 (s) + P8,12 (s) P12,18(s) ) + P46(s) P6,18 C2(s)
=
P59(s) (pg,15(s) + P9,13 (s) P13,19 ('s)) + P57(s) P7,19
C3(s ) = P35(s) (P48(s) P8,12(s) P12,3(s) + P46(s) P63 _(I0)
C4(s) = P24(s)
[(1-P48(s)
P84(s))
- P46(s) P60 (s) |
(s),
(s})
,
Cs(S) = P24(S) [P59(s) Pg, 13(s) P13,2 (s) + P57(s) P72 -(ll)(s) ] , Ci(o) - C i , and d Ci(s) cl = i ds
;
i = 0,I,...,5
s=o
Special caset
If the two units are similar
NI(8) - Q01QI2 E(Q24[Q47 + Q46Q69 ] + ^~23w35 ^(9))((I-Q01QI0)( 1-Q24Q42 )QI2Q23Q30Q01 ) + (Q24[Q47 +
+ QI2 [ Q24 Q46 Q61
Q46Q69 ] +
^(5) Q23 ~31
(5))( Q23 Q39
] )~
" Q01QI2[Q24(Q47 + Q46Q69 ) + ^~23~39 ^(5)][(I-Q01QI0)
(I -
(5) + Q24Q42} + Q12(Q23Q31 Q24Q46Q61 - Q23Q30Q01 ) ] MR 30/(~--J
1164
S.S. ELIASand S. W. LABIB
2 Dl(S) = [(I - Q01QI0)(I - Q24Q42)
- Q23Q30Q01QI2] 2
[QI2(Q24Q46Q61
-
+ Q23Q(~I))]
Therefore,
(5) Q01Q12[Q23Q39 + Q24(Q47 + Q46 Q69 ) ] = [Q ~(5) (I-Q01Q10)(I-Q24Q42)-Q12 23~31 +Q24Q46Q61+Q23Q30Q01 ]
~l(S)
and the mean time to
system
failure
for
the
first time
is given by : (5) E(T) = P12[~2 + P23 ~3 + P24(~4 + P46~6)]-(P23P31 +P24P46P61)~o + (1-P24P42)(go
+ ~l)/P12[P2 3 p~5)+ 9 P 2 4 ( 1 - P 4 2 - P46 P61 )]
which coincide with the results given in [3]. 4.
POINTWISE
AVAILABILITY
We have define
AND
Ai(t)
system is up at time
t
generative state
at t=0
Si
STEADY
STATE
AVAILABILITY
as the probability that the
given that the system enters rei.e.
Ai(t) = P(System is up, i.e., available at time Eo
=
t = o /
Si )
and Mi(t) is the probability that the system is up initially in state t
Si
and
continues in that state at time
or in a non-regenerative
state.
Therefore,
Mo(t) = Vl(t)
,
Ml(t)
M2(t) = Gll(t)Dl(t)
,
M3(t) = Gl2(t) D2(t)
,
M4(t) = V2(t) ~l(t)
,
M5(t) = Vl(t) H2(t)
,
= V2(t)
,
M6(t) = G21 (t) IV2 (t) +
t o
L2 (t-u) dV2(u)
]
,
M7(t) = G22 (t) [VI (t) +
t o
£i (t-u) dVl(U)
]
,
M8(t) = Hl(t)Gl2(t)D2(t) Ml2(t) = L2 (t) G21 (t)
, '
S9(t) = H2(t)Gll(t)
Dl(t)
Ml3(t) = L1 (t) G22 (t)
,
1165
Two-dissimilar unit cold standby redundant ~ s ~ m
By probabilistic argument we have : Ao(t) = Mo(t) + q02 (t) @
A2(t)
,
Al(t) = Ml(t) + q13 (t) @
A3(t)
,
A2(t) = M2(t) + q20 (t) @
Ao(t) + q24(t)
A4(t),
A3(t) = M3(t) + q31 (t) @
Al(t) + q35(t)
A5(t),
A4(t) = M4(t) + q46 (t) @
A6(t) + q48(t)
A8(t),
A5(t) " M5(t) + q57 (t) @
A7(t) + q59(t)
A9(t),
A6(t) " M6(t) + q60 (t) @
Ao(t ) + q63 _(i0) (t) @
A3(t)
+ q67(10'18)(t) @
A7(t) '
A7(t) = M7(t) + q71(t)
Al(t) + q72
+ q76(ll'19)(t) Q
A4(t) + qs,12(t) @
@
Al2(t)
AI4 (t) :
A9(t) = M9(t) + q95 (t) @ + q9,15(t)
A2(t)
A6(t) '
A8(t) = M8(t) + q84(t) @ + q8,14(t) @
(t)
A5(t) + q9,13(t) @
Al3(t)
Als(t)'
Al2(t) = Ml2(t) + q12,3(t) Q
A3(t ) + q12,7 (18) (t) @
A7(t)
A13(Z) = Ml3(t) + q13,2(t) @
A2(t) + q13,6(19)(t) @
A6(t),
Al4(t) = q14,16(t) @
Al6(t)
Als(t)
= q15,17(t) @
Al7(t)
Al6(t)
= q16,7 (t) @
A 7 (t)
'Al7(t)
= q17,6 (t) @
A 6 (t)
,
Taking Laplace transform of these equations and solving for A* (s) we have : o
A~(s)u
=
Na(S) / Da(S)
(3)
| 166
S.S. ELIASand S. W. LABm
where Na(S)
=
* K?
M i
i=o
Da(S) = R(s)-Jo(S)Ro(S)-Jl(S)Rl(S)+J2(s)R2(s)+J3(s)R3(s)+J4(s)R4(s ........ ,
.
Ko
(4)
(l_q(10,18) (11,19) 67 q76 )[(l-q13q31)(l-q48 q84)(l-q59 q95 ) q24 q48 q8,12 q12,3 q35 q59 q9,13 q13,2 ]
_(ll) [( )( )(q46 _(10,18) b) - q72 q24 l-q13q31 I-q59 q95 q67 - q48 (q59 q67 -(10'18)a - q57 )(q48 q8,12 q12,3 q35 )] (I0) q35[ ( )( a (I1,19)) + q63 1-q48q84 q59 q57 q76
+
q59qg,13q13,2q24 (q48 q76 -(ii'19) b - q46 ) ] (10,18) - q71q13q35 [(1-q48q84)(q57 - q59q67 a) + q59qg,13q13,2q24 (q46 q67-(10'18) - q48 b) ] _(i0) -
q24 q35 q63
_(ll)
[
q72
q
q46
-
57
a bq48 q59 ]
where a
_ = -q9,15 q15,17 q17,6
b =
K1
'
_(18) -q8,14 q14,16 q16,7 - q8,12q12,7
~ q02q24 _
(19) qg,13q13,6
l_q59q9511- - (10,181 _(11,191)( q67 q76 q48q8,12q12,3q31 )
(l_q59q95). (i0) . (11,19) _ . . (10,18) |q63 q31(q76 q48D-q46)-q711q46q67 -bq48)]
• (10,18 )q59 a)+q~10) + q71q35[q48q8,12q12,3~q57-q67 3 (q46q57_abq48q59)]~ K*
2 = q02(l-q48 q84 ) C6 ;
C6
= (l-q13q31)(l-q59q95)('~-q67(10'18)-(ll'lg)q76) + q35 [_(I0)(q63qs~ (11,19) ) (q57 q57 q76 q71 ql3
K~
-
q02 q24 (i- q59 q95 ) C7 ;
(10,18) a ) ] , q59 q67
-
)
Two-dissimilar unitcoldstandbyredundantsystem =
(I
C7
q48q8,].2q12,3 +
K4* K~
=
_(10,18)_(].1,19))+ _(10)(q ¢ (11,19)b) -q67 q76 ~63 46-|48q76
(10,18) b ) , qTl q13 ( q46 q67 - q48
q02 q24
=
1167
C6
,
q02 q24 q35 C7 '
K*
q02q24 [(l-q13q31)(l-q59q95)(q46-q48
6
(11,19) q76
b)
(11,19) + q35 [q48q8,12 q12,3 (q57 q76 - q59 a - q71 q13 (q46 q57 - q48 q59 ab) ] , K* 7
q02q24 El_ql3q31) (l-q59q95)t,q 46q67 (10,18) - q48 b) (10,18) + q35[q48q8, 12 q12,3(q57-q59 q67
a)
+
(i0) (q46 q57 - q48 q5 9 a b)]~ q63 K*
=
q02 q24 q48
K*
9
=
q02 q24 q35 q59
K*
=
K*
=
8
12 13
C6
' C7
q02 q24 q48 q8,12
' C
,
6
C
q02 q24 q35 q59 q9,13 +
(10,18)
(ii)
Jo(S) =
q60 q02
Jl(S) =
q71 q13 +
J2(s) =
(11,19) q60 q02 q76
J3 (s) =
q71 q13 q(170'18) + q(6130) '
J4(s)
=
_(10) _(11) q60 q02 q71 q13 - q63 g72
R(s)
=
(I-
q67
7
q72
'
q76(11'19) q63(10) , +
_(ii)
g72
-(10'18)-(11'19))[(1-
q67
q76
'
'
q02q20
)(
l-q13q31)(l-q48q84)(l-
q59 q95)-q24q48qs,12q12,3q35q59q9,13q13,2 ] ' Ro(S ) . q24[q46(l-q13q31)(l-q59q95)-q48q 8,12q12,3q35q59 a ] ,
1168
S.S. ELIAS and S. W. LABIB
Rl(S) = q35 [q57(l-q02q20)(l-q48q84 ) - q59q9,13q13,2q24q48 b]'
R2(s) =
q24q48[(l-q13q31)(l-q59q95 )b- q8,12q12,3q35
=
--
)a
q57 ]
--
43(s)
q35q59 [(I q02q20)(l-q48q84
q9,13q13,2q24
R 4 (s) =
q2 4 q35 [q46 q57 - q48 q59 ab ]
The steady state availability
q46 | '
is given by : Na(S)
AoC-) =
~
AoCtl =
lim
S ~O
.... D'(S)
sA*ls) ~
a
(5) s=o
where N a (0)
13 ~--O= Ti ~i
=
.
=
TI
K?
;
l
I
qij =
Pij
and t
D'a (s) I
=
R'
S=O
- J o R'o
- J'o R o - J l
R1
!
-
43 ÷J34 where R(o),
, J4(O)
stand for
and
R'(s)
Special case:
M 8*
=
K* 0
+ K* = 1
M 9*
,
+
J2
44 ÷J44
t
R'O
,6,,
,
R~
, J'O
R2' (6)
* MI3
=
,,*
,
J~
..... J~ (s) I S=O
If two units are similar,
* MI2
#
then :
and
[(i-~'9))[( l-q24q42)-q12q24q46q61]
= (I~ q33 -(5, -
R'
J , R~(s) I S=O S=O
- q24a]~ C8
R1
Jl
R , R o, "''' R4' Jo' Jl .... ' J4 stand for ooo
_ (5 q31)q12[~23
C8 ;
))[( l-~lqlo) (l-q24q42)
+ q12 q24 q46 q61 ]
q12 (q30 q01- q31)(q23 + q24 a) ,
K2* + K3* =
(5,9)) q01 (i -q33
K~ + K~ =
qol ql2 (1-q33-(5'9))c8
'
( i-q24 q42) C8 '
Two-dissimilar unit cold standby redundant system
K~ + K* =
7
K*
8
+ K* =
*
(q
q01 q12
a
23- q24
(5,9)
9
q01 q12 q24 (1-q33
* -
) C8
,
) c8
'
(1 _ ( 5 , 9 ) )
KI2 + K13- q01 q12 q24 q46
C8
-g33
1169
•
Therefore , (9) ÷ q2,~q47q78%31 1 % 1 M*~-c"(~I M~1 Nacs~ = E %2 [q 2~+ %4q46%3 +
(5,9)) (1 - q33
[ (1
+
q12 (q01 (M*2 + q24 M~4
- q24 q42 +
) ( M* 0
+
M*) 1
q01
+
q24 q46 M*6 ) -
-M: q24q46q61 )]~ E('+ (5'9))[(l-q01q10) (1-q24 q42 ) z q33 + q12q24q46q61 ] - q12(q30 q01 - q31 ) ( q23 (9) q24 q46 q63 - q24 q47 q78 q83)~
'
and Da(S) =
-
~(1-q(5'9))[(1-q q )(1-q q ) - q q q q ] 33 01 I0 24 42 12 24 46 61 %
%2'%1 ÷ %o%1 ) (%3 ÷ q24q46q63 + % , % 7 % 8 % 3 ~ (1 + _ ( 5 , 9 ) ) [
g33
(l-qOlqlo)(l-q24q42
- q12(q30qOl-q31)(q23-q24q46q63
) +
[-"
q12q24q46q61 ]
- q24 q47 q78 q83
Then (9)+
A:(S) i
12[q23 + q24q46q63 (5)M*]
q31
O
+
(i
q24 q47 q78 q83
][
qOl
M* -
3
_(5,9)) [( ) (M: + *) + -q33 1-q24 q42 qolM1 q24 4
q24q46M~ )-
M*
/
E.. (5,9) )[ (l-q01ql0 )( i-q24 q42 )-q12 q24 q46 q61 ]%1-q33 q12(q23 + q24q46q63 + q24q47q78q83)(q31
+ q30 q01)~
The steady state availiability is given by : Na(S)
Ao(~) N a ( s ) ljlo =
=
D a(s) s=ol
)
(l-q(5~91)[(l-q24q42)~l+P12(~2+P24~4+P24P46
~6 ) ]
1170
S.S. ELIASand S. W. LABm
+ P12 [i - P24 P42 - P24 P46 P61 ] P3 + [(1- P24P42)(1 - P33 _(5,9) -P31 (5)P121-P30P12P24P46P61 . ]~O' D'(s)i - Na(S) I + P P P (I-p(5'9)) ( P7 s-o s=o 12 24 46 33
+
~8 ) "
which agree with the corresponding results obtained in [3]
5. E X P B C T E D BUSY PERIOD OF SERVER FOR R E P A I R
IN (O f t~
Let BJk(t)-the probability that at time
t
the server is busy
with repair , according to the distribution Gjk (t) given that the system entered regenerative state S i at
t =o.
By probabilistic argument we have the following relations for BiJk(t). For j = I,
k = 1
B101(t'" q02 (t, Q
Bll(t)2
'
Bl11(t'" qz3 (t, Q
BII3 (t)
,
B~ l(t) = q20 (t) Q
Bll(t)0 + q24 (t) Q
B141(t)+ GI1 (t' '
B~l(t) = q 3 1 ( t ) Q
Bll(t) + (t) Q 1 q35
Bll(t) 5
Bn(t)4 : q46 (t) (~ ezzctl6 + q48(t) (~) BiZet)8 BZl(t)5 " ~57 (t) (~BlZ(t)7 + q59 (t) (~)Bzzlt}9 BZi(t) :6
q60(t) (~BlZct) l ( t+-(10)(t)(~B~ "63 ~ o q 10'lS)(t) 67
B~z(t) - qTl(t) ~
Q
Bll(t) 7
BnIt) ÷ q72 -(n)(t) 1
q(11'19)(t) 76
Q
(t) 8,14
Q
Q
Bll.t) 2 t
Q
11 B12(t)
B~ l(t)
B~l(t~ " qe4 (t~ C) Bll(t~4 q
,
÷ qs,n(t)
Bll(t) 14
,
Two-dissimilar unit cold standby redundant system
B191(t) " q95(t) G
Blsl(t) + q9,13 (t)
%,lSCt> ~
Blltt)12 = ql2,3(t> Bll(t> 13
ffi q13,2
1171
11 B13(t)
Q
+
B~t> + ~11 ct~ , 15
(~
Bll~t~3 + ql2-ClB>,7(t~ (~) ell(t~ ,
(t) ~
8llct> +_~19> ~ 2
q13,6
Bll(t)14 = q14,16 (t)
Q
Bll(t)16
'
811(t)15 = qlS,l~ (t>
~)
sll(t)17
'
Bll(t) (~t) 16 = q16,7
Q
Bll(t) 7
,
Bll(t) (t) 17 = q17,6
Q
Bll(t) 6
Taking Laplace-Stieltjes
8Zl.t~ 6 t
,
transform and simplifying,
we obtain ~-11. mo ~s), as follows : ~ll(s) o
=
Mll(s) / Da(S)
(7)
where
Mll(s) and
Da
=
Gll(S) (K 2
+
K9 )
as given in (4). In the long run, the fraction of time for which system
is under repair according to the distribution
Gll(t)
is
given by : ~11 Bo
lim t )oo Mllls) D,a(S)
=
B II 't)
lim s >o
(8)
1 smo
where Mll(s) and For
I
s=o D~(o) J - 1
=
'
T2
+
T9
as given in 16) ,
ii s B o (s)
k = 2
Bl2(t) = q02 (t) Q
B12(t)2
'
Bl2(t)l " q13 (t) Q
B12(t)3
'
1172
S.S. ELIAS and S. W. LABIB
,,;,,,:, .-q,o,t, ® ,,,,,t, +o
B121t)4 = %6~t)
,t, @
C) ~12c~)6+ %8 (t)
B12(t)5 : q57(t) @
.
~
B12(t)7 + q59 (t) @
Bn~t)8 B12(t)9
I~o)itl ® B12(t)6 = q60(t)@ Bl2(t) + q63
c~o,.~(t) Q ~ , t , , B~ 3 (t)+q67 -
BT12(t) = q71(t) ~2(t)+q7211(t)~B12(t)+p(ll'lg)(t ) 2 76
Bn(t) 8
= %4
~t) ~Bl2(t) + 4
q8,14 (t) @
(tl ~
%,n
q9,15 (t) @
nct)
B12(t)14 + G12 (t) , Bl2(t) 13
B12(t)15 '
B12(t)12 = ql2,3(t) @
B12(t)3 + q12,7-(18)(t)@
Bl2(t) = (t) @ 13 q13,2
Bl2(t)+ _(19) 2 q13,6
B12(t)14 = q14,1~t)
@
Bl2(t) = (t) @ 15 q15,17 Bl2(t) (t) 16 = q16,7
@
Bl2(t) (t) @ 17 = q17,6
B12(t)'6
Bn
(t) @ Bl2(t) + 5 q9,13
B~2(t) = q 9 5 ( t ) @
~
@
B12(t)7 ' Bl2(t), 6
B12(t)16 ' Bl2(t)
,
17
Bl2(t)
,
7
Bl2(t) , 6
Taking Laplace-Stieltjes transform and simplifying, we obtain ~12(s), as follows : o ~12(s)
=
Ml2(s) / D (s)
MI2(s)
=
GI2(S) (K~
o
(9)
a
where +
K 8*)
and Da(S) as given in (4). In the long run, the fraction of time for which the
Two-dissimilar unitcoldstandbyredundantsystem
1173
server is busy according to the distribution law Gl2(t) is given by : B 12 o
=
lim t--->~ MI2(s)
=
D'(s)
BO12( t )
lim S s-->o
=
~ 1 2 (S)
(i0)
I
s
o
where
and
Ml2(s)
[ s=o
=
D'(s)
i s=o
as in
For
T3
j = 2
+
T8
(6)
,
B21(t)0 = q02(t) Q
B21(t)2
B21(t) (t) Q 1 -- q13
B~l(t)
.
k = 1 '
B21(t) = q20(t) , Q B201(t) +
(t) Q
B21(t) 4 '
B~l(t) + q35(t) Q
B215 (t) ,
B241(t) = q 4 6 ( t ) Q
B261(t) +
B21(t) , 8
B251(t) = q57(t) Q
B~l(t) + q 5 9 ( t ) Q
B21(t) , 9
B261(t) = q60 (t) Q
B21(t)+0 q63-(10)(t)Q
B231(t) +
B21(t) = 3
q31(t) Q
q(10'lS) (t) Q 67 B21(t) = 7
q71(t) ~
(t) Q
B21(t) + - (t) 7 G21 '
B21(t) 12
21 B14(t) ,
B21(t)5 4 q9,!3 (t) Q
q9,15 (t) G
+
B21(t), 6
B21(t)4 + q8,12 (t) Q
q8,14 (t) Q B21(t)9 = q95 (t) Q
q48
B21(t)+I ~72-(ii)(t)C)B221(t)
q(ll'lg)(t) ~ 76 B21(t)8 = q84 (t) Q
q24
B21(t) + 13
B21(t)15 '
G
B21(t) = (t) Q 12 q12,3
B21(t) + -(18)(t) 3 q12,7
21 (t) Q Bl3(t) = q13,2
B21(t) + -(19)(t) Q 2 q13,6
B21(t)
7
+
B21(t) , 6
21
(t),
1174
S.S. ELIAS and S. W. LABIB
B21Ct'14= %4,1~t' G
~211t'16
B21(t)15 = q15,1~ t)
Q
B21(t)17 '
B21(t) (t) 16 = q16,7
Q
B21(t) + - (t) 7 G21 '
B21(t) = (t) 17 q17,6
@
B21(t) 6
Taking Laplace-Stieltjes transform, we obtain~21(s), as follows : ~21(s')
=
M21(s) / Da(S)
(ii)
where M21(s)
=
G21(s)
and
Da(S)
[K;
K* + q8,14 q14,16 8
+
K* ], 12
as given in (4).
In the long run, the fraction of time for which the server is busy according to the distribution law G21(t) is given by :
B~l
=
lim
s
s
"~_21,
~o
)o
=
~)
M 2~s) D ~ (s)
I
~z21
s=o
where M21(s) ~=o
and
D'a(S)
J
=
T6 +
as in
P8,14 T8 + TI2
(6)
s=o
For
j
=
2
,
k=2,
B22 (t) '
B~2(t) = q02(t) B~2(t) = ql3(t)
B 22 (
B~2(t) = q20 ct, @
Bnct'0 * %4(tI Q
4 t,
B~2(t) = q31 ~t, Q
B2~It'l* %s ct, @ ,~nIt,s '
Bnct~4 = q4~t~, (~
B=2ct~ * q48(tl (~
snctl8
'
B~2(t~ =qs~ (t~ C)B22't)~ *qs9 (t) ( ~ B22(t~9 ' '~ 2't~ =%o `tl (~)Bn(tl *o q(10';18)(t) @ 67
'~,3-(l°l(t~ C ) ' ~ 2(t~ * B 22 (t) 7 '
Two-dissimilar unit cold standby redundant system
B~ 2(t) = q71 (t) Q
B22(t) + i
q76(ll'19)(t) Q
B22 8 ~t~ = q84(t~ ~
q72(ll)(t) Q
B~2(t)
B22(t~4 + q8,1~t~ ~ n
BI4 (t) Q
+
B22(t)6 + G22-(t),
q8,14(t) B22(t) = q95 (t)
1175
Bn(t~12 +
, B22(t) 13
B22(t)5 + q 9 , 1 3 ( t ) ~
+
q9,15 (t)
Q
B22(t)15 '
(t~
~
Bn(t)
B22(t)13 = ql3,2(t)
~
B22(t)2 + q13,6-(19)(t)Q B22(t)6 + G22(t) '
B22(t~ 12
= q12,3
+
3
(18~,.,
(~
q12,7 It;
B22(t) (t) 14 = q14,16
Q
B22(t) 16
,
B22(t) (t) 15 = q15,17
Q
B22(t)
,
B22(t) (t) 16 = q16,7
~)
B22(t) 7
Q
B22(t) + - (t) 6 G22 "
B22(t) , 7
17 ,
p
B22(t) = (t) 17 q17,6
Taking Laplace-Stieltjes transform, we obtainB22(s) as follows : ~22(S) o
=
M22(S) / Da(S)
(13)
where
and
M'2(S) = G22(s)(K~7 + -q9,15 q15 ' 17 K*9 + K*13 ). Da(S) as given in (4). In the long run, the fraction of time for which the
server is busy according to the distribution law
G22(t)
is given by z B ss o
=
lim s B~'(s) S'--->O
=
M''(s)
I
D~(s)
(14)
S=O
where M2S(s) and
i
=
s=0
D~(s) as in (6)
T
+
7
P
T
9,15
+
9
T
,
13
I176
S.S. ELIASand S. W. LABIB 6.
COST
ANALYSIS:
(I) The e x p e c t e d
up-time
of the s y s t e m
in
(o,t]
is
t ~up(t)
=
~ o
Ao(U)
du
so that ~*up(S) (2) The e x p e c t e d
=
down-time
A* (s)/s o
(15)
of the s y s t e m
~dn(t)
=
• Udn(S)
=
i6
(o,t]
is
:
t - Nup(t)
so that
(3) The e x p e c t e d
s12
busy period
~ *up (s)
of the s e r v e r
(16)
for repair
in (o,t]
t ~ll(t) P
=
~ o
B 11 o
*ll(s) ~b "
= ~ll~s) o '
(u) du
so that / s
(17) '
also ~12(s)
= ~12(s)/So
• 21 ~b (s)
=
~'22(s)
= ~22(s)/S~o
'
(18)
~21(s)/s o
(19)
and
Denote which
by G(t)
is e q u a l
revenue
the e x p e c t e d
to the d i f f e r e n c e
and the e x p e c t e d
total
total
(20) gain
between
cost
incurred
in
the e x p e c t e d
(o,t] total
in the same interval.
Therefore, Glt)
= Zl~uplt)
ii, t 12(t - z 2 ~b ~ ) - Z3 Ub
) - Z4~ b21 (t) - Z5~ p2 2 ( t ) (21)
where Z1
is the r e v e n u e
Z i, i = 2,3,4,5,
per unit
are the cost
ver is b u s y a c c o r d i n g 22,
,
per unit time
to the d i s t r i b u t i o n
for w h i c h
the ser-
Gij(t) , i j = l l , 1 2 , 2 1 ,
respectively. The e x p e c t e d
given G
up-time
by
= t =
total
cost per unit
time
~n s t e a d y
state
:
lim >~
G(t) t
= s
lim >o
s G*(s)
ZIAo - Z2 BIIo - Z3 B12o - Z4 Bo21 _ Z5 B22o
(22)
is
Two-dissimilar unit cold standby redundant system
REFERENCES I.
G. S. Mokaddis, Reliab,
2.
S.M. Gupta, Reliab.
3.
(1989).
N.K. Jaiswal and L.R. Goel, Microelectron.
23, 329-331
S.M. Gupta, Reliab.
S.S. Elias and S.W. Labib, Microelectron
29, 511-515
(1983).
D.K. Pandey and Renu Gupta, Microelectron,
26, 847-850
(1986).
1177