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Complex system reliability with exponential repair time distributions under head-of-line-repair-discipline
Microelectronics and Reliability
Pergamon Press 1973. Vol. 12, pp. 151-158.
Printed in Great Britain
COMPLEX SYSTEM RELIABILITY W I T H EXPONENTIAL REPAIR TIME D I S T R I B U T I O N S U N D E R HEAD-OF-LINEREPAIR-DI SCIPLINE P. P. GUPTA Senior Lecturer in Mathematics, M.M. College Modinagar, Meerut, (U.P.) India Almtract--In this paper the behaviour of a complex system, composed of two classes of components Lx and L~, has been investigated. It has been assumed that the repair of the failed components in classes L1 and L~ is carried out under the "head-of-line-repair-discipline". Laplace transforms of the various state probabilities have been obtained. Inversions have also been carried out to obtain time-dependent probabilities. In the end, the behaviour of this system under steady state has also been examined. Key words: reliability, maintenance, mathematical models, component failure analysis. I. I N T R O D U C T I O N EARLIER workers Govil et al. (a~ have treated the problem of evaluating the reliability of a complex system,
consisting of two classes of components in which class L 1 consists of N components in series and class L 2 consists of only one component, under the head-of-line-repair-discipline. I n practical situations, sometimes it becomes necessary that the class L 2 may have more than one component, say M, in parallel redundancy to obtain a higher order of reliability. Keeping this in view, in this paper a complex system with two classes of components in which class L 1 consists of N components connected in series and class L~ consists of M identical components connected in parallel redundancy under the head-of-line-repair-discipline, has been considered. It has been assumed that a failure in L~ brings about the complete breakdown of the system, whereas a failure of two components in class L 2 causes the system to work in reduced efficiency state. Let the components in class L t have constant failure and repair rates ;~t, X2, • • •, Xiv and ~1, 03, • • •, ~lv, respectively. Also, assume that X' be the constant failure rate of any one of the components of class Lz and let 9 be the constant repair rate of any two components of L~, taken together, with which this system is brought to the state of normal efficiency from the state of reduced efficiency. It has been assumed that the components in the classes L t and Lz behave independently, i.e. the parameters of the components in L I are assumed to be unchanged by a non-disabling failure. I n the end, the behaviour of this system under steady state has been examined and from this the behaviour of the complex system, consisting of one component in series and two components in parallel, has also been obtained.
(~/)--1 ~
2. N O T A T I O N S mean failure rate of ith component in class L 1 (i = 1, 2 , . . . ,
N)
N i~l
(X')-1 ~--- mean failure rate of any one component in class L~ ~
--= constant repair rate of the ith component of class L 1 ~ constant repair rate of any two components of class L2, taken together. 151
152
P.P. GUPTA 3. A S S U M P T I O N S
The following assumptions have been made in formulating the mathematical model: (1) Failure is detected immediately. (2) The repair of the components of class L 2 is only undertaken when the system works in a state of reduced efficiency. (3) The repair of the components of the two classes is carried out in accordance with head-of-linerepair-discipline. (z) 4. DIFFERENCE-DIFFERENTIAL E Q U A T I O N S G O V E R N I N G THE B M V I O U R OF THE COMPLEX SYSTEM
Define :
PO,M(t) = the probability that at time t, the system is operating in the state of normal efficiency. Po,M_l(t) = the probability that at time t, the system is in operable state wherein one component of class L 2 has already failed. Po,M_~(t) = the probability that at time t, the system is operating in reduced efficiency state due to the failure of any two components of class L z. PF~,M(t) ~- the probability that at time t, the system is in the failed state due to the failure of ith component of class L I and all the M components of class L 2 are in the operable state. PV~,M-I(t) =- the probability that at time t, the system is in the failed state due to the failure of ith component of class L x and ( M - - l ) components of class L~ are in operable state. qe~,M_2(t) =_ the probability that at time t, the system is in the failed state due to the failure of ith component of class L 1 and any two components of class L2, such that class L 2 is under repair and class L 1 is awaiting repair. By elementary probability considerations and continuity arguments, we get the following differencedifferential equations: N
+X+MX'
PO,M(t) = 9×Po, M-2(t) + ~
PF,,M(t)×~
(1)
i=1 N
+X+(M--1)X'
Po,M_t(t)= MX'×Po,M(t ) + ~¢ Pl~,M_l(t)×v~
(2)
i=1
+?,+~
Po,M-2(t) = (M--1)X' Po,M-I(t)
gt + ~ PFi,M(t) = 7,~Po,M(t)+~ × qF~,M_2(t)
(3) (4)
[~-~ -~-,,]PF,,M-I(t) = x,× Po,M-l(t)
(5)
[~t +91 qF,.M-2 (t) = z'×P°'M-2(t)"
(6)
These equations are to be solved under the conditions that initially the system is operating in the state of normal efficiency, i.e. P0,M(0) = 1 . . . (7) and Po,M-j(O) = Po.M-,(O) = PF,,M(O) = PF,.M-I(O) = qV,.M_~(O)= O. (8) Let the Laplace transform of the function f(t) be denoted by f(s), i.e.
COMPLEX SYSTEM RELIABILITY WITH EXPONENTIAL REPAIR TIME DISTRIBUTIONS 153 QO
f(s) = f e-stf(t) dt; Re(s) )0.
(9)
0
"Faking the Laplace transform of equations (1) through (6) we get: N
Po,M(S)X ( s + ; ~ + M x ' )
1+9
=
X
Po,M-2(s) + ~./P~i,M(S)X ~
(10)
i=1 N
Po,m-t(s)x
(s+;~+M-lX') = MX'
Po,M(S) + ~ PF~,M-x(S)X ~l
(11)
i=l
Po,M-2(s)X (s+X+~) = (M-- 1)X' Po,M-l(s)
(12)
Pv~,M(O x (' +'~0 = x~Po,M(~)+,P x q~,M-,(*)
(13) (14) (is)
Equations (15), (14), (13) and (12) imply:
qF~,M_2(S)= ~
(16)
~o,M-~(S)
(17)
Pe,,M(S) = S--+~i xi Po,M(s) + s! + ~, 4~ ,M_~(~)
(18)
(M--I)x' & ' ~ - ~ ( ' ) - , + x + ~ &,M-,(,)
(19)
using relation (17) in (11) and solving, we get
Po,~-~(s) =
Mx' Po,,~(O
(20)
,, x,,~,1-
s+x+(M--1)x'-
s+~d Making use of relations (16), (18) through (20) in equation (10), we get i=l
(s+X+~){s+x+(M--1)x'--
~
Xc~,~
Po,~(~) = N
--~M(M--1)X '2 1+ i~_a (s+'O,)(s+~)J]"
(21)
Using (21), relation (20) gives Po M-l(s) - '
where
Mx'(s+~+,) D
(22)
154
P.P. GUPTA N
N
N
--~M(M--1)X" {1+ i~ ' (s+,~-~+~p)JJ By substituting the value of Po,M-~(s) in (19), we get M(M--1)X '~
D
(23)
M(s + X+~)X'X~(s + ~) -1
(24)
Po,M-~(S) --
Also equation (17) gives
Pv.M-I($) =
D
Further putting the value of Po,M-~(s) in (16), we get
qFt,M-'2(s) =
M(M-- 1)x'~;q(s+~) -1 D
(25)
Finally, substituting the values of i5o,M($) and qF~,M_~(s) in (13), we get
PF~.m(s) = (s+~l,)D1 X,(s+~.+9) s+x+(M--1)X'-- ~ s - ~ d + ~pM(M--1)X s + , i=1
The Laplace transform of the probability that the system is in the failed state (Paown(S))can be given by Paown(S) = 2~ P~'f'M(S)+PF~M-I(S)+qF*M'-2 = i=1
;~(S+;~+,) s+X+(M--1)Y-- i,7_1s--~*" X i=1
x~--~+
=
x'~xiX s--~t+l +M(sq-),+~)~.~.,~]~-
s+q~
1 --
S
-- Po,M($)--
Po,M--I(S)--
(27)
P0,M- 2($/.
(28/
Relations (21) through (26) can further be written in the form
Po,M(S)= (s+z+~)(s+M--lX')
]-[(s+~j) + s
]-[(s+~j)
j=l
/=1
x i=1
N
X(s+~){l-[(s+vll)}]g-~))
(29)
j=l
po,~_~($)= Mx'($+x+~) ]-I'($+~j) ($+~)g-~
(30)
j=l
Po,M-z(s) = ~
1[M(M--1)~,'Z(s+9)X {
}1
N
]7" (s+'¢~) j=l
,
(31)
j=l
P~,,,M(s) =g--~) X~(s+~)(s+x+~) x (s+M--lx')
~" (s+~) + s j=~
f i (s+~) j=l
"=
~
+
COMPLEX SYSTEM RELIABILITY WITH EXPONENTIAL REPAIR TIME DISTRIBUTIONS
N
155
N
/=1
(33)
j=~
j*i
(34)
j=! where g(s)=
(s+M~')
(s+:ql) +
(s+~ 1 s i=
.N
X (S-~-X@~?)(S-[-~){(s+M--1 X ' ) ( H
(s~-~]))~-s(
j=l
N H
N =
N
}(
(
N
t(
-~M(M-1)x'~ 1"]"(,+~,) x (,+x+~) 1"[ (,+~j) - s j=l
~i~1
(s-~-~,)) ( i~ 1 ~ / / - -
j=t
N
{
×
j=l
j=l
i=1
r l (s+~j) j=l
I( ~ ~ ~'i}] i=l
Let [3~'s (k = 0,1,2 . . . . . 2 N + 3 ) be the distinct roots of the (2N+4)th degree equation g(s) = 0, where 13o = 0, and [31:/:[3~:/:[3s:/: . . . :/:[3zN+a:/:0. On partial fractions and subsequent inversion of relations (29) through (34), we get
2N+3
[~,.(t~ r
N
Po,M(t)=k~=O~[(~k+~+~)(~k+~ )
=
N
"l"l'([3k+~,)
× ([3z+M--1;C)
rl(13~+~j)
j=l
+
j=l
N j=t
,
i=l
Po,M-I(t)= ~ _,--~,|MX'(~k+X+~)(~k+~) × 17" (~+~J) k=o g tPk] k
(36)
j=~
_ 2N+3 ~_ e.,,[ { }1 ~ (~k+~)M(M--1)X '~ × 1"[ (~k-t-~j) ~,.(tl
N
k=0 g ( ~ )
j=l
P0,M-,(t)-- ~
2N+3 t~ ~,,x
o
N
~]" (~,~+'01)}{ X ,~, ~e~,,.,[ M(ff,~+x++)X';q { j=~
"~Fi,M-lt ] = ~ g ~ k ) ~=o
2N+3
j#l
(
N
i=~
1
(38)
N
, ~(~-t-~)(~+~,-t-~)× k=o g (l~k)
(~+M--lX')
) (~ ~' t} ~
N
+M(M-1)~x"x,×
.=
,~,.,,_.~o=
r] (~+~)
~o~
g'(~g) =
[~ ~]
y=~
j=l
(39)
N
)](
t
1"[ (~+~j), j=~
j~-i
,
(40)
g(s
S
(
1"[ (~+~)
n ~+~,~
M,M-,~'~,
where
+
j=l
N
1-[ (~+~) ×
1
1-[ (ff~+~l) × (~3~+~) j=~
~(tl
PF~,M (t)-- ~
+~
N
(37)
f~k
156
P.P. GUPTA 5. BEHAVIOUR OF M
COMPLEX SYSTEM U N D E R STEADY STATE
Employing the corollary of Abel's theorem in Laplace transforms, viz. lim lim s-*O sf(s)= t~oof(t ) = f say (provided the limit on the right exists), the various state probabilities when the statistical equilibrium has been reached may be obtained from relations (21) through (26) as (x+9)(M-- 1)~.' Po,M - -
(41)
H
Po,M-I --
M?,'(?,+~?) H
(42)
Po,M-2 --
M(M--1)Y 2 H
(43)
Mx'(x+~)(x~/~O
PF,,M-1 =
H
(44)
M(M-- 1)).'2).1q~ -1 qF*,m-2 = PV,,M ----
(45)
H
(),+ q0(M-- 1)X'(:;q/~,)+M(M-- 1)X'z(;q/'~,) H
(46)
where H = [MX' {(x+~)(1 + ~ ~)+(M--1)x'} +(X+~)(M--1)X'× i=1
i=1 X~ +M(M--1) X,2X+M(M_I)X, ~ ~ ~ . × 1+ __ ~ 9 i=1 If the complex system is of such nature that class L 1 consists of one component and the class L 2 consists of two components, i.e. N = 1 and M ----2, then we have
z'(z~+,~) Po,2--
A
'
where A = [ 2x'{(~,x+9) (1 + ~ ) +?,'} +(?'l+~)x' (1 + ~ ) + ~ ~""~-1+2x'2(~a/'~l)], Po,1 --
2x'(xl+,t,) A
'
(48)
2x'2 Po,o- A ' P&,I =
(47)
(49)
2z'(Xl/',~l)(x~+~) A
'
(50)
2X'2),lq~-I
qFl,o =
PF1,2 =
A
'
[(Xl/~l)(Xl+ ~)x'+2z'2(xd~0] A
(51) (52)
COMPLEX SYSTEM RELIABILITY W I T H EXPONENTIAL REPAIR TIME DISTRIBUTIONS 157 T o depict the results numerically and graphically P0,M has been plotted against M in the steady state for N = 1, )'l = 0.03, ~z = 0"3, ),' = 0-02, q~ = 0.4, and the results are tabulated in the following table: S. No.
M
Po,M
1
2
0'29
2
3
0"34
3
4
0.345
4
5
0"363
5
6
0"364
6
7
0.362
7
8
0"357
8
9
0"353
9
10
0.348
10
11
0-343
11
12
0.338
0.8
0.7
0-6
0"5
0-4
Po,M 0.3
0.2
0"1
0
I
1
I
2
I
5
I
4
I
5
I¸ I
6
7
I
8
I
9
i
I0
I
I
11 IZ
A critical examination of the above table and graph indicates that optimum reliability in the steady state may be obtained by using 6 components in parallel redundancy. I n other words, to have a higher order reliability in the steady state, it is desirable that for such a system only 6 components may be used in parallel redundancy.
158
P.P.
Acknowledgements--The author is most grateful to Dr. S. S. Srivastava, Controller of Systems, and Dr. R. C. Garg, Senior Scientific Officer, Grade I, Research and Development Organization, Ministry of Defence, New Delhi, for their keen interest in the preparation of this paper. T h e author is also grateful to Dr. A. K. Govil, Senior Scientific Officer, Directorate of Aeronautics, Research and Development Organization, Ministry of Defence, New Delhi, for suggesting the problem and later on giving concrete comments but for which the paper could not have taken the present shape. He is also indebted to Prof. J. N. Kapur, Vice-Chancellor, Meerut University, for his constant encouragement and illuminating comments during the preparation of this paper.
GUPTA REFERENCES 1. R. C. GARG, Analytical study of a complex system having two types of components. Naval Res. Logistics Quart. 10 (1963). 2. KEILSON and A. KOOHARIAN, On time dependent queuing process. Ann. math. Statist. 31 (1960). 3. A. K. GOVIL and S. KUKAR, Stochastic behaviour of a complex system under priority repair. Computing 6, 200-213 (1970). 4. A. K. GOVIL, Operational behaviour of a complex system under priority repair disciplines. Trabajos Estad. Y De Investigacion Operativa (Spain). X X I , 81-100 (1970). 5. D. V. WIDDER, Laplace Transform. Princeton University Press, New Jersey (1941).
Pergamon Press 1973. Vol. 12, pp. 151-158.
Printed in Great Britain
COMPLEX SYSTEM RELIABILITY W I T H EXPONENTIAL REPAIR TIME D I S T R I B U T I O N S U N D E R HEAD-OF-LINEREPAIR-DI SCIPLINE P. P. GUPTA Senior Lecturer in Mathematics, M.M. College Modinagar, Meerut, (U.P.) India Almtract--In this paper the behaviour of a complex system, composed of two classes of components Lx and L~, has been investigated. It has been assumed that the repair of the failed components in classes L1 and L~ is carried out under the "head-of-line-repair-discipline". Laplace transforms of the various state probabilities have been obtained. Inversions have also been carried out to obtain time-dependent probabilities. In the end, the behaviour of this system under steady state has also been examined. Key words: reliability, maintenance, mathematical models, component failure analysis. I. I N T R O D U C T I O N EARLIER workers Govil et al. (a~ have treated the problem of evaluating the reliability of a complex system,
consisting of two classes of components in which class L 1 consists of N components in series and class L 2 consists of only one component, under the head-of-line-repair-discipline. I n practical situations, sometimes it becomes necessary that the class L 2 may have more than one component, say M, in parallel redundancy to obtain a higher order of reliability. Keeping this in view, in this paper a complex system with two classes of components in which class L 1 consists of N components connected in series and class L~ consists of M identical components connected in parallel redundancy under the head-of-line-repair-discipline, has been considered. It has been assumed that a failure in L~ brings about the complete breakdown of the system, whereas a failure of two components in class L 2 causes the system to work in reduced efficiency state. Let the components in class L t have constant failure and repair rates ;~t, X2, • • •, Xiv and ~1, 03, • • •, ~lv, respectively. Also, assume that X' be the constant failure rate of any one of the components of class Lz and let 9 be the constant repair rate of any two components of L~, taken together, with which this system is brought to the state of normal efficiency from the state of reduced efficiency. It has been assumed that the components in the classes L t and Lz behave independently, i.e. the parameters of the components in L I are assumed to be unchanged by a non-disabling failure. I n the end, the behaviour of this system under steady state has been examined and from this the behaviour of the complex system, consisting of one component in series and two components in parallel, has also been obtained.
(~/)--1 ~
2. N O T A T I O N S mean failure rate of ith component in class L 1 (i = 1, 2 , . . . ,
N)
N i~l
(X')-1 ~--- mean failure rate of any one component in class L~ ~
--= constant repair rate of the ith component of class L 1 ~ constant repair rate of any two components of class L2, taken together. 151
152
P.P. GUPTA 3. A S S U M P T I O N S
The following assumptions have been made in formulating the mathematical model: (1) Failure is detected immediately. (2) The repair of the components of class L 2 is only undertaken when the system works in a state of reduced efficiency. (3) The repair of the components of the two classes is carried out in accordance with head-of-linerepair-discipline. (z) 4. DIFFERENCE-DIFFERENTIAL E Q U A T I O N S G O V E R N I N G THE B M V I O U R OF THE COMPLEX SYSTEM
Define :
PO,M(t) = the probability that at time t, the system is operating in the state of normal efficiency. Po,M_l(t) = the probability that at time t, the system is in operable state wherein one component of class L 2 has already failed. Po,M_~(t) = the probability that at time t, the system is operating in reduced efficiency state due to the failure of any two components of class L z. PF~,M(t) ~- the probability that at time t, the system is in the failed state due to the failure of ith component of class L I and all the M components of class L 2 are in the operable state. PV~,M-I(t) =- the probability that at time t, the system is in the failed state due to the failure of ith component of class L x and ( M - - l ) components of class L~ are in operable state. qe~,M_2(t) =_ the probability that at time t, the system is in the failed state due to the failure of ith component of class L 1 and any two components of class L2, such that class L 2 is under repair and class L 1 is awaiting repair. By elementary probability considerations and continuity arguments, we get the following differencedifferential equations: N
+X+MX'
PO,M(t) = 9×Po, M-2(t) + ~
PF,,M(t)×~
(1)
i=1 N
+X+(M--1)X'
Po,M_t(t)= MX'×Po,M(t ) + ~¢ Pl~,M_l(t)×v~
(2)
i=1
+?,+~
Po,M-2(t) = (M--1)X' Po,M-I(t)
gt + ~ PFi,M(t) = 7,~Po,M(t)+~ × qF~,M_2(t)
(3) (4)
[~-~ -~-,,]PF,,M-I(t) = x,× Po,M-l(t)
(5)
[~t +91 qF,.M-2 (t) = z'×P°'M-2(t)"
(6)
These equations are to be solved under the conditions that initially the system is operating in the state of normal efficiency, i.e. P0,M(0) = 1 . . . (7) and Po,M-j(O) = Po.M-,(O) = PF,,M(O) = PF,.M-I(O) = qV,.M_~(O)= O. (8) Let the Laplace transform of the function f(t) be denoted by f(s), i.e.
COMPLEX SYSTEM RELIABILITY WITH EXPONENTIAL REPAIR TIME DISTRIBUTIONS 153 QO
f(s) = f e-stf(t) dt; Re(s) )0.
(9)
0
"Faking the Laplace transform of equations (1) through (6) we get: N
Po,M(S)X ( s + ; ~ + M x ' )
1+9
=
X
Po,M-2(s) + ~./P~i,M(S)X ~
(10)
i=1 N
Po,m-t(s)x
(s+;~+M-lX') = MX'
Po,M(S) + ~ PF~,M-x(S)X ~l
(11)
i=l
Po,M-2(s)X (s+X+~) = (M-- 1)X' Po,M-l(s)
(12)
Pv~,M(O x (' +'~0 = x~Po,M(~)+,P x q~,M-,(*)
(13) (14) (is)
Equations (15), (14), (13) and (12) imply:
qF~,M_2(S)= ~
(16)
~o,M-~(S)
(17)
Pe,,M(S) = S--+~i xi Po,M(s) + s! + ~, 4~ ,M_~(~)
(18)
(M--I)x' & ' ~ - ~ ( ' ) - , + x + ~ &,M-,(,)
(19)
using relation (17) in (11) and solving, we get
Po,~-~(s) =
Mx' Po,,~(O
(20)
,, x,,~,1-
s+x+(M--1)x'-
s+~d Making use of relations (16), (18) through (20) in equation (10), we get i=l
(s+X+~){s+x+(M--1)x'--
~
Xc~,~
Po,~(~) = N
--~M(M--1)X '2 1+ i~_a (s+'O,)(s+~)J]"
(21)
Using (21), relation (20) gives Po M-l(s) - '
where
Mx'(s+~+,) D
(22)
154
P.P. GUPTA N
N
N
--~M(M--1)X" {1+ i~ ' (s+,~-~+~p)JJ By substituting the value of Po,M-~(s) in (19), we get M(M--1)X '~
D
(23)
M(s + X+~)X'X~(s + ~) -1
(24)
Po,M-~(S) --
Also equation (17) gives
Pv.M-I($) =
D
Further putting the value of Po,M-~(s) in (16), we get
qFt,M-'2(s) =
M(M-- 1)x'~;q(s+~) -1 D
(25)
Finally, substituting the values of i5o,M($) and qF~,M_~(s) in (13), we get
PF~.m(s) = (s+~l,)D1 X,(s+~.+9) s+x+(M--1)X'-- ~ s - ~ d + ~pM(M--1)X s + , i=1
The Laplace transform of the probability that the system is in the failed state (Paown(S))can be given by Paown(S) = 2~ P~'f'M(S)+PF~M-I(S)+qF*M'-2 = i=1
;~(S+;~+,) s+X+(M--1)Y-- i,7_1s--~*" X i=1
x~--~+
=
x'~xiX s--~t+l +M(sq-),+~)~.~.,~]~-
s+q~
1 --
S
-- Po,M($)--
Po,M--I(S)--
(27)
P0,M- 2($/.
(28/
Relations (21) through (26) can further be written in the form
Po,M(S)= (s+z+~)(s+M--lX')
]-[(s+~j) + s
]-[(s+~j)
j=l
/=1
x i=1
N
X(s+~){l-[(s+vll)}]g-~))
(29)
j=l
po,~_~($)= Mx'($+x+~) ]-I'($+~j) ($+~)g-~
(30)
j=l
Po,M-z(s) = ~
1[M(M--1)~,'Z(s+9)X {
}1
N
]7" (s+'¢~) j=l
,
(31)
j=l
P~,,,M(s) =g--~) X~(s+~)(s+x+~) x (s+M--lx')
~" (s+~) + s j=~
f i (s+~) j=l
"=
~
+
COMPLEX SYSTEM RELIABILITY WITH EXPONENTIAL REPAIR TIME DISTRIBUTIONS
N
155
N
/=1
(33)
j=~
j*i
(34)
j=! where g(s)=
(s+M~')
(s+:ql) +
(s+~ 1 s i=
.N
X (S-~-X@~?)(S-[-~){(s+M--1 X ' ) ( H
(s~-~]))~-s(
j=l
N H
N =
N
}(
(
N
t(
-~M(M-1)x'~ 1"]"(,+~,) x (,+x+~) 1"[ (,+~j) - s j=l
~i~1
(s-~-~,)) ( i~ 1 ~ / / - -
j=t
N
{
×
j=l
j=l
i=1
r l (s+~j) j=l
I( ~ ~ ~'i}] i=l
Let [3~'s (k = 0,1,2 . . . . . 2 N + 3 ) be the distinct roots of the (2N+4)th degree equation g(s) = 0, where 13o = 0, and [31:/:[3~:/:[3s:/: . . . :/:[3zN+a:/:0. On partial fractions and subsequent inversion of relations (29) through (34), we get
2N+3
[~,.(t~ r
N
Po,M(t)=k~=O~[(~k+~+~)(~k+~ )
=
N
"l"l'([3k+~,)
× ([3z+M--1;C)
rl(13~+~j)
j=l
+
j=l
N j=t
,
i=l
Po,M-I(t)= ~ _,--~,|MX'(~k+X+~)(~k+~) × 17" (~+~J) k=o g tPk] k
(36)
j=~
_ 2N+3 ~_ e.,,[ { }1 ~ (~k+~)M(M--1)X '~ × 1"[ (~k-t-~j) ~,.(tl
N
k=0 g ( ~ )
j=l
P0,M-,(t)-- ~
2N+3 t~ ~,,x
o
N
~]" (~,~+'01)}{ X ,~, ~e~,,.,[ M(ff,~+x++)X';q { j=~
"~Fi,M-lt ] = ~ g ~ k ) ~=o
2N+3
j#l
(
N
i=~
1
(38)
N
, ~(~-t-~)(~+~,-t-~)× k=o g (l~k)
(~+M--lX')
) (~ ~' t} ~
N
+M(M-1)~x"x,×
.=
,~,.,,_.~o=
r] (~+~)
~o~
g'(~g) =
[~ ~]
y=~
j=l
(39)
N
)](
t
1"[ (~+~j), j=~
j~-i
,
(40)
g(s
S
(
1"[ (~+~)
n ~+~,~
M,M-,~'~,
where
+
j=l
N
1-[ (~+~) ×
1
1-[ (ff~+~l) × (~3~+~) j=~
~(tl
PF~,M (t)-- ~
+~
N
(37)
f~k
156
P.P. GUPTA 5. BEHAVIOUR OF M
COMPLEX SYSTEM U N D E R STEADY STATE
Employing the corollary of Abel's theorem in Laplace transforms, viz. lim lim s-*O sf(s)= t~oof(t ) = f say (provided the limit on the right exists), the various state probabilities when the statistical equilibrium has been reached may be obtained from relations (21) through (26) as (x+9)(M-- 1)~.' Po,M - -
(41)
H
Po,M-I --
M?,'(?,+~?) H
(42)
Po,M-2 --
M(M--1)Y 2 H
(43)
Mx'(x+~)(x~/~O
PF,,M-1 =
H
(44)
M(M-- 1)).'2).1q~ -1 qF*,m-2 = PV,,M ----
(45)
H
(),+ q0(M-- 1)X'(:;q/~,)+M(M-- 1)X'z(;q/'~,) H
(46)
where H = [MX' {(x+~)(1 + ~ ~)+(M--1)x'} +(X+~)(M--1)X'× i=1
i=1 X~ +M(M--1) X,2X+M(M_I)X, ~ ~ ~ . × 1+ __ ~ 9 i=1 If the complex system is of such nature that class L 1 consists of one component and the class L 2 consists of two components, i.e. N = 1 and M ----2, then we have
z'(z~+,~) Po,2--
A
'
where A = [ 2x'{(~,x+9) (1 + ~ ) +?,'} +(?'l+~)x' (1 + ~ ) + ~ ~""~-1+2x'2(~a/'~l)], Po,1 --
2x'(xl+,t,) A
'
(48)
2x'2 Po,o- A ' P&,I =
(47)
(49)
2z'(Xl/',~l)(x~+~) A
'
(50)
2X'2),lq~-I
qFl,o =
PF1,2 =
A
'
[(Xl/~l)(Xl+ ~)x'+2z'2(xd~0] A
(51) (52)
COMPLEX SYSTEM RELIABILITY W I T H EXPONENTIAL REPAIR TIME DISTRIBUTIONS 157 T o depict the results numerically and graphically P0,M has been plotted against M in the steady state for N = 1, )'l = 0.03, ~z = 0"3, ),' = 0-02, q~ = 0.4, and the results are tabulated in the following table: S. No.
M
Po,M
1
2
0'29
2
3
0"34
3
4
0.345
4
5
0"363
5
6
0"364
6
7
0.362
7
8
0"357
8
9
0"353
9
10
0.348
10
11
0-343
11
12
0.338
0.8
0.7
0-6
0"5
0-4
Po,M 0.3
0.2
0"1
0
I
1
I
2
I
5
I
4
I
5
I¸ I
6
7
I
8
I
9
i
I0
I
I
11 IZ
A critical examination of the above table and graph indicates that optimum reliability in the steady state may be obtained by using 6 components in parallel redundancy. I n other words, to have a higher order reliability in the steady state, it is desirable that for such a system only 6 components may be used in parallel redundancy.
158
P.P.
Acknowledgements--The author is most grateful to Dr. S. S. Srivastava, Controller of Systems, and Dr. R. C. Garg, Senior Scientific Officer, Grade I, Research and Development Organization, Ministry of Defence, New Delhi, for their keen interest in the preparation of this paper. T h e author is also grateful to Dr. A. K. Govil, Senior Scientific Officer, Directorate of Aeronautics, Research and Development Organization, Ministry of Defence, New Delhi, for suggesting the problem and later on giving concrete comments but for which the paper could not have taken the present shape. He is also indebted to Prof. J. N. Kapur, Vice-Chancellor, Meerut University, for his constant encouragement and illuminating comments during the preparation of this paper.
GUPTA REFERENCES 1. R. C. GARG, Analytical study of a complex system having two types of components. Naval Res. Logistics Quart. 10 (1963). 2. KEILSON and A. KOOHARIAN, On time dependent queuing process. Ann. math. Statist. 31 (1960). 3. A. K. GOVIL and S. KUKAR, Stochastic behaviour of a complex system under priority repair. Computing 6, 200-213 (1970). 4. A. K. GOVIL, Operational behaviour of a complex system under priority repair disciplines. Trabajos Estad. Y De Investigacion Operativa (Spain). X X I , 81-100 (1970). 5. D. V. WIDDER, Laplace Transform. Princeton University Press, New Jersey (1941).