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xed amount E
a
and E
b
of a certain
substratum (for instance, energy) and be able to distribute it evenly between their
protective elements, so that for each fraction of R
a
-and R
b
-elements there is
E
1a
=E
a
/N
ra
(0) and E
1b
=E
b
/N
rb
(0) respectively. Then the ef
Now let the systems H and N have a
cients of
the protective elements must increase with the increase in the E
1a
and E
1b
portions,
since in this case the probabilities of D
a
and D
b
increase. Therefore let D
a
¼
1
exp
F
a
F
1a
ciency coef
n
o
, where, F
a
,F
b
,
, D
a
¼ 1
exp
F
b
F
1b
are indepen-
dent of the number of protective elements. This gives then the following expres-
sions for the coef
ʱ
and
β
cients of effectiveness of the protective elements:(
10.48
)
d
a
¼
h
a
ln
ð
1
D
a
Þ
¼
h
a
F
a
E
1a
¼ G
a
N
a
Ra
ð
0
Þ;
d
b
¼
h
b
ln
ð
1
D
b
Þ
¼
h
b
F
b
E
1b
¼ G
b
N
b
ð
10
:
48
Þ
Rb
ð
0
Þ;
where, G
a
¼
h
a
F
a
E
a
;
G
b
¼
h
b
F
b
E
b
. From Eqs. (
10.47
) and (
10.48
) we obtain the
following transcendental equations for the amount of the necessary protective
elements in systems H and N:
N
Ra
ð
0
Þ
¼N
Ra
ð
0
Þ
G
1
f
ln½G
a
M
b
ð
0
Þ=
f
1
a
ln N
Ra
ð
0
Þg;
ð
10
:
49
Þ
a
N
Rb
ð
0
Þ
¼N
Rb
ð
0
Þ
G
1
f
ln½G
b
M
a
ð
0
Þ=
f
2
b
ln N
Rb
ð
0
Þg;
ð
10
:
50
Þ
b
f
2
¼ ln½eG
b
ð
1
h
b
Þ
N
b
ð
0
Þ
N
b
where f
1
¼ ln½eG
a
ð
1
h
a
Þ
N
a
ð
0
Þ
N
a
Rb
ð
0
Þ
An analysis of Eqs. (
10.49
) and (
10.50
) reveals that in this case the number
N
ra
(0) is very sensitive with respect to changes in the quantity ln[(1
Ra
;
− ʸ
a
)N
a
(0)].
And this is natural, since with the increase in the number of R
a
-elements their
effectiveness sharply decreases. It is clear that there exists a certain optimal level for
the number of protective elements. This level is de
ned by the assigned surviv-
ability of the system. It is better to have a small number of R
a
-elements of high
effectiveness, than a large number of R
a
-elements of low effectiveness. When k =3,
from Eqs. (
10.37
)
(
10.39
) we have:
-
Q
1
¼
X
3
m
a
ð
i
1
Þ
exp½
a
N
Ra
ð
i
1
Þ;
ð
10
:
51
Þ
i¼1
where,
N
Ra
ðÞ
¼N
Ra
ðÞ
m
Ra
ðÞ
exp
d
a
N
Ra
ð
½
;
N
ra
ðÞ
¼N
ra
ðÞ
m
Ra
ðÞ
exp
d
a
N
Ra
ðÞ
½
m
Ra
ðÞ
exp
d
a
N
Ra
ðÞ
½
ð
10
:
52
Þ
m
a
ðÞþ
m
a
ðÞþ
m
a
ðÞþ
m
Ra
ðÞþ
m
Ra
ðÞþ
m
Ra
ðÞ
¼M
b
ðÞ
From Eqs. (
10.51
) and (
10.52
) we obtain that the function Q
1
reaches its
maximum value when m
a
ð
0
Þ
¼m
Ra
ð
2
Þ
¼m
Ra
ð
1
Þ
¼0, with this maximum value
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