12
0
1
2
1
12
12
12
12...
12
n
n
n
n
n
n
n
XX
X
H
XX
X
H
H
H
XX
X
H
XX
X
H
XX
X
==
, (15)
with Х
i
being an event of operable condition for the i-th element, whereаs
i
X
failure event of the i-th element.
Matrix of probabilities of such conditions Р = {Р
m
} may be matched with
conditions matrix Н:
()()
()()
(
)
0
1
12
12...
n
n
P
PH
PH
PH
PH
PH
=
, (16)
Since the conditions {Н
m
} form a complete group of incompatible events the sum
of P matrix elements is equal to 1, as below,
()
()
()
()
(
)
0
1
12
12...
1
n
n
m
P
PH
PH
PH
PH
PH
=
=
+
++
+
++
, (17)
In cases of unidentified failure definition another interpretation of reliability
indicator is applicable. Here the reliability definition is connected with technical
system capability to operate efficiently with changes in technical condition for a certain
period of time under certain conditions. In such a case reliability indicator is
represented as an efficiency correlation between actual technical system condition and
changed technical system condition. This indicator is named as an efficiency decrease
ratio or a relative efficiency of an aggregate (complicated item) [9].
Conditional efficiency alteration value Ф = {Ф
m
} may be proposed depending on
probable system conditions {Н
m
}. Suppose, that such relation is described in the form
of a certain matrix:
()()
()()
(
)
0
1
12
12...
T
n
n
H
H
H
H
H
=
, (18)
Applying matrixes Н, Р, Ф, discrete probability distribution principle for
conditional efficiency may be formulated
()
mm
PP
=
, (19)
Specifying a certain permissible level of conditional efficiency decrease Ф
b
enables to identify reliability indicator as a sum of operable conditions:
1
1
m
m
m
RP
=
=
, (20)
with index m
l
to be found from condition Ф
m
> Ф
b
.
If such a border cannot be specified the technical system reliability may be
identified as a ratio between unconditional efficiency indicator to newer system
efficiency, i.e.
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