Environmental Engineering Reference
In-Depth Information
∞
=
∫
T
P t dt
() ,
[1.38]
0
0
taking into account the fact that
P
(0) = 1,
P
(∞) = 0.
From [1.38] it follows that the mean operating time to failure is
geometrically interpreted as the area under the curve
P
(
t
) (Fig. 1.5).
It is obvious that with increasing test sample
N
→∞ the arithmetic mean
operating time (estimate
T
0
) converges in probability to the EV of the
operating time to failure.
The mathematical expectation of the operating time
T
0
is the
mathematically the expected operating time to failure of similar elements,
i.e. the average operating time to first failure.
In practice, the
conditional mean operating time
is of interest:
1)
the mean useful operating time
(
T
≤
) determined under the condition
that when the operating time
t
1
is reached all remaining objects are removed
from service;
2)
the mean duration of impending work
(
0
tt
1
T
), provided that the item
0
tt
>
1
was working smoothly in the interval (0,
t
1
).
Reasons for using these indicators:
1. Highly reliable objects, usually operated over a shorter period than
T
0
(
t
ser
<
T
0
), i.e. replaced due to obsolescence before the end of the operating
time
T
0
.
2. Frequently, the test period for these facilities is reduced (carried out
for the operating time corresponding to their obsolescence), so
T
0
in this
case is understood as the mean time which would occur in reality, if FR
remained the same as in the initial period of testing.
The mean net operating time
T
≤
(by analogy with
T
0
):
0
tt
1
t
1
=
∫
T
P t dt
() .
0|
tt
≤
1
0
The mean duration of impending work
T
0|
tt
>
1
1. 5
Mean operating time to failure.
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