Environmental Engineering Reference
In-Depth Information
∆
t
), failed in the operating time period [
t, t +
∆
t
], to the product of the
number
N
of efficiently working object at time
t
by the duration of the
operating time period ∆
t
:
D
ntt t
(,
+D
)
ˆ
()
l=
t
.
[1.33]
Nt t
()
D
Comparing [1.25] and [1.33] it may be noted that the failure rate
characterises slightly better the reliability of the object at the operating time
t
, since it shows the failure rate related to the acutal number of working
objects at the operating time
t
.
Probabilistic definition of the failure rate is obtained by multiplying and
dividing the right-hand side of expression [1.33] by
N
D
nt t t N nt t t N
( ,
+D
)
D
( ,
+D
)
ˆ
()
l=
t
=
.
Nt t N
()
D
N t Nt
D
()
The estimate of the failure rate
T
≤
(
t
) is
0
tt
1
ˆ
(,
Qtt t
+D
)
1
ˆ
()
l=
t
,
()
D
t
Pt
where at ∆
t
→ 0 and
N
→∞
ˆ
(,
Qtt t
+D
)
1
dQt
()
1
ft
()
l=
( )
t
lim
=
.
=
[1.34]
ˆ
D
t
dt Pt Pt
()
()
Pt
()
D→
t
0
Possible changes of the failure rate λ(
t
) are shown in Fig. 1.4.
1.2.3 Relationship of reliability indicators
Since the failure rate λ(
t
) is a more complete characteristic of reliability, it
is interesting to express c.d.f.
P
(
t
) through FR.
Using the expression for the failure rate
l=
()
t ft Pt
()/ (),
we write
dP
(
t
)/
dt
= -λ (
t
)
P
(
t
).
Separating the variables (multiplying both sides by
dt
/
P
(
t
)), we obtain
dP
(
t
)/
P
(
t
) = -λ (
t
)
dt
.
Integrating from 0 to
t
and taking into account that at
t
= 0 c.d.f. of the
object is
P
(0) = 1 leads to
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