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estimated from the data.
Φnor
0
0.2
0.4
0.6
0.8
0 10000 20000 30000 40000 50000 60000 70000
Fraction Failing
Cumulative Flight Hours
file location:c:\splidauser\FinalRptFig1.sgr
Figure 5. Cumulative Probability Function
In practice, figure 5 can also be presented on a different scale. In this example, the x axis is time
plotted on a log scale, and the y axis is the quantile of the distribution (see figure 6), because
both the curve and the data should be along a straight line when the distribution function fits the
data well. Figure 6 was often used for this diagnostic reason. In general, both figures 5 and 6
were used to project reliability statistics as products of age.
Cumulative Flight Hours
.0001
.0005
.001
.002
.005
.01
.02
.05
.1
.2
.3
.4
.5
.6
.7
.8
5000 20000 35000 50000 65000
Fraction Failing
file location: c:\splidauser\FinalRptFig3.sgr
Figure 6. Cumulative Probability—Quantile Display
10
To examine the product life distribution, the following density distribution function was used.
( ; , ) 1 nor log( )
f t t
t
μ
μ σ ϕ
σ σ
= ⎡⎢⎣ − ⎤⎥⎦
(5)
φnor is the probability density function for the standard normal distribution. Figure 7 shows the
life distribution of the products; in this example, it is a bell-like shape distribution skewed to the
right. As shown, most components had an average product life in the middle portion. Many
experienced life spans longer than the mode. Median lifetime, for example, or any quantile of
the distribution can be obtained. They are life predictions on when a certain percent of the unit
will have failed. These predictions are critical in manufacturing a process for forecasting the
needs for spares and repairs.
10000 30000 50000 70000 90000 110000
Cumulative Flight Hours
0.0e0
5.0e-6
1.0e-5
1.5e-5
2.0e-5
2.5e-5
3.0e-5
Probability of Failing
file location:c:\splidauser\FinalRptFig2.sgr
Figure 7. Probability Density Function
Most importantly, to examine the instantaneous risk that the products were likely to fail or be
removed, a hazard/removal curve was used.
( ) ( ; , ) ( ; , )
( ; , ) 1 ( ; , )
h t f t f t
S t F t
μ σ μ σ
μ σ
= =
− μ σ
(6)
It was proportional to, as in the next given time or age, the probability of failing. This function
had a practical meaning because of its direct relationship with the manufacturing process and
maintenance strategies. Using the same example, the hazard/removal curve is presented in
figure 8. It was heavily used in the study. The curve in figure 8 can have many different shapes:
it can be flat, indicating a random removal pattern, or it can be rising, indicating a wear-out
phase. One commonly known shape is the bathtub shape, where infant mortality is observed on
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the left-hand side, as a burn-in phase, and natural rising/aging is observed on the right-hand side,
as a wear-out phase.
Cumulative Flight Hours
0 10000 20000 30000 40000 50000 60000 70000
0.0e0
2.0e-5
4.0e-5
6.0e-5
8.0e-5
1.0e-4
Hazard Function
file location:c\splidauser\FinalRptFig4.sgr
Figure 8. Hazard/Removal Function
3.7.2 Models, Diagnostics, and Challenges.
The functions described in section 3.7.1 and mathematical techniques used to estimate μ and σ2
are called maximum likelihood optimization techniques. In real life, more practical and
complicated situations arise.
For example, key contributing factors, such as different aircraft models, operators, designs, and
upgrades, may affect the performance of the components and, thus, need to be investigated. In
addition, the exact failure times, as described in equations 1, 2, and 3, are not always known.
The failure times can be observed in four ways: (1) during routine maintenance (exact failure
time uncertain to the left, left censored), (2) in service/pilot report (exact failure time),
(3) between inspections (failure time uncertain within interval, interval censored), and (4) in
future time (failure time uncertain to the right, right censored). It is necessary to choose the best
model.
To resolve these issues, the methodology allowed key contributing factors to have separate sets
of parameters (different μ’s and σ2’s), as shown in equation 7; failure time uncertainties to be
described, as in equation 8; and a diagnostic tool, as described in equation 9. They are briefly
　
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