(7)The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for a priori estimation.
This section of the on-line reference includes the following subsections:
Eqn. 2 may be linearized by taking the natural log of both sides.
(7)
Consequently, plotting 
versus T
on log-log paper will result in a straight line with a negative slope,
such that:
Similarly, Eqn. 4 can also be linearized by taking the natural log of both sides:
(8)
Plotting 
versus T on log-log paper will result
in a straight line with a positive slope such that:
Two ways of determining these curves are as follows:
Predict the 
and 
of the system from its reliability block
diagram and available component failure intensities. Plot this value on
log-log plotting paper at T = 1. From past experience and
from past data for similar equipment, find values of 
, the slope of the improvement lines for 
or 
. Modify this 
as necessary. If a better design effort is expected and a
more intensive research, test and development or a TAAF program is to
be implemented, then a 15% improvement in the growth rate may be attainable.
Consequently, the available value for ,
should be adjusted by this amount.
The value to be used will then be 
. A line is then drawn
through point 
and T = 1 with the just determined
slope , keeping in mind that is negative for
the 
curve. This line should be extended
to the design, development and test time scheduled to be expended to achieve
the failure intensity goal to see if this goal will indeed be achieved
on schedule. It is also possible to find the design, development and test
time to achieve the goal that may be earlier than the delivery date or
later. If earlier, then either the reliability program effort can be judiciously
and appropriately trimmed; or if it is an incentive contract, full advantage
is taken of the fact that the failure intensity goal can be exceeded with
the associated increased profits to the company. A similar approach may
be used for the MTBF growth model, where 
is plotted at T = 1,
and a line is drawn through the point 
and
T
= 1 with slope to obtain the MTBF growth line. If values
are not available, consult Table 4.1, which gives actual values for various types of equipment. These have been obtained
from the literature or by MTBF growth tests. It may be seen from Table
4.1 that values range between 0.24 and 0.65. The lower
values reflect slow early growth and the higher values reflect fast early
growth.
During the design, development
and test phase and at specific milestones, the 
is calculated from generated total failures and T values. These values of 
or 
are plotted
above the corresponding T values on log-log paper. A straight
line is drawn favoring these points to minimize the distance between these
points and the line, thus establishing the improvement or growth model
and its parameters, graphically. If needed, linear regression analysis
techniques can be used to determine these parameters.
Table 4.1 - Sample values
for the slope ()
for various equipment
|
Equipment |
|
Slope |
|
Computer system |
Actual |
0.24 |
|
Mainframe computer |
|
0.50 |
|
Aerospace electronics |
All malfunctions |
0.57 |
|
Attack radar |
|
0.60 |
|
Rocket engine |
|
0.46 |
|
Afterburning turbojet |
|
0.35 |
|
Complex hydromechanical system |
|
0.60 |
|
Aircraft generator |
|
0.38 |
|
Modern dry turbojet |
|
0.48 |
A complex system's reliability growth is being monitored and the data set is given in Table 4.2. Do the following:
Plot the cumulative MTBF growth curve.
Write the equation of this growth curve.
Write the equation of the instantaneous MTBF growth model.
Plot the instantaneous MTBF growth curve.
Table 4.2 - Cumulative test hours and the corresponding observed failures for the complex system of Example 1
|
Given the data in the second and third columns
of Table 4.2, the cumulative MTBF, 
, values
are calculated in the fourth column. The information in the second and
fourth columns is then plotted. Figure 4.2 shows the cumulative MTBF while
Figure 4.3 shows the instantaneous MTBF. It can be seen that a straight
line represents the MTBF growth very well on log-log scales.
Figure 4.2: Cumulative MTBF plot for Example 1
Figure 4.3: Instantaneous MTBF plot for Example 1
By changing the x-axis scaling, you are able to extend the line to T = 1. You can get the value of b from the graph by positioning the cursor at the point where the line meets the y-axis. Then read the value of the y-coordinate position at the bottom left corner. In this case, b is approximately 14 hr. Figure 4.4 illustrates this.
Figure 4.4: Cumulative
MTBF plot for b 
14 hr at T = 1
Figure 4.5: Cumulative and Instantaneous MTBF vs. Time plot
Another way of determining b is to calculate by using two points on the fitted straight line and substituting
the corresponding 
and T values into:
(9)
Then substitute this and choose a set of values for 
and T1
into Eqn. 4
and solve for b. The slope of the line, , may also be found from Eqn. 8 or from:
(10)
Using the plot in Figure 4.2, at T1
= 200 hr, 
hr, and at T2 =
3,500 hr, 
hr; and b
= 14 hr when T = 1
from Figure 4.4. Substituting the first set of values, b = 14 hr and ln 1
= 0, into Eqn. 9 yields:
Substituting the second set of values, b = 14 hr and ln 1 = 0, into Eqn. 9 yields:
Averaging these two a values yields a better estimate
of 
.
Now the equation for the cumulative MTBF growth curve is:
(11)
The equation for the instantaneous MTBF growth curve using Eqn. 6 is:
(12)
Eqn. 12 is plotted in Figures 4.3 and 4.5. In Figure 4.5, you can see
that a parallel shift upward of the cumulative MTBF, 
, line by a distance of 
gives
the instantaneous MTBF, or the 
, line.
The parameters can also be estimated using a mathematical approach. To do this, apply least squares analysis on Eqn. 8:
(13)
And for simplicity in the calculations, let:
Therefore, Eqn. 13 becomes:
Assume that a set of data pairs (X1, Y1),
(X2, Y2),...,
(XN, YN) were obtained and plotted.
Then according to the Least Squares
Principle, which minimizes the vertical distance between the data
points and the straight line fitted to the data, the best fitting straight
line to this data set is the straight line 
such that:
And where 
and 
are the least squares estimates of a and c.
To obtain 
and 
, let:
Differentiating F with respect to a and c yields:
(14)
and:
(15)
Set Eqns. 14 and 15 equal to zero:
and:
Solve the equations simultaneously:
and:
Now substituting back, in terms of 
ln
Ti,
ln b and , yields:
(16)
where:
(17)
Using the data from Table 4.2, estimate the parameters of the MTBF model using least squares.
From Table 4.2:
From Eqn. 17:
Also from Eqn. 16:
Therefore, Eqn. 4:
(18)
The equation for the instantaneous MTBF growth curve using Eqn. 6 is:
For the data given in columns 1 and 2 of Table 4.3, estimate the Duane parameters using least squares.
Table 4.3 - Failure Times data
|
Failure Number |
Failure Time (hr) |
ln (Ti) |
ln (Ti)2 |
mc |
|