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Reliability Basics | ||||||||||||||||

Analyzing Growth Success/Failure and
Reliability Data Using the Lloyd-Lipow Model
This continues the series of articles, initiated with the
Reliability Basics article in Issue 81 of
A test program is conducted in
where: -
*R*= the estimated reliability during the_{k}*k*^{th}stage of testing. -
*R*= the ultimate reliability that would be attained if the new prototypes were continuously improved and tested (_{∞ }*i.e.*when*k*→ ∞). -
α = is a parameter that modifies that rate of growth.
Note that the actual observed reliability during the
If the data set consists of reliability data, then R'k is the observed reliability.
The
parameters of the model can be estimated using the maximum likelihood method
or the regression method. For
more information about the parameter estimation of the Lloyd-Lipow model,
click
here. The confidence bounds on the parameters and
Example with Success/Failure Data A one-shot system undergoes reliability growth development testing for a total of 68 trials. Delayed corrective actions are incorporated after the 14th, 33rd and 48th trials. From trial 49 to trial 68 the configuration is not changed. The goal of the developmental growth program is to demonstrate that the system achieved an 80% reliability with 90% confidence. The test results are shown next.
The
next figure shows the data set entered in
Note:
The parameters of the Lloyd-Lipow model estimated using the maximum likelihood method are:
α = 0.1725
The following plot shows the reliability growth plot across the different configurations (stages of development).
The
achieved reliability at the end of the 4
The amount of testing and improvements
performed up to this stage did not demonstrate the reliability goal.
However, the ultimate reliability that can be potentially achieved if
improvements continue with the same growth rate, ^{th} new configuration.
Example with Reliability Data This example deals with the estimated reliabilities obtained for 10 stages of the development of an automotive product. The reliability values were calculated from times-to-failure data obtained through an accelerated test. The goal of the reliability growth program is to achieve an 80% reliability with 80% confidence.
The
following figure shows the data entered in
The parameters of the Lloyd-Lipow model estimated using the maximum likelihood method are:
α = 0.6145
The following plot shows the reliability growth plot across the different stages of development.
The achieved reliability at the end of the 10
RGA,
and showed its application to model success/failure data as well as data
sets that consist of reliability values at different times or stages.
Examples were also presented to illustrate how the model is fitted and how
to calculate some metrics useful in demonstrating a reliability goal or
predicting the additional amount of testing needed to achieve the
reliability goal.
1- Lloyd, D. K., Lipow, M., "Reliability Growth Models,"
2- Nelson, Wayne, | ||||||||||||||||

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