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Lloyd and Lipow (1962) considered a situation in which a test program is conducted in  <math>N</math>  stages. Each stage consists of a certain number of trials of an item undergoing testing and the data set is recorded as successes or failures. All tests in a given stage of testing involve similar items. The results of each stage of testing are used to improve the item for further testing in the next stage. For the  <math>{{k}^{th}}</math>  group of data, taken in chronological order, there are  <math>{{n}_{k}}</math>  tests with  <math>{{S}_{k}}</math>  observed successes. The reliability growth function is then [6]:
#REDIRECT [[Lloyd-Lipow]]
 
::<math>{{R}_{k}}={{R}_{\infty }}-\frac{\alpha }{k}</math>
 
where:
 
:<math>R_k =</math> the actual reliability during the <math>k^{th}</math> stage of testing
 
:<math>R_{\infty} =</math> the ultimate reliability attained if <math>k\to{\infty}</math>
 
:<math>\alpha>0 =</math> modifies the rate of growth
 
Note that essentially,  <math>{{R}_{k}}=\tfrac{{{S}_{k}}}{{{n}_{k}}}</math> . If the data set consists of reliability data, then  <math>{{S}_{k}}</math>  is assumed to be the observed reliability given and  <math>{{n}_{k}}</math>  is considered 1.
<br>
{{parameter estimation llyd-lipow}}
 
{{confidence bounds lloyd-l}}
 
==General Examples==
===Example 4===
A 15-stage reliability development test program was performed. The grouped per configuration data set that was obtained is given in Table 6.3. Do the following:
<br>
<br>
:1) Fit the Lloyd-Lipow model to the data using MLE.
:2) What is the maximum reliability attained as the number of test stages approaches infinity?
:3) What is the maximum achievable reliability with a 90% confidence level?
<br>
{|style= align="center" border="2"
|+'''Table 6.3 - Grouped per Configuration data for Example 4'''
!Stage, <math>k</math>
!Number of Tests (<math>n_k</math>)
!Number of Successes (<math>S_k</math>)
|-
|1|| 10|| 3
|-
|2|| 10|| 3
|-
|3|| 10|| 4
|-
|4|| 10|| 5
|-
|5|| 10|| 5
|-
|6|| 12|| 6
|-
|7|| 12|| 5
|-
|8|| 12|| 7
|-
|9|| 14|| 8
|-
|10|| 14|| 8
|-
|11|| 14|| 10
|-
|12|| 14|| 12
|-
|13|| 14|| 11
|-
|14|| 14|| 12
|-
|15|| 14|| 12
|}
<br>
====Solution to Example 4====
<br>
<br>
:1) Figure figLLSe11 displays the entered data and the estimated Lloyd-Lipow parameters.
:2) The maximum achievable reliability as the number of test stages approaches infinity is equal to the value of  <math>R</math> . Therefore,  <math>R=0.7157</math> .
:3) The maximum achievable reliability with a 90% confidence level can be estimated by viewing the confidence bounds on the parameters in the QCP, as shown in Figure QCPex4. The lower bound on the value of  <math>R</math>  is equal to  <math>0.6691</math> .
<math></math>
[[Image:rga6.4.png|thumb|center|400px|Estimated Lloyd-Lipow parameters using MLE.]]
<br>
<br>
[[Image:rga6.5.png|thumb|center|400px|Confidence bounds on the parameters.]]
<br>
 
===Example 5===
<br>
Given the reliability data in Table 6.4, do the following:
<br>
<br>
:1) Fit the Lloyd-Lipow model to the data using least squares analysis.
:2) Plot the Lloyd-Lipow reliability with 90% 2-sided confidence bounds.
:3) Determine how many months of testing are required to achieve a reliability goal of 90%.
:4) Determine what is the attainable reliability if the maximum duration of testing is 30 months.
 
<br>
{|style= align="center" border="2"
|+'''Table 6.4 - Reliability data'''
!Time(months)
!Reliability(%)
|-
|1|| 33.35
|-
|2|| 42.50
|-
|3|| 58.02
|-
|4|| 68.50
|-
|5|| 74.20
|-
|6|| 80.00
|-
|7|| 82.30
|-
|8|| 89.50
|-
|9|| 91.00
|-
|10|| 92.10
|}
 
<br>
<br>
====Solution to Example 5====
<br>
<br>
:1) Figure figLLSe21 displays the estimated parameters.
:2) Figure figLLSe22 displays Reliability vs. Time plot with 90% 2-sided confidence bounds.
:3) Figure figLLSe23 shows the number of months of testing required to achieve a reliability goal of 90%.
:4) Figure figLLSe24 displays the reliability achieved after 30 months of testing.
 
<br>
[[Image:rga6.6.png|thumb|center|300px|Estimated Lloyd-Lipow parameters using least squares.]]
<br>
[[Image:rga6.7.png|thumb|center|300px|Reliability vs. Time plot with 90% 2-sided confidence bounds.]]
<br>
[[Image:rga6.8.png|thumb|center|300px|Number of months required to achieve a reliability of 90%.]]
<br>
[[Image:rga6.9.png|thumb|center|300px|Maximum attainable reliability for a testing duration of 30 months.]]
<br>
 
===Example 6===
Find the Lloyd-Lipow model that represents the data in Table 6.5 using MLE and plot it along with 95% 2-sided confidence bounds. Does the model follow the data?
 
<br>
{|style= align="center" border="2"
|+'''Table 6.5 - Sequential data'''
!Run Number
!Result
|-
|1|| F
|-
|2|| F
|-
|3|| S
|-
|4|| S
|-
|5|| S
|-
|6|| F
|-
|7|| S
|-
|8|| F
|-
|9|| F
|-
|10|| S
|-
|11|| S
|-
|12|| S
|-
|13|| F
|-
|14|| S
|-
|15|| S
|-
|16|| S
|-
|17|| S
|-
|18|| S
|-
|19|| S
|-
|20|| S
|}
<math></math>
[[Image:rga6.10.png|thumb|center|400px|Estimated Lloyd-Lipow parameters using MLE]]
<br>
<br>
 
====Solution to Example====
Figures figLLSe41 and figLLSe42 demonstrate the solution. As it can be seen from the plot in Figure figLLSe42, the model does not seem to follow the data. You may want to consider another model for this data set.
<br>
<br>
[[Image:rga6.11.png|thumb|center|300px|Reliability vs. Time plot with 95% 2-sided confidence bounds.]]
<br>
 
===Example 7===
Find the Lloyd-Lipow model that represents the data in Table 6.6 using least squares. This data set includes information about the failure mode that was responsible for each failure, so that the probability of each failure mode reccurring is taken into account in the analysis.
<br>
Table 6.6 - Sequential with mode data
<br>
{|style= align="center" border="1"
!Run Number
!Result
!Mode
|-
|1|| S||
|-
|2|| F|| 1
|-
|3|| F|| 2
|-
|4|| F|| 3
|-
|5|| S||
|-
|6|| S||
|-
|7|| S||
|-
|8|| F|| 3
|-
|9|| F|| 2
|-
|10|| S||
|-
|11|| F|| 2
|-
|12|| S||
|-
|13|| S||
|-
|14|| S||
|-
|15|| S||
|}
 
 
====Solution to Example 7====
Figure figLLSe51 shows the analysis.
<br>
 
[[Image:rga6.12.png|thumb|center|400px|Estimated Lloyd-Lipow parameters using least squares.]]

Latest revision as of 02:30, 27 August 2012

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