Template:Test-find-test rga
Test-Find-Test
Test-find-test is the case where all corrective actions are delayed until after the test. Therefore, there are no BC modes when analyzing test-find-test data. This scenario is also called the Crow-AMSAA Projection model, but for the purposes of RGA 7 it is simply a special case of the Crow Extended model.
Suppose a system is subjected to development testing for a period of time, [math]\displaystyle{ T }[/math] . The system can be considered as consisting of two types of failure modes: A modes and BD modes. It is assumed that all BD modes are in series and fail independently according to the exponential distribution. Also assume that the rate of occurrence of A modes follows an exponential distribution with failure intensity [math]\displaystyle{ {{\lambda }_{A}} }[/math] . The system MTBF is constant throughout the test phase since all of the corrective actions are delayed until after the completion of the test. After the delayed fixes have been implemented, the system MTBF will then jump to a higher value.
Let [math]\displaystyle{ K }[/math] denote the total number of BD modes in the system and let [math]\displaystyle{ {{\lambda }_{i}} }[/math] denote the failure intensity for the [math]\displaystyle{ {{i}^{th}} }[/math] BD mode, such that [math]\displaystyle{ i }[/math] = [math]\displaystyle{ 1,2,\ldots ,K }[/math] . Then, at time equal to zero, the system failure intensity [math]\displaystyle{ r(0) }[/math] is:
- [math]\displaystyle{ r(0)={{\lambda }_{A}}+{{\lambda }_{BD}} }[/math]
where:
- [math]\displaystyle{ {{\lambda }_{BD}}=\underset{i=1}{\overset{K}{\mathop{\sum }}}\,{{\lambda }_{i}} }[/math] .
During the test [math]\displaystyle{ (0,T) }[/math] , a random number of [math]\displaystyle{ M }[/math] distinct BD modes will be observed, such that [math]\displaystyle{ M\le K }[/math] . Denote the effectiveness factor (EF) for the [math]\displaystyle{ {{i}^{th}} }[/math] BD mode as [math]\displaystyle{ {{d}_{i}} }[/math] , [math]\displaystyle{ i }[/math] = [math]\displaystyle{ 1,2,\ldots ,K }[/math] . The effectiveness factor [math]\displaystyle{ {{d}_{i}} }[/math] is the percent decrease in [math]\displaystyle{ {{\lambda }_{i}} }[/math] after a corrective action has been made for the [math]\displaystyle{ {{i}^{th}} }[/math] BD mode. That is, the corrective action for the [math]\displaystyle{ {{i}^{th}} }[/math] BD mode removes [math]\displaystyle{ 100\times {{d}_{i}} }[/math] percent of the failure rate and [math]\displaystyle{ 100\times (1-{{d}_{i}}) }[/math] percent remains. The failure intensity for the [math]\displaystyle{ {{i}^{th}} }[/math] BD failure mode after a corrective action is [math]\displaystyle{ (1-{{d}_{i}}){{\lambda }_{i}} }[/math] . If corrective actions are taken on the [math]\displaystyle{ M }[/math] BD modes observed by time [math]\displaystyle{ T }[/math] , then the system failure intensity is reduced from [math]\displaystyle{ r(0) }[/math] to:
- [math]\displaystyle{ \begin{align} & r\left( T \right)= & {{\lambda }_{A}}+\underset{i=1}{\overset{M}{\mathop \sum }}\,\left( 1-{{d}_{i}} \right){{\lambda }_{i}}+({{\lambda }_{BD}}-\underset{i=1}{\overset{M}{\mathop \sum }}\,{{\lambda }_{i}}) \\ & = & {{\lambda }_{A}}+{{\lambda }_{BD}}-\underset{i=1}{\overset{M}{\mathop \sum }}\,{{d}_{i}}{{\lambda }_{i}} \end{align} }[/math]
where:
- • [math]\displaystyle{ \underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}}){{\lambda }_{i}} }[/math] is the failure intensity for the [math]\displaystyle{ M }[/math] modes after the corrective actions
- • [math]\displaystyle{ ({{\lambda }_{BD}}-\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{\lambda }_{i}}) }[/math] is the remaining failure intensity for all unseen BD modes
All [math]\displaystyle{ M }[/math] BD modes observed by test time [math]\displaystyle{ T }[/math] may not be fixed by time [math]\displaystyle{ T }[/math] so the actual failure intensity at time [math]\displaystyle{ T }[/math] may not be [math]\displaystyle{ r(T) }[/math] . However, [math]\displaystyle{ r(T) }[/math] can be viewed as the achieved failure intensity at time [math]\displaystyle{ T }[/math] if all fixes were updated and incorporated into the system. All of the fixes for the BD modes found during the test are incorporated as delayed fixes at the end of the test phase. Therefore, the system failure intensity is constant at [math]\displaystyle{ r(0)={{\lambda }_{A}}+{{\lambda }_{BD}} }[/math] through the test phase and will then jump to a lower value [math]\displaystyle{ r(T) }[/math] after the delayed fixes have been implemented. Let [math]\displaystyle{ {{N}_{A}} }[/math] and [math]\displaystyle{ {{N}_{BD}} }[/math] be the total number of A and BD failures observed during the test [math]\displaystyle{ (0,T) }[/math] and let [math]\displaystyle{ N={{N}_{A}}+{{N}_{BD}} }[/math] . In addition, there are [math]\displaystyle{ M }[/math] distinct BD modes observed during the test. After implementing the [math]\displaystyle{ M }[/math] fixes, the failure intensity for the system at time [math]\displaystyle{ T }[/math] (after the jump) is given by the function [math]\displaystyle{ r(T) }[/math] .
[math]\displaystyle{ r(0) }[/math] is actually the demonstrated failure intensity, which is based on actual system performance of the hardware tested and not of some future configuration. A demonstrated reliability value should be determined at the end of each test phase. The demonstrated failure intensity is:
- [math]\displaystyle{ {{\widehat{\lambda }}_{D}}(T)=r(0)=\frac{{{N}_{A}}+{{N}_{BD}}}{T} }[/math]
The demonstrated MTBF is given by:
- [math]\displaystyle{ M\widehat{T}B{{F}_{D}}={{[{{\widehat{\lambda }}_{D}}(T)]}^{-1}} }[/math]
The detailed procedure for estimating [math]\displaystyle{ r(T) }[/math] is given in Crow, L.H., An Extended Reliability Growth Model for Managing and Assessing Corective Actions and is reviewed here.
Let [math]\displaystyle{ E[\cdot ] }[/math] denote the expected value:
- [math]\displaystyle{ E[r(T)]={{\lambda }_{A}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,(1-{{d}_{i}}){{\lambda }_{i}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,{{d}_{i}}{{\lambda }_{i}}{{e}^{-{{\lambda }_{i}}T}} }[/math]
Under realistic assumptions [math]\displaystyle{ E[r(T)] }[/math] also may be expressed as:
- [math]\displaystyle{ E[r(T)]={{\lambda }_{A}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,(1-{{d}_{i}}){{\lambda }_{i}}+\overline{d}h(T) }[/math]
where [math]\displaystyle{ \overline{d} }[/math] is the mean effectiveness factor and [math]\displaystyle{ h(T) }[/math] is the instantaneous rate at which a new BD mode will occur at time [math]\displaystyle{ T }[/math] . The maximum likelihood estimate for the [math]\displaystyle{ h(T) }[/math] is:
- [math]\displaystyle{ h(T)={{\lambda }_{BD}}{{\beta }_{BD}}{{T}^{{{\beta }_{BD}}-1}} }[/math]
And, [math]\displaystyle{ \overline{d}h(T) }[/math] is the bias term, such that:
- [math]\displaystyle{ B(T)=\overline{d}h(T) }[/math]
Estimation of Bias Term
Let [math]\displaystyle{ {{X}_{1}}\lt {{X}_{2}}\lt \ldots \lt {{X}_{M}}\lt T }[/math] denote the cumulative test times for the first occurrences of BD modes. Then, the maximum likelihood estimates of [math]\displaystyle{ {{\lambda }_{BD}} }[/math] and [math]\displaystyle{ {{\beta }_{BD}} }[/math] are:
- [math]\displaystyle{ {{\hat{\beta }}_{BD}}=\frac{M}{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln \left( \tfrac{T}{{{X}_{i}}} \right)} }[/math]
- [math]\displaystyle{ {{\hat{\lambda }}_{BD}}=\frac{M}{{{T}^{{{{\hat{\beta }}}_{BD}}}}} }[/math]
The intensity function [math]\displaystyle{ h(t) }[/math] for [math]\displaystyle{ t\gt 0 }[/math] is estimated by:
- [math]\displaystyle{ h(t)={{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{t}^{{{\widehat{\beta }}_{BD}}-1}} }[/math]
In particular, the maximum likelihood estimate for the rate of occurrence for the distinct BD modes at time [math]\displaystyle{ T }[/math] is:
- [math]\displaystyle{ \begin{align} & \widehat{h}(T)= & {{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{T}^{{{\widehat{\beta }}_{BD}}-1}} \\ & = & \frac{M{{\widehat{\beta }}_{BD}}}{T} \end{align} }[/math]
Furthermore, the maximum likelihood estimate of the bias term [math]\displaystyle{ B(T) }[/math] is given by:
- [math]\displaystyle{ B(T)=\overline{d}\frac{M{{\widehat{\beta }}_{BD}}}{T} }[/math]
The unbiased estimate of [math]\displaystyle{ {{\beta }_{BD}} }[/math] is:
- [math]\displaystyle{ {{\bar{\beta }}_{BD}}=\frac{M-1}{M}{{\hat{\beta }}_{BD}} }[/math]
Thus the unbiased estimate of the bias term is given by:
- [math]\displaystyle{ B(T)=\overline{d}\frac{M{{{\bar{\beta }}}_{BD}}}{T} }[/math]
The mean [math]\displaystyle{ \overline{d} }[/math] is given by:
- [math]\displaystyle{ \overline{d}=\frac{1}{M}\underset{i=1}{\overset{M}{\mathop \sum }}\,{{d}_{i}} }[/math]
Therefore, the projected failure intensity [math]\displaystyle{ r(T) }[/math] is then estimated at the end of the test phase by:
- [math]\displaystyle{ \widehat{r}(T)=\left( \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right)+\overline{d}\left( \frac{M}{T}{{\overline{\beta }}_{BD}} \right) }[/math]
The projected MTBF is:
- [math]\displaystyle{ M\widehat{T}B{{F}_{P}}={{[r(T)]}^{-1}} }[/math]
Reliability Growth Potential
The failure intensity [math]\displaystyle{ r(T) }[/math] will depend on the management strategy that determines the classification of the A and BD failure modes. The engineering effort applied to the corrective actions determines the effectiveness factors. In addition, [math]\displaystyle{ r(T) }[/math] depends on [math]\displaystyle{ h(t) }[/math] , which is the rate at which problem failure modes are being seen during testing. [math]\displaystyle{ h(t) }[/math] drives the opportunity to take corrective actions based on the seen failure modes and it is an important factor in the overall reliability growth rate. The reliability growth potential is the limiting value of [math]\displaystyle{ r(T) }[/math] as [math]\displaystyle{ T }[/math] increases. This limit is the maximum MTBF that can be attained with the current management strategy. The maximum MTBF will be attained when all [math]\displaystyle{ K }[/math] BD modes have been observed and fixed with EFs [math]\displaystyle{ {{d}_{i}} }[/math] . In terms of failure intensity, the growth potential is expressed by the following equation:
- [math]\displaystyle{ {{r}_{GP}}={{\lambda }_{A}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,(1-{{d}_{i}}){{\lambda }_{i}} }[/math]
In terms of the MTBF, the growth potential is given by:
- [math]\displaystyle{ MTB{{F}_{GP}}=1/{{r}_{GP}} }[/math]
The procedure for estimating the growth potential is as follows. Suppose that the system is tested for a period of time [math]\displaystyle{ T }[/math] and that [math]\displaystyle{ N }[/math] failures have been observed. According to the management strategy, [math]\displaystyle{ {{N}_{A}} }[/math] of these failures are A modes and [math]\displaystyle{ {{N}_{BD}} }[/math] of these failures are BD modes. For the BD modes, there will be [math]\displaystyle{ M }[/math] distinct fixes. As before, [math]\displaystyle{ {{N}_{i}} }[/math] is the total number of failures for the [math]\displaystyle{ {{i}^{th}} }[/math] BD mode and [math]\displaystyle{ {{d}_{i}} }[/math] is the corresponding assigned EF. From this data, the growth potential failure intensity is estimated by:
- [math]\displaystyle{ {{\widehat{r}}_{GP}}(T)=\left( \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right) }[/math]
The growth potential MTBF is estimated by:
- [math]\displaystyle{ M\widehat{T}B{{F}_{GP}}={{[{{\widehat{r}}_{GP}}]}^{-1}} }[/math]
Example 1
Consider the data in Table 9.1. A system was tested for [math]\displaystyle{ T=400 }[/math] hours. There were a total of [math]\displaystyle{ N=42 }[/math] failures and all corrective actions will be delayed until after the end of the 400 hour test. Each failure has been designated as either an A failure mode (the cause will not receive a corrective action) or a BD mode (the cause will receive a corrective action). There are [math]\displaystyle{ {{N}_{A}}=10 }[/math] A mode failures and [math]\displaystyle{ {{N}_{BD}}=32 }[/math] BD mode failures. In addition, there are [math]\displaystyle{ M=16 }[/math] distinct BD failure modes, which means 16 distinct corrective actions will be incorporated into the system at the end of test. The total number of failures for the [math]\displaystyle{ {{j}^{th}} }[/math] observed distinct BD mode is denoted by [math]\displaystyle{ {{N}_{j}} }[/math] and the total number of BD failures during the test is [math]\displaystyle{ {{N}_{BD}}=\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{N}_{j}} }[/math] . These values and effectiveness factors are given in Table 9.2.
| Table 9.1 - Test-find-test data | ||||||
| [math]\displaystyle{ i }[/math] | [math]\displaystyle{ {{X}_{i}} }[/math] | Mode | [math]\displaystyle{ i }[/math] | [math]\displaystyle{ {{X}_{i}} }[/math] | Mode | |
|---|---|---|---|---|---|---|
| 1 | 15 | BD1 | 22 | 260.1 | BD1 | |
| 2 | 25.3 | BD2 | 23 | 263.5 | BD8 | |
| 3 | 47.5 | BD3 | 24 | 273.1 | A | |
| 4 | 54 | BD4 | 25 | 274.7 | BD6 | |
| 5 | 56.4 | BD5 | 26 | 285 | BD13 | |
| 6 | 63.6 | A | 27 | 304 | BD9 | |
| 7 | 72.2 | BD5 | 28 | 315.4 | BD4 | |
| 8 | 99.6 | BD6 | 29 | 317.1 | A | |
| 9 | 100.3 | BD7 | 30 | 320.6 | A | |
| 10 | 102.5 | A | 31 | 324.5 | BD12 | |
| 11 | 112 | BD8 | 32 | 324.9 | BD10 | |
| 12 | 120.9 | BD2 | 33 | 342 | BD5 | |
| 13 | 125.5 | BD9 | 34 | 350.2 | BD3 | |
| 14 | 133.4 | BD10 | 35 | 364.6 | BD10 | |
| 15 | 164.7 | BD9 | 36 | 364.9 | A | |
| 16 | 177.4 | BD10 | 37 | 366.3 | BD2 | |
| 17 | 192.7 | BD11 | 38 | 373 | BD8 | |
| 18 | 213 | A | 39 | 379.4 | BD14 | |
| 19 | 244.8 | A | 40 | 389 | BD15 | |
| 20 | 249 | BD12 | 41 | 394.9 | A | |
| 21 | 250.8 | A | 42 | 395.2 | BD16 | |
| Table 9.2 - Effectiveness factor for each unique BD mode | |||
| BD Mode | Number [math]\displaystyle{ {{N}_{j}} }[/math] | First Occurrence | EF [math]\displaystyle{ {{d}_{i}} }[/math] |
|---|---|---|---|
| 1 | 2 | 15.0 | .67 |
| 2 | 3 | 25.3 | .72 |
| 3 | 2 | 47.5 | .77 |
| 4 | 2 | 54.0 | .77 |
| 5 | 3 | 54.0 | .87 |
| 6 | 2 | 99.6 | .92 |
| 7 | 1 | 100.3 | .50 |
| 8 | 3 | 112.0 | .85 |
| 9 | 3 | 125.5 | .89 |
| 10 | 4 | 133.4 | .74 |
| 11 | 1 | 192.7 | .70 |
| 12 | 2 | 249.0 | .63 |
| 13 | 1 | 285.0 | .64 |
| 14 | 1 | 379.4 | .72 |
| 15 | 1 | 389.0 | .69 |
| 16 | 1 | 395.2 | .46 |
- 1) Determine the projected MTBF and failure intensity.
- 2) Determine the growth potential MTBF and failure intensity.
- 3) Determine the demonstrated MTBF and failure intensity.
Solution
- 1) From Eqns. (Extend1) and (Extend2), the maximum likelihood estimates of [math]\displaystyle{ {{\beta }_{BD}} }[/math] and [math]\displaystyle{ {{\lambda }_{BD}} }[/math] are determined to be:
- [math]\displaystyle{ \begin{align} & {{{\hat{\beta }}}_{BD}}= & \frac{M}{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln (\tfrac{T}{{{X}_{i}}})} \\ & = & 0.7970 \\ & {{{\hat{\lambda }}}_{BD}}= & 0.1350 \end{align} }[/math]
The unbiased estimate of [math]\displaystyle{ \beta }[/math] is:
- [math]\displaystyle{ \begin{align} & {{\overline{\beta }}_{BD}}= & \frac{M-1}{M}{{{\hat{\beta }}}_{BD}} \\ & = & 0.7472 \end{align} }[/math]
Based on the data in Table 9.2, [math]\displaystyle{ \overline{d}=\tfrac{1}{M}\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{d}_{i}}= }[/math] [math]\displaystyle{ 0.72125 }[/math] . Therefore, [math]\displaystyle{ B(T)=\overline{d}\tfrac{M{{\overline{\beta }}_{BD}}}{T}=0.0215 }[/math] . From Eqn. (extended), the projected failure intensity due to incorporating the 16 corrective actions is:
- [math]\displaystyle{ \begin{align} & r(T)= & \left( \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right)+\overline{d}\left( \frac{M}{T}{{\overline{\beta }}_{BD}} \right) \\ & = & 0.0661 \end{align} }[/math]
The projected MTBF is:
- [math]\displaystyle{ M\widehat{T}B{{F}_{P}}={{[r(T)]}^{-1}}=15.127 }[/math]
- 2) To estimate the maximum reliability that can be attained with this management strategy, use the following calculations.
- [math]\displaystyle{ {{N}_{A}}/T=0.0250 }[/math]
- [math]\displaystyle{ \frac{1}{T}\underset{i=1}{\overset{16}{\mathop \sum }}\,(1-{{d}_{i}}){{N}_{i}}=0.0196 }[/math]
From Eqn. (extendedGP), the growth potential failure intensity is estimated by:
- [math]\displaystyle{ \begin{align} & {{\widehat{r}}_{GP}}(T)= & \left( \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right) \\ & = & 0.0250+0.0196 \\ & = & 0.0446 \end{align} }[/math]
The growth potential MTBF is:
- [math]\displaystyle{ M\widehat{T}B{{F}_{GP}}={{[{{\widehat{r}}_{GP}}]}^{-1}}=22.4467 }[/math]
- 3) From Eqn. (extendeddfi), the demonstrated failure intensity and MTBF are estimated by:
- [math]\displaystyle{ \begin{align} & {{\widehat{\lambda }}_{D}}(T)= & \frac{{{N}_{A}}+{{N}_{BD}}}{T} \\ & = & \frac{42}{400} \\ & = & 0.1050 \end{align} }[/math]
- [math]\displaystyle{ \begin{align} & M\widehat{T}B{{F}_{D}}= & {{[{{\widehat{\lambda }}_{D}}(T)]}^{-1}} \\ & = & 9.5238 \end{align} }[/math]
Figure extendedpic1 shows the demonstrated, projected and growth potential MTBF. Figure extendedpic2 shows the demonstrated, projected and growth potential failure intensity.