Template:Growth potential and projections crow rga

Growth Potential and Projections

The failure intensity left in the system will depend on the management strategy that determines the classification of the A, BC and BD failure modes. The engineering effort applied to the corrective actions determines the effectiveness factors. In addition, the failure intensity depends on [math]\displaystyle{ h(t) }[/math] , which is the rate at which unique BD failure modes are being discovered during testing. The rate of discovery 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 the failure intensity as time [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 BD modes have been observed and fixed.
If all seen BD modes are corrected by time [math]\displaystyle{ T }[/math] , that is, no deferred corrective actions at time [math]\displaystyle{ T }[/math] , then the Growth Potential is the maximum attainable based on the Type BD designation of the failure modes, the corresponding assigned effectiveness factors and the remaining A modes in the system. This is called the Nominal Growth Potential. If some seen BD modes are not corrected at the end of the current test phase then the prevailing growth potential is below the maximum attainable with the Type BD designation of the failure modes and the corresponding assigned effectiveness factors. The Crow-AMSAA (NHPP) model is used to estimate the current demonstrated MTBF or [math]\displaystyle{ MTB{{F}_{D}}. }[/math] The demonstrated MTBF does not take into account any type of projected improvements. Refer to Chapter 5 for more details on the Crow-AMSAA (NHPP) model. The corresponding current demonstrated failure intensity is:

[math]\displaystyle{ {{\lambda }_{D}}=\lambda \beta {{T}^{\beta -1}} }[/math]

or:

[math]\displaystyle{ {{\lambda }_{D}}=\frac{1}{MTB{{F}_{D}}} }[/math]

The nominal growth potential factor is:

[math]\displaystyle{ {{\lambda }_{NGPFactor}}=\underset{i=1}{\overset{M}{\mathop \sum }}\,\left( 1-{{d}_{Ni}} \right)\frac{{{N}_{i}}}{T} }[/math]

where:

• [math]\displaystyle{ M }[/math] is the total number of distinct unfixed BD modes at time [math]\displaystyle{ {{T}_{j}} }[/math] .

• [math]\displaystyle{ {{d}_{Ni}} }[/math] is the assigned (nominal) EF for the [math]\displaystyle{ {{i}^{th}} }[/math] unfixed BD mode at time [math]\displaystyle{ {{T}_{j}}. }[/math]

• [math]\displaystyle{ {{N}_{i}} }[/math] is the total number of failures over (0, [math]\displaystyle{ {{T}_{j}} }[/math] ) for the distinct unfixed BD mode [math]\displaystyle{ i }[/math] .

The nominal growth potential factor signifies the failure intensity of the [math]\displaystyle{ M }[/math] modes after corrective actions have been implemented for them, using the nominal values for the effectiveness factors.
Similarly, the actual growth potential factor is:

[math]\displaystyle{ {{\lambda }_{AGPFactor}}=\underset{i=1}{\overset{M}{\mathop \sum }}\,\left( 1-{{d}_{Ai}} \right)\frac{{{N}_{i}}}{T} }[/math]

where [math]\displaystyle{ {{d}_{Ai}} }[/math] is the actual EF for the [math]\displaystyle{ {{i}^{th}} }[/math] unfixed BD mode at time [math]\displaystyle{ {{T}_{j}}. }[/math]
The actual growth potential factor signifies the failure intensity of the [math]\displaystyle{ M }[/math] modes after corrective actions have been implemented for them, using the actual values for the effectiveness factors.
Based on the definition of BD modes for the Crow Extended - Continuous Evaluation model, the estimate of [math]\displaystyle{ p }[/math] at time [math]\displaystyle{ {{T}_{j\text{ }}} }[/math] is calculated as follows:

[math]\displaystyle{ p=\frac{\text{Total number of distinct unfixed BD modes at time }{{T}_{j}}}{\text{Total number of distinct BD modes at time }{{T}_{j}}\text{ (both fixed and unfixed)}} }[/math]

The unfixed BD mode failure intensity at time [math]\displaystyle{ {{T}_{j}} }[/math] is:

[math]\displaystyle{ \lambda_{BDunfixed} = \frac{\text{Total number of unfixed BD failures at time} T_j}{T_j} }[/math]

Similar to the Crow Extended model, the discovery function at time [math]\displaystyle{ T }[/math] for the Crow Extended - Continuous Evaluation model is calculated using all the first occurrences of the all the BD modes, both fixed and unfixed. [math]\displaystyle{ h(t) }[/math] is the unseen BD mode failure intensity and is also the rate at which new unique BD modes are being discovered.

[math]\displaystyle{ \begin{align} & \widehat{h}(T|BD)= & {{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{T}^{{{\widehat{\beta }}_{BD}}-1}} \\ & = & \frac{M{{\widehat{\beta }}_{BD}}}{T} \end{align} }[/math]

where:

• [math]\displaystyle{ {{\widehat{\beta }}_{BD}} }[/math] is the unbiased estimated of [math]\displaystyle{ \beta }[/math] for the Crow-AMSAA (NHPP) model based on the first occurrence of [math]\displaystyle{ M }[/math] distinct BD modes.

• [math]\displaystyle{ {{\widehat{\lambda }}_{BD}} }[/math] is the unbiased estimated of [math]\displaystyle{ \lambda }[/math] for the Crow-AMSAA (NHPP) model based on the first occurrence of [math]\displaystyle{ M }[/math] distinct BD modes.

[math]\displaystyle{ {{\widehat{\beta }}_{BD}} }[/math] and [math]\displaystyle{ {{\widehat{\lambda }}_{BD}} }[/math] are also known as the Rate of Discovery Parameters.
The nominal growth potential failure intensity is:

[math]\displaystyle{ \lambda_{NGP} = \lambda_D - \lambda_{BD\text{unfixed}} + \lambda_{NGP\text{Factor}} + d_N\cdot p\cdot h(T) - d_N h(T) }[/math]

and the nominal growth potential MTBF is:

[math]\displaystyle{ MTB{{F}_{NGP}}=\frac{1}{{{\lambda }_{NGP}}} }[/math]

The nominal projected failure intensity at time [math]\displaystyle{ T }[/math] is:

[math]\displaystyle{ {{\lambda }_{NP}}={{\lambda }_{NGP}}+{{d}_{N}}h(T) }[/math]

and the nominal projected MTBF at time [math]\displaystyle{ T }[/math] is:

[math]\displaystyle{ MTB{{F}_{NP}}=\frac{1}{{{\lambda }_{NP}}} }[/math]

The actual growth potential failure intensity is:

[math]\displaystyle{ \lambda{AGP} = \lambda_D - \lambda_{BD\text{unfixed}} + \lambda{AGP\text{Factor} + d_A\cdot p\cdot h(T) - d_A h(T)} }[/math]

and the actual growth potential MTBF is:

[math]\displaystyle{ MTB{{F}_{AGP}}=\frac{1}{{{\lambda }_{AGP}}} }[/math]

The actual projected failure intensity at time is:

[math]\displaystyle{ {{\lambda }_{AP}}={{\lambda }_{AGP}}+{{d}_{A}}\cdot h\left( T \right) }[/math]

and the actual projected MTBF at time is:

[math]\displaystyle{ MTB{{F}_{AP}}=\frac{1}{{{\lambda }_{AP}}} }[/math]

In terms of confidence intervals and goodness-of-fit tests, the calculations are the same as for the Crow Extended model, described in Chapter 9.