Template:Actual idealized growth curve

Actual Idealized Growth Curve

The actual idealized growth curve differs from the nominal idealized curve in that it takes into account the average fix delay that might occur in each test phase. The actual idealized growth curve is continuous and goes through each of the test phase target MTBFs.

Introduction

In developmental reliability growth testing, the objective is to test a system, find problem failure modes, incorporate corrective actions and therefore increase the reliability of the system. This process is continued for the duration of the test time. If the corrective actions are effective then the system mean time between failures (MTBF) or mean trials between failures (MTrBF) will move from an initial low value to a higher value. Typically, the objective of reliability growth testing is not to just increase the MTBF/MTrBF, but to increase it to a particular value called the goal or requirement. Therefore, determining how much test time is needed for a particular system is generally of particular interest in reliability growth testing.

The Duane postulate is based on empirical observations, and it reflects a learning curve pattern for reliability growth. This learning curve pattern forms the basis of the Crow-AMSAA (NHPP) model. The Duane postulate is also reflected in the Crow Extended model in the form of the discovery function [math]\displaystyle{ h(t)\,\! }[/math].

The discovery function is the rate in which new, distinct problems are being discovered during reliability growth development testing. The Crow-AMSAA (NHPP) model is a special case of the discovery function. Consider that when a new and distinct failure mode is first seen, the testing is stopped and a corrective action is incorporated before the testing is resumed. In addition, suppose that the corrective action is highly effective that the failure mode is unlikely to be seen again. In this case, the only failures observed during the reliability growth test are the first occurrences of the failure modes. Therefore, if the Crow-AMSAA (NHPP) model and the Duane postulate are accepted as the pattern for a test-fix-test reliability growth testing program, then the form of the Crow-AMSAA (NHPP) model must be the form for the discovery function, [math]\displaystyle{ h(t)\,\! }[/math].

To be consistent with the Duane postulate and the Crow-AMSAA (NHPP) model, the discovery function must be of the same form. This form of the discovery function is an important property of the Crow extended model and its application in growth planning. As with the Crow-AMSAA (NHPP) model, this form of the discovery function ties the model directly to real-world data and experiences.

Growth Planning Models

There are two types of reliability growth planning models available in RGA:

• Continuous Reliability Growth Planning

• Discrete Reliability Growth Planning

Growth Planning Inputs

The following parameters are used in both the continuous and discrete reliability growth models.

Management Strategy Ratio & Initial Failure Intensity

When a system is tested and failure modes are observed, management can make one of two possible decisions, either to fix or to not fix the failure mode. Therefore, the management strategy places failure modes into two categories: A modes and B modes. The A modes are all failure modes such that, when seen during the test, no corrective action will be taken. This accounts for all modes for which management determines to be not economical or otherwise justified to take a corrective action. The B modes are either corrected during the test or the corrective action is delayed to a later time. The management strategy is defined by what portion of the failures will be fixed.

Let [math]\displaystyle{ {{\lambda }_{I}}\,\! }[/math] be the initial failure intensity of the system in test. [math]\displaystyle{ {{\lambda }_{A}}\,\! }[/math] is defined as the A mode's initial failure intensity and [math]\displaystyle{ {{\lambda }_{B}}\,\! }[/math] is defined as the B mode's initial failure intensity. [math]\displaystyle{ {{\lambda }_{A}}\,\! }[/math] is the failure intensity of the system that will not be addressed by corrective actions even if a failure mode is seen during testing. [math]\displaystyle{ {{\lambda }_{B}}\,\! }[/math] is the failure intensity of the system that will be addressed by corrective actions if a failure mode is seen during testing.

Then, the initial failure intensity of the system is:

[math]\displaystyle{ \begin{align} {{\lambda }_{I}}={{\lambda }_{A}}+{{\lambda }_{B}} \end{align}\,\! }[/math]

The initial system MTBF is:

[math]\displaystyle{ {{M}_{I}}=\frac{1}{{{\lambda }_{I}}}\,\! }[/math]

Based on the initial failure intensity definitions, the management strategy ratio is defined as:

[math]\displaystyle{ msr=\frac{{{\lambda }_{B}}}{{{\lambda }_{A}}+{{\lambda }_{B}}}\,\! }[/math]

The [math]\displaystyle{ msr\,\! }[/math] is the portion of the initial system failure intensity that will be addressed by corrective actions, if seen during the test.

The failure mode intensities of the type A and type B modes are:

[math]\displaystyle{ \begin{align} {{\lambda }_{A}}= & \left( 1-msr \right)\cdot {{\lambda }_{I}} \\ {{\lambda }_{B}}= & msr\cdot {{\lambda }_{I}} \end{align}\,\! }[/math]

Effectiveness Factor

When a delayed corrective action is implemented for a type B failure mode, in other words a BD mode, the failure intensity for that mode is reduced if the corrective action is effective. Once a BD mode is discovered, it is rarely totally eliminated by a corrective action. After a BD mode has been found and fixed, a certain percentage of the failure intensity will be removed, but a certain percentage of the failure intensity will generally remain. The fraction decrease in the BD mode failure intensity due to corrective actions, [math]\displaystyle{ d\,\! }[/math], [math]\displaystyle{ \left( 0\lt d\lt 1 \right),\,\! }[/math] is called the effectiveness factor (EF).

A study on EFs showed that an average EF, [math]\displaystyle{ d\,\! }[/math], is about 70%. Therefore, about 30% (i.e., [math]\displaystyle{ 100(1-d)%\,\! }[/math]) of the BD mode failure intensity will typically remain in the system after all of the corrective actions have been implemented. However, individual EFs for the failure modes may be larger or smaller than the average. This average value of 70% can be used for planning purposes, or if such information is recorded, an average effectiveness factor from a previous reliability growth program can be used.

MTBF Goal

When putting together a reliability growth plan, a goal MTBF/MTrBF [math]\displaystyle{ {{M}_{G}}\,\! }[/math] (or goal failure intensity [math]\displaystyle{ {{\lambda }_{G}}\,\! }[/math] ) is defined as the requirement or target for the product at the end of the growth program.

Growth Potential

The failure intensity remaining in the system at the end of the test will depend on the management strategy given by the classification of the type A and type B 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 problem 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/MTrBF will be attained when all type B modes have been observed and fixed.

If all the discovered type B 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 with the type B designation of the failure modes and the corresponding assigned effectiveness factors. This is called the nominal growth potential. In other words, the nominal growth potential is the maximum attainable growth potential assuming corrective actions are implemented for every mode that is planned to be fixed. In reality, some corrective actions might be implemented at a later time due to schedule, budget, engineering, etc.

If some of the discovered type B 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 B designation of the failure modes and the corresponding assigned effectiveness factors.

If all type B failure modes are discovered and corrected with an average effectiveness factor, [math]\displaystyle{ d\,\! }[/math], then the maximum reduction in the initial system failure intensity is the growth potential failure intensity:

[math]\displaystyle{ {{\lambda }_{GP}}={{\lambda }_{A}}+\left( 1-d \right){{\lambda }_{B}}\,\! }[/math]

The growth potential MTBF/MTrBF is:

[math]\displaystyle{ {{M}_{GP}}=\frac{1}{{{\lambda }_{GP}}}\,\! }[/math]

Note that based on the equations for the initial failure intensity and the management strategy ratio (given in the Management Strategy and Initial Failure Intensity section), the initial failure intensity is equal to:

[math]\displaystyle{ {{\lambda }_{I}}=\frac{{{\lambda }_{GP}}}{1-d\cdot msr}\,\! }[/math]

Growth Potential Design Margin

The Growth Potential Design Margin ( [math]\displaystyle{ GPDM\,\! }[/math] ) can be considered as a safety margin when setting target MTBF/MTrBF values for the reliability growth plan. It is common for systems to degrade in terms of reliability when a prototype product is going into full manufacturing. This is due to variations in materials, processes, etc. Furthermore, the in-house reliability growth testing usually overestimates the actual product reliability because the field usage conditions may not be perfectly simulated during testing. Typical values for the [math]\displaystyle{ GPDM\,\! }[/math] are around 1.2. Higher values yield less risk for the program, but require a more rigorous reliability growth test plan. Lower values imply higher program risk, with less safety margin.

During the planning stage, the growth potential MTBF/MTrBF, [math]\displaystyle{ {{M}_{GP}},\,\! }[/math] can be calculated based on the goal MTBF, [math]\displaystyle{ {{M}_{G}},\,\! }[/math] and the growth potential design margin, [math]\displaystyle{ GPDM\,\! }[/math].

[math]\displaystyle{ {{M}_{GP}}=GPDM\cdot {{M}_{G}}\,\! }[/math]

or in terms of failure intensity:

[math]\displaystyle{ {{\lambda }_{GP}}=\frac{{{\lambda }_{G}}}{GPDM}\,\! }[/math]

Actual Failure Intensity Function

Consider a test plan consisting of [math]\displaystyle{ i }[/math] phases. Taking into account the fix delay within each phase, we expect the actual failure intensity to be different (i.e. shifted) from the nominal failure intensity. This is because fixes are not incorporated instantaneously, thus growth is realized at a later time compared to the nominal case. Specifically, the actual failure intensity will be estimated as follows:

Test Phase 1
For the first phase of a test plan, the actual idealized curve failure intensity, [math]\displaystyle{ {{r}_{AI}}(t) }[/math] , is [math]\displaystyle{ : }[/math]

[math]\displaystyle{ {{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)t \right]}^{(\beta -1)}}\text{ for }0\lt t\le {{T}_{1}} }[/math]

Note that the end time of Phase 1, [math]\displaystyle{ {{T}_{1}}, }[/math] must be greater than [math]\displaystyle{ {{L}_{1}}+{{t}_{0}} }[/math] . That is, [math]\displaystyle{ {{T}_{1}}\gt {{L}_{1}}+{{t}_{0}} }[/math] . The actual idealized curve initialization time for Phase 1, [math]\displaystyle{ T_{0}^{AIC}, }[/math] is calculated from:

[math]\displaystyle{ {{r}_{AI}}(T_{0}^{AIC})={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC} \right]}^{(\beta -1)}} }[/math]

Where [math]\displaystyle{ {{r}_{AI}}(T_{0}^{AIC})={{r}_{I}}. }[/math] Therefore, using Eqn. (lambdaqualsominal):

[math]\displaystyle{ {{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC} \right]}^{(\beta -1)}}={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)} }[/math]

Solving Eqn. (ActIhase1) for [math]\displaystyle{ T_{0}^{AIC} }[/math] we get:

[math]\displaystyle{ T_{0}^{AIC}=\frac{{{t}_{0}}}{\left( \tfrac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)} }[/math]

Test Phase [math]\displaystyle{ i }[/math]
For any test phase [math]\displaystyle{ i }[/math] , the actual idealized curve failure intensity is given by:

[math]\displaystyle{ {{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)(t-{{T}_{i-1}}) \right]}^{(\beta -1)}} }[/math]

where [math]\displaystyle{ {{T}_{i-1}}\le t\le {{T}_{i}} }[/math] and [math]\displaystyle{ {{T}_{i}} }[/math] is the test time of each corresponding test phase.
The actual idealized curve MTBF is:

[math]\displaystyle{ {{M}_{AI}}=\frac{1}{{{r}_{AI}}(t)} }[/math]

Actual Time to Reach Goal

The actual time to reach the target MTBF or failure intensity goal, [math]\displaystyle{ {{t}_{AC,G}}, }[/math] can be found by solving Eqn. (ActIhase):

[math]\displaystyle{ \begin{align} & {{r}_{AI}}({{t}_{AC,G}})= & {{\lambda }_{A}}+(1-d){{\lambda }_{B}} \\ & & +d\lambda \beta {{\left[ {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)({{t}_{AC,G}}-{{T}_{i-1}}) \right]}^{(\beta -1)}} \end{align} }[/math]

Since the actual idealized growth curve depends on the phase durations and average fix delays, there are three different cases that need to be treated differently in order to determine the actual time to reach the MTBF goal. The cases depend on when the actual MTBF that can be reached within the specific phase durations and fix delays becomes equal to the MTBF goal. This can be determined by solving Eqn. (ActIhase) for phases [math]\displaystyle{ 1 }[/math] through [math]\displaystyle{ i }[/math] , then solving in terms of actual MTBF using Eqn. (MTBFctualq) for each phase and finding the phase during which the actual MTBF becomes equal to the goal MTBF. The three cases are presented next.

Case 1: MTBF goal is met during the last phase
If [math]\displaystyle{ {{T}_{F}} }[/math] indicates the cumulative end phase time for the last phase and [math]\displaystyle{ {{L}_{F}} }[/math] indicates the fix delay for the last phase, then we have:

[math]\displaystyle{ \begin{align} & {{r}_{G}}= & {{\lambda }_{A}}+(1-d){{\lambda }_{B}} \\ & & +d\lambda \beta {{\left[ {{T}_{F-1}}-{{L}_{F-1}}+\left( \frac{{{T}_{F}}-{{L}_{F}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)({{t}_{AC,G}}-{{T}_{F-1}}) \right]}^{(\beta -1)}} \end{align} }[/math]

Starting to solve for [math]\displaystyle{ {{t}_{AC,G}} }[/math] yields:

[math]\displaystyle{ {{\left[ \frac{{{r}_{G}}-{{\lambda }_{A}}-(1-d){{\lambda }_{B}}}{d\lambda \beta } \right]}^{\tfrac{1}{\beta -1}}}={{T}_{F-1}}-{{L}_{F-1}}+\left( \frac{{{T}_{F}}-{{L}_{i}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)({{t}_{AC,G}}-{{T}_{F-1}}) }[/math]

We can substitute the left term by using Eqn. (tgoalominal), thus we have:

[math]\displaystyle{ {{t}_{N,G}}={{T}_{F-1}}-{{L}_{F-1}}+\left( \frac{{{T}_{F}}-{{L}_{F}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)({{t}_{AC,G}}-{{T}_{i-1}}) }[/math]

Therefore:

[math]\displaystyle{ {{t}_{AC,G}}=\frac{{{t}_{N,G}}-{{T}_{F-1}}+{{L}_{F-1}}}{\left( \tfrac{{{T}_{F}}-{{L}_{F}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)}+{{T}_{F-1}} }[/math]

Case 2: MTBF goal is met before the last phase
Eqn. (toalctual) still applies, but in this case [math]\displaystyle{ {{T}_{F}} }[/math] and [math]\displaystyle{ {{L}_{F}} }[/math] are the time and fix delay of the phase during which the goal is met.

Case 3: MTBF goal is met after the final phase
If the goal MTBF, [math]\displaystyle{ {{M}_{G}}, }[/math] is met after the final test phase, then the actual time to reach the goal is not calculated, since additional phases have to be added with specific duration and fix delays. The reliability growth program needs to be re-evaluated with the following options:

• Add more phase(s) to the program.

• Re-examine the phase duration of the existing phases.

• Investigate whether there are potential process improvements in the program that can

reduce the average fix delay for the phases.
Other alternative routes for consideration would be to investigate the rest of the inputs in the model:

• Change the management strategy.

• Consider if further program risk can be acceptable, and if so, reduce the growth potential design margin.

• Consider if it is feasible to increase the effectiveness factors of the delayed fixes by using more robust engineering redesign methods.

Note that each change of input variables into the model can significantly influence the results.
With that in mind, any alteration in the input parameters should be justified by actionable decisions that will influence the reliability growth program. For example, increasing the average effectiveness factor value should be done only when there is proof that the program will pursue a different, more effective path in terms of addressing fixes.