User:David J. Groebel/MySandbox

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 reflects a learning curve pattern for reliability growth. This identical learning curve pattern forms the basis of the Crow (AMSAA) model. The Duane Postulate is also reflected in the Crow Extended model in the form of the discovery function $$h(t)\,\!$$. The discovery function is the rate in which new distinct problems are being discovered during reliability growth development testing. The Crow (AMSAA) model is a special case of the discovery function. Suppose that when a new distinct failure mode is first seen the testing is stopped and a corrective action is incorporated before the testing is resumed. In addition, consider that the corrective action is highly effective so that the failure mode is unlikely to be seen again. In this case, the only failures observed during the reliability growth test are first occurrences of failure modes. Therefore, if the Crow (AMSAA) 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) model must be the form for the discovery function $$h(t)\,\!$$.

The Duane postulate is based on empirical observations, and it reflects a learning curve pattern for reliability growth. This identical 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 $$h(t)\,\!$$.

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, $$h(t)\,\!$$.

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 that it is 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 $${{\lambda }_{I}}\,\!$$ be the initial failure intensity of the system in test. $${{\lambda }_{A}}\,\!$$ is defined as the A mode's initial failure intensity and $${{\lambda }_{B}}\,\!$$ is defined as the B mode's initial failure intensity. $${{\lambda }_{A}}\,\!$$ is the failure intensity of the system that will not be addressed by corrective actions even if a failure mode is seen during test. $${{\lambda }_{B}}\,\!$$ 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:


 * $$\begin{align}

{{\lambda }_{I}}={{\lambda }_{A}}+{{\lambda }_{B}} \end{align}\,\!$$

The initial system MTBF is:


 * $${{M}_{I}}=\frac{1}\,\!$$

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


 * $$msr=\frac{{{\lambda }_{A}}+{{\lambda }_{B}}}\,\!$$

The $$msr\,\!$$ 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:


 * $$\begin{align}

{{\lambda }_{A}}= & \left( 1-msr \right)\cdot {{\lambda }_{I}} \\ {{\lambda }_{B}}= & msr\cdot {{\lambda }_{I}} \end{align}\,\!$$

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, $$d\,\!$$, $$\left( 0<d<1 \right),\,\!$$ is called the effectiveness factor (EF).

A study on EFs showed that an average EF, $$d\,\!$$, is about 70%. Therefore, about 30%, (i.e., $$100(1-d)%\,\!$$), 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 $${{M}_{G}}\,\!$$ (or goal failure intensity $${{\lambda }_{G}}\,\!$$ ) 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 $$h(t)\,\!$$, 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 $$T\,\!$$ 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 $$T\,\!$$, that is, no deferred corrective actions at time $$T\,\!$$, 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, $$d\,\!$$, then the maximum reduction in the initial system failure intensity is the growth potential failure intensity:


 * $${{\lambda }_{GP}}={{\lambda }_{A}}+\left( 1-d \right){{\lambda }_{B}}\,\!$$

The growth potential MTBF/MTrBF is:


 * $${{M}_{GP}}=\frac{1}\,\!$$

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:


 * $${{\lambda }_{I}}=\frac{1-d\cdot msr}\,\!$$

Growth Potential Design Margin
The Growth Potential Design Margin ( $$GPDM\,\!$$ ) 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 $$GPDM\,\!$$ 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, $${{M}_{GP}},\,\!$$ can be calculated based on the goal MTBF, $${{M}_{G}},\,\!$$ and the growth potential design margin, $$GPDM\,\!$$.


 * $${{M}_{GP}}=GPDM\cdot {{M}_{G}}\,\!$$

or in terms of failure intensity:


 * $${{\lambda }_{GP}}=\frac{GPDM}\,\!$$

Continuous Reliability Growth Planning
The use of the Duane postulate as a reliability growth planning model poses two significant drawbacks: The first drawback is that the Duane postulate's MTBF is zero at time equal to zero. This was addressed in MIL-HDBK-189 by specifying a time $${{T}_{i}}\,\!$$ where growth starts after $${{T}_{i}}\,\!$$ and the Duane postulate applies [13]. However, determining $${{T}_{i}}\,\!$$ is subjective and is not a desirable property of the MIL-HDBK-189. The second drawback is that the MTBF for the Duane postulate increases indefinitely to infinity, which is not realistic.

Therefore, the desirable features of a planning model are:

All of these desirable features are included in the planning model discussed in this chapter, which is based on the Crow extended model.
 * 1) The discovery function must have the form of the Crow-AMSAA (NHPP) model and the Duane postulate.
 * 2) The start time $${{T}_{i}}\,\!$$ is not required as an input.
 * 3) An upper bound on the system MTBF is specified in the model.

The Crow extended model for reliability growth planning is a revised and improved version of the MIL-HDBK-189 growth curve [13]. MIL-HDBK-189 can be considered as the growth curve based on the Crow-AMSAA (NHPP) model. Using MIL-HDBK-189 for reliability growth planning assumes that the corrective actions for the observed failure modes are incorporated during the test and at the specific time of failure. However, in actual practice, fixes may be delayed until after the completion of the test or some fixes may be implemented during the test while others are delayed and some are not fixed at all. The Crow extended model for reliability growth planning provides additional inputs that accounts for specific management strategies and delayed fixes with specified effectiveness factors.

Nominal Idealized Growth Curve
During developmental testing, management should expect that certain levels of reliability will be attained at various points in the program in order to have assurance that reliability growth is progressing at a sufficient rate to meet the product reliability requirement. The idealized curve portrays an overall characteristic pattern, which is used to determine and evaluate intermediate levels of reliability and construct the program planned growth curve. Note that growth profiles on previously developed, similar systems provide significant insight into the reliability growth process and are valuable in the construction of idealized growth curves.

The nominal idealized growth curve portrays a general profile for reliability growth throughout system testing. The idealized curve has the baseline value $${{\lambda }_{I}}\,\!$$ until an initialization time, $${{t}_{0}},\,\!$$ when reliability growth occurs. From that time and until the end of testing, which can be a single or, most commonly, multiple test phases, the idealized curve increases steadily according to a learning curve pattern until it reaches the final reliability requirement, $${{M}_{F}}\,\!$$. The slope of this curve on a log-log plot is the growth rate of the Crow extended model [13].

Nominal Failure Intensity Function
The nominal idealized growth curve failure intensity as a function of test time $$t\,\!$$ is:


 * $${{r}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{t}^{\left( \beta -1 \right)}}\text{ for }t\ge {{t}_{0}}\,\!$$

and:


 * $${{r}_{NI}}(t)={{\lambda }_{I}}\text{ for }t\le {{t}_{0}}\,\!$$

where $${{\lambda }_{I}}\,\!$$ is the initial system failure intensity, $$t\,\!$$ is test time and $${{t}_{0}}\,\!$$ is the initialization time, which is discussed in the next section.

It can be seen that the first equation for $${{r}_{NI}}(t)\,\!$$ is the failure intensity equation of the Crow extended model.

Initialization Time
Reliability growth can only begin after a type B failure mode occurs, which cannot be at a time equal to zero. Therefore, there is a need to define an initialization time that is different from zero. The nominal idealized growth curve failure intensity is initially set to be equal to the initial failure intensity, $${{\lambda }_{I}},\,\!$$ until the initialization time, $${{t}_{0}}\,\!$$ :


 * $${{r}_{NI}}({{t}_{0}})={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)}\,\!$$

Therefore:


 * $${{\lambda }_{I}}={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)}\,\!$$

Then:


 * $${{t}_{0}}={{\left[ \frac{{{\lambda }_{I}}-{{\lambda }_{A}}-(1-d){{\lambda }_{B}}}{d\lambda \beta } \right]}^{\tfrac{1}{\beta -1}}}\,\!$$

Using the equation for initial failure intensity:


 * $$\lambda_{I}=\lambda_{A} + \lambda_{B}\,\!$$

we substitute $${{\lambda }_{I}}\,\!$$ to get:


 * $${{t}_{0}}={{\left[ \frac{{{\lambda }_{A}}+{{\lambda }_{B}}-{{\lambda }_{A}}-(1-d){{\lambda }_{B}}}{d\cdot \lambda \cdot \beta } \right]}^{\tfrac{1}{\beta -1}}}\,\!$$

Then:


 * $${{t}_{0}}={{\left( \frac{{{\lambda }_{B}}}{\lambda \cdot \beta } \right)}^{\tfrac{1}{\beta -1}}}\,\!$$

The initialization time, $${{t}_{0}},\,\!$$ allows for growth to start after a type B failure mode has occurred.

Nominal Time to Reach Goal
Assuming that we have a target MTBF or failure intensity goal, we can solve the equation for the nominal failure intensity to find out how much test time, $${{t}_{N,G}}\,\!$$, is required (based on the Crow extended model and the nominal idealized growth curve) to reach that goal:


 * $${{t}_{N,G}}={{\left[ \frac{{{r}_{G}}-{{\lambda }_{A}}-(1-d){{\lambda }_{B}}}{d\cdot \lambda \cdot \beta } \right]}^{\tfrac{1}{\beta -1}}}\,\!$$

Note that when $${{\lambda }_{I}}<{{r}_{G}}\,\!$$ or, in other words, the initial failure intensity is lower than the goal failure intensity, then there is no need to solve for the nominal time to reach the goal because the goal is already met. In this case, no further reliability growth testing is needed.

Growth Rate for Nominal Idealized Curve
The growth rate for the nominal idealized curve is defined in the same context as the growth rate for the Duane postulate [8]. The nominal idealized curve has the same functional form for the growth rate as the Duane postulate and the Crow-AMSAA (NHPP) model.

For both the Duane postulate and the Crow-AMSAA (NHPP) model, the average failure intensity is given by:


 * $$C(t)=\frac{\lambda {{t}^{\beta }}}{t}=\lambda {{t}^{(\beta -1)}}\,\!$$

Also, for both the Duane postulate and the Crow-AMSAA (NHPP) model, the instantaneous failure intensity is given by:


 * $$\begin{align}

r(t)=\lambda \beta {{t}^{(\beta -1)}} \end{align}\,\!$$

Taking the difference, $$D(t),\,\!$$ between the average failure intensity, $$C(t)\,\!$$ and the instantaneous failure intensity, $$r(t)\,\!$$, yields:


 * $$\begin{align}

D(t)=\lambda {{t}^{(\beta -1)}}-\lambda \beta {{t}^{(\beta -1)}} \end{align}\,\!$$

Then:


 * $$\begin{align}

D(t)=\lambda {{t}^{(\beta -1)}}[1-\beta ] \end{align}\,\!$$

For reliability growth to occur, $$D(t)\,\!$$ must be decreasing.

The growth rate for both the Duane postulate and the Crow-AMSAA (NHPP) model is the negative of the slope of $$\log (D(t))\,\!$$ as a function of $$\log (t)\,\!$$ :


 * $$\begin{align}

{{\log }_{e}}(D(t))=\text{constant}-(1-\beta ){{\log }_{e}}(t) \end{align}\,\!$$

The slope is negative under reliability growth and equals:


 * $$\begin{align}

\text{slope}=-(1-\beta ) \end{align}\,\!$$

The growth rate for both the Duane postulate and the Crow-AMSAA (NHPP) model is equal to the negative of this slope:


 * $$\begin{align}

\text{Growth Rate}=(1-\beta ) \end{align}\,\!$$

The instantaneous failure intensity for the nominal idealized curve is:


 * $$\begin{align}

{{r}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{(t)}^{(\beta -1)}} \end{align}\,\!$$

The cumulative failure intensity for the nominal idealized curve is:


 * $$\begin{align}

{{C}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda {{(t)}^{(\beta -1)}} \end{align}\,\!$$

therefore:


 * $$\begin{align}

{{D}_{NI}}(t)=[{{C}_{NI}}(t)-{{r}_{NI}}(t)]=\lambda {{t}^{(\beta -1)}}[1-\beta ] \end{align}\,\!$$

and:


 * $$\begin{align}

{{\log }_{e}}({{D}_{NI}}(t))=\text{constant}-(1-\beta ){{\log }_{e}}(t) \end{align}\,\!$$

Therefore, in accordance with the Duane postulate and the Crow-AMSAA (NHPP) model, $$a=1-\beta \,\!$$ is the growth rate for the reliability growth plan.

Lambda - Beta Parameter Relationship
Under the Crow-AMSAA (NHPP) model, the time to first failure is a Weibull random variable. The MTTF of a Weibull distributed random variable with parameters $$\beta \,\!$$ and $$\eta \,\!$$ is:


 * $$MTTF=\eta \cdot \Gamma \left( 1+\frac{1}{\beta } \right)\,\!$$

The parameter lambda is defined as:


 * $$\lambda =\frac{1}\,\!$$

Using the equation for lambda in the MTTF relationship, we have:


 * $$MTB{{F}_{B}}=\frac{\Gamma \left( 1+\tfrac{1}{\beta } \right)}\,\!$$

or, in terms of failure intensity:


 * $${{\lambda }_{B}}=\frac{\Gamma \left( 1+\tfrac{1}{\beta } \right)}\,\!$$

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.

Fix Delays and Test Phase Target MTBF
Fix delays reflect how long it takes from the time a problem failure mode is discovered in testing, to the time the corrective action is incorporated into the system and reliability growth is realized. The consideration of the fix delay is often in terms of how much calendar time it takes to incorporate a corrective action fix after the problem is first seen. However, the impact of the delay on reliability growth is reflected in the average test time it takes between finding a problem failure mode and incorporating a corrective action. The fix delay is reflected in the actual idealized growth curve in terms of test time.

In other words, the average fix delay is calendar time converted to test hours. For example, say that we expect an average fix delay of two weeks. If in two weeks the total test time is 1,000 hours, then the average fix delay is 1,000 hours. If in the same two weeks the total test time is 2,000 hours (maybe there are more units available or more shifts), then the average fix delay is 2,000 hours.

There can be a constant fix delay across all test phases or, as a practical matter, each test phase can have a different fix delay time. In practice, the fix delay will generally be constant over a particular test phase. $${{L}_{i}}\,\!$$ denotes the fix delay for phase $$i=1,...,P,\,\!$$ where $$P\,\!$$ is the total number of phases in the test. The RGA software allows for a maximum of ten test phases.

Actual Failure Intensity Function
Consider a test plan consisting of $$i\,\!$$ 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, $${{r}_{AI}}(t)\,\!$$, is :


 * $${{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}} \right)t \right]}^{(\beta -1)}}\text{ for }0{{L}_{1}}+{{t}_{0}}\,\!$$.

The actual idealized curve initialization time for Phase 1, $$T_{0}^{AIC},\,\!$$ is calculated from:


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

where $${{r}_{AI}}(T_{0}^{AIC})={{r}_{I}}.\,\!$$

Therefore, using the equation for the initialization time, we have:


 * $${{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC} \right]}^{(\beta -1)}}={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)}\,\!$$

By obtaining the initial failure intensity for $$T_{0}^{AIC}\,\!$$, we get:


 * $$T_{0}^{AIC}=\frac{\left( \tfrac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)}\,\!$$

Test Phase $$i\,\!$$

For any test phase $$i\,\!$$, the actual idealized curve failure intensity is given by:


 * $${{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)(t-{{T}_{i-1}}) \right]}^{(\beta -1)}}\,\!$$

where $${{T}_{i-1}}\le t\le {{T}_{i}}\,\!$$ and $${{T}_{i}}\,\!$$ is the test time of each corresponding test phase.

The actual idealized curve MTBF is:


 * $${{M}_{AI}}=\frac{1}{{{r}_{AI}}(t)}\,\!$$

Actual Time to Reach Goal
The actual time to reach the target MTBF or failure intensity goal, $${{t}_{AC,G}},\,\!$$ can be found by solving for the actual idealized curve failure intensity:


 * $$\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}\,\!$$

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 for the actual idealized curve failure intensity for phases $$1\,\!$$ through $$i\,\!$$, and then solving in terms of actual idealized curve MTBF 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 $${{T}_{F}}\,\!$$ indicates the cumulative end phase time for the last phase, and $${{L}_{F}}\,\!$$ indicates the fix delay for the last phase, then we have:


 * $$\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}\,\!$$

Starting to solve for $${{t}_{AC,G}}\,\!$$ yields:


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

We can substitute the left term by solving for the nominal time to reach the goal; thus, we have:


 * $${{t}_{N,G}}={{T}_{F-1}}-{{L}_{F-1}}+\left( \frac{{{T}_{F}}-{{L}_{F}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)({{t}_{AC,G}}-{{T}_{i-1}})\,\!$$

therefore:


 * $${{t}_{AC,G}}=\frac{{{t}_{N,G}}-{{T}_{F-1}}+{{L}_{F-1}}}{\left( \tfrac{{{T}_{F}}-{{L}_{F}}-{{T}_{F-1}}+{{L}_{F-1}}}{{{T}_{F}}-{{T}_{F-1}}} \right)}+{{T}_{F-1}}\,\!$$

Case 2: MTBF goal is met before the last phase

The equation for $${{t}_{AC,G}}\,\!$$ that was derived for case 1 still applies, but in this case $${{T}_{F}}\,\!$$ and $${{L}_{F}}\,\!$$ 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, $${{M}_{G}},\,\!$$ 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.

Discrete Reliability Growth Planning
To be consistent with the Duane Postulate and the Crow (AMSAA) 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 of the Crow Extended Discrete Planning model. As with the Crow (AMSAA) model, this form of the discovery function ties the model directly to real-world data and experiences. Therefore, a desirable feature of a discrete reliability growth planning model is that $$\left( 0,t \right)=\lambda {{\left( t \right)}^{\beta }}\,\!$$. This implies that the probability, $$f\,\!$$, of a new distinct failure mode occurring at trial $$t\,\!$$ is given by:


 * $$f=\lambda {{t}^{\beta }}-\lambda {{\left( t-1 \right)}^{\beta }}\,\!$$

Let $${{f}_{A}}$$ be the Type A initial failure probability and let $${{f}_{B}}$$ be the Type B initial failure probability.

A system failure occurs at the first event of a Type A mode or a Type B mode. Only one event causes a system failure. That is, Type A failures and Type B failures are disjointed.

The initial system failure probability $${{f}_{I}}$$ is:


 * $$\begin{align}

{{f}_{I}}={{f}_{A}}+{{f}_{B}} \end{align}\,\!$$

The Type A failure probability is the failure probability in that part of the system that will not be addressed by corrective actions even if a failure mode is seen during test. The Type B failure probability is the failure probability in that part of the system that will be addressed by corrective actions if a failure mode is seen during test.

When a failure mode in the Type B part of the system is seen during test a corrective action will be implemented, and for discrete trials, always at a later time after the failure mode is first seen. When a corrective action is implemented for a Type B failure mode the failure probability for that mode is reduced if the corrective action is effective. The fraction decrease in the Type B failure modes due to corrective actions is d, where The Average Effectiveness Factor (EF) is d and the corrective action may be implemented before the next trial or at a later date.

The management strategy ratio is:


 * $$msr=\frac{{{f}_{A}}+{{f}_{B}}}\,\!$$

The msr is the fraction of the initial system failure probability that will be addressed by corrective actions, if seen during the test.

If all Type B failure modes are seen and corrected with an average EF d, then the maximum reduction in the initial system failure probability is the Growth Potential failure probability:


 * $${{f}_{GP}}={{f}_{A}}+\left( 1-d \right){{f}_{B}}\,\!$$

The initial system Mean Trials Between Failure (MTrBF) is:


 * $${{M}_{I}}=\frac{1}\,\!$$

The Growth Potential MTrBF is:


 * $${{M}_{GP}}=\frac{1}\,\!$$

Nominal Idealized Growth Curve
The nominal idealized growth curve portrays a general profile for reliability growth throughout system testing. It represents a best case scenario where fix delay is not taken into account. The Crow Extended Discrete Planning Model Nominal Idealized Growth Curve failure probability as a function of test trials t is:


 * $${{f}_{NI}}(t)={{f}_{A}}+(1-d){{f}_{B}}+d\left[ \lambda {{t}^{\beta }}-\lambda {{(t-1)}^{\beta }} \right]\text{ for }t\ge {{t}_{0}}\,\!$$


 * $${{f}_{NI}}(t)={{f}_{I}}\text{ for }t<{{t}_{0}}\,\!$$

where $${{t}_{0}}\,\!$$ is the initialization time, and $${t}\,\!$$ is the number of test trials.

It is virtually impossible to justify with any model that reliability growth starts at time zero because growth can only start when a Type B failure mode occurs. The Extended Discrete Planning Model Nominal Idealized Growth Curve failure probability is initially set equal to the initial failure probability $${{f}_{I}}\,\!$$ until the time $${{t}_{0}}\,\!$$ when:


 * $${{f}_{NI}}({{t}_{0}})={{f}_{A}}+(1-d){{f}_{B}}+d\left[ \lambda {{t}^{\beta }}-\lambda {{({{t}_{0}}-1)}^{\beta }} \right]={{f}_{I}}\,\!$$

The Initialization time $${{t}_{0}}\,\!$$ allows for the probability of a Type B failure mode to be greater than zero before growth starts. $${{t}_{0}}\,\!$$ is then solved numerically.

The mean time to the first failure mode is given by the mean of a sequence of independent Bernoulli Trials, each trial with a different probability of a new failure mode given by:


 * $${{g}_{i}}=\lambda {{i}^{\beta }}-\lambda {{(i-1)}^{\beta }}\text{ for }i=1,2,...\,\!$$

Then the mean trial to the first Type B mode is given by:


 * $$MTrB{{F}_{B}}=\sum\limits_{k=1}^{\infty }{k\centerdot {{g}_{k}}}\centerdot \left[ \prod\limits_{j=1}^{k-1} \right]\,\!$$

where:


 * $$MTrB{{F}_{B}}=\frac{1}\,\!$$


 * $${{g}_{k}}=\lambda \left[ \mathop{k}^{\beta }-{{\left( k-1 \right)}^{\beta }} \right]\,\!$$


 * $${{P}_{k}}=1-{{g}_{k}}\text{ such that }k=1,2,...,\infty\,\!$$

$$\lambda\,\!$$ is then solved numerically.

Nominal Time to Reach Goal
The nominal time to reach the goal is calculated by setting the nominal idealized failure intensity, $${{f}_{NI}}\,\!$$, equal to the failure intensity goal, $${{f}_{G}}\,\!$$. Therefore,


 * $${{f}_{G}}={{f}_{A}}+(1-d){{f}_{B}}+d\left[ \lambda {{t}_{N,G}}^{\beta }-\lambda {{\left( {{t}_{N,G}}-1 \right)}^{\beta }} \right]\,\!$$

$${{t}_{N,G}}\,\!$$ is solved numerically.

Actual Idealized Growth Curve
The Actual Idealized Growth Curve is a continuous function that incorporates the test phase average fix delay times and goes through each of the test phase target MTrBF.

Fix Delay
The fix delay reflects how long it takes from the time a problem failure mode is discovered in test to the time the corrective action is incorporated into the system and reliability growth is realized. The consideration of the fix delay is often in terms of how much calendar time does it take for incorporate a corrective action fix after the problem is first seem. However, the impact of the fix delay on reliability growth is reflected in the average test time it takes between finding a problem failure mode and incorporating a corrective action.

There can be a constant fix delay across all test phases or, as a practical matter, each test phase can have a different fix delay time. In practice, the fix delay will be considered constant over a test phase. $${{L}_{i}}\,\!$$ denotes the fix delay for phase $$i=1,...,P,\,\!$$ where $$P\,\!$$ is the total number of phases in the test. The RGA software allows for a maximum of ten test phases.

Actual Failure Intensity Function
Test Phase 1

The actual failure intensity function for test phase 1 is given by:


 * $${{f}_{AI}}(t)={{f}_{A}}+(1-d){{f}_{B}}+d\lambda \left[ {{\left( \left( \frac{{{T}_{1}}-{{L}_{1}}} \right)t \right)}^{\beta }}-{{\left( \left( \frac{{{T}_{1}}-{{L}_{1}}} \right)t-1 \right)}^{\beta }} \right]\text{ for }0{{L}_{1}}+{{t}_{0}}\,\!$$.

The actual idealized curve initialization time for Phase 1, $$T_{0}^{AIC}\,\!$$, is calculated from:


 * $${{f}_{AI}}(T_{0}^{AIC})={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \left[ {{\left( \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC} \right)}^{\beta }}-{{\left( \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC}-1 \right)}^{\beta }} \right]\,\!$$

By obtaining the initial failure intensity for $$T_{0}^{AIC}\,\!$$:


 * $$T_{0}^{AIC}=\frac{\left( \tfrac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)}\,\!$$

Test Phase i

For any test phase $$i\,\!$$, the actual idealized curve failure intensity is given by:


 * $${{f}_{AI}}(t)={{f}_{A}}+(1-d){{f}_{B}}+d\lambda \left[ {{\left( {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)(t-{{T}_{i-1}}) \right)}^{\beta }}-{{\left( {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)(t-{{T}_{i-1}})-1 \right)}^{\beta }} \right]\,\!$$

where $${{T}_{i-1}}\le t\le {{T}_{i}}\,\!$$ and $${{T}_{i}}\,\!$$ is the test time of each corresponding test phase.

Actual Time to Reach Goal
The actual time to reach the goal is calculated using the actual failure intensity equation by solving for $${{t}_{AC,G}}\,\!$$ such that the $${{f}_{AI}}(t)\,\!$$ is equal to the failure intensity goal, $${{f}_{G}}\,\!$$. Therefore:


 * $${{f}_{G}}={{f}_{A}}+(1-d){{f}_{B}}+d\lambda \left[ {{\left( {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)({{t}_{AC,G}}-{{T}_{i-1}}) \right)}^{\beta }}-{{\left( {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)({{t}_{AC,G}}-{{T}_{i-1}})-1 \right)}^{\beta }} \right]\,\!$$.

$${{t}_{AC,G}}\,\!$$ is solved numerically where $${{T}_{F}}\,\!$$ and $${{L}_{F}}\,\!$$ correspond to the phase in which the goal is met.

If the goal, $${{M}_{G}}\,\!$$, 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 can 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.

Example
A missile system will be going through a reliability growth development program. The goal for the system is to reach a reliability requirement of 0.9 at the end of the program. This corresponds to a goal MTrBF, $${{R}_{G}}\,\!$$, of 10 trials. The new missile system is similar to an existing missile system, but this will not be an upgrade. Therefore, to help design the reliability growth plan for the new system, parameters from the existing version will be used.

The previous missile system met its reliability goal of $${{R}_{G}=0.95}\,\!$$ which corresponds to a MTrBF = 20. There were over 600 trials of this missile system during developmental testing. This large number of trials were used for testing, but also for training. In addition, given the complexity of the technology that was utilized, additional trials were needed. The system also had an initial reliability of $${{R}_{I}=0.84}\,\!$$. Additional parameters associated with the previous system are given below:


 * Average Effectiveness Factor is $$EF=0.85\,\!$$
 * Discovery Beta is $$\beta=0.70\,\!$$
 * Management Strategy Ratio is $$msr=0.972\,\!$$
 * Growth Potential Design Margin is $$GPDM=1.8\,\!$$

For the new program, a $$msr=0.97\!$$ will be used, along with an average $$EF=0.8\,\!$$. For initial planning purposes, the average fix delay will be assumed to be equal to zero. Therefore, the nominal idealized curve will equal the actual idealized curve.

Determine the following:  The growth potential MTrBF and failure intensity. The initial failure intensity and reliability. The growth potential reliability. The type A and type B initial failure intensity. The parameter $$\lambda \,\!$$ of the Crow extended model. The type B failure mode discovery function. The initialization time, $${{t}_{0}},\,\!$$ for the nominal failure intensity function.</li> The nominal idealized failure intensity function.</li> The nominal time to reach the MTrBF goal.</li> Use RGA to generate the reliability growth plan for the new missile design.</li> </li> </ol>

Solution

<ol> The growth potential MTrBF is:


 * $$\begin{align}

{{MTrBF}_{GP}}= & {{MTrBF}_{G}}\cdot GPDM \\ = & 10\cdot 1.8 \\ = & 18 \end{align}\,\!$$

</li>  The growth potential failure intensity is equal to:
 * $$\begin{align}

{{f}_{GP}}= & \frac{1} \\ = & \frac{1}{18} \\ = & 0.0556 \end{align}\,\!$$

Therefore, the initial failure intensity is:


 * $$\begin{align}

{{f}_{I}}= & \frac{1-d\cdot msr} \\ = & \frac{0.0556}{\left( 1-0.85\cdot 0.97 \right)} \\ = & 0.2480 \end{align}\,\!$$

The initial reliability is then:


 * $$\begin{align}

R_{I}=&1-f_{I} \\ =& 1-0.2480 \\ =& 0.7520 \end{align}\,\!$$

</li> The growth potential reliability is equal to:


 * $$\begin{align}

R_{GP}=&1-f_{GP} \\ =& 1-0.0556 \\ =& 0.9444 \end{align}\,\!$$

</li> The type A failure mode intensity, $${{f}_{A}},\,\!$$ is:


 * $$\begin{align}

f_{A}=&(1-msr)f_{I} \\ =& (1-0.97)0.2480 \\ =& 0.0074 \end{align}\,\!$$

The type B failure mode intensity, $${{f}_{B}},\,\!$$ is:


 * $$\begin{align}

{{f}_{B}}= & msr\cdot {{f}_{I}} \\ = & 0.97\cdot 0.2480 \\ = & 0.2406 \end{align}\,\!$$

</li>  The initial MTrBF is:


 * $$\begin{align}

{{MTrBF}_{GP}}= & \frac{1} \\ = & \frac{1}{0.2480} \\ = & 4.032 \end{align}\,\!$$

$$\lambda \,\!$$ is estimated such that:


 * $$\begin{align}

MTrB{{F}_{B}}= & \sum\limits_{k=1}^{\infty }{k\centerdot {{g}_{k}}}\centerdot \left[ \prod\limits_{j=1}^{k-1} \right] \\ = & 4.0322 \end{align}\,\!$$

where:


 * $${{g}_{k}}=\lambda \left[ \mathop{k}^{\beta }-{{\left( k-1 \right)}^{\beta }} \right]\,\!$$

Therefore:

$$\lambda =0.4145\,\!$$

</li> The type B failure mode discovery function is:


 * $$h\left( t \right)=\lambda \beta {{t}^{\beta -1}}\,\!$$

Since $$\lambda \,\!$$ and $$\beta \,\!$$ are known:


 * $$\begin{align}

h\left( t \right)= & 0.4145\cdot 0.7{{t}^{0.7-1}} \\ = & 0.2901{{t}^{-0.3}} \end{align}\,\!$$

</li> The initialization time, $${{t}_{0}},\,\!$$ is calculated using the nominal idealized equation and setting this equal to initial failure intensity.


 * $$\begin{align}

{{f}_{NI}}({{t}_{0}})= {{f}_{A}}+(1-d){{f}_{B}}+d\left[ \lambda {{t}^{\beta }}-\lambda {{({{t}_{0}}-1)}^{\beta }} \right]= & {{f}_{I}} \\ = & 0.2480 \end{align}\,\!$$

Then, using numerical methods:


 * $$\begin{align}

{{t}_{0}}=2.3964 \end{align}\,\!$$

</li> The nominal idealized growth curve is given by the following equation:
 * $${{f}_{NI}}\left( t \right)=\left\{ \begin{matrix}

{{\lambda }_{I}} & t\le {{t}_{0}} \\ \begin{matrix} {{f}_{A}}+(1-d){{f}_{B}}+d\left[ \lambda {{t}^{\beta }}-\lambda {{({{t}}-1)}^{\beta }} \right] \\ \end{matrix} & t>{{t}_{0}} \\ \end{matrix} \right\}\,\!$$

or:


 * $${{f}_{NI}}\left( t \right)=\left\{ \begin{matrix}

0.2480 & t\le 2.3964 \\ =0.0556+0.8\left[ 0.4145{{t}^{0.7}}-0.4145 {{({{t}}-1)}^{0.7 }} \right] & t>2.3964 \\ \end{matrix} \right\}\,\!$$

</li> The nominal time to reach the MTrBF goal is estimated by solving for the value of $$t\,\!$$ such that the nominal idealized curve $${{f}_{NI}}\,\!$$ is equal to $${{f}_{I}}\,\!$$. Therefore, solve for $$t\,\!$$ such that:


 * $${{f}_{NI}}({{t}_{0}})={{f}_{A}}+(1-d){{f}_{B}}+d\left[ \lambda {{t}^{\beta }}-\lambda {{({{t}_{0}}-1)}^{\beta }} \right]=0.2480$$

Using numerical methods, the nominal time to reach the goal $$t=247.6606\,\!$$ trials.

</li>  The reliability growth plan in RGA is shown next.



The plot of the reliability growth plan is shown below.

</li> </ol>