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.