Template:Nominal idealized growth curve

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 [math]\displaystyle{ {{\lambda }_{I}} }[/math] until an initialization time, [math]\displaystyle{ {{t}_{0}}, }[/math] when reliability growth occurs. From that time and until the end of testing, which can be one or, most commonly, multiple test phases, the idealized curve increases steadily according to a learning curve pattern until it reaches the final reliability requirement, [math]\displaystyle{ {{M}_{F}} }[/math] . 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 [math]\displaystyle{ t }[/math] is:

[math]\displaystyle{ {{r}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{t}^{\left( \beta -1 \right)}}\text{ for }t\ge {{t}_{0}} }[/math]

and:

[math]\displaystyle{ {{r}_{NI}}(t)={{\lambda }_{I}}\text{ for }t\le {{t}_{0}} }[/math]

where [math]\displaystyle{ {{\lambda }_{I}} }[/math] is the initial system failure intensity, [math]\displaystyle{ t }[/math] is test time and [math]\displaystyle{ {{t}_{0}} }[/math] is the initialization time, which is discussed in the next section.
It can be seen that Eqn. (Nominal FI) 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 for an initialization time, different than zero, to be defined. The nominal idealized growth curve failure intensity is initially set equal to the initial failure intensity, [math]\displaystyle{ {{\lambda }_{I}}, }[/math] until the initialization time, [math]\displaystyle{ {{t}_{0}} }[/math] :

[math]\displaystyle{ {{r}_{NI}}({{t}_{0}})={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)} }[/math]

Therefore:

[math]\displaystyle{ {{\lambda }_{I}}={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)} }[/math]

Then:

[math]\displaystyle{ {{t}_{0}}={{\left[ \frac{{{\lambda }_{I}}-{{\lambda }_{A}}-(1-d){{\lambda }_{B}}}{d\lambda \beta } \right]}^{\tfrac{1}{\beta -1}}} }[/math]

Using Eqn. (lambda initial) to substitute [math]\displaystyle{ {{\lambda }_{I}} }[/math] we have:

[math]\displaystyle{ {{t}_{0}}={{\left[ \frac{{{\lambda }_{A}}+{{\lambda }_{B}}-{{\lambda }_{A}}-(1-d){{\lambda }_{B}}}{d\cdot \lambda \cdot \beta } \right]}^{\tfrac{1}{\beta -1}}} }[/math]

Then:

[math]\displaystyle{ {{t}_{0}}={{\left( \frac{{{\lambda }_{B}}}{\lambda \cdot \beta } \right)}^{\tfrac{1}{\beta -1}}} }[/math]

The initialization time, [math]\displaystyle{ {{t}_{0}}, }[/math] 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 Eqn. (Nominal FI) to find out how much test time, , is required, (based on the Crow Extended model and the nominal idealized growth curve) to reach that goal:

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

Note that when [math]\displaystyle{ {{\lambda }_{I}}\lt {{r}_{G}} }[/math] or, in other words, the initial failure intensity is lower than the goal failure intensity, 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:

[math]\displaystyle{ C(t)=\frac{\lambda {{t}^{\beta }}}{t}=\lambda {{t}^{(\beta -1)}} }[/math]

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

[math]\displaystyle{ r(t)=\lambda \beta {{t}^{(\beta -1)}} }[/math]

Taking the difference, [math]\displaystyle{ D(t), }[/math] between the average failure intensity, [math]\displaystyle{ C(t) }[/math] , and the instantaneous failure intensity, [math]\displaystyle{ r(t) }[/math] , yields:

[math]\displaystyle{ D(t)=\lambda {{t}^{(\beta -1)}}-\lambda \beta {{t}^{(\beta -1)}} }[/math]

Then:

[math]\displaystyle{ D(t)=\lambda {{t}^{(\beta -1)}}[1-\beta ] }[/math]

For reliability growth to occur, [math]\displaystyle{ D(t) }[/math] must be decreasing. The growth rate for both the Duane Postulate and the Crow-AMSAA (NHPP) model is the negative of the slope of [math]\displaystyle{ \log (D(t)) }[/math] as a function of [math]\displaystyle{ \log (t) }[/math] :

[math]\displaystyle{ {{\log }_{e}}(D(t))=\text{constant}-(1-\beta ){{\log }_{e}}(t) }[/math]

The slope is negative under reliability growth and equals:

[math]\displaystyle{ \text{slope}=-(1-\beta ) }[/math]

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

[math]\displaystyle{ \text{Growth Rate}=(1-\beta ) }[/math]

The instantaneous failure intensity for the nominal idealized curve is:

[math]\displaystyle{ {{r}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{(t)}^{(\beta -1)}} }[/math]

The cumulative failure intensity for the nominal idealized curve is:

[math]\displaystyle{ {{C}_{NI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda {{(t)}^{(\beta -1)}} }[/math]

Therefore:

[math]\displaystyle{ {{D}_{NI}}(t)=[{{C}_{NI}}(t)-{{r}_{NI}}(t)]=\lambda {{t}^{(\beta -1)}}[1-\beta ] }[/math]

and:

[math]\displaystyle{ {{\log }_{e}}({{D}_{NI}}(t))=\text{constant}-(1-\beta ){{\log }_{e}}(t) }[/math]

Therefore, in accordance with the Duane Postulate and the Crow-AMSAA (NHPP) model, [math]\displaystyle{ a=1-\beta }[/math] 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 [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \eta }[/math] is:

[math]\displaystyle{ MTTF=\eta \cdot \Gamma \left( 1+\frac{1}{\beta } \right) }[/math]

The parameter lambda is defined as:

[math]\displaystyle{ \lambda =\frac{1}{{{\eta }^{\beta }}} }[/math]

Using Eqn. (lambda eta relationship), the MTTF relationship shown in Eqn. (Weibull MTTF) becomes:

[math]\displaystyle{ MTB{{F}_{B}}=\frac{\Gamma \left( 1+\tfrac{1}{\beta } \right)}{{{\lambda }^{\left( \tfrac{1}{\beta } \right)}}} }[/math]

Or, in terms of failure intensity:

[math]\displaystyle{ {{\lambda }_{B}}=\frac{{{\lambda }^{\left( \tfrac{1}{\beta } \right)}}}{\Gamma \left( 1+\tfrac{1}{\beta } \right)} }[/math]