Crow-AMSAA (NHPP)

Dr. Larry H. Crow [17] noted that the Duane Model could be stochastically represented as a Weibull process, allowing for statistical procedures to be used in the application of this model in reliability growth. This statistical extension became what is known as the Crow-AMSAA (NHPP) model. This method was first developed at the U.S. Army Materiel Systems Analysis Activity (AMSAA). It is frequently used on systems when usage is measured on a continuous scale. It can also be applied for the analysis of one shot items when there is high reliability and a large number of trials.

Test programs are generally conducted on a phase by phase basis. The Crow-AMSAA model is designed for tracking the reliability within a test phase and not across test phases. A development testing program may consist of several separate test phases. If corrective actions are introduced during a particular test phase, then this type of testing and the associated data are appropriate for analysis by the Crow-AMSAA model. The model analyzes the reliability growth progress within each test phase and can aid in determining the following:


 * Reliability of the configuration currently on test
 * Reliability of the configuration on test at the end of the test phase
 * Expected reliability if the test time for the phase is extended
 * Growth rate
 * Confidence intervals
 * Applicable goodness-of-fit tests

Background
The reliability growth pattern for the Crow-AMSAA model is exactly the same pattern as for the Duane postulate, that is, the cumulative number of failures is linear when plotted on ln-ln scale. Unlike the Duane postulate, the Crow-AMSAA model is statistically based. Under the Duane postulate, the failure rate is linear on ln-ln scale. However, for the Crow-AMSAA model statistical structure, the failure intensity of the underlying non-homogeneous Poisson process (NHPP) is linear when plotted on ln-ln scale.

Let $$N(t)\,\!$$ be the cumulative number of failures observed in cumulative test time $$t\,\!$$, and let $$\rho (t)\,\!$$ be the failure intensity for the Crow-AMSAA model. Under the NHPP model, $$\rho (t)\Delta t\,\!$$ is approximately the probably of a failure occurring over the interval $$[t,t+\Delta t]\,\!$$ for small $$\Delta t\,\!$$. In addition, the expected number of failures experienced over the test interval $$[0,T]\,\!$$ under the Crow-AMSAA model is given by:


 * $$E[N(T)]=\int_{0}^{T}\rho (t)dt\,\!$$

The Crow-AMSAA model assumes that $$\rho (T)\,\!$$ may be approximated by the Weibull failure rate function:


 * $$\rho (T)=\frac{\beta }{{T}^{\beta -1}}\,\!$$

Therefore, if $$\lambda =\tfrac{1},\,\!$$ the intensity function, $$\rho (T),\,\!$$ or the instantaneous failure intensity, $${{\lambda }_{i}}(T)\,\!$$, is defined as:


 * $${{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0\,\!$$

In the special case of exponential failure times, there is no growth and the failure intensity, $$\rho (t)\,\!$$, is equal to $$\lambda \,\!$$. In this case, the expected number of failures is given by:


 * $$\begin{align}

E[N(T)]= & \int_{0}^{T}\rho (t)dt \\ = & \lambda T  \end{align}\,\!$$

In order for the plot to be linear when plotted on ln-ln scale under the general reliability growth case, the following must hold true where the expected number of failures is equal to:


 * $$\begin{align}

E[N(T)]= & \int_{0}^{T}\rho (t)dt \\ = & \lambda {{T}^{\beta }} \end{align}\,\!$$

To put a statistical structure on the reliability growth process, consider again the special case of no growth. In this case the number of failures, $$N(T),\,\!$$ experienced during the testing over $$[0,T]\,\!$$ is random. The expected number of failures, $$N(T),\,\!$$ is said to follow the homogeneous (constant) Poisson process with mean $$\lambda T\,\!$$ and is given by:


 * $$\underset{}{\overset{}{\mathop{\Pr }}}\,[N(T)=n]=\frac{n!};\text{ }n=0,1,2,\ldots \,\!$$

The Crow-AMSAA model generalizes this no growth case to allow for reliability growth due to corrective actions. This generalization keeps the Poisson distribution for the number of failures but allows for the expected number of failures, $$E[N(T)],\,\!$$ to be linear when plotted on ln-ln scale. The Crow-AMSAA model lets $$E[N(T)]=\lambda {{T}^{\beta }}\,\!$$. The probability that the number of failures, $$N(T),\,\!$$ will be equal to $$n\,\!$$ under growth is then given by the Poisson distribution:


 * $$\underset{}{\overset{}{\mathop{\Pr }}}\,[N(T)=n]=\frac{n!};\text{ }n=0,1,2,\ldots \,\!$$

This is the general growth situation, and the number of failures, $$N(T)\,\!$$, follows a non-homogeneous Poisson process. The exponential, "no growth" homogeneous Poisson process is a special case of the non-homogeneous Crow-AMSAA model. This is reflected in the Crow-AMSAA model parameter where $$\beta =1\,\!$$. The cumulative failure rate, $${{\lambda }_{c}}\,\!$$, is:


 * $$\begin{align}

{{\lambda }_{c}}=\lambda {{T}^{\beta -1}} \end{align}\,\!$$

The cumulative $$MTB{{F}_{c}}\,\!$$ is:


 * $$MTB{{F}_{c}}=\frac{1}{\lambda }{{T}^{1-\beta }}\,\!$$

As mentioned above, the local pattern for reliability growth within a test phase is the same as the growth pattern observed by Duane. The Duane $$MTB{{F}_{c}}\,\!$$ is equal to:


 * $$MTB{{F}_}=b{{T}^{\alpha }}\,\!$$

And the Duane cumulative failure rate, $${{\lambda }_{c}}\,\!$$, is:


 * $${{\lambda }_}=\frac{1}{b}{{T}^{-\alpha }}\,\!$$

Thus a relationship between Crow-AMSAA parameters and Duane parameters can be developed, such that:


 * $$\begin{align}

{{b}_{DUANE}}= & \frac{1} \\ {{\alpha }_{DUANE}}= & 1-{{\beta }_{AMSAA}} \end{align}\,\!$$

Note that these relationships are not absolute. They change according to how the parameters (slopes, intercepts, etc.) are defined when the analysis of the data is performed. For the exponential case, $$\beta =1\,\!$$, then $${{\lambda }_{i}}(T)=\lambda \,\!$$, a constant. For $$\beta >1\,\!$$, $${{\lambda }_{i}}(T)\,\!$$ is increasing. This indicates a deterioration in system reliability. For $$\beta <1\,\!$$, $${{\lambda }_{i}}(T)\,\!$$ is decreasing. This is indicative of reliability growth. Note that the model assumes a Poisson process with the Weibull intensity function, not the Weibull distribution. Therefore, statistical procedures for the Weibull distribution do not apply for this model. The parameter $$\lambda \,\!$$ is called a scale parameter because it depends upon the unit of measurement chosen for $$T\,\!$$, while $$\beta \,\!$$ is the shape parameter that characterizes the shape of the graph of the intensity function.

The total number of failures, $$N(T)\,\!$$, is a random variable with Poisson distribution. Therefore, the probability that exactly $$n\,\!$$ failures occur by time $$T\,\!$$ is:


 * $$P[N(T)=n]=\frac{n!}\,\!$$

The number of failures occurring in the interval from $${{T}_{1}}\,\!$$ to $${{T}_{2}}\,\!$$ is a random variable having a Poisson distribution with mean:


 * $$\theta ({{T}_{2}})-\theta ({{T}_{1}})=\lambda (T_{2}^{\beta }-T_{1}^{\beta })\,\!$$

The number of failures in any interval is statistically independent of the number of failures in any interval that does not overlap the first interval. At time $${{T}_{0}}\,\!$$, the failure intensity is $${{\lambda }_{i}}({{T}_{0}})=\lambda \beta T_{0}^{\beta -1}\,\!$$. If improvements are not made to the system after time $${{T}_{0}}\,\!$$, it is assumed that failures would continue to occur at the constant rate $${{\lambda }_{i}}({{T}_{0}})=\lambda \beta T_{0}^{\beta -1}\,\!$$. Future failures would then follow an exponential distribution with mean $$m({{T}_{0}})=\tfrac{1}{\lambda \beta T_{0}^{\beta -1}}\,\!$$. The instantaneous MTBF of the system at time $$T\,\!$$ is:


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

Note About Applicability
The Duane and Crow-AMSAA models are the most frequently used reliability growth models. Their relationship comes from the fact that both make use of the underlying observed linear relationship between the logarithm of cumulative MTBF and cumulative test time. However, the Duane model does not provide a capability to test whether the change in MTBF observed over time is significantly different from what might be seen due to random error between phases. The Crow-AMSAA model allows for such assessments. Also, the Crow-AMSAA allows for development of hypothesis testing procedures to determine growth presence in the data (where $$\beta <1\,\!$$ indicates that there is growth in MTBF, $$\beta =1\,\!$$ indicates a constant MTBF and $$\beta >1\,\!$$ indicates a decreasing MTBF). Additionally, the Crow-AMSAA model views the process of reliability growth as probabilistic, while the Duane model views the process as deterministic.

Parameter Estimation
The parameters for the Crow-AMSAA (NHPP) model are estimated using maximum likelihood estimation (MLE). The probability density function (pdf) of the $${{i}^{th}}\,\!$$ event given that the $${{(i-1)}^{th}}\,\!$$ event occurred at $${{T}_{i-1}}\,\!$$ is:


 * $$f({{T}_{i}}|{{T}_{i-1}})=\frac{\beta }{\eta }{{\left( \frac{{{T}_{i}}}{\eta } \right)}^{\beta -1}}\cdot {{e}^{-\tfrac{1}\left( T_{i}^{\beta }-T_{i-1}^{\beta } \right)}}\,\!$$

The likelihood function is:


 * $$L={{\lambda }^{n}}{{\beta }^{n}}{{e}^{-\lambda {{T}^{*\beta }}}}\underset{i=1}{\overset{n}{\mathop \prod }}\,T_{i}^{\beta -1}\,\!$$

where $${{T}^{*}}\,\!$$ is the termination time and is given by:


 * $${{T}^{*}}=\left\{ \begin{matrix}

{{T}_{n}}\text{ if the test is failure terminated} \\ T>{{T}_{n}}\text{ if the test is time terminated} \\ \end{matrix} \right\}\,\!$$

Taking the natural log on both sides:


 * $$\Lambda =n\ln \lambda +n\ln \beta -\lambda {{T}^{*\beta }}+(\beta -1)\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln {{T}_{i}}\,\!$$

And differentiating with respect to $$\lambda \,\!$$ yields:


 * $$\frac{\partial \Lambda }{\partial \lambda }=\frac{n}{\lambda }-{{T}^{*\beta }}\,\!$$

Set equal to zero and solve for $$\lambda \,\!$$ :


 * $$\widehat{\lambda }=\frac{n}\,\!$$

Now differentiate with respect to $$\beta \,\!$$ :


 * $$\frac{\partial \Lambda }{\partial \beta }=\frac{n}{\beta }-\lambda {{T}^{*\beta }}\ln {{T}^{*}}+\underset{i=1}{\overset{n}{\mathop \sum }}\,\ln {{T}_{i}}\,\!$$

Set equal to zero and solve for $$\beta \,\!$$ :


 * $$\widehat{\beta }=\frac{n}{n\ln {{T}^{*}}-\underset{i=1}{\overset{n}{\mathop{\sum }}}\,\ln {{T}_{i}}}\,\!$$

Biasing and Unbiasing of Beta
The equation above returns the biased estimate of $$\beta \,\!$$. The unbiased estimate of $$\beta \,\!$$ can be calculated by using the following relationships. For time terminated data (meaning that the test ends after a specified number of failures):


 * $$\bar{\beta }=\frac{N-1}{N}\hat{\beta }\,\!$$

For failure terminated data (meaning that the test ends after a specified test time):


 * $$\bar{\beta }=\frac{N-2}{N}\hat{\beta }\,\!$$

Confidence Bounds
The RGA software provides two methods to estimate the confidence bounds for the Crow Extended model when applied to developmental testing data. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. See the Crow-AMSAA Confidence Bounds chapter for details on how the confidence bounds are calculated.

Grouped Data
For analyzing grouped data, we follow the same logic described previously for the Duane model. If the $$E[N(T)]\,\!$$ equation from the Crow-AMSAA (NHPP) Model section above is linearized:


 * $$\begin{align}

\ln [E(N(T))]=\ln \lambda +\beta \ln T \end{align}\,\!$$

According to Crow [9], the likelihood function for the grouped data case, (where $${{n}_{1}},\,\!$$ $${{n}_{2}},\,\!$$ $${{n}_{3}},\ldots ,\,\!$$ $${{n}_{k}}\,\!$$ failures are observed and $$k\,\!$$ is the number of groups), is:


 * $$\underset{i=1}{\overset{k}{\mathop \prod }}\,\underset{}{\overset{}{\mathop{\Pr }}}\,({{N}_{i}}={{n}_{i}})=\underset{i=1}{\overset{k}{\mathop \prod }}\,\frac{{{(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })}^{{{n}_{i}}}}\cdot {{e}^{-(\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })}}}{{{n}_{i}}!}\,\!$$

And the MLE of $$\lambda \,\!$$ based on this relationship is:


 * $$\widehat{\lambda }=\frac{n}{T_{k}^{\widehat{\beta }}}\,\!$$

where $$n \,\!$$ is the total number of failures from all the groups.

The estimate of $$\beta \,\!$$ is the value $$\widehat{\beta }\,\!$$ that satisfies:


 * $$\underset{i=1}{\overset{k}{\mathop \sum }}\,{{n}_{i}}\left[ \frac{T_{i}^{\widehat{\beta }}\ln {{T}_{i}}-T_{i-1}^{\widehat{\beta }}\ln {{T}_{i-1}}}{T_{i}^{\widehat{\beta }}-T_{i-1}^{\widehat{\beta }}}-\ln {{T}_{k}} \right]=0\,\!$$

See the Crow-AMSAA Confidence Bounds for details on how confidence bounds for grouped data are calculated.

Crow Discrete Reliability Growth Model
The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called discrete or attribute data). Suppose system development is represented by $$i\,\!$$ configurations. This corresponds to $$i-1\,\!$$ configuration changes, unless fixes are applied at the end of the test phase, in which case there would be $$i\,\!$$ configuration changes. Let $${{N}_{i}}\,\!$$ be the number of trials during configuration $$i\,\!$$ and let $${{M}_{i}}\,\!$$ be the number of failures during configuration $$i\,\!$$. Then the cumulative number of trials through configuration $$i\,\!$$, namely $${{T}_{i}}\,\!$$, is the sum of the $${{N}_{i}}\,\!$$ for all $$i\,\!$$, or:


 * $${{T}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{N}_{i}}\,\!$$

And the cumulative number of failures through configuration $$i\,\!$$, namely $${{K}_{i}}\,\!$$, is the sum of the $${{M}_{i}}\,\!$$ for all $$i\,\!$$, or:


 * $${{K}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{M}_{i}}\,\!$$

The expected value of $${{K}_{i}}\,\!$$ can be expressed as $$E[{{K}_{i}}]\,\!$$ and defined as the expected number of failures by the end of configuration $$i\,\!$$. Applying the learning curve property to $$E[{{K}_{i}}]\,\!$$ implies:


 * $$E\left[ {{K}_{i}} \right]=\lambda T_{i}^{\beta }\,\!$$

Denote $${{f}_{1}}\,\!$$ as the probability of failure for configuration 1 and use it to develop a generalized equation for $${{f}_{i}}\,\!$$ in terms of the $${{T}_{i}}\,\!$$ and $${{N}_{i}}\,\!$$. From the equation above, the expected number of failures by the end of configuration 1 is:


 * $$E\left[ {{K}_{1}} \right]=\lambda T_{1}^{\beta }={{f}_{1}}{{N}_{1}}\,\!$$


 * $$\therefore {{f}_{1}}=\frac{\lambda T_{1}^{\beta }}\,\!$$

Applying the $$E\left[ {{K}_{i}}\right]\,\!$$ equation again and noting that the expected number of failures by the end of configuration 2 is the sum of the expected number of failures in configuration 1 and the expected number of failures in configuration 2:


 * $$\begin{align}

E\left[ {{K}_{2}} \right] = & \lambda T_{2}^{\beta } \\ = & {{f}_{1}}{{N}_{1}}+{{f}_{2}}{{N}_{2}} \\ = & \lambda T_{1}^{\beta }+{{f}_{2}}{{N}_{2}} \end{align}\,\!$$


 * $$\therefore {{f}_{2}}=\frac{\lambda T_{2}^{\beta }-\lambda T_{1}^{\beta }}\,\!$$

By this method of inductive reasoning, a generalized equation for the failure probability on a configuration basis, $${{f}_{i}}\,\!$$, is obtained, such that:


 * $${{f}_{i}}=\frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}\,\!$$

For the special case where $${{N}_{i}}=1\,\!$$ for all $$i\,\!$$, the equation above becomes a smooth curve, $${{g}_{i}}\,\!$$, that represents the probability of failure for trial by trial data, or:


 * $${{g}_{i}}=\lambda \cdot {{i}^{\beta }}-\lambda \cdot {{\left( i-1 \right)}^{\beta }}\,\!$$

In this equation, $$i\,\!$$ represents the trial number. Thus an equation for the reliability (probability of success) for the $${{i}^{th}}\,\!$$ configuration is obtained:


 * $$\begin{align}

{{R}_{i}}=1-{{f}_{i}} \end{align}\,\!$$

The equation for the reliability for the $${{i}^{th}}\,\!$$ trial is:


 * $$\begin{align}

{{R}_{i}}=1-{{g}_{i}} \end{align}\,\!$$

Maximum Likelihood Estimators for Discrete Model
This section describes procedures for estimating the parameters of the Crow-AMSAA model for success/failure data. An example is presented illustrating these concepts. The estimation procedures provide maximum likelihood estimates (MLEs) for the model's two parameters, $$\lambda \,\!$$ and $$\beta \,\!$$. The MLEs for $$\lambda \,\!$$ and $$\beta \,\!$$ allow for point estimates for the probability of failure, given by:


 * $${{\hat{f}}_{i}}=\frac{\hat{\lambda }T_{i}^-\hat{\lambda }T_{i-1}^}=\frac{\hat{\lambda }\left( T_{i}^-T_{i-1}^ \right)}\,\!$$

And the probability of success (reliability) for each configuration $$i\,\!$$ is equal to:


 * $${{\hat{R}}_{i}}=1-{{\hat{f}}_{i}}\,\!$$

The likelihood function is:


 * $$\underset{i=1}{\overset{k}{\mathop \prod }}\,\left( \begin{matrix}

{{N}_{i}} \\ {{M}_{i}} \\ \end{matrix} \right){{\left( \frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }} \right)}^}{{\left( \frac{{{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta }} \right)}^{{{N}_{i}}-{{M}_{i}}}}\,\!$$

Taking the natural log on both sides yields:


 * $$\begin{align}

& \Lambda = & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ \ln \left( \begin{matrix}  {{N}_{i}}  \\   {{M}_{i}}  \\ \end{matrix} \right)+{{M}_{i}}\left[ \ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] \\ & & +\underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ ({{N}_{i}}-{{M}_{i}})\left[ \ln ({{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] \end{align}\,\!$$

Taking the derivative with respect to $$\lambda \,\!$$ and $$\beta \,\!$$ respectively, exact MLEs for $$\lambda \,\!$$ and $$\beta \,\!$$ are values satisfying the following two equations:


 * $$\begin{align}

& \underset{i=1}{\overset{K}{\mathop \sum }}\,{{H}_{i}}\times {{S}_{i}}= & 0 \\ & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{U}_{i}}\times {{S}_{i}}= & 0 \end{align}\,\!$$


 * where:


 * $$\begin{align}

{{H}_{i}}= & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ T_{i}^{\beta }\ln {{T}_{i}}-T_{i-1}^{\beta }\ln {{T}_{i-1}} \right] \\ {{S}_{i}}= & \frac{\left[ \lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta } \right]}-\frac{{{N}_{i}}-{{M}_{i}}}{\left[ {{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta } \right]} \\ {{U}_{i}}= & T_{i}^{\beta }-T_{i-1}^{\beta }\, \end{align}\,\!$$

Discrete Model for Mixed Data
In the RGA software, the Discrete Data > Mixed Data option gives a data sheet that can have input data that is either a configuration in groups or individual trial by trial, or a mixed combination of individual trials and configurations of more than one trial. The calculations use the same mathematical methods described in the Grouped Data section.

Mixed Data Confidence Bounds
Bounds on Average Failure Probability for Mixed Data The process to calculate the average unreliability confidence bounds for mixed data is as follows:
 * 1)	Calculate the average failure probability $$({{t}_{1}},{{t}_{2}})\,\!$$.
 * 2)	There will exist a $${{t}^{*}}\,\!$$ between $${{t}_{1}}\,\!$$ and $${{t}_{2}}\,\!$$ such that the instantaneous unreliability at $${{t}^{*}}\,\!$$ equals the average unreliability $$({{t}_{1}},{{t}_{2}})\,\!$$. The confidence intervals for the instantaneous unreliability at $${{t}^{*}}\,\!$$ are the confidence intervals for the average unreliability $$({{t}_{1}},{{t}_{2}})\,\!$$.

Bounds on Average Reliability for Mixed Data The process to calculate the average reliability confidence bounds for mixed data is as follows:
 * 1)	Calculate confidence bounds for average unreliability $$({{t}_{1}},{{t}_{2}})\,\!$$ as described above.
 * 2)	The confidence bounds for reliability are 1 minus these confidence bounds for average unreliability.