Duane Model

In 1962, J. T. Duane published a report in which he presented failure data of different systems during their development programs [8]. While analyzing the data, it was observed that the cumulative MTBF versus cumulative operating time followed a straight line when plotted on log-log paper, as shown next.



Based on that observation, Duane developed his model as follows. If $$N(T)\,\!$$ is the number of failures by time $$T\,\!$$, the observed mean (average) time between failures, $$MTB{{F}_{c}},\,\!$$ at time $$T\,\!$$ is:


 * $$MTB{{F}_{c}}=\frac{T}{N(T)}\,\!$$

The equation of the line can be expressed as:


 * $$\begin{align}

y=mx+c \end{align}\,\!$$

Setting:


 * $$\begin{align}

y= & \ln (MTB{{F}_{c}}) \\ x= & \ln (T) \\ m= & \alpha \\ c= & \ln b \end{align}\,\!$$

yields:


 * $$\begin{align}

\ln (MTB{{F}_{c}})=\alpha \ln (T)+\ln b \end{align}\,\!$$

Then equating $$MTB{{F}_{c}}\,\!$$ to its expected value, and assuming an exact linear relationship, gives:


 * $$\begin{align}

E(MTB{{F}_{c}})=b{{T}^{\alpha }} \end{align}\,\!$$

or:


 * $$\begin{align}

MTB{{F}_{c}}=b{{T}^{\alpha }} \end{align}\,\!$$

And, if you assume a constant failure intensity, then the cumulative failure intensity, $${{\lambda }_{c}}\,\!$$, is:


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

or:


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

Also, the expected number of failures up to time $$T\,\!$$ is:


 * $$\begin{align}

E(N(T))= & {{\widehat}_{c}}\cdot T \\ = & \frac{1}{b}{{T}^{1-\alpha }} \end{align}\,\!$$

where:


 * $${{\widehat}_{c}}\,\!$$ is the average estimate of the cumulative failure intensity, failures/hour.
 * $$T\,\!$$ is the total accumulated unit hours of test and/or development time.
 * $$1/b\,\!$$ is the cumulative failure intensity at $$T = 1\,\!$$ or at the beginning of the test, or the earliest time at which the first $${{\widehat{{\bar{\lambda }}}}}\,\!$$ is predicted, or the $${{\widehat{{\bar{\lambda }}}}}\,\!$$ for the equipment at the start of the design and development process.
 * $$\alpha\,\!$$is the improvement rate in the $${{\widehat{{\bar{\lambda }}}}}, 0\le \alpha \le 1\,\!$$.

The corresponding $$MTB{{F}_{c}}\,\!$$, or $${{\hat{m}}_{c}}\,\!$$, is equal to:


 * $${{\hat{m}}_{c}}=b{{T}^{\alpha }}\,\!$$

where $$b=\,\!$$ cumulative MTBF at $$T=1\,\!$$ or at the beginning of the test, or the earliest time at which the first $$\hat{m}\,\!$$ can be determined, or the $$\hat{m}\,\!$$ predicted at the start of the design and development process ( $$b>0\,\!$$ ).

The cumulative MTBF, $${{\hat{m}}_{c}}\,\!$$, and $${{\widehat{{\bar{\lambda }}}}_{c}}\,\!$$ tell whether $$m\,\!$$ is increasing or $$\lambda \,\!$$ is decreasing with time, utilizing all data up to that time. You may want to know, however, the instantaneous $${{\hat{m}}_{i}}\,\!$$ or $${{\widehat{{\bar{\lambda }}}}_{i}}\,\!$$ to see what you are doing at a specific instant or after a specific test and development time. The instantaneous failure intensity, $${{\lambda }_{i}}\,\!$$, is:


 * $$\begin{align}

{{\lambda }_{i}}= & \frac{d(E(N(T)))}{dT} \\ = & \frac{1}{b}(1-\alpha ){{T}^{-\alpha }} \\ = & (1-\alpha ){{\lambda }_{c}} \end{align}\,\!$$

Similarly, using the equation for the expected number of failures up to time $$T\,\!$$, this procedure yields:


 * $$\begin{align}

{{m}_{i}}= & \frac{1}{1-\alpha }b{{T}^{\alpha }} \\ = & \frac{1}{1-\alpha }{{{\hat{m}}}_{c}},:\ \ \alpha \not{=}1 \end{align}\,\!$$

where $$\alpha =1\,\!$$ implies infinite MTBF growth.

As shown in these derivations, the instantaneous failure intensity improvement line is obtained by shifting the cumulative failure intensity line down, parallel to itself, by a distance of $$(1-\alpha )\,\!$$. Similarly, the current or instantaneous MTBF growth line is obtained by shifting the cumulative MTBF line up, parallel to itself, by a distance of $$\tfrac{1}{1-\alpha }\,\!$$, as illustrated in the figure below.



Parameter Estimation
The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for a priori estimation.

Graphical Method
The Duane model for the cumulative failure intensity is:
 * $${{\widehat}_{c}}=\frac{1}{b}{{T}^{-\alpha }}\,\!$$

This equation may be linearized by taking the natural log of both sides:


 * $$\ln \left( {{\widehat}_{c}} \right)=\ln \left( \frac{1}{b} \right)-\alpha \ln (T)\,\!$$

Consequently, plotting $$\widehat\,\!$$ versus $$T\,\!$$ on log-log paper will result in a straight line with a negative slope, such that:


 * $$\ln \left ({\frac{1}{b}} \right)\,\!$$ is the y-intercept at $$T = 1\,\!$$
 * $$\frac{1}{b}\,\!$$ is the cumulative failure intensity at $$T = 1\,\!$$
 * $$\alpha\,\!$$ is the slope of the straight line on the log-log plot

Similarly, the corresponding MTBF of the cumulative failure intensity can also be linearized by taking the natural log of both sides:


 * $$\begin{align}

\hat{m_{c}}&=bT^{\alpha } \\ \ln \hat{m_{c}}&=\ln b+\alpha \ln T \\ \end{align}\,\!$$

Plotting $$\hat{m}\,\!$$ versus $$T\,\!$$ on log-log paper will result in a straight line with a positive slope such that:


 * $$\ln{b}\,\!$$ is the y-intercept at $$T = 1\,\!$$
 * $$b\,\!$$ is the cumulative mean time between failure at $$T = 1\,\!$$
 * $$\alpha\,\!$$ is the slope of the straight line on the log-log plot

Two ways of determining these curves are as follows:


 * 1. Predict the $${{\widehat}_{0}}\,\!$$ and $$\hat{m}=\,\!$$ $$\tfrac{1}\,\!$$ of the system from its reliability block diagram and available component failure intensities. Plot this value on log-log plotting paper at $$T=1.\,\!$$ From past experience and from past data for similar equipment, find values of $${{\alpha }_{1}}\,\!$$, the slope of the improvement lines for $$\widehat\,\!$$ or $$\hat{m}\,\!$$. Modify this $$\alpha \,\!$$ as necessary. If a better design effort is expected and a more intensive research, test and development or TAAF program is to be implemented, then a 15% improvement in the growth rate may be attainable. Consequently, the available value for slope $$\alpha \,\!$$, and $${{\alpha }_{1}}\,\!$$, should be adjusted by this amount. The value to be used will then be $$\alpha =1.15{{\alpha }_{1}}.\,\!$$ A line is then drawn through point $${{\widehat}_{0}}\,\!$$ and $$T=1\,\!$$ with the just determined slope $$\alpha \,\!$$, keeping in mind that $$\alpha \,\!$$ is negative for the $$\widehat\,\!$$ curve. This line should be extended to the design, development and test time scheduled to be expended to see if the failure intensity goal will indeed be achieved on schedule. It is also possible to find that the design, development and test time to achieve the goal may be earlier than the delivery date or later. If earlier, then either the reliability program effort can be judiciously and appropriately trimmed; or if it is an incentive contract, full advantage is taken of the fact that the failure intensity goal can be exceeded with the associated increased profits to the company. A similar approach may be used for the MTBF growth model, where $${{\hat{m}}_{0}}=\tfrac{1}\,\!$$ is plotted at $$T=1\,\!$$, and a line is drawn through the point $${{\hat{m}}_{0}}\,\!$$ and $$T=1\,\!$$ with slope $$\alpha \,\!$$ to obtain the MTBF growth line. If $$\alpha \,\!$$ values are not available, consult the table below, which gives actual $$\alpha \,\!$$ values for various types of equipment. These have been obtained from literature or by MTBF growth tests. It may be seen from the following table that $$\alpha \,\!$$ values range between 0.24 and 0.65. The lower values reflect slow early growth and the higher values reflect fast early growth.


 * 2. During the design, development and test phase and at specific milestones, the $$\widehat=\tfrac{1}\,\!$$ is calculated from the total failures and $$T\,\!$$ values. These values of $$\widehat\,\!$$ or $$\hat{m}\,\!$$ are plotted above the corresponding $$T\,\!$$ values on log-log paper. A straight line is drawn favoring these points to minimize the distance between the points and the line, thus establishing the improvement or growth model and its parameters graphically. If needed, linear regression analysis techniques can be used to determine these parameters.

Least Squares (Linear Regression)
The parameters can also be estimated using a mathematical approach. To do this, linearize the MTBF of the cumulative failure intensity by taking the natural log of both sides, and then apply least squares analysis. For example:


 * $$\begin{align}

\hat{m_{c}}&=bT^{\alpha } \\ \ln \hat{m_{c}}&=\ln b+\alpha \ln T \\ \end{align}\,\!$$

For simplicity in the calculations, let:


 * $$\begin{align}

\ln ({{m}_{ci}})= & {{Y}_{i}} \\ \ln (b)= & a \\ \alpha = & c \\ \ln ({{T}_{i}})= & {{X}_{i}} \end{align}\,\!$$

Therefore, the equation becomes:


 * $${{Y}_{i}}=\widehat{a}+\widehat{c}{{X}_{i}}\,\!$$

Assume that a set of data pairs $$({{X}_{1}},{{Y}_{1}})\,\!$$, $$({{X}_{2}},{{Y}_{2}})\,\!$$,..., $$({{X}_{N}},{{Y}_{N}})\,\!$$ were obtained and plotted. Then according to the Least Squares Principle, which minimizes the vertical distance between the data points and the straight line fitted to the data, the best fitting straight line to this data set is the straight line $$Y=\widehat{a}+\widehat{c}X\,\!$$ such that:


 * $$\underset{i=1}{\overset{N}{\mathop \sum }}\,{{(\widehat{a}+\widehat{c}{{X}_{i}}-{{Y}_{i}})}^{2}}=\underset{(a,c)}{\mathop{min}}\,\underset{i=1}{\overset{N}{\mathop \sum }}\,{{(a+c{{X}_{i}}-{{Y}_{i}})}^{2}}\,\!$$

where $$\widehat{a}\,\!$$ and $$\widehat{c}\,\!$$ are the least squares estimates of $$a\,\!$$ and $$c\,\!$$. To obtain $$\widehat{a}\,\!$$ and $$\widehat{c}\,\!$$, let:


 * $$F=\underset{i=1}{\overset{N}{\mathop \sum }}\,{{(a+c{{X}_{i}}-{{Y}_{i}})}^{2}}\,\!$$

Differentiating $$F\,\!$$ with respect to $$a\,\!$$ and $$c\,\!$$ yields:


 * $$\frac{\partial F}{\partial a}=2\underset{i=1}{\overset{N}{\mathop \sum }}\,(a+c{{X}_{i}}-{{Y}_{i}})\,\!$$


 * and:


 * $$\frac{\partial F}{\partial c}=2\underset{i=1}{\overset{N}{\mathop \sum }}\,(a+c{{X}_{i}}-{{Y}_{i}}){{X}_{i}}\,\!$$

Set those two equations equal to zero:


 * $$\underset{i=1}{\overset{N}{\mathop \sum }}\,(a+c{{X}_{i}}-{{Y}_{i}})=\underset{i=1}{\overset{N}{\mathop \sum }}\,(\widehat-{{Y}_{i}})=-\underset{i=1}{\overset{N}{\mathop \sum }}\,({{Y}_{i}}-\widehat)=0\,\!$$


 * and:


 * $$\underset{i=1}{\overset{N}{\mathop \sum }}\,(a+c{{X}_{i}}-{{Y}_{i}}){{X}_{i}}=\underset{i=1}{\overset{N}{\mathop \sum }}\,(\widehat-{{Y}_{i}}){{X}_{i}}=-\underset{i=1}{\overset{N}{\mathop \sum }}\,({{Y}_{i}}-\widehat){{X}_{i}}=0\,\!$$

Solve the equations simultaneously:


 * $$ \begin{align}

\widehat{a}& =\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{Y}_{i}}}N-\widehat{c} \frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{X}_{i}}}N \\ & = \overline{Y} - \hat{c}\overline{X} \\ \end{align}\,\!$$

and:


 * $$\widehat{c}=\frac{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{X}_{i}}{{Y}_{i}}-\tfrac{\left( \underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{X}_{i}}\underset{i=1}{\overset{N}{\mathop{\sum }}}\,{{Y}_{i}} \right)}{N}}{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,X_{i}^{2}-\tfrac{N}}\,\!$$

Now substituting back $$\ln ({{m}_{ci}})={{Y}_{i}},\,\!$$ $$\ln (b)=a,\,\!$$ $$\alpha =c\,\!$$ and $$\ln ({{T}_{i}})={{X}_{i}},\,\!$$ we have:


 * $$\widehat{b}={{e}^{\tfrac{1}{n}\left[ \underset{i=1}{\overset{n}{\mathop{\sum }}}\,\ln ({{m}_{ci}})-\alpha \underset{i=1}{\overset{n}{\mathop{\sum }}}\,\ln ({{T}_{i}}) \right]}}\,\!$$


 * where:


 * $$\widehat{\alpha }=\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,\ln ({{T}_{i}})\ln ({{m}_{ci}})-\tfrac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,\ln ({{T}_{i}})\underset{i=1}{\overset{n}{\mathop{\sum }}}\,\ln ({{m}_{ci}})}{n}}{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{\left[ \ln ({{T}_{i}}) \right]}^{2}}-\tfrac{n}}\,\!$$

Maximum Likelihood Estimators
L. 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. The Crow-AMSAA model, which is described in the Crow-AMSAA (NHPP) chapter, provides a complete Maximum Likelihood Estimation (MLE) solution to the Duane model.

Confidence Bounds
Least squares confidence bounds can be computed for both the model parameters and metrics of interest for the Duane model.

Parameter Bounds
Apply least squares analysis on the Duane model:


 * $$\ln ({{\hat{m}}_{c}})=\ln (b)+\alpha \ln (t)\,\!$$

The unbiased estimator of $${{\sigma}^{2}}\,\!$$ can be obtained from:


 * $${{\sigma }^{2}}=Var\left[ \ln {{m}_{c}}(t) \right]=\frac{SSE}{(n-2)}\,\!$$

where:


 * $$SSE=\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left[ \ln {{{\hat{m}}}_{c}}({{t}_{i}})-\ln {{m}_{c}}({{t}_{i}}) \right]}^{2}}\,\!$$

Thus, the confidence bounds on $$\alpha \,\!$$ and $$b\,\!$$ are:


 * $$C{{B}_{\alpha }}=\hat{\alpha }\pm {{t}_{n-2,\alpha /2}}SE(\hat{\alpha })\,\!$$


 * $$C{{B}_{b}}=\hat{b}{{e}^{\pm {{t}_{n-2,\alpha /2}}SE\left[ \ln (\hat{b}) \right]}}\,\!$$

where $${{t}_{n-2,\alpha /2}}\,\!$$ denotes the percentage point of the $$t\,\!$$ distribution with $$n-2\,\!$$ degrees of freedom such that $$P\{{{t}_{n-2}}\ge {{t}_{\alpha /2,n-2}}\}=\alpha /2\,\!$$ and:


 * $$SE(\hat{\alpha })=\frac{\sigma }{\sqrt}\,\!$$


 * $$SE\left[ \ln (\hat{b}) \right]=\sigma \cdot \sqrt{\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{(\ln {{t}_{i}})}^{2}}}{n\cdot {{S}_{xx}}}}\,\!$$


 * $${{S}_{xx}}=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{(\ln {{t}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,\ln ({{t}_{i}}) \right)}^{2}}\,\!$$

Other Bounds
Confidence bounds also can be obtained on the cumulative MTBF and the cumulative failure intensity:


 * $$C{{B}_}={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var\left[ \ln ({{{\hat{m}}}_{c}}) \right]}}}\,\!$$


 * $$\begin{align}

{{[{{\lambda }_{c}}(t)]}_{L}}= & \frac{1} \\ {{[{{\lambda }_{c}}(t)]}_{U}}= & \frac{1} \end{align}\,\!$$

When $$n\,\!$$ is large, the approximate $$100(1-\alpha )%\,\!$$ confidence bounds for instantaneous MTBF are given by:


 * $$\begin{align}

{{m}_{i}}{{(t)}_{L}}= & \frac \\ {{m}_{i}}{{(t)}_{U}}= & \frac \end{align}\,\!$$

and
 * $$\lambda_{i}(t) = \frac {1}{m_{i}(t)}\,\!$$

therefore, the confidence bounds on the instantaneous failure intensity are:


 * $$\begin{align}

{{[{{\lambda }_{i}}(t)]}_{L}}= & \frac{1} \\ {{[{{\lambda }_{c}}(t)]}_{U}}= & \frac{1} \end{align}\,\!$$