Issue 81, November 2007

Typical developmental testing reliability growth analysis involves the use of times-to-failure data obtained from the tested system(s) during the development program. Many products or systems, such as missiles and air bags, may not produce times-to-failure data because they are considered one time/usage (or one-shot) systems. Such scenarios require models other than the ones used to analyze times-to-failure data. In this article, we cover the Crow-AMSAA Discrete reliability growth model that is found in RGA 6.

Success/Failure Data
Success/failure data are also referred to as discrete or attribute data. They are obtained from a test for units in which there are only two possible outcomes: success or failure. An example of this is a missile that gets fired once and either succeeds or fails. The data types available in RGA 6 for success/failure data entry are:

• Grouped per Configuration

• Sequential

• Sequential with Mode

Grouped per Configuration data are obtained from a test that incorporates design changes after multiple trials of units of a single configuration. Sequential data represent the sequential results of prototypes that are improved throughout the test program. The Sequential with Mode data type is the same as the Sequential data type with the addition that the cause of failure (failure mode) is noted for each failure.

There are multiple models that can be used to analyze success/failure data, such as the Crow-AMSAA (NHPP) Discrete model, standard Gompertz, Lloyd Lipow, modified Gompertz, Duane and the logistic model.

In this article we will address the Crow-AMSAA (NHPP) Discrete model. In RGA 6, this model is one of the models used to analyze the Grouped per Configuration and the Sequential data types. The Crow-AMSAA model is common in the analysis of times-to-failure data, but to use it in the context of failure/success data, some modifications are needed. The procedure is explained next.

The Crow-AMSAA (NHPP) Discrete Model
Suppose a reliability growth program 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:

• Ni = the number of trials during configuration i.

• Ti = the cumulative number of trials through configuration i.

• Mi = the number of failures during configuration i.

• Ki = the cumulative number of failures through configuration i.

The expected number of failures by the end of configuration i is:

Applying the properties of the learning curve model, log(E[K_i]) is assumed to be linear when plotted against log(T_i).

The failure probability on a configuration basis, fi, is:

In case N_i = 1 for all i, a fix is incorporated after each trial and the data set becomes "trial by trial" data. In this case, the above failure probability calculation becomes a smooth curve represented by the probability of failure, gi:

The reliability for the ith configuration is:

And in case N_i = 1 for all i:

The parameters of the Crow-AMSAA Discrete model can be estimated using maximum likelihood estimation (MLE). For more information, click here.

The Discrete Crow-AMSAA (NHPP) Model for Sequential Data
The expected number of failures up to a trial, T, is:

E[N(T)]=λT^β

The log(E[N(T)] is linear when plotted against log(T_i).

The cumulative MTBF is:

The instantaneous MTBF is:

The instantaneous failure intensity is:

FI_i(T)=λβT^β-1

The parameters of the Crow-AMSAA Discrete model can be estimated using maximum likelihood estimation (MLE).

Example 1 - Grouped per Configuration
A one-shot system undergoes reliability growth development testing for a total of 68 trials. Delayed corrective actions are incorporated after the 14th, 33rd and 48th trials. From trial 49 to trial 68 the configuration is not changed. In each new configuration, new units are built incorporating the design changes.

• Configuration 1: 14 trials and 5 failures.

• Configuration 2: 19 trials and 3 failures.

• Configuration 3: 15 trials and 4 failures.

• Configuration 4: 20 trials and 4 failures.

The Data Folio used to analyze these data is created in RGA 6 as follows.

The data are entered in RGA 6 as follows.

The estimated parameters are:

β = 0.7801
λ = 0.5954

The growth rate is estimated to be 1 - β = 0.2199.

The estimated reliability at each configuration is shown next.

The achieved reliability at the end of the growth program is R = 0.8096.

Example 2 -  Sequential Data
In this example, a one-shot system undergoes reliability growth development testing conducted in a sequential fashion. A total of 20 different units are tested with improvements implemented throughout the test program.

The Data Folio used to analyze these data is created in RGA 6 as follows.

The results of the test, as entered in RGA 6, are shown next.

The estimated parameters are:

β = 0.4618
λ = 1.2415

The growth rate is estimated to be 1 - β = 0.5382.

The demonstrated MTBF is 8.74 trials (i.e. the demonstrated failure intensity at the end of the test is 0.1143 failures per unit). Because we are dealing with one unit at the 20^th trial, this metric can be interpreted as the probability of failure. Therefore, the demonstrated reliability is 0.8857.

The model can be used to estimate the number of additional test trials to demonstrate a certain MTBF goal. For example, if the MTBF goal is 10 trials (i.e. the reliability goal is 0.9) then the required additional testing trials is 25.64 - 20 = 5.64 trials. The following figure shows the calculation for the estimate of the total necessary trials.

Conclusion
This article explained a process for analyzing failure/success data using the Crow-AMSAA Discrete model. The model was used to estimate the reliability throughout the test and estimate additional trials needed to demonstrate a certain reliability goal.

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