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Analyzing Success/Failure Data Using the
Crow Discrete Reliability Growth Model
[Editor's Note: This article has been updated since its original publication to reflect a more recent version of the software interface.] 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.
Success/Failure Data
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, 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
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 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
The Data Folio used to analyze these data is created in RGA as follows.
The data are entered in RGA as follows.
The estimated parameters are:
β = 0.7801 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 The Data Folio used to analyze these data is created in RGA as follows.
The results of the test, as entered in RGA, are shown next.
The estimated parameters are:
β
= 0.4618 The growth rate is estimated to be 1 - β = 0.5382. The demonstrated reliability at the 20^{th} trial is 88.57%.
The model can be used to estimate the number of additional test trials to demonstrate a certain reliability goal. For example, if the goal is 90% reliability, 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.
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