Parameter Estimation for the Logistic Model

Logistic Model Example 1

Using the reliability growth data given in Table 8.1, do the following:

1. Find a Gompertz curve that represents the data and plot it with the raw data.

2. Find a Logistic reliability growth curve that represents the data and plot it with the raw data.

Table 8.1 - Development time versus observed reliability data and predicted reliabilities

Solution to Logistic Model Example 1

1. The entered data and the estimated parameters using the Standard Gompertz model are shown in Figure 8.1.

Therefore:

 
(3)

Figure 8.1: Estimated Logistic parameters for Example 1

The values of predicted reliabilities are plotted in Figure 8.2.

Figure 8.2.: Gompertz Reliability vs. Time plot

Notice how the Standard Gompertz model is not really capable of handling the S-shaped characteristics of this data.

2. The least squares estimators of the Logistic growth curve parameters are [9]:

   (4)

   (5)

   where:

   (6)

   (7)

   (8)

   (9)

   In this example N = 9, which gives:

   From Eqns. 6 and 7:

And from Eqns. 4 and 5:

Therefore, the Logistic reliability growth curve that represents this data set is given by:

(10)

Figure 8.3 shows the Reliability vs. Time plot. The plot shows that the observed data set is estimated well by the Logistic reliability growth curve, except in the region closely surrounding the inflection point of the observed reliability. This problem can be overcome by using the Modified Gompertz model, which was presented in the Gompertz Models chapter of this on-line reference.

Figure 8.3: Logistic Reliability vs. Time plot

Logistic Model Example 2

A prototype was tested under a success/failure pattern. The test consisted of 15 runs and the data from the test is presented in Table 8.2a. Find the Logistic model that best fits the data set and plot it along with the reliability observed from the raw data.

Table 8.2a - Prototype Sequential test data

Solution to Logistic Model Example 2

The first run is ignored because it was a success and the reliability at that point was 100%. This failure will be ignored throughout the analysis because it is considered that the test starts when the reliability is not equal to zero or one. The test essentially begins at time 1, and is now considered as time 0 with N = 14. The observed reliability is shown in column 3 of Table 8.2b. Keep in mind that the observed reliability values still account for the initial suspension.

Table 8.2b - Prototype observed reliability values

Eqn. 9 becomes:

and:

Now, from the least squares estimators, Eqns. 6 and 7 are:

Therefore:

The Logistic reliability model that best fits the data is given by:

Figure 8.4 shows the Reliability vs. Time plot.

Figure 8.4: Logistic Reliability vs. Time plot

Logistic Model Example 3

Some equipment underwent testing in different stages. For example, the testing may have been performed in subsequent days, weeks or months with an unequal number of units tested every day. Each group was tested and several failures occurred. The data set is given in columns 1 and 2 of Table 8.3. Find the Logistic model that best fits the data and plot it along with the reliability observed from the raw data.

Table 8.3 - Grouped per Configuration data

Solution to Logistic Model Example 3

The observed reliability is and column 4 of Table 8.3 shows the values for each group. Then with N = 9, Eqn. 9 becomes:

and:

Now from the least squares estimators, Eqns. 6 and 7 give:

Therefore:

The Logistic reliability model that best fits the data is given by:

Figure 8.5 shows the Reliability vs. Time plot.

Figure 8.5: Logistic Reliability vs. Time plot displaying the intervals

See Also:
Logistic

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