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Crow-AMSAA (NHPP) Model

Grouped Data Confidence Bounds for the Crow-AMSAA Model

Grouped Data for the Crow-AMSAA Model

Grouped Data Confidence Bounds Examples for the Crow-AMSAA Model

Crow-AMSAA Example 5

A new helicopter system is under development. System failure data has been collected on five helicopters during the final test phase. The actual failure times cannot be determined since the failures are not discovered until after the helicopters are brought into the maintenance area. However, total flying hours are known when the helicopters are brought in for service and every two weeks, each helicopter undergoes a thorough inspection to uncover any failures that may have occurred since the last inspection. Therefore, the cumulative total number of flight hours and the cumulative total number of failures for the five helicopters are known for each two-week period. The total number of flight hours from the test phase is 500, which was accrued over a period of 12 weeks (6 2-week intervals). For each 2-week interval, the total number of flight hours and total number of failures for the five helicopters were recorded. The grouped data set is displayed in Table 5.3.

Table 5.3 - Grouped data for a new helicopter system

Interval

Interval Length

Failures In Interval

1

0 - 62

12

2

62 -100

6

3

100 - 187

15

4

187 - 210

3

5

210 - 350

18

6

350 - 500

16

  1. Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
  2. Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods.
Solution to Crow-AMSAA Example 5
  1. Obtain the estimator of β using Eqn. (33). Using RGA, the value of $\widehat{\beta }$ is 0.81361. Now plug this value into Eqn. (32) and $\widehat{\lambda }$ is: MATHFisher Matrix confidence bounds can be obtained on the parameters $\widehat{\beta }$ and $\widehat{\lambda }$ at the 90% confidence level by:

MATHand: MATH

Crow confidence bounds can also be obtained on the parameters $\widehat{\beta }$ and $\widehat{\lambda }$ at the 90% confidence level, as:

MATHand: MATH

  1. The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for T = 500 hr are:

MATHand: MATH

Figures 5.10 and 5.11 show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF.

Figure

Figure 5.10: Cumulative MTBF with 2-sided 90% Fisher Matrix confidence bounds

Figure

Figure 5.11: Instantaneous MTBF with 2-sided 90% Fisher Matrix confidence bounds

The Crow confidence bounds for the cumulative and instantaneous MTBF at the 90% 2-sided confidence level and for T = 500 hours are:

MATH and: MATH

Figures 5.12 and 5.13 show plots of the Crow confidence bounds for the cumulative and instantaneous MTBF.

Figure

Figure 5.12: Cumulative MTBF with 2-sided 90% Crow confidence bounds

Figure

Figure 5.13: Instantaneous MTBF with 2-sided 90% Crow confidence bounds