## Wqb0

2 (for where we take q1 = 0.5, b-i^ = 0.7, q2 = 0.6, b2 = 0.9, q3 = 0.5, b3 discrete time Weibull distribution, see, e.g., [Bra01], [BGX01]).

The empirical estimator and confidence interval at level 95% for the BMP-failure rate and the RG-failure rate of the system, for the total time of observation M = 5000, are given in Figure 2.

Figure 3 gives a comparison between failure rates estimators obtained for different sample sizes. We see that, as M increases, the estimators approach the true value. We also notice that the failure rates become constant as time

- true value of BMP-failure rate empirical estimator 95% confidence interval

— true value of RG-failure rate * empirical estimator

- 95% confidence interval

*********************************************************:

*********************************************************: — true value of RG-failure rate * empirical estimator

- 95% confidence interval

- true value of BMP-failure rate empirical estimator 95% confidence interval 50 200 250 300 0 50 100 150 200 250 300

50 200 250 300 0 50 100 150 200 250 300

Fig. 2. Failure rates estimators and confidence interval at level 95%

increases, that is the semi-Markov system approaches a Markov one as time increases.

 — true value of BMP- failure rate * empirical estimator M =1000 ■-• empirical estimator M =2000 • empirical estimator M 5000 0 50

— true value of RG-failure rate * empirical estimator: M=1000

- empirical estimator: M=2000 ■ empirical estimator: M=5000

— true value of RG-failure rate * empirical estimator: M=1000

- empirical estimator: M=2000 ■ empirical estimator: M=5000 150 200 250 300 0 50 100 150 200 250 300

0 50

150 200 250 300 0 50 100 150 200 250 300

### Fig. 3. Failure rates estimators consistency

Figure 4 presents the failure rates variances estimators for some different values of the sample size. As in Figure 3, we can note the consistency of variances estimators.

### References

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