Examples

 Example 1. The input data for this example is generated by Monte Carlo method for Kijima model 1 with restoration factor q=0 and Weibull underlying function with shape parameter beta=2.0 and scale parameter eta=1.0. The mean time to repair is 0.02. We obtained the following input data for 5 components with suspension time for each component 4.0: 1.15 1.89 2.39 3.05 4.0 0 0.078 1.92 3.62 4.0 0 1.66 2.39 3.07 3.68 4.0 0 0.351 1.58 2.64 3.08 3.79 4.0 0 1.089 2.39 2.48 3.36 4.0 0 Below you can see the 3 steps of calculation. 1. Using the rank regression method and analyzing only first events (first failures) of each components the Weibull function parameters were calculated: shape parameter beta=0.8018113 and scale parameter eta=1.0404179. This result is used as input for next step of calculation. 2. Maximum likelihood method is applied calculating Weibull parameters taking into account only first events for each component as above. We obtained the result: beta=1.3033577 eta=0.9280154. This result is used as initial input for the next calculation step. 3. The general renewal process is calculated using MLE. The obtained result: beta=1.9533635, eta=1.0611752, restoration factor q=0.028360577, which is close to initial parameters of simulated g-renewal process. All steps were performed automatically. Variance-covariance matrix and confidence bounds were calculated using Fisher matrix. Output:
Calculation #1. Tue Feb 09 20:06:40 EST 2016

Entered number of components: 5
Distribution function is: Weibull
1. Rank regression method.
Probability function parameter beta=0.8018113 eta=1.0404179
2. MLE method.
Probability function parameter beta=1.3033577 eta=0.9280154
3. Kijima model. MLE method. Likelihood=-15.051945
Probability function parameter beta=1.9533635 eta=1.0611752
Restoration factor q=0.028360577
Mean time to repair is 0.02
Prediction method is: MLE
Kijima Model 1 is used in the calculation
Confidence level is: 90.0%
Var(beta)=0.2692863; Covar(beta, eta)=0.0626471; Covar(beta, q)=0.033972017; Var(eta)=0.036251675;
Covar(eta, q)=0.013108807; Var(q)=0.008231301

 Example 2. The input data for this example is generated by Monte Carlo method for Kijima model 1 with restoration factor q=1 and Log-Normal underlying function with parameter mu=2.0 and parameter sigma=1.0. The mean time to repair is 1.0. We obtained the data for 5 components. The observation of first two components finished at their last failure. Corresponding set of numbers (times) is finished with -1. Last 3 components were suspended at time 60. 13.67 21.08 25.26 32.34 -1 1.001 45.96 -1 37.27 60.0 0 5.69 10.69 15.33 25.06 27.79 55.11 60.0 0 2.70 7.95 10.35 25.00 54.91 56.91021 60.0 0 Below you can see the 4 steps of calculation. The first 3 steps are similar to steps described above. Step 4 shows result of selected Residual sum of squares method. It took about 10 seconds to calculate this example. Output:
Calculation #1. Wed Feb 10 06:40:16 EST 2016

Entered number of components: 5
Distribution function is: LogNormal
1. Rank regression method.
Probabilty function parameter mu=1.7932256 sigma=1.6168088
2. MLE method.
Probability function parameter mu=1.7932256 sigma=1.2531306
3. Kijima model. MLE method. Likelihood=-66.81926
Probability function parameter mu=1.7897998 sigma=1.1043798
Restoration factor q=0.48222998
4. Kijima model. Residual sum of squares method.
Probability function parameter mu=1.8322077 sigma=1.2121247
Restoration factor q=0.5537386; Error =0.12154537

Mean time to repair is 1.0
Prediction method is: Residual sum of squares
Kijima Model 1 is used in the calculation
Confidence level is: 90.0%
Var(mu)=0.17098467; Covar(mu, sigma)=-0.016717048; Covar(mu, q)=-0.059069663;
Var(sigma)=-0.007902209; Covar(sigma, q)=0.07389288; Var(q)=-4.728893E-5