User has to input the end time of interval where g-renewal process parameters will be calculated, then input the number N of components observed. Each component should have a set of event times ending with integer number 0 or -1. Number 0 indicates that the component suspended at the time corresponding to the previous number. Number -1 corresponds to the case when observation finished at the last event which is the previous number. All N sets of event times should be stored in a text file (with .txt extension). If it is convenient, the user can prepare input data in Excel format and then copy and paste the data in a .txt file and then use it in the calculation.

The following example of input data is the result of observation of 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

Each number in the file should be separated by white space or new line ("Enter"). Note that two integer numbers without separator will be read as one number, for example "23" and "67" will be read as "2367". Two float numbers without separator could be corrupted, for example "0.34" and "3.5" typed as "0.343.5" can be read as "0.343" and "0.5". We check the format of input numbers and in the most cases one will get an error message if the format is not valid, but it is the responsibility of the user to prepare the input data properly. We also check for consistency of the input set of numbers. One can use "Browse" to select the required file and click "Upload" to submit.

On the next page user has to select the underlying distribution function (Weibull, Normal or Log-Normal), then enter the Mean repair time, and then select the prediction method: Maximum likelihood estimator (MLE) or Residual sum of squares. Kijima model 1 or 2 and restriction for restoration factor should be selected. Then user should input the confidence level (for example 0.9) and select the type of confidence bounds (Two-sided, Upper or Lower sided). For Weibull function it takes a couple of seconds to calculate. If you do not need to calculate confidence bounds, select N/A. Calculation of confidence bounds is the most time consuming part and requires about 10-20 seconds if Normal or Log-Normal function is selected, therefore we recommend to start the new task without confidence bounds calculation. If calculation takes more than 1 minute, it is likely that input is not correct or mission time is too large. Please check your input file and value of mean time to repair entered or try another type of underlying probability function.

The running time depends on the type of underlying probability distribution function, selected method of calculation and value of cumulative intensity function (CIF) at mission time. Here we provide the expected running time for CIF=10 at mission time when confidence interval is selected for calculation.
Weibull function, MLE method: running time is 2-4 sec.
Weibull function, residual sum method: running time is 5-10 sec.
(Log)Normal function, MLE method: running time is 5-10 sec.
(Log)Normal function, residual sum method: running time is 10-20 sec.
The running time is proportional to CIF. The running time without confidence bounds calculation is much less.

Note that we used an additional assumption about Normal and Log-Normal CDF function in the model: the value of Normal (Log-Normal) probability function should be small at time t=0 (t=1).

To find maximum likelihood or minimum of residual sum of squares, we use modifications of the Newton–Raphson or Gauss–Newton iterative method respectively. Iterations can lead to several local maximum (minimum) values, therefore we provide user the option to enter the initial value of restoration factor which is 0.1 by default. In addition, calculation is performed with initial value of restoration factor equal to 0.5 of selected restriction value q. By default each calculation is performed for two initial values of q: 0.1 and 0.5. Then the better solution is selected as the result. By entering several initial values, user can find global maximum or minimum. Each calculation is performed for two initial values of restoration factor: the one selected by user (0.1 by default) and the value corresponding to the middle of the interval (right side is 1 by default).

After clicking “Calculate” you will get the result on the same page. You can change calculation options on this page without uploading the same input file if the session has not timed out.