Examples

Example 1.
This example from paper [1] shows the possibility of data extrapolation using the model. We consider the Kijima model with increasing restoration factor. We have simulated a statistical sample of 100 trajectories (components) from a general renewal model with the underlying Weibull distribution (scale parameter eta=1 and shape parameter beta= 2) and with Kijima I restoration factor depending on failure number, i, in the following functional form: qi = qi-1 +0.5(1- qi-1). Specifically, with q1 = 0.2, q2 = 0.6, q3 = 0.8, ... The estimation interval was truncated at 2 units of time, corresponding to ~3 failures of the system. The input file is here
Output:
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=2.1008 and scale parameter eta=1.0342. 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=2.0406 eta=1.0086. 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=1.9314783, eta=, restoration factors q=0.26468012, q*=0.39835262.
Standard deviation (residual error) is 0.0430098.
All steps were performed automatically.

The output file is here

Red curve in the figure corresponds to the exact solution. Black solid line represents obtained approximation; dash-dotted curve is the solution obtained by Kijima model with constant restoration factor. The example shows that the model can be used for data extrapolation of recurrent event process.

Example 2.
This example shows the possibility of Unavailability calculation as Failure Intensity (FI) function multiplied by mean time to repair. We consider constant failure time as deterministically given. We have simulated a statistical sample of 400 trajectories (components) from a general renewal model with the underlying Weibull distribution (scale parameter eta=1 and shape parameter beta= 1). The estimation interval was truncated at 3 units of time. Constant mean to repair was 0.1. The input file with failure times is here
Output:
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=0.9700 and scale parameter eta=1.0131. 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=0.9964 eta=0.9996. 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=0.99506193, eta=1.0017928, restoration factors q=q=0.83793837, q*=0.0046279356.
Standard deviation (residual error) is 0.013663. The result is very close to exact solution.
All steps were performed automatically.

The output file is here

Black solid line represents obtained approximation of normalized Unavailability function. Dashed curve is the corresponding solution obtained by raw Monte Carlo method with 1000,000 trials. The difference can be observed only when time is less than mean time to repair μ. The red curve in the figure corresponds to the exact solution when the probability of repair time is exponentially distributed with the same constant mean time. One can see that at the time greater than 3μ the result does not depend on repair time distribution.
We considered the same example with the exponential distribution function for failure time distribution. The input file with failure times is here. In case when the failure times are not defined the following approach can be used. Consider the input with repair times equals to 0. The result is shown by blue curve with output file. The red curve in the figure corresponds to the exact solution. The example shows that the model can be used for Unavailability calculation. Note that in some cases for approximation of the derivative of the function it is required more parameters (for example if FI function has maximum). The following calculation provides this possibility based on paper [2]. The result is shown by dash-dotted line.
References
1. A. Yevkin, V. Krivtsov, A generalized model for recurrent failures prediction, Reliability Engineering and System Safety 204 (2020).
2. A. Yevkin, V. Krivtsov, Modeling Recurrent Failure Processes using Padé Approximants, RAMS 2025. Proceedings of Reliability and Maintainability Symposium, RAMS 2025, Florida, US.