Simulation with RGA Models

When analyzing developmental systems for reliability growth, and conducting data analysis of fielded repairable systems, it is often useful to experiment with various what if scenarios or put together hypothetical analyses before data sets become available in order to plan for the best way to perform the analysis. With that in mind, the RGA software offers applications based on Monte Carlo simulation that can be used in order to:



Better understand reliability growth concepts. Experiment with the impact of sample size, test time and growth parameters on analysis results. Construct simulation-based confidence intervals. Better understand concepts behind confidence intervals. Design reliability demonstration tests.   There are two applications of the Monte Carlo simulation in the RGA software. One is called Generate Monte Carlo Data and the other is called SimuMatic.

=Generate Monte Carlo Data= Monte Carlo simulation is a computational algorithm in which we randomly generate input variables that follow a specified probability distribution. In the case of reliability growth and repairable system data analysis, we are interested in generating failure times for systems that we assume to have specific characteristics. In our applications we want the inter-arrival times of the failures to follow a non-homogeneous Poisson process with a Weibull failure intensity, as specified in the Crow-AMSAA (NHPP) model.

The first time to failure, $${{t}_{1}},\,\!$$ is assumed to follow a Weibull distribution. It is obtained by solving for $${{t}_{1}}\,\!$$ :


 * $$R(t_1)=e^{(-\frac{t_1}{\eta})^\beta}= Uniform (0,1)\,\!$$

where:


 * $$\eta ={{\left( \frac{1}{\lambda } \right)}^{\tfrac{1}{\beta }}}\,\!$$

Solving for $${{t}_{1}}\,\!$$ yields:


 * $$t_{1} = \eta \left [ -ln(Uniform(0,1)) \right ]^{\frac {1}{\beta}}$$

The failure times are then obtained based on the conditional unreliability equation that describes the non-homogeneous Poisson process (NHPP):


 * $$F(t_{i}\ | \ t_{i-1}) = 1 -e^{-\lambda\left[t_{i}^{\beta}-t_{i-1}^{\beta}\right]} = Uniform(0,1)$$

and then solving for $${{t}_{i}}\,\!$$ yields:


 * $$t_{i}=\left[-\frac {ln(1-Uniform(0,1))}{\lambda}+t_{i-1}^{\beta} \right ]^{\frac{1}{\beta}}$$

To access the data generation utility, choose Home > Tools > Generate Monte Carlo Data. There are different data types that can be generated with the Monte Carlo utility. For all of them, the basic parameters that are always specified are the beta $$(\beta )\,\!$$ and lambda $$(\lambda )\,\!$$ parameters of the Crow-AMSAA (NHPP) model. That does not mean that the generated data can be analyzed only with the Crow-AMSAA (NHPP) model. Depending on the data type, the Duane, Crow extended and power law models can also be used. They share the same growth patterns, which are based on the $$\beta \,\!$$ and $$\lambda \,\!$$ parameters. In the case of the Duane model, $$\beta =1-\alpha \,\!$$, where $$\alpha \,\!$$ is the growth parameter for the Duane model. Below we present the available data types that can generated with the Monte Carlo utility.


 * Failure Times: The data set is generated assuming a single system. There is a choice between a time terminated test, where the termination time needs to be specified, or a failure terminated test, where the number of failures needs to be specified. The generated failure times data can then be analyzed using the Duane or Crow-AMSAA (NHPP) models, or the Crow extended model if classifications and modes are entered for the failures.


 * Grouped Failure Times: The data is generated assuming a single system. There is a choice between a time terminated test, where the termination time needs to be specified, or a failure terminated test, where the number of failures needs to be specified. In addition, constant or user-defined intervals need to be specified for the grouping of the data. The generated grouped data can then be analyzed using the Duane or Crow-AMSAA (NHPP) models, or the Crow Extended model if classifications and modes are entered for the failures.


 * Multiple Systems - Concurrent: In this case, the number of systems needs to be specified. There is a choice between a time terminated test, where the termination time needs to be specified, or a failure terminated test, where the number of failures needs to be specified. The generated folio contains failure times for each of the systems. The data can then be analyzed using the Duane or Crow-AMSAA (NHPP) models, or the Crow Extended model if classifications and modes are entered for the failures.


 * Repairable Systems: In this case, the number of systems needs to be specified. There is a choice between a time terminated test, where the termination time needs to be specified, or a failure terminated test, where the number of failures needs to be specified. The generated folio contains failure times for each of the systems. The data can then be analyzed using the power law model, or the Crow extended model if classifications and modes are entered for the failures.

The next figure shows the Monte Carlo utility and all the necessary user inputs.

The seed determines the starting point from which the random numbers will be generated. The use of a seed forces the software to use the same sequence of random numbers, resulting in repeatability. In other words, the same failure times can be generated if the same seed, data type, parameters and number of points/systems are used. If no seed is provided, the computer's clock is used to initialize the random number generator and a different set of failure times will be generated at each new request.

Monte Carlo Data Example
=SimuMatic= Reliability growth analysis using simulation can be a valuable tool for reliability practitioners. With this approach, reliability growth analyses are performed a large number of times on data sets that have been created using Monte Carlo simulation.

The RGA software's SimuMatic utility generates calculated values of beta and lambda parameters, based on user specified input parameters of beta and lambda. SimuMatic essentially performs a number of Monte Carlo simulations based on user-defined required test time or failure termination settings, and then recalculates the beta and lambda parameters for each of the generated data sets. The number of times that the Monte Carlo data sets are generated and the parameters are re-calculated is also user defined. The final output presents the calculated values of beta and lambda, and allows for various types of analysis.

To access the SimuMatic utility, choose Insert > Tools > Add SimuMatic. For all of the data sets, the basic parameters that are always specified are the beta $$(\beta )\,\!$$ and lambda $$(\lambda )\,\!$$ parameters of the Crow-AMSAA (NHPP) model or the power law model.


 * Failure Times: The data set is generated assuming a single system. There is a choice between a time terminated test, where the termination time needs to be specified, or a failure terminated test, where the number of failures needs to be specified. SimuMatic will return the calculated values of $$\beta \,\!$$ and $$\lambda \,\!$$ for a specified number of data sets.


 * Grouped Failure Times: The data set is generated assuming a single system. There is a choice between a time terminated test, where the termination time needs to be specified, or a failure terminated test, where the number of failures needs to be specified. In addition, constant or user-defined intervals need to be specified for the grouping of the data. SimuMatic will return the calculated values of $$\beta \,\!$$ and $$\lambda \,\!$$ for a specified number of data sets.


 * Multiple Systems - Concurrent: In this case, the number of systems needs to be specified. There is a choice between a time terminated test, where the termination time needs to be specified, or a failure terminated test, where the number of failures needs to be specified. SimuMatic will return the calculated values of $$\beta \,\!$$ and $$\lambda \,\!$$ for a specified number of data sets.


 * Repairable Systems: In this case, the number of systems needs to be specified. There is a choice between a time terminated test, where the termination time needs to be specified, or a failure terminated test, where the number of failures needs to be specified. SimuMatic will return the calculated values of $$\beta \,\!$$ and $$\lambda \,\!$$ for a specified number of data sets.

The next figure shows the Main tab of the SimuMatic window where all the necessary user inputs for a multiple systems - concurrent data set have been entered. The Analysis tab allows you to specify the confidence level for simulation-generated confidence bounds, while the Results tab gives you the option to compute for additional results, such as the instantaneous MTBF given a specific test time.



The next figure shows the generated results based on the inputs shown above. The data sheet called "Sorted" allows us to extract conclusions about the simulation-generated confidence bounds because the lambda and beta parameters and any other additional output are sorted by percentage.



The following plot shows the simulation-confidence bounds for the cumulative number of failures based on the input parameters specified.