Template:Generate monte carlo data

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 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 Eqn. (R(t1) =uniform) for $${{t}_{1}}$$  yields:

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

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

To access the data generation utility, either choose Tools > Generate Monte Carlo Data or click the Generate Monte Carlo Data icon on the Tools toolbar. 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, Crow-AMSAA (NHPP) 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, Crow-AMSAA (NHPP) 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, Crow-AMSAA (NHPP) 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. Figure Montearlo 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.

Example 1
A reliability engineer wants to experiment with different testing scenarios as the reliability growth test of the company's new product is being prepared. From the reliability growth test data of a similar product that was developed previously, the beta and lambda parameters are $$\beta =0.5$$  and  $$\lambda =0.75.$$  Three systems are to be used to generate a representative data set of expected times-to-failure for the upcoming test. The purpose is to explore different test durations in order to demonstrate an MTBF of 200 hours.

Solution
In the Monte Carlo window, the parameters are set to $$\beta =0.5$$  and  $$\lambda =0.75.$$  Since we have three systems, the Multiple Systems - Concurrent option is selected and the number of systems is set to 3. Initially, the test is set to be time terminated with 2000 operating hours per system, for a total of 6000 operating hours. Figure concurrent monte carlo shows the Monte Carlo generation window for this example. $$$$

Figure folio monte carlo shows the generated failure times data. In this folio, the Advanced Systems View is used, so the data sheet shows the times-to-failure for system 2.

The data can then be analyzed just like a regular folio in RGA 7. In this case, we are interested in analyzing the data with the Crow-AMSAA (NHPP) model to calculate the demonstrated MTBF at the end of the test. In Figure folio monte carlo, in the results area of the folio, it can be seen that the demonstrated MTBF at the end of the test is 189.83 hours. Since that does not meet the requirement of demonstrating an MTBF of 200 hours, we can either generate a new Monte Carlo data set with different time termination settings, or access the Quick Calculation Pad in this folio and find the time for which the demonstrated (instantaneous) MTBF becomes 200 hours, as shown in Figure QCP monte carlo. From the figure it can be seen that, based on this specific data set, 6651.38 total operating hours are needed to show a demonstrated MTBF of 200 hours.

Note that since the Monte Carlo routine generates random input variables that follow the NHPP based on the specific $$\beta $$  and  $$\lambda $$, if the same seed is not used the failure times will be different the next time you run the Monte Carlo routine. Also note that since input variables are pulled from an NHPP with the expected values of $$\beta $$  and  $$\lambda ,$$  it should not be expected that the calculated parameters of the generated data set will match exactly the input parameters that were specified. In this example, the input parameters were set as $$\beta =0.5$$  and  $$\lambda =0.75$$  and the data set based on the Monte Carlo generated failure times yielded calculated Crow-AMSAA (NHPP) parameters of  $$\beta =0.4939$$  and  $$\lambda =0.8716.$$  The next time a data set is generated with a random seed, the calculated parameters will be slightly different, since we are essentially pulling input variables from a predefined distribution. The more simulations that are run, the more the calculated parameters will converge with the expected parameters. In RGA 7, the total number of generated failures with the Monte Carlo utility has to be less than 64,000.