Monte Carlo Simulation of Failure Distributions

Monte Carlo simulation, a procedure for mimicking observations on a random variable, pennits verification of results tliat ordinarily would require difficult inatliematical calculations or extensive experimentation. 

Monte Carlo simulation uses computer programs called random number generators. A random number may be defined as a nmnber selected from tlie interval (0, 1) in such a way tliat tlie probabilities that the number comes from any two subintervals of equal lengtli are equal. For example, the probability tliat tlie number is in tlie subinter al (0.1, 0.3) is the same as the probability tliat tlie nmnber is in tlie subinterval (0.5, 0.7). Random numbers thus defined are observations on a random variable X having a uniform distribution on tlie interval (0, 1). Tliis means tliat tlie pdf of X is specified by 

This pdf assigns equal probability to subintervals of equal lengtli in tlie interval 

As a sample example of Monte Carlo simulation, consider a pump whose time to failure T, measured in years, lias an exponential distribution with pdf specified by 

It is desired to estimate tlie average life of the pump on the basis of 15 simulated values of T. 

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The probabilities of failure for these two limit states were calculated using a level 3 Monte Carlo simulation and the level 2 algorithms of Rackwitz and Horne and Price. The flow diagram for the computer program for the Monte Carlo simulation is shown in Fig. 5.9. Various types of probability distribution were used. The level 2 algorithms were used only with normal distributions for the basic variables. The data for the level 2 algorithms and the Monte Carlo simulations using normal distributions only are shown in Table 5.1. In order to estimate... 

To do so, we first provide in Section 2 a brief overview of Markov Chains and Monte-Carlo simulations. Section 3 presents the structure of our Markov model of failures and replacement as well as its imder-lying assumptions. In Section 4, we run Monte-Carlo simulations of the model and generate probability distributions for the lifecycle cost and utility of the two considered architectures that serve as a basis for our comparative analysis. Important trends and invariants are identified and discussed. For example, changes in average lifecycle cost and utility resulting from fractionation are observed, as well as reductions in cost risk. We conclude this work in Section 5. 

In tlie case of random variables assumed to be normally distributed, Monte Carlo simulation is facilitated by use of a table of the normd distribution (Table 20.5.2). Consider, for example, a series system consisting of two electrical components, A and B. Component A lias a time to failure Ta, assumed to be nomially distributed with mean 100 hours and standard deviation 20 hours. Component B has a time to failure Tb, assumed to be normally distributed witli mean 90 hours and stimdiud deviation 10 hours. Tlie system fails whenever component A or component B fails. Tlierefore, tlie time to failure of the system Ts is tlie minimum of time to failure of components A and B. 

Performing estimation and risk analysis in the presence of uncertainty requires a method that reproduces the random nature of certain events (such as failures in the context of reliability theory). A Monte-Carlo simulation addresses this issue by running a model many times and picking values from a predefined probability distribution at each run (Mun 2006). This process allows the generation of output distributions for the variables of interest, from which several statistical measures (such as mean, variance, skewness) can be computed and analyzed. 

To validate further the results obtained from the MLE procedure, we conduct a simulation to determine the sateUite probabUity of failure from the failure of the sateUite. The subsystems are in series the failure of one subsystem leads to the total failure of the satellite -and they are modeled with the Weibull distributions given in the previous subsection. The induced satellite reUabUity is determined using a Monte Carlo simulation and this result is compared with the nonparametric satelhte reliabiUty derived in Castet Saleh (2009). This comparison is showed in Fig. 4. 

In this article, we propose to evaluate the failure probability of a fluid bearing using a Design Of Experiments (DOE) followed by a Monte Carlo Simulation and FORM. The performance function of the fluid bearing chosen for the example is evaluated thanks to the calculation of the pressure distribution obtained by Finite Element Method (FEM). 

The Monte Carlo simulation method (MCS) is a general simulation technique, i.e. it is applicable to linear and non linear problems indifferently. Moreover, its efficiency is independent of the number of random variables involved in the problem under analysis. The basic idea behind MCS is to generate N samples of 0 which are distributed according to h 0). Then, the failure probability can be estimated as ... 

Through abundant investigation home and abroad of failure mechanism, we know that the life for structural and electron-backstreaming failure modes submit Normal and Weibull distribution separately. In order to make the data more abundant, Monte-Carlo simulation method is used to obtain the solution of competing failure model and the procedure is as follows ... 

Adachi and Ellingwood (2008) proposed a simple method to determine pipeline failure. For a section of pipeline, Eq. 4 can be used to determine the probability of at least one repair. Then a random number is sampled from a uniformly distributed random variable between 0 and 1. If the random number is below this probability, then that section of pipeline is assumed to have failed. This is repeated for all sections of pipeline within a group and then for all groups to simulate a pipeline system response. To reflect the probability distribution, however, multiple pipeline system responses should be generated using a Monte Carlo simulation technique. 

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