Failure Probability Distributions

There are a number of probability distributions to model failures. The distribution types can be found in various sources (Henley and Kumamoto (1992), Hoover (1989), Law and Kelton (1982), Rubinstein (1981), Savic (1989)). The typical ones are listed as follows  

In this Chapter, only two particular types of distributions (i.e. Exponential and Normal distributions) are briefly described. 

The item failure rate is relatively high. Such failure is usually due to factors such as defective manufacture, incorrect installation, learning curve of equipment user, etc. Design should also aim at having a short initial period . 

In this period of an item, the failure rate is constant. Failures appear to occur purely by chance. This period is known as the useful life of the item. 

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The Burchell model s prediction of the tensile failure probability distribution for grade H-451 graphite, from the "SIFTING" code, is shown in Fig. 23. The predicted distribution (elosed cireles in Fig. 23) is a good representation of the experimental distribution (open cireles in Fig. 23)[19], especially at the mean strength (50% failure probability). Moreover, the predicted standard deviation of 1.1 MPa con ares favorably with the experimental distribution standard deviation of 1.6 MPa, indicating the predicted normal distribution has approximately the correct shape. 

As described above, the code "SIFTING" requires several microstructural inputs in order to ealculate a failure probability distribution. We are thus able to assess the physieal soundness of the Burchell model by determining the change in the predicted distribution when microstructural input parameters, such as particle or pore size, are varied in the "SIFTING" code. Each microstructural input parameter... 

Fig. 23. Predicted and experimental tensile failure probability distributions for grade H-451 graphite.

The behavioural nature and probability of such systematic errors of events are not easily predictable or quantifiable in numerical terms because they do not relate to the normal properties of reliability or wear-out typically modelling by failure probability distributions. For this reason modelling their probability density function is very difficult. They relate to a lack of knowledge (and thus uncertainty) in the existence of the fault and the resulting behaviour. Hence, systematic failures are not random. Systematic failures are repeatable though, although knowledge of the internal and external conditions required to repeat them may be difficult to detennine for some unintuitive faults. 

The system description takes the form a fault tree which has top event representing system LOTC. Along with this fault tree, the failure probability distributions and associated parameters for each of the fault tree basic events must be known. 

In order to demonstrate the calculation of importance measures for a system to which TLD is applied consider the example system depicted in Fig. 6. The fault tree for this system is given in Fig. 7, with top event LOTC. Assume, for simplicity, that the system components fail according to exponential failure probability distributions and have failure rates as shown in Table 1. 

Fig. 14.8 Calculated failure probability distribution after 31 months IP/DP box irradiation...

Therefore, the life time is applied as appropriate specification parameter, which defines the time at which a failure probability of x % is reached. Depending on the application a specification of IL B, B or 5,0 value is common. It is assumed that the same failure probability of 63.2% shall be reached at time 50,000 h. Figure 9 shows the corresponding failure probability distributions F(t) for shape parameter b =, 1, 2, A and 5. It can be seen that the failure probability of 10% is... 

FIG. 9 Cumulative failure probability distributions as a function of the number of thermal cycles for 1206 resistors mounted with several Pb-free solder alloys and eutectic Sn-Pb. (a) 0 to lOO C thermal cycling conditions, (b) —55 to 125°C thermal cycling conditions. 

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