Page 249 - Design of Simple and Robust Process Plants

6.2 Basic Theory of Reliability 235
Table 6.1. Probability of failures for serial and parallel systems for two probabilities of component
failures 0.05 and 0.1, it is assumed that the failures are mutually exclusive.

System Probability of Probability of Probability of Probability of
failure 0.05 failure 0.1 failure 0.05 failure 0.1
Eq. (9) Eq. (9) Approx. Eq. (10) Approx. Eq. (10)
Serial components
1 0.05 0.1 0.05 0.1
20.0975 0.19 0.1 0.2
3 0.143 0.271 0.15 0.3
4 0.185 0.344 0.20.4
5 0.226 0.41 0.25 0.5
Parallel components Eq. (11) Eq. (11)
20.0025 0.01
3 0.000125 0.001
4 0.00000620.0001
5 0.0000003 0.00001
Eq. (12) Eq. (12)
2out of 3 0.00725 0.028

4. More parallel components leads to higher system reliability
5. The two-out-of-three systems have a system failure probability between a
single-component system and a one-out-of-two component system. These
systems are often selected as they are also used for error detection in case
deviations between measurements are notified.

6.2.4.2 Monte Carlo simulations (Dubi, 1998)
Practical problems in reliability engineering cannot be, or are difficult to be, solved
analytically. The reasons behind this are: the size of the problem becomes too large
and the distribution functions of the input data are difficult to handle. Therefore,
the analytical problem-solving techniques are limited to smaller problems where
average values are used as input.
The best practice for large problems is to build a probabilistic model of the prob-
lem. The model is based on the RBD structure, while the input data are randomly
selected from its probabilistic distributions by a random number generator, often
called RAN. The random generator needs to be adapted to the distribution functions
as applicable to the different components. The model is run in time for a large num-
ber of cases, each case with a new random selected set of input data. Each run pro-
vides a success or failure of the system model, and the cause of a failure. Based on
the accumulated probabilistic results the reliability of the system with its distribu-
tion is calculated. Reliability programs are equipped with a clock to monitor the data
over time. The type and number of failures over the mission time is recorded, as
will be the related down-times. The down-times are converted in system availability.
Such a Monte Carlo simulation can easily be adopted, which makes the evaluation