In many cases, it is not easy to recognize which components are in series and which are in parallel in a complex system. The network shown in Figure 4.15 is a good example of such a complex system.
Figure 4.15: Example of a complex system.
The system in Figure 4.15 cannot be broken down into a group of series and parallel systems. This is primarily due to the fact that component C has two paths leading away from it, whereas B and D have only one. Several methods exist for obtaining the reliability of a complex system including:
The decomposition method.
The event space method.
The path-tracing method.
The decomposition method is an application of the law of total probability. It involves choosing a "key" component and then calculating the reliability of the system twice: once as if the key component failed (R = 0) and once as if the key component succeeded (R = 1). These two probabilities are then combined to obtain the reliability of the system, since at any given time the key component will be failed or operating. Using probability theory, the equation is:
Consider three units in series.
A is the event of Unit 1 success.
B is the event of Unit 2 success.
C is the event of Unit 3 success.
s is the event of system success.
First, select a "key" component for the system. Selecting Unit 1, the probability of success of the system is:
If Unit 1 is good, then:
That is, if Unit 1 is operating, the probability of the success of the system is the probability of Units 2 and 3 succeeding.
If Unit 1 fails, then:
That is, if Unit 1 is not operating, the system has failed since a series system requires all of the components to be operating for the system to operate.
Thus the reliability of the system is:
Consider the following system:
A is the event of Unit 1 success.
B is the event of Unit 2 success.
C is the event of Unit 3 success.
s is the event of system success.
Selecting Unit 3 as the "key" component, the system reliability is:
If Unit 3 survives, then:
That is, since Unit 3 represents half of the parallel section of the system, then as long as it is operating, the entire system operates.
If Unit 3 fails, then the system is reduced to:
The reliability of the system is given by:
Or:
The event space method is an application of the mutually exclusive events axiom. All mutually exclusive events are determined and those that result in system success are considered. The reliability of the system is simply the probability of the union of all mutually exclusive events that yield a system success. Similarly, the unreliability is the probability of the union of all mutually exclusive events that yield a system failure. This is illustrated in the following example.
Consider the following system, with reliabilities R1, R2 and R3 for a given time.
A is the event of Unit 1 success.
B is the event of Unit 2 success.
C is the event of Unit 3 success.
The mutually exclusive system events are:
System events 
, 
and 
result
in system failure. Thus the probability of failure of the system is:
Since events , and are mutually
exclusive, then:
And:
Combining terms yields:
Since:
Then:
This is of course the same result as the one obtained previously using the decomposition method.
If R1 = 99.5%, R2 = 98.7% and R3 = 97.3%, then:
Or:
With the path-tracing method, every path from a starting point to an ending point is considered. Since system success involves having at least one path available from one end of the RBD to the other, as long as at least one path from the beginning to the end of the path is available, then the system has not failed. One could consider the RBD to be a plumbing schematic. If a component in the system fails, the "water'" can no longer flow through it. As long as there is at least one path for the "water" to flow from the start to the end of the system, the system is successful. This method involves identifying all of the paths the "water" could take and calculating the reliability of the path based on the components that lie along that path. The reliability of the system is simply the probability of the union of these paths. In order to maintain consistency of the analysis, starting and ending blocks for the system must be defined.
Obtain the reliability equation of the following system.
The successful paths for this system are:
The reliability of the system is simply the probability of the union of these paths:
Thus:
Obtain the reliability equation of the following system.
Assume starting and ending blocks that cannot fail, as shown next.
The paths for this system are:
The probability of success of the system is given by:
Or:
(7)
Note that BlockSim requires that all diagrams start from a single block and end on a single block. To meet this requirement for this example, we arbitrarily added a start and an end block, as shown in Figure 4.16. These blocks can be set to a "cannot fail condition," or R = 1, thus not affecting the outcome. However, when the analysis is performed in BlockSim, the returned equation will include terms for the non-failing blocks, as shown in Figure 4.17 and Eqn. (8).
Figure 4.16: BlockSim representation of the RBD for Example 12.
Figure 4.17: BlockSim solution to the RBD in Figure 4.16.
(8)
Note that since RS = RE = 1, the system equation, Eqn. (8), can be reduced to:
This is equivalent to Eqn. (7). The reason that BlockSim includes all items regardless of whether they can fail or not is because BlockSim only recomputes the equation when the system structure has changed. What this means is that the user can alter the failure characteristics of an item without altering the diagram structure. For example, a block that was originally set not to fail can be re-set to a failure distribution and thus it would need to be used in subsequent analysis.
For this example:
(a) Determine the reliability equation of the system shown in Figure
4.7 using the decomposition method.
(b) Determine the reliability equation of the same system using BlockSim.
To obtain the solution:
(a) Choose A as the "key" component, then:
(b) Using BlockSim:
Figure 4.18: Diagram for Example 13.
Figure 4.19: BlockSim solution to Example 13.
See Also:
RBDs and Analytical System Reliability
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