System Reliability Basics

# System Reliability Basics

### Example (Parallel System)

The next simplest type of system is the parallel system. This one is the exact opposite of the series system: if even one component in a parallel system is still working, the system still functions.

The structure function for the parallel system is given by
$$\phi_{\text{parallel}}(\mathbf{x}) = 1-\prod_{i=1}^{n}(1-x_{i})$$

To see this, remember that the component state is 1 when the component functions, and 0 when it fails. A system is the same way. In a parallel system, the system fails if all of the components fail. Each component failure contributes a $(1-x_{i})$, and we must subtract this product from 1, because we want to know when it functions, not fails.

These two are the basic system reliability topologies. We can make ones that are much more complex, but we have a theorem that shows any system’s block diagram can be re-structured into a series system of parallel subsystems, or a parallel system of series subsystems.
From Leemis (Reliability: Probabilistic Models and Statistical Methods, Lawrence M. Leemis 2nd ed.) ,

Theorem (Decomposition of Systems into Series/Parallel Subsystems).
Let $P_{1},...,P_{s}$ be the minimal path sets for a system. Then
$$\phi(\mathbf{x}) = 1-\prod_{i=1}^{s}\left(1-\prod_{j \in P_{i}}x_{j}\right)$$
where $x_{j}$ is the component state vector.

What are path sets? The path sets are the sets of components that form a complete path through the block diagram.

If we look at the figure above, we have lots of possible paths from left to right. The sets (1,3), (1,4), (2,4), and (2,5) all provide paths through the diagram. These are also minimum path sets in this case. Focusing on any particular path set, if I drop one component, the path disappears. The theorem gives the mathematical structure function corresponding to a parallel system of the series subsystems that the minimal path sets generate.

In other words, the system is functioning if Components 1 and 3 are functioning, or Components 1 and 4 are functioning, or Components 2 and 4, etc.

Last, here’s an example of how we can take a given block diagram and rearrange it in terms of a parallel system of series subsystems according to the theorem.

In the figure above, the top diagram represents a topology called a bridge system. The right side is the alternative version where I’ve arranged this into a parallel system of series subsystems. To test your knowledge, try the following exercise.

Exercise. Write the structure function for the bridge system.

Solution. $\phi(\mathbf{x}) = 1-(1-x_{1}x_{3}x_{5})(1-x_{1}x_{4})(1-x_{2}x_{3}x_{4})(1-x_{2}x_{5})$

Why do we actually care about this? Engineers will use block diagrams to help them design reliable systems. We can prove mathematically that the series system is the least reliable, and the parallel system is the most reliable, but that doesn’t mean we should always put every component in parallel. An airplane with more landing gear than it needs can cause the airplane to weigh too much, for example.

This results in complex system designs, particularly for mechanical and electronic devices. These designs may become too difficult to study visually via block diagrams, but the structure function helps. Studying the form of the structure function in terms if the components can help us determine where a critical component is, for example. A critical component is a component that will fail the whole system if it fails. In a series system, every component is critical. In a parallel system, no one is. Since most system designs are mixtures of these two, identification of a critical component isn’t always simple with the block diagram. The structure function allows us to simulate various components failing (having state 0), and quickly seeing its effect on the system.

Upon identification of a critical component, an engineer has a couple options