# An Efficient Library for Reliability Block Diagram Evaluation
## 2 Reliability Block Diagrams
### Quantitative Evaluation Using RBDs
#### 2.2.1 Quantitative Evaluation: General Formulas  
The following general formulas can be used to quantitatively evaluate the probability that the state of an RBD block composed by \(N\) components is equal to _success_:  
* **Series**. The probability of _success_ of the series block \(p_{series}\) is computed as: \[p_{series}=\prod_{i=1}^{N}p_{i}\] (2)
* **Parallel**. The probability of _failure_ of the parallel block \(q_{parallel}\) is computed as: \[q_{parallel}=\prod_{i=1}^{N}q_{i}=\prod_{i=1}^{N}(1-q_{i})\] (3) The probability of _success_ of the parallel block \(p_{parallel}\) is thus computed as: \[p_{parallel}=1-\prod_{i=1}^{N}(1-p_{i})\] (4)
* **K-out-of-N (KooN)**. In order to compute the probability of _success_ of a _KooN_ block, we can use one of the following approaches: 1. Let \(C(N,i,j)\) be the \(j\)-th unique combination of \(i\) out of \(N\) components correctly working. For a given couple \(<\!i,N\!\!>\), the number of unique combinations is equal to the binomial coefficient \((\! {N\choose i})\). We define \(path(N,i,j)\) as the specific realization of one of the possible system states for which \(i\) components out of \(N\) are correctly working while the other \((N-i)\) have failed: the exact set of  
Figure 3: Layout of RBD bridge block. the working components is selected through the usage of unique combination \(C(N,i,j)\). Its probability of occurrence is:  
\[P_{path(N,i,j)}=\prod_{l\in C(N,i,j)}p_{l}\cdot\prod_{m\
otin C(N,i,j)}q_{m} \tag{5}\]  
The state of a Koo\(N\) block is equal to _success_ if and only if the current system state is satisfied by one path of at least \(K\) working components. The probability of _success_ of the Koo\(N\) block can be defined as:  
\[p_{KooN}=\sum_{i=K}^{N}\sum_{j=1}^{\binom{N}{i}}P_{path(N,i,j)} \tag{6}\]
2. Observe that the probability of _success_ of a system with 0 or more components out of \(I\) in _success_ state is equal to 1 and observe that the probability of _success_ of a system with \(J\) or more components out of \(I\) with \(J>I\) in _success_ state is equal to 0. A recursive approach for evaluating the probability of _success_ of a Koo\(N\) system is derived by conditioning on the state of the \(N\)-th component [3]. The \(N\)-th component can assume only two states, _success_ with probability \(p_{N}\) and _failed_ with probability \(q_{N}\). Let us assume that the \(N\)-th component is correctly working: for a Koo\(N\) system to be correctly operating, we need at least \(K-1\) working components out of the remaining \(N-1\). If, on the other hand, the \(N\)-th component is failed, we need at least \(K\) working components out of the remaining \(N-1\) in order to have a correctly operating Koo\(N\) system. The probability of _success_ of a Koo\(N\) block can then be recursively computed as: \[p_{KooN}=q_{N}\cdot p_{Koo(N-1)}+p_{N}\cdot p_{(K-1)oo(N-1)}\] (7) \[p_{000l}=1\] \[p_{jool}=0\quad\forall J>I\]
* **Bridge**. In order to compute the probability of _success_ of a bridge block, we apply the same decompositional approach used in the second set of formulas to compute the probability of _success_ of a Koo\(N\) block. Let us analyze the bridge block by conditioning the status of component \(E\). If \(E\) is failed, the state of the block is equal to _success_ if either \(A\) and \(B\) or \(C\) and \(D\) are correctly operating, i.e., if the parallel of two series \(A\), \(B\) and \(C\), \(D\) is satisfied. The probability of occurrence of this first event is equal to the probability of _failure_ of \(E\). On the other hand, if \(E\) is correctly operating, the state of the block is equal to _success_ if at least one component between \(A\) and \(C\) is correctly operating and if at least one component between \(B\) and \(D\) is correctly operating, i.e., if the series of two parallel \(A\), \(C\) and \(B\), \(D\) is satisfied. The probability of occurrence of this second event is equal to the probability of _success_ of \(E\).