## 3 Reliability Models of WSNs
### Region Model  
As defined in Section 3.1, a region has one or more sensor nodes. In the case the region has just one node, the region and path models are the same. Otherwise, if the region has more than one sensor node, it is necessary to combine all path models of the sensor nodes belonging to the region. Before combining them, it is necessary to assess whether these paths shared nodes or not. For example, the region in Figure 10a has three nodes, namely \(A\), \(B\) and \(C\); consequently, it has three path models (A\(\rightarrow\)D\(\rightarrow\)E, B\(\rightarrow\)D\(\rightarrow\)E and C\(\rightarrow\)D\(\rightarrow\)E), which share two nodes (_D_ and _E_). Figure 10b shows the corresponding reliability model to this region when we combine the path models. Additionally, the sensor nodes inside of region (_A_, \(B\) and _C_) are considered as the sources and sensor node \(E\) is considered as the target. Assuming that the WSN shown in Figure 10a uses a hierarchical algorithm (see Section 2.1) and the sensor node \(D\) is the Cluster Head (CH) of sensor nodes \(A\), \(B\) and \(C\), all nodes in that region (_A_, \(B\) and _C_) should send their packets to the node \(D\) (CH). This means that all sensor nodes have one thing in common (node _D_) and this fact should be represented in the reliability model. If this common point fails, the region will fail independently from the existing paths. Therefore, lets first label the basic  
Figure 10: Example of a (**a**) region and its (**b**) region model. Figure 9: Example of a (**a**) network with multipath and (**b**) its RBD model. blocks: _Node A, Node B, Node C, Node D, Node E, Link \((A-D)\), Link \((B-D)\), Link \((C-D)\) and Link \((DE)\)_. The RBD is represented by three minimal paths : \(MP_{1}=\{Node\ A,Link\ (A-D),\ Node\ D,\ Link\ (D-E),Node\ E\}\), \(MP_{2}=\{Node\ B,Link\ (B-D),Node\ D,Link\ (D-E),\ Node\ E\}\) and \(MP_{3}=\{Node\ C,Link\ (C-D),Node\ D,Link\ (D-E),Node\ E\}\). This RBD is depicted in Figure 10b. In some specific scenarios, as shown in Figure 11a, the operational behavior may not be represented by a series-parallel RBD composition. In such cases \"pivoting\" or SDP [22, 23] methods should be applied compute the measures of interest. For instance, the node \(A\) sends packets in multicast to nodes \(D\) and \(E\), creating a no-series-parallel RBD composition--as shown in Figure 11b. To solve this situation, we need use only the SDP method, but it is necessary first to define minimal paths (see Section 2.3.1) as shown in Figure 11c. As these minimal paths have elements in common (_Node A_, _Node D_ and _Node E_), the region reliability using SDP method is calculated as follows (see Section 2.3.2):  
\[ \begin{array}{l}R_{Region}=\ R_{MP_{1}}+\ \overline{R_{MP_{1}}}\ R_{MP_{2}} +\ \overline{R_{MP_{1}}}\ \overline{R_{MP_{2}}}\ R_{MP_{3}}\ +\ \overline{R_{MP_{1}}}\ \overline{R_{MP_{2}}}\ \overline{R_{MP_{3}}}\ R_{MP_{4}}\\ \\ =R_{A}\ R_{L\ (AE)}\ R_{E}\\ \\ +\ \overline{R_{A}\ R_{L\ (AE)}\ R_{E}}\ R_{A}\ R_{L\ (AD)}\ R_{D}\ R_{L(DE)}\ R_{E}\\ \\ +\ \overline{R_{A}\ R_{L\ (AE)}\ R_{E}}\ R_{A}\ R_{L\ (AD)}\ R_{D}\ R_{L(DE)}\ R_{E}\ R_{B}\ R_{L(BD)}\ R_{D}\ R_{L(DE)}\ R_{E}\\ \\ +\ \overline{R_{A}\ R_{L\ (AE)}\ R_{E}}\ R_{A}\ R_{L\ (AD)}\ R_{D}\ R_{L(DE)}\ R_{E}\ R_{B}\ R_{L(BD)}\ R_{D}\ R_{L(DE)}\ R_{E}\ R_{C}\ R_{L\ (CE)}\ R_{D}\ R_{L\ (DE)}\ R_{L\ (DE)}\ R\\ \\ =R_{E}(R_{A}\ R_{L\ (AE)}+\ R_{D}\ R_{L\ (DE)}(R_{C}\ R_{L\ (CD)}+R_{B}\ R_{L\ (BD)}+\ R_{A}\ R_{L\ (AD)}))\end{array}  \tag{8}\]  
where \(MP\) represents a minimal path illustrated in Figure 11c and \(R_{x}\) represents the reliability of a node \(x\) (_i.e._, \(R_{A}\) is reliability of node \(A\)), link (_i.e._, \(R_{L(AE)}\) is reliability of link between node \(A\) and \(E\)), minimal path (_i.e._, \(R_{MP_{1}}\) is reliability of minimal path \(MP_{1}\)) or region (\(R_{Region}\)). Applying the SDP method (see Section 2.3.2), we obtained the expression \(R_{E}\left(R_{A}\ R_{L\ (AE)}+\ R_{D}\ R_{L\ (DE)}(R_{C}\ R_{L\ (CD)}+R_{B}\ R_{L\ (BD)}+R_{A}\ R_{L\ (AD)})\right)\).