# Location-Based Lattice Mobility Model for Wireless Sensor Networks
## 3 Problem Statement and Lattice Mobility Model Formulation
For example, once we validate that a sensor node is bounded from above, then we may prove the existence of the supremum without determining its value. The infimum and supremum sensor node is the lattice node in the region '\(R\)', which is unique, if it exists. Additionally, if both the supremum and infimum exist, then \(\text{Rinfmum}\leq\text{RSupmum}\). Figure 2: (**a**) Selection of the supremum node based on the highest residual energy using the upper bound. (**b**) Selection of the infimum node based on the shortest distance using the greatest lower bound. (**c**) Selection of the lattice node based on the shortest distance and highest residual energy using greatest lower bound and upper bound, respectively (satisfying both conditions of supremum and infimum). **Proof. ** Assume that a sensor node '\(K\)' is the supremum in the region 'R'. Then \(K\geq\gamma\); thus, a sensor node '\(K\)' is the upper bound in the region '\(R\)'. Similarly, if '\(K\)' is the infimum sensor node in the region 'R', then \(K\leq\gamma\). Thus, '\(K\)' is the greatest lower bound in the region '\(R\)'. It can be expressed as follows:  
If \(K\geq\gamma\), then \(K=\gamma\). If Rinfimum and RSupmum exist, then the region '\(R\)' is nonempty. Thus, select '\(K\)' sensor node that is the nearest node with the highest residual energy. It can be written as:  
\(K\in R\), &Rinfimum \(\leq K\leq\) RSupmum. Rinfimum and RSupmum are the greatest lower bound and upper bound respectively in the region '\(R\)'. It follows that Rinfimum \(\leq\) RSupmum. If RSupmum \(\in R\), then it is represented as Rhigh and called the highest residual energy of the node '\(K\)' in region '\(R\)'. If Rinfimum \(\in R\), then it is represented as Rlow and called the shortest distance of node '\(K\)' from location-determining node '\(L\)' in the region '\(R\)'. \(\Box\)  
Based on the concept of infimum and supremum, LMM searches for the unique node with the highest residual energy and the shortest distance from the source node to the destination node. The idea of infimum and supremum is applied with LMM to support three types of mobility patterns: dynamic medium mobility patterns, walking mobility patterns, and vehicular mobility patterns. Dynamic medium mobility patterns are used when the sensor nodes are in a medium (e.g., wind, water, or other fluid) and typically supports two or three dimensions. Walking mobility patterns are suitable for people; movements in these patterns are two dimensional. These patterns are characterized by restricted speeds, a chaotic nature, and obstacle avoidance. Vehicular mobility patterns support vehicles that are equipped with sensor nodes. The vehicles communicate by exchanging traffic conditions with each other. The movement of the vehicle is measured in only one direction. **Theorem 2. **_The greatest lower bound is constructed on the infimum. Let \(V=(a,t)\) be the speed of the mobile sensor node at the given time '\(t\)' and at the location \(X=(X_{1},\ X_{2})\)._  
**Proof. ** In the greatest lower bound, \(D=(a,\ t)\) is the node distribution along the direction '\(\Delta di\)'. Each sensor node has three nearest neighbor nodes as:  
\[a_{i}+R.\ \Delta di(\ \gamma=1,\ 2,\ 3) \tag{1}\]  
where '\(R\)' is the region of the network. Thus, different direction '\(\Delta di\)' of neighbor node '\(a\)' in the region '\(R\)' can be defined as follows:  
\[\Delta d_{i}=\left\{ \begin{array}{ll}1&i=1\\ \
icefrac{{\pi}}{{2}}\cos{(i)},\
icefrac{{\pi}}{{2}}\sin{(i)}&i=2,3\end{array} \right. \tag{2}\]  
Greatest lower bound (Infimum) can be governed by following equation:  
\[D_{i}(a_{i}+R.\ \Delta di,\ t+R)-D_{i}(a,t)=\tau_{i}(V).[{D_{l}}^{\beta}(a,t)-D_{i}(a,t)](i=1,2,3) \tag{3}\]  
where '\(D_{l}^{\beta}\)' is the distribution of the nodes that can be predicted by the value of \(D_{i}(a,t)\); we set \({D_{l}}^{\beta}(a,t)\) to be an average speed of \(v=(a,t)\) :  
\[{D_{l}}^{\beta}(a,t)=\frac{V(a,t)}{3}(i=1,2,3) \tag{4}\]  
where '\(i\)' indicates the number of the mobile sensor nodes having an average speed.