# Location-Based Lattice Mobility Model for Wireless Sensor Networks
## 3 Problem Statement and Lattice Mobility Model Formulation
The greatest lower bound requires that speed of mobile sensor node is measured during the progression. Mathematically, it entails that the summation of the left-hand side of Equation (3) over all three neighborhoods must be equal to 1:\[\sum\limits_{i=1}^{3}\tau_{i}(V).[D_{i}^{\beta}(a,t)-D_{i}(a,t)]=1 \tag{5}\]  
Substituting Equation (5) into Equation (4), we get:  
\[V=(a,t)=\sum\limits_{i=1}^{3}\tau_{i}(V).\ D_{i}(a,t)\sum\limits_{i=1}^{3}\frac {\tau_{i}(V)}{3}=\sum\limits_{i=1}^{3}D_{i}^{\beta}(a,t) \tag{6}\]  
\(\square\)  
Thus, we summarize the result of the theorem based on the greatest lower bound (infimum) in the algorithm given as:  
```
(a). For all neighbor nodes \(D_{i}(a,\ K+1)=D_{i}(a,\ K)+\tau_{i}(V). [\frac{V(a,\ K)}{3}-D_{i}(a,\ K)](i=1,2,3)\) (b). For each neighbor node \(D_{i}(a+\Delta di,\ K+1)=D_{i}(a,\ K+1)\ (i=1,2,3)\) (c). For each border node \(D_{i}(a+\Delta di,\ K+1)=D_{i}(a+\Delta d_{i+1},K+1)(i=1,2,3)\) Here, boarder node functions as head node to collect the data from other nodes that are within reach. (d). Update the speed for all moving nodes \(V=(a,K+1)=\sum\limits_{i=1}^{3}\tau_{i}(V).\ D_{i}(a,K+1)/\sum\limits_{i=1}^{3} \frac{\tau_{i}(V)}{3}\)
```  
**Algorithm 1**The greatest lower bound (Infimum). The goal of this contribution is to focus on the walking mobility patterns because no existing mobility model supports walking patterns well. Hence, RWMM and the group mobility model have few capable features that can be modified to support many scenarios [23]. However, both are closer to ad-hoc and wireless networks than WSNs. LMM helps increase throughput, increase energy efficiency and reduce latency. Let us define how to use LMM in region based WSNs to handle walking mobility patterns:  
Let '\(K\)' be a sensor node that reaches location '\(L\)'. The probability of determining the location '\(PK\) (\(L\))' of the sensor node can be defined as:  
\[L=K_{1}x_{1}+K_{2}x_{2}+K_{3}x_{3}+K_{4}x_{4}+,\ldots,K_{n} \tag{7}\]  
Equation (7) shows the number of hops from originating node to destination node for determining the location of the desired node. Assume that the sensor nodes in the lattice are periodic with a periodic location '\(L\)'. Then no boundary condition exists, so the probability '\(P_{K}(L)\)' of determining the location of the sensor node can be calculated using the following recurrence formula:  
\[P_{K+1}(L)=\sum\limits_{L}T(L-\hat{L})P_{K}(\hat{L}) \tag{8}\]  
where \(T(L-\hat{L})\) is the probability of the sensor node when it moves from location '\(L\)' to location '\(L\)'. We introduce a generating function for the moving sensor node as follows:  
\[\psi(L,\ T_{n})=\sum\limits_{K=0}^{\infty}T_{n}^{\ K}P_{K}(L) \tag{9}\]  
where '\(T_{n}\)' is the trajectory of the sensor node and '\(\psi\)' is the state of the node in WSNs. We express the case in which the sensor node starts moving from the origin to another destination as  
\[(L,\ T_{n})=-T_{n}\sum_{L}T(L-\dot{L})\psi(\dot{L},\ T_{n})=\delta_{L,0} \tag{10}\]  
\[P_{0}(L)=\gamma_{L,0}=\left\{ \begin{array}{c}1,\ L=0,\\ 0,\ \text{otherwise}.\end{array} \right. \tag{11}\]  
where \(\delta\) is the Kronecker delta. We measure the walking patterns in two dimensions, which involve five types of Bravais lattices: rectangular, oblique, hexagonal, rhombic, and square. These five Bravais lattices help determine the location of a sensor node. We apply the Fourier transform to Equation (10) to determine the accurate location of the moving sensor node.