# Location-Based Lattice Mobility Model for Wireless Sensor Networks
## 3 Problem Statement and Lattice Mobility Model Formulation  
The limitations postured by the mobility of WSNs greatly interrupt the smooth communication process. Moreover, it is quite difficult to concurrently address the problems of node and sink mobility. The key question is how to enable WSNs to efficiently maintain QoS provisioning and extend the network lifetime with respect to imposed mobility limitations for robust disaster/earthquake salvage operations. In a WSN, the mobility behavior in the mobile sensor node can be categorized by a two-phase process--an active and an inactive phase. During an active phase, the compressed data are transmitted to other nodes or sinks. During an inactive phase, a mobile sensor node moves to a new location, then collects and locally compresses data. To frame this mobility behavior, we introduce the LMM to control the random movement of sensor nodes over WSNs. The idea of the lattice comes from a moderately ordered set in which every pair of elements is based on a supremum (known as a join or least upper bound) and an infimum (known as a meet or greatest lower bound). For example, the network '\(n\)' consists of different regions and supremum can be considered as a subset '\(S_{s}\)' or a network region-1 '\(n\)', but it could be partially ordered set of another region '\(R\)'. Thus, nodes in '\(S_{s}\)' are referred to as least element in R. The nodes in another region '\(R\)' are greater than or equal to all the existing elements of '\(S_{s}\)'. As a result, supremum is also referred to as the least upper bound (LUB). On other hand, infimum is the subset '\(S_{s1}\)' or a network region-2 that is partially ordered set of another region '\(R\)', but the nodes of '\(S_{s}\)' are greatest elements in '\(R\)' that are less than or equal to all elements of '\(S_{s1}\)'. As a result, infimum is referred to as greatest lower bound (GLB). Thus, an idea of supremum and infimum is applied in the LMM, which helps the sensor nodes maintain connectivity and choose the node with more energy and the shortest distance during the communication. We choose the specific node for targeting the mobility-aware node based on the least distance and high residual energy using the concept of infimum and supremum, as depicted in Figure 1. We formulate the problem using supremum and infimum. The supremum is used to represent and sample the maximum energy of senor node in the region '\(R\)' and infimum is used for determining the shortest distance of node available at 1-hop neighborhood. Let '\(K\)' be a sensor node that wants to find the location and distance of communicating node. Thus, node '\(K\)' uses infimum features to determine the 1-hop neighbor distance node that is called as the location-determining node '\(L\)'. '\(L\)' is the closest node to '\(K\)' in the region '\(R\)' in the network '\(N\)'. In addition, we initially pick three nodes, including '\(K\)' in the same region with maximum energy. These nodes are '\(K\)', '\(L\)' and '\(M\)' with the highest residual energy. The nodes are picked using supremum features for determining the highest energy available in the node at 1-hop neighbor distance. Based on infimum and supremum, the next node is selected to be responsible for forwarding the data and finding the location of desired destination node explained in Figure 1.  
The chosen node is called a lattice node. Figure 2 gives a graphical representation of determining the lattice node. Figure 1: Lattice node selection process based on infimum and supremum properties. Figure 2: \(Cont\). In Figure 2b, the final lattice node is the one that satisfies both conditions of supremum and infimum. If the sensor node '\(K\)' is already proved with maximum energy (supremum), then this claim can furthermore be satisfied using Theorem 1. **Theorem 1**. : _Every nonempty search of '\(K\)' node with below infimum and above supremum in the region '\(R\)' is bounded._  
Proof. : This theorem provides the foundation for many existence results in realistic scenarios and studies.