# Improvement and Performance Evaluation When Implementing a Distance-Based Registration
## 3 DBR with an LA Increasing on a Cell-by-Cell Basis  
In this study, instead of defining the system on a ring-by-ring basis, \(D\), we analyzed DBR by defining it as a continuously increasing number of cells \(N\). In other words, previous ring-based DBR has defined the center of the central cell of the LA as the center of the LA. Thus, the number of cells comprising the LA is 1, 7 (=1 + 6), 19 (=1 + 6 + 12), and so on. In this case, \(N\) = 2-6 (between \(D\) = 1 and 2), \(N\) = 8-18 (between \(D\) = 2 and 3), and so on cannot be determined. In this study, to consider all cases, we did not simply limit the center of the LA to the center of the cell. Instead, we tried to have an LA consisting of a variety of cells by newly defining the center of the LA to be possible at any point in the LA. This study dealt with cases where \(N\) = 1-12 for convenience of analysis. Note that previous studies [11; 15; 16] have shown that the optimal \(D\) with minimal signaling cost is mostly 2. In this study, on the other hand, we found that the signaling cost appeared to be smaller for \(N\) = 1-12 compared to \(D\) = 2. As one example for the new definition of the center of the LA, let us look at the case when the number of cells, \(N\), is 4 in DBR. To explain the proposed method, let us denote two cells close to the center of the LA (C), as 0 and the remaining two cells as 1 for convenience as shown in Figure 2 below. In a ring-by-ring DBR, when entering a new cell, the distance between the center of the entered cell and the center of the central cell of the LA is calculated and the location registration is performed if the calculated distance is greater than \(D\). However, if \(N\) = 4, location registration cannot be defined by such a datum. Therefore, if \(N\) = 4, the UE calculates the distance between the center of the entered cell and the newly defined center of the LA (C). If \(N\) = 4, the center of LA, C, is defined as the center of the two cells marked with 0. If the calculated distance is greater than the reference distance \(D_{4}\), then location registration is performed. What value should the reference distance \(D_{4}\) have in this case? The reference distance \(D_{4}\) should have a value between the lower bound (\(L_{4}\)) and the upper bound (\(U_{4}\)). As shown in Figure 3, \(L_{4}\) is the distance between the center of the current LA (C) and the center  
Figure 2: Center of location area (\(N\) = 4). of the farthest cell from C. In addition, \(U_{4}\) is the distance between the center of the current LA (C) and the center of the nearest cell (from C) that is not included in the current LA. \[L_{4}<D_{4}<U_{4}\]  
when \(N=4\), assuming one side is 1 in a hexagonal cell, then \(L_{4}=\frac{3}{2}\) and \(U_{4}=\frac{\sqrt{21}}{2}\) can be obtained. \[\frac{3}{2}<D_{4}<\frac{\sqrt{21}}{2}=2.29\]  
For example, if we set \(D_{4}=2\), a DBR for \(N=4\) can be defined. As one more example, we can look at a more complex case of \(N=8\). To explain the proposed method, let us denote one cell containing the center of the LA (C) as 0 for convenience, the farthest cells from C as 4, and six cells surrounding 0 as 1, 2, and 3 by two cells each as shown in Figure 4 below. Figure 4: Center of location area, lower bound, and upper bound of reference distance (\(N=8\)). Figure 3: Lower bound and upper bound of reference distance (\(N=4\)). In a ring-by-ring DBR, if \(N=8\), location registration cannot be defined. If \(N=8\), the UE calculates the distance between the center of the entered cell and the newly defined center of the LA (C). If \(N=8\), the center of the LA, C, is defined as the midpoint between the center of the cell 0 and the vertex (of cell 0) directed for the farthest cell (4). If the calculated distance is greater than the reference distance \(D_{8}\), then the location registration is performed. If \(N=8\), the reference distance \(D_{8}\) should have the value between the lower bound (\(L_{8}\)) and the upper bound (\(U_{8}\)).