# Improvement and Performance Evaluation When Implementing a Distance-Based Registration
## 6 Numerical Results  
This section analyzes registration and paging costs in DBR using the mathematical model presented in Section 4. The environment used in the analysis is assumed to be the same as in previous studies [6; 9; 10; 15; 16]:  
* UE's sojourn time in a cell follows an exponential distribution with mean \(1/\lambda_{m}\)
* Arrival process of incoming calls to UE follows a Poisson process with rate \(\lambda_{c}\)
* Registration cost for one registration, \(U\), is 4
* Paging cost for one cell, \(V\), is 1 As a basis for comparison, call-to-mobility ratio (CMR) is used. It represents the mobility of UEs and arrival characteristics of calls. CMR stands for \(\lambda_{\text{c}}/\lambda_{\text{m}}\). A CMR of 1/3 means that the UE crosses the cell boundary an average of three times between call arrivals. The smaller the CMR, the greater the mobility of the UE between call arrivals, and the more frequent the location registration occurs. Figure 9 shows registration cost, paging cost, and total signaling cost when CMR is 1 according to a change in the number of cells in the LA. In general, the larger the number of cells in an LA, the smaller the UE's location registration and the smaller the location registration cost. On the other hand, the larger the number of cells in an LA, the higher the number of cells that need to be paged and the higher the paging cost. Therefore, the optimal \(N\) that the total signaling cost is minimal can be obtained using the trade-off between the registration cost and the paging cost. In the same analytical environment, the optimal LA size was \(D=2\) (\(N=7\)) in previous studies [11; 15; 16]. However, in this study, assuming an LA is increasing on a cell-by-cell basis, the number of cells with a minimum total signaling cost appears to be 4-7. This means that the optimal LA can be constructed in an environment that is not possible if only a ring-by-ring DBR is considered. Therefore, for an accurate evaluation of DBR, the performance should be evaluated for the LA increasing on a cell-by-cell basis as shown in this study. Figure 10 shows the total signaling cost according to changes in CMR. If the CMR is 0.4, the total signaling cost represents the minimum value when \(N\) is 7. If the CMR is 1, the total signaling cost represents the minimum value when \(N\) is 4. This is because when CMR increases, paging will occur frequently, resulting in a larger role of paging cost in the total signaling cost and a smaller optimal \(N\). Figure 9: Total signaling cost for various numbers of cells in an LA (CMR = 1). Figure 11 shows total signaling cost when the registration cost U changes. When U is 6 or 8, the total signaling cost is minimal when \(N\) is 7. When U becomes smaller, the optimal \(N\) with the minimal total cost also becomes smaller. As shown in the figure, when U is 4, the total signaling cost is minimal when \(N\) is 4. When U is 2, the total signaling cost is minimal when \(N\) is 1. In other words, the smaller the registration cost U, the smaller the LA with the minimal total cost. As the registration cost U decreases, the smaller the proportion of the registration cost. Conversely, the larger the paging cost, the smaller the number of cells in the LA with the minimal cost. In general, the smaller the proportion of registration cost, the smaller the number of cells in the LA. Similarly, the greater the proportion of registration cost, the larger the optimal number of cells in the LA and the larger the total cost. Figure 12 shows the total signaling cost when variance (Var) of cell sojourn time varies. As shown in the figure, even if the cell sojourn time is the same, the total signaling cost decreases as the variance increases. As the variance increases, the number of registrations decreases while the optimal \(N\) increases which minimizes the total signaling cost. Figure 11: Total signaling cost for various Us (CMR = 1/3). Figure 10: Total signaling cost for various CMRs. Figure 13 shows the total signaling cost when cell sojourn time E(Tm) changes.