# Analyzing Zone-Based Registration under 2-Step Paging in Mobile Communication Network
## 4 Numerical Results  
To compare the performances of 1 ZR and 2 ZR in various environments using the analytical model presented in Section 3, the following factors are assumed [9; 10; 11; 12]:  
* The zone is composed of cells of the same square. * It is assumed that the probability of moving to the zone visited immediately before, \(q\), is 0.3. * The time spent in a cell follows an exponential distribution with a mean of \(1/\lambda_{m}\). * The number of incoming calls to a UE follows a Poisson distribution with rate \(\lambda_{c}\). * The location registration cost is four times the paging cost per cell (\(U=4,\,V=1\)). For 12R and 22R, the total signaling cost for various numbers of cells in a zone when simultaneous paging (1 SP) or 2-step paging (2 SP) are performed is shown in Figure 11. When the number of cells in the zone is \(n=4\), all the cases have the minimal cost, but among them, 12R under 2-step paging has the lowest signaling cost. Specifically, it can be seen that 12R under simultaneous paging exhibits worse performance than 22R, but 12R under 2-step paging has better performance than 22R. Note that it can be seen that the simulation results for proposed 12R with 2 SP are generally consistent with the numerical results using proposed mathematical models and equations. Figure 12 shows the paging cost, location registration cost, and total signaling cost of 12R and 22R. By examining the total signaling cost of both methods, we can see that the cost is the smallest when n is 4 and the total signaling cost of 12R is less than that of 2 ZR. The main reason for this is that, in the case of 12R, selecting 2 SP increases the probability of success in the first paging, so the number of cells to be paged is significantly reduced compared to 22R. Figure 11: Total signaling cost for various numbers of cells in a zone (\(\lambda_{m}=4\), \(\lambda_{c}=2\), \(q=0.3\)). Figure 12: Total signaling cost for various numbers in a zone (\(\lambda_{m}=2.5\), \(\lambda_{c}=1\), \(q=0.3\)). When the number of cells in the zone is \(n=1\), the total signaling cost of 1 ZR is higher than that of 2 ZR due to the very high registration cost. However, a decrease in the rate of registration cost in 1 ZR and an increase in the rate of paging cost in 2 ZR demonstrate steepness as the number of cells in the zone increases. Consequently, the total signaling cost of 1 ZR is less than that of 2 ZR in most of the cases when \(n\) is large. Figure 13 shows the total signaling cost for two values of \(q\) under different numbers of entering cells in unit time (\(\lambda_{m}\)). The total signaling costs of 1 ZR and 2 ZR increase as the number of entering cells increases because the probability of location registration increases. Generally, when \(q=0.5\) in the case of 2 ZR, the total signaling cost is lower than that of \(q=0.3\), because as \(q\) increases, the probability of location registration decreases. By contrast, in the case of 1 ZR, when \(q=0.5\), the total signaling cost is large compared to \(q=0.3\), because location registration occurs frequently due to the high probability of moving to the previous cell. As a result, when \(q=0.3\), the total signaling cost is always low compared to \(q=0.5\). 1 ZR shows a lower total signaling cost than 2 ZR when the number of entering cells is small. However, as the number of entering cells increases, the total signaling cost of 1 ZR increases more steeply and exceeds that of 2 ZR due to frequent location registration. In conclusion, 1 ZR with small \(q\) shows the best performance when the number of entering cells is small, and 2 ZR with large \(q\) shows the best performance when the number of entering cells is large. Figure 14 shows the total signaling cost of 1 ZR for two values of \(q\) under simultaneous paging or 2-step paging. From the figure, it can be seen that 2-step paging is superior to simultaneous paging and that the difference is very large. We can also see that the larger \(q\) is associated with the larger total signaling cost.