## 2 Zone-Based Registration Scheme  
In this study, we assumed that the number of cells in each zone was \(n\) and that the zone had a hexagonal shape. Moreover, the number of zones included in each RA was defined as a 1Z system when it had one zone, a 2Z system when it had two zones, and a 3Z system when it had three zones. Figure 1 shows the location registration process due to UE's movement for each system. UE moved from 3\(\rightarrow\)2\(\rightarrow\)1, changed direction, and then moved back to zone 1\(\rightarrow\)2\(\rightarrow\)3. First, in 1Z, a new registration was required whenever UE left the zone, resulting in four registrations in total. In 2Z, since the UE and network stored two zones in the database (DB), the registration procedure did not occur if it returned from zone 1 to zone 2. Accordingly, if the incoming call arrived when the UE remained in zone 2, then the UE would be connected with two paging processes. That is, in 2Z, since the registration process was not carried out when the UE moved from 1\(\rightarrow\)2, the network database (Network_DB) knew that the UE was in zone 1. Thus, a call connection was made possible by paging zone 2 without receiving a response after paging zone 1 while the UE remained in zone 2. Finally, in a 3Z system, no registration was required when moving among three zones belonging to the same RA. Similarly, if the UE stayed in zone 2, then two paging messages occurred, while when it was in zone 3, three paging procedures were needed. Note that in DB = \(\boxed{a|b|c}\)\(a\) was the most recently visited (or registered) zone, \(b\) was the previously registered zone, and \(c\) was the oldest visited zone. Unlike in previous studies [16; 17], when a hexagonal zone was assumed, the 3Z scheme should be analyzed in consideration of two different types, as shown in Figure 1. Type I was defined as having three zones registered sequentially without forming a cluster. By contrast, Type II was a case where three zones formed a cluster with the number of surfaces in contact being three. To analyze the performance of 3Z, we assumed the following mobility models:  
1. The number of cells in each zone was assumed to be n. As analyzed in [27; 28], it should be analyzed assuming UE's movement between cells. However, when n was large (usually 7 or more), the shape of the zone became close to a hexagonal shape. Thus, a mobility model between zones was adopted. Note that in 1Z, registration would be required if the zone changed, not when the cell changed. 2. Adopting a random-walk mobility model [16; 17; 26], the UE moved to one of the six neighboring zones, where the probability of it returning to the previously visited zone was \(\theta\). Thus, the probability of it moving to one of the other five adjacent zones was equal to \((1-\theta)/5\). 3. The number of calls to be processed per unit of time followed a Poisson distribution with a mean of \(\lambda\). Due to the additive nature of the Poisson process [26], the \(\lambda\) could be expressed as the sum of the number of outgoing calls (\(\lambda_{o}\)) and incoming calls (\(\lambda_{i}\)) treated during unit time (\(\lambda=\lambda_{o}+\lambda_{i}\)). 4. Based on the above assumption (2), the time interval, \(T_{c}\), between two calls followed an exponential distribution with a mean of \(1/\lambda\). 5. The time, \(T_{m}\), that the UE stayed in each zone followed a general distribution with an average of \(\mathrm{E}(T_{m})=1/\mu\). When a call occurred to the UE, the UE's location information in the network database could be modified using an implicit registration process [1; 15] without any additional registration cost. The call-to-mobility ratio (CMR) was then defined as CMR = \(\lambda/\mu\). The case of CMR = 1 implied a situation in which the UE entered another zone once on average until the next call occurred. 6.