## 2 Zone-Based Registration Scheme
The probability, \(m\), that no call would occur until the UE entered a new zone was derived as follows: \[m=\mathrm{P}(T_{c}>T_{m})=\int_{0}^{\infty}\int_{t_{m}}^{\infty}\lambda e^{- \lambda t_{c}}f_{m}(t_{m})dt_{c}dt_{m}=f_{m}^{*}(\lambda)\] (1)  
Figure 1: Zone-based registration based on the number of zones (\(\blacktriangle\) Registration, \(\bigcirc\) No registration). where \(f_{m}^{*}(s)\) was the Laplace-Stieltjes transform for \(T_{m}=\left(\int_{0}^{\infty}e^{-st}f_{m}(t)dt\right)\). Thus, \(1-m=\mathrm{P}(T_{c}\leq T_{m})\) was the probability of a call occurring before leaving the zone, in which case the implicit registration was performed after the call was connected. Note that if a call occurred while UE stayed in the zone, the implicit registration was processed. If UE moved to another new zone before the call occurred, registration was required. 7. The time remaining in the zone after a call occurrence was defined as a random variable \(R_{m}\). The \(f_{r}(t)\) was a probability density function for \(R_{m}\), which could be easily obtained using the random observer property [8; 26]: \[f_{r}(t)=\mu\int_{t}^{\infty}f_{m}(\tau)d\tau=\mu(1-F_{m}(t))\] (2) Further, the Laplace-Stieltjes transform for \(R_{m}\) was \[f_{r}^{*}(s)=\int_{0}^{\infty}e^{-st}f_{r}(t)dt=\int_{0}^{\infty}e^{-st}\mu(1- F_{m}(t))dt=\frac{\mu}{s}(1-f_{m}^{*}(s))\] (3)
8. After the call processing, the UE would stay in the zone during \(R_{m}\). Using Equations (2) and (3), we could derive the probability, \(m^{\prime}\), that another call would not happen again while the UE stayed in the zone: \[m^{\prime}=\mathrm{P}(T_{c}>R_{m})=\int_{0}^{\infty}\int_{r_{m}}^{\infty} \lambda e^{-\lambda t_{c}}f_{r}(r_{m})dt_{c}dr_{m}=\frac{\mu}{\lambda}(1-f_{ m}^{*}(\lambda))\] (4) Likewise, \(1-m^{\prime}=\mathrm{P}(T_{c}\leq R_{m})\) denoted the probability of another call connection before \(R_{m}\), and Network_DB could be updated by implicit registration. 9. Compared to times (\(T_{m}\) and \(R_{m}\)) of staying in the zone, the call duration was negligibly small [4; 5; 9; 16; 17; 21]. Under this assumption, we could obtain \(m\) and \(m^{\prime}\) normally. It is now necessary to define the states for the two types described in Figure 1. To define the status of UE, three different cases for UE's movements were considered for each of the two types as follows:  
Case 1: Three different zones form one RA, and the UE travels inside these zones. In particular, in Type I, the most recently registered zone belongs to the boundary of RA. As depicted in Figure 2, four states can be defined for Type I, whereas six states can be defined for Type II. Here, states \(E_{ij}^{I}\) and \(E_{ij}^{II}\) represent a case of moving from zone \(i\) to zone \(j\) for Types I and II, respectively. For example, state \(E_{23}^{I}\) indicates the state in which the UE remains in zone 3 after the UE moves to 3\(\rightarrow\)2\(\rightarrow\)1, stores three zones as RA, and then moves to 1\(\rightarrow\)2\(\rightarrow\)3 without an additional registration. In this state, if a call arrives to the UE, then three paging processes are required (1\(\rightarrow\)2\(\rightarrow\)3). That is, after paging zones 1 and 2, a response is not received. However, after zone 3 is paged, the call can be connected. Figure 2: Moving without additional registration in a 3\(Z\) system. (**a**) Type I; (**b**) Type II. Case 2: This is a case in which an incoming (or outgoing) call is generated while UE stays in the RA. Hence, UE's location information can be modified in the network database without requiring an additional registration procedure. This process is called implicit registration [1; 15]. Figure 3 shows the states in which implicit registration occurs for each type. States \(I_{ij}^{I}\) and \(I_{ij}^{II}\) are defined as the states in which the call is processed while the UE resides in zone \(j\) after arriving from zone \(i\) for each type.