# Analyzing Zone-Based Registration Using a Three Zone System: A Semi-Markov Process Approach
## 3 New Mathematical Model
### Semi-Markov Process Model for 3Z
#### 3.2.2 Definition of States by the Call Occurrence of Each Zone in a 3Z System  
Location registration is divided into two main categories [9; 10; 11]. The first category is if the UE moves to a new zone that is not a previously stored zone, then it registers the location. The second category is if a call occurs from/to the UE in the current zone, the network updates the location. In the case of the first category, a location registration message is actually sent to the network. In the case of the second category, a separate location registration message is not delivered to the network, but the network can infer the UE's zone through the call processing messages. The first category is referred to as regular registration (RR) and the second category is referred to as implicit registration (IR).  
In the case of the first category, when the UE enters a new zone, the corresponding state is defined and the time it stays in that state is the total sojourn time in that zone. On the other hand, in the case of the second category, when a UE is currently in a zone and a call occurs from/to the UE, the corresponding state is defined so that the time it takes to stay in that state is typically less than the total sojourn time in that zone. In other words, we introduce state I to distinguish cases in the second category from cases in the first category since the time in each state is different.  
We define the following states with respect to the second category:  
\[\text{I}_{ij}\;(i=1,\,2,\,3;j=1,\,2)\]  
I indicates that UE receives/generates a call to register its location by IR. The first subscript \(i\) (\(=1\), \(2\), \(3\)) represents the UE's immediately previous zone, the second subscript \(j\) (\(=1,\,2\)) indicates the UE's current zone.  
It is sufficient to define only states I\({}_{21}\), I\({}_{12}\) and I\({}_{32}\) concerning call occurrence. For example, if UE with a state of E\({}_{21}\), E\({}_{12}\) and E\({}_{32}\) receives/generates a call, the UE's zone remains the same but the states are changed to I\({}_{21}\), I\({}_{12}\) and I\({}_{32}\) respectively.  
Figure 5 presents a collection of examples for states I\({}_{ij}\).  
Figure 5: Examples of states I\({}_{ij}\): (a) I\({}_{12}\); (b) I\({}_{21}\); (c) I\({}_{32}\).  
Note that if UE with a state of E\({}_{23}\) or C\({}_{23}\) receives/generates a call, its state will not be I\({}_{23}\) but I\({}_{21}\) since the last zone is the center (\(i=2\)) and the current zone is the edge zone where it registered its location most recently (\(j=1\)) by IR, which means that I\({}_{23}\) is not necessary.  
For another example, let us say in the case of the UE shown in Figure 6, its state becomes I\({}_{12}\) when a call occurs since its state was E\({}_{12}\) just before the call occurred. If the UE returns to zone X, its state will be C\({}_{23}\) since the last zone is the center (\(i=2\)) and the current zone is the edge zone where it did not register its location most recently (\(j=3\)). In addition, if the above UE with a state of C\({}_{23}\) returns to zone Y, its state will be C\({}_{32}\) since the last zone is the edge zone where it did not register its location most recently (\(i=3\)) and the current zone is the center (\(j=2\)).  
On the other hand, if the UE in Figure 6 returns to zone Z, its state will be C\({}_{21}\) since the last zone is the center (\(i=2\)) and the current zone is the edge zone where it registered its location most recently (\(j=1\)). In addition, if the above UE with a state of C\({}_{21}\) returns to zone Y, its state will be C\({}_{12}\) since the last zone is the edge zone where it registered its location most recently (\(i=1\)) and the current zone is the center (\(j=2\)).  
Figure 7 presents a collection of examples for states C\({}_{ij}\).