In this section, we propose a new mathematical model based on a semi-Markov process theory, to analyze the accurate performance of the 3Z system from the viewpoint of the total signaling cost on radio channels.

#### 3.2. Semi-Markov Process Model for 3Z

Many studies on location registration assume that sojourn time in a zone follows an exponential distribution. On the other hand, our model applies to any general distributions for sojourn time in a zone only if its Laplace–Stieltjes transform can be obtained.

Furthermore, many studies employ a symmetric random-walk mobility model that gives the same probability of moving to any of the neighboring zones. However, it is easy to see that these assumptions are not realistic when considering the mobility of the actual UE. Normally, the next zone of the UE is related to the current zone. Therefore the assumption of a symmetric random walk is not realistic and does not reflect true UE behavior. Therefore, the probability of returning to the previous zone is assumed to be

θ to reflect the dependency between the current zones and the next zone [

1,

10,

11]. In a square zone environment, as shown in

Figure 1 and

Figure 2,

θ would normally have a value greater than 0.25. Note that

θ = 0.25 means a symmetric random walk model.

To illustrate the semi-Markov process model, we take into consideration the composition of the three zones for a 3Z system.

#### 3.2.1. Definition of State by the Position of Each Zone in a 3Z System

Among the three zones, let us call the two zones at each end edge zones and the middle zone center zone and mark them as E and C, respectively. Sometimes, it is simpler to mark the two edge zones as ① and ③, and the center zone as ②. Note that the edge zone, marked as ① (E_{1}), is where location registration occurred more recently than the other edge zone (E_{3}).

For example, let us assume UE moves in the order of zone X (and registers) → zone Y (and registers) → zone Z (and registers) → zone Y, as shown in

Figure 3. In this situation, zone X and zone Z are the edge zones, and zone Z is where location registration occurred most recently, we define this as state 1.

Let us define the state where we apply our semi-Markov process model. We define the following states with respect to the regular case.

E and C indicate that the most recently registered zones are an edge or center zones, respectively. The first subscript i (= 1, 2, 3) represents the UE’s immediately previous zone, the second subscript j (= 1, 2, 3) indicates the UE’s current zone.

For example, for the case of the UE shown in

Figure 3, its state is E

_{12} since the current zone is the center (

j = 2) and the last zone is the edge zone where it registered its location (

i = 1). If the UE in

Figure 3 returns to zone X, its state will be E

_{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 E

_{23} returns to zone Y, its state will be E

_{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 3 returns to zone Z, its state will be E

_{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).

Figure 4 presents a collection of examples for states E

_{ij}.

Example for C_{ij} (i = 1, 2, 3; j = 1, 2, 3) will be given in the next subsection for better understanding.

#### 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:

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}.

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}.

#### 3.2.3. Paging Procedure

In this study, it is assumed that sequential paging to the three zones was performed. When an incoming call occurred, the paging procedure was completed if there was a response from the paged zone. If there was no response, paging was performed again for the remaining zones.

The paging order should be determined to minimize paging costs. In this study, the most recently registered zone is paged first. If there is no response, the center zone is paged next. If there is no response again, the remaining edge zone is paged. Note that if the UE is in state I_{ij}, the first paging attempt always succeeds.

#### 3.2.4. States for Semi-Markov Process and Transition Diagram

A summary of all the defined states is in

Table 1.

The zone registered most recently by the UE can be the edge or center zone as shown in the table above. Additionally, all the states of the UE can be defined by classifying its state with respect to the last and current zones.

If the UE’s current zone is an edge zone, regardless of whether or not the edge zone is 1 or 3, the last zone is unconditionally a center zone. As a result, these states are expressed as E_{21} or E_{23}. If the state is E_{23}, this means that the UE’s most recently registered location is in an edge zone and it moved to the other edge zone that is already in the zone list. If the status is E_{21}, this means that the UE’s most recently registered location is in an edge zone and it moved to another zone that is already in the zone list and its current zone is an edge zone in which it registered its location most recently.

Through a time-consuming process, a state transition diagram for all the states was obtained, as shown in

Figure 8.

#### 3.2.5. Calculation of State Transition Probabilities

Note that, in the state transition diagram, every transition probability contains

m or

m’. The first

m is the probability that the UE is in a zone and will move to the neighboring zone before a call to/from the UE occurs.

m can be obtained using the following equation:

where

${f}_{m}^{\ast}\left(s\right)$ is the Laplace–Stieltjes transform for

${T}_{m}$ (=

${{\displaystyle \int}}_{t=0}^{\infty}{e}^{-st}{f}_{m}\left(t\right)dt)$.

Next, let us derive m’ = P(T_{c} > R_{m}), which is the probability that the UE, whose state changed due to a call without movement to a new zone, will move to the neighboring zone before another call from/to the UE occurs.

The density function of

R_{m},

f_{r}(

t) comes from the random observer property [

1],

The Laplace–Stieltjes transform for the distribution is as follows:

#### 3.2.6. Calculation of Sojourn Time in Each State

It is necessary to note that the sojourn time of state I_{ij} (i = 1, 2, 3; j = 1, 2, 3) is different from the sojourn time of state E_{ij} (i = 1, 2, 3; j = 1, 2, 3) or state C_{ij} (i = 1, 2, 3; j = 1, 2, 3). The sojourn time of state E_{ij} (i = 1, 2, 3; j = 1, 2, 3) or state C_{ij} (i = 1, 2, 3; j = 1, 2, 3) is the interval from the time a UE enters a zone until the time it leaves. On the other hand, the sojourn time of state I_{ij} (i = 1, 2, 3; j = 1, 2, 3) is the interval from the time a call to/from a UE occurs in a zone, until the time it leaves.

In the transition diagram, the UE in state E_{21} (the state that the UE registered when moving from the center zone to an edge zone) can transit to one of three states: (i) If it moves to a new zone (with probability (1 – θ)P(T_{c} > T_{m}) = (1 – θ)m), its state becomes E_{21} again. (ii) If it returns to the last zone (with probability θP(T_{c} > T_{m}) = θm), its state becomes E_{12}. (iii) If a call from/to the UE occurs before the UE moves to a neighboring zone (with probability P(T_{c} < T_{m}) = 1 – P(T_{c} > T_{m}) = 1 – m), UE’s state changes to state I_{21}.

We should consider the sojourn time of each state to be different, in order to derive an accurate probability of the first paging failure.

Now, let us derive the sojourn time of state I

_{ij} (

i = 1, 2, 3;

j = 1, 2, 3). Since the sojourn time of state I

_{ij} (

i = 1, 2, 3

; j = 1, 2, 3) can be expressed as:

and its mean can be derived as follows:

Next, let us derive the sojourn time of state E

_{ij} (

i = 1, 2, 3

; j = 1, 2, 3) or state C

_{ij} (

i = 1, 2, 3

; j = 1, 2, 3). Since the sojourn time of state E

_{ij} (

i = 1, 2, 3;

j = 1, 2, 3) or state C

_{ij} (

i = 1, 2, 3

; j = 1, 2, 3) can be expressed as:

and its mean can be derived as follows:

#### 3.2.7. Calculation of Steady-State Probabilities

To get the steady-state probability

$\tilde{\pi}$ considering the different sojourn times, we first calculate the steady-state probability

**π** for the usual Markov chain with transition probability

**P**. This can be obtained by using the following balanced equations [

20]:

Then, the final steady-state probability of the semi-Markov process can be obtained as below [

6]: