論文 / 著書情報
Article / Book Information
題目(和文)
Title(English)

Periodic Handover Skipping in Cellular Networks and Its Stochastic
Geometry Analysis

著者(和文)

徳山喜一

Author(English)

Kiichi Tokuyama

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第12655号,
授与年月日:2024年3月26日,
学位の種別:課程博士,
審査員:三好 直人,金森 敬文,髙邉 賢史,中野 張,府川 和彦

Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第12655号,
Conferred date:2024/3/26,
Degree Type:Course doctor,
Examiner:,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Science Tokyo Research Repository)

Periodic Handover Skipping in
Cellular Networks and Its
Stochastic Geometry Analysis

Kiichi Tokuyama
Thesis submitted forthe Degree of Doctor of Science
Department of Mathematical and Comuting Sceience
School of Computing
Tokyo Institute of Technology2024Preface
Stochastic geometry is the study of random spatial patterns. Such random
structures have been recognized to play a key role in various fields such ascosmology, ecology, cell biology, engineering, and data sciences. One substantial essence in the field of stochastic geometry is the theory of spatial
point processes, which are the tools to represent the random allocation ofpoints in a certain given space. The spatial point processes have been applied to various contexts in the real world, for example, random arrivals ofcustomers, distribution of Earthquake location on the whole world.
This thesis studies applications of the stochastic geometry to the handover
management problem in wireless cellular communication networks. Handover
management has attracted attention of research in the context of wirelesscellular communication networks. One crucial problem of handover management is to deal with increasing handovers experienced by mobile users. To
address this problem, handover skipping techniques have been studied in recent years. In this thesis, we propose yet another handover skipping scheme,
called periodic handover skipping. In the proposed scheme, handovers of amobile user are controlled by a certain fixed period of time, which we callskipping period. The skipping period can be managed as a system parameter,
thereby enabling flexible operation of handover skipping.
We first investigate the periodic handover skipping scheme on a basicmodel of a single-tier cellular network, where base station in the cellular network are deployed according to a homogeneous Poisson point process. Homogeneous Poisson point processes are often associated with point patterns
that do not have any interaction between points. We provide a tractable
framework for analyzing the periodic handover skipping scheme. Under arandom walk model of user mobility, we derive the analytical expressions of
the two performance metrics, the handover rate and the expected downlink
data rate. Moreover, by using these two metrics, we construct a utility metricrepresenting transmission performance, regarding the trade-off relation between the handover rate and the data rate. Based on this utility metric, we
conduct performance comparison between two scenarios where the periodic
1

handover skipping scheme is introduced and not introduced.
We next investigate the optimal skipping period on the same single-tiercellular network model, and consider maximizing the utility metric by controlling the skipping period introduced in the periodic handover skipping
scheme. Moreover, we attempt to derive an approximate expression of the
optimal skipping period. We also conduct numerical comparison with someother handover skipping techniques.
We further study the periodic handover skipping scheme on a two-tier cellular network based on a homogeneous Poisson point process and a PoissonPoisson cluster process. Poisson-Poisson cluster processes are often used tomodel clustered nodes in a network, for example, small base stations deployed organically to complement the capacity of the cellular networks atuser hotspots. Inside the network, we consider a random walk-based UEwith the periodic handover skipping technique. Based on the system model,
we provide analytical results for both the handover rate and the expected
data rate, which are derived via the exact and the approximate analyses,
respectively. In addition, we conduct numerical experiments to verify theefficiency of our proposed model.

2

Acknowledgements
First of all, I would like to express my gratitude to my supervisor, Professor
Naoto Miyoshi, who instructs me in doctor course. He has given me a lot of
userful advice and taught me how to write a thesis of research. I would also
like to express my gratitude to Professor Toru Ohira, my supervisor in mastercourse. He has taught me ABC of research. Without their support, I could
not complete this thesis. I would also like to express my gratitude to thelaboratory members. Thanks to them, I had a good time in the laboratory.
I am grateful to Associate Professor Tatsuaki Kimura of Doshisha University. He was Senior Lecturer of Osaka University until an year ago, andgave a lot of valuable advice on how to write a paper by English.
Last but not least, I would like to thank my father for giving me theopportunity to go on to the doctoral course and for keeping to watch overme in every situation. In addition to the people mentioned above, I was able
to complete this thesis thanks to the support of many people. I would liketo express my deep appreciation here.

3

Contents
Preface1

Acknowledgements

3

1 Introduction
6
1.1 Basics of stochastic geometry and spatial point processes . . . 6
1.2 Poisson point process . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Poisson-Poisson cluster process . . . . . . . . . . . . . . . . . 8
1.4 Handover management problem in cellular networks . . . . . . 9
1.4.1 Handover skipping . . . . . . . . . . . . . . . . . . . . 9
1.4.2 Periodic handover skipping . . . . . . . . . . . . . . . . 10
1.5 Organization of this thesis . . . . . . . . . . . . . . . . . . . . 11
2 Data Rate and Handover Rate Analysis of Periodic Handover
Skipping in Homogeneous Networks
13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Related work . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Network model . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Periodic handover skipping . . . . . . . . . . . . . . . . 17
2.2.3 Mobility model . . . . . . . . . . . . . . . . . . . . . . 18
2.2.4 Performance metrics . . . . . . . . . . . . . . . . . . . 18
2.3 Performance analysis and evaluation . . . . . . . . . . . . . . 19
2.3.1 Expected downlink data rate analysis . . . . . . . . . . 20
2.3.2 Handover rate analysis . . . . . . . . . . . . . . . . . . 25
2.3.3 Numerical evaluation of performance metrics . . . . . . 26
2.3.4 Utility metric . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Proofs of Lemmas and Corollary . . . . . . . . . . . . . . . . . 29
2.4.1 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . 29
4

2.5

2.4.2 Proof of Lemma 2.2 . . . . . . . . . . . . . . . . . . . . 30
2.4.3 Proof of Corollary 2.1 . . . . . . . . . . . . . . . . . . 31
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Optimal Skipping Period
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Related work . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Contribution . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . .
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Utility metric . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Results of analysis . . . . . . . . . . . . . . . . . . . .
3.3 Analysis of optimal skipping period . . . . . . . . . . . . . . .
3.3.1 Approximation of utility metric . . . . . . . . . . . . .
3.3.2 Analytical expression of optimal skipping period . . . .
3.4 Numerical evaluation . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Accuracy of approximate utility metric . . . . . . . . .
3.4.2 Numerical evaluation of 𝑠∗ . . . . . . . . . . . . . . . .
3.5 Comparison with other handover skipping schemes . . . . . . .
3.6 Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . . .
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34
34
34
36
36
36
36
37
39
39
39
40
40
40
42
44
46

4 An Extension to Heterogeneous Networks Using PoissonPoisson Cluster Process47
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.1 Network model . . . . . . . . . . . . . . . . . . . . . . 49
4.2.2 Mobility model for a moving user equipment . . . . . . 50
4.2.3 Performance metrics . . . . . . . . . . . . . . . . . . . 50
4.3 Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.1 Handover rate analysis . . . . . . . . . . . . . . . . . . 51
4.3.2 Data rate analysis . . . . . . . . . . . . . . . . . . . . . 54
4.4 Evaluation of transmission performance . . . . . . . . . . . . . 56
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Conclusion
60
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Directions of future works . . . . . . . . . . . . . . . . . . . . 61

5

Chapter 1
Introduction
1.1

Basics of stochastic geometry and spatial
point processes

Stochastic geometry is the study of random spatial patterns. At the heart ofthe subject lies the study of random point patterns. Such random structures
have been known to play a key role in several branches of natural sciences(cosmology, ecology, cell biology) and engineering (material sciences, networks) for several decades. Their use is currently expanding to new fields
like data sciences. One key essence in the field of stochastic geometry is thetheory of spatial point processes, which are the tools to represent the random
allocation of points in a certain given space. The spatial point processes havebeen applied to various contexts in the real world, e.g., random arrivals ofcustomers, distribution of Earthquake location on the whole world.
To define the spatial point processes, we first start to define a counting
measure. In this thesis, we especially consider the given space as R𝑑 .
Deﬁnition 1.1. A counting measure 𝜇 is a locally ﬁnite measure on (R𝑑 ,B (R𝑑 )) such that 𝜇(𝐵) ∈ N for all 𝐵 ∈ B𝑐 (R𝑑 ), where B𝑐 (R𝑑 ) denotes
the set of relatively compact sets in B (R𝑑 ), i.e., B𝑐 (R𝑑 ) := {𝐵 ∈ B (R𝑑 );the closure of 𝐵 is compact}.
Intuitively, a counting measure 𝜇 on (R𝑑 , B (R𝑑 )) represents a sample ofrandom allocation of points on the space R𝑑 , since 𝜇(𝐵) ∈ N could represent
the number of points included by 𝐵 ∈ B (R𝑑 ) where the pattern of the points
is characterized by 𝜇.
Using the above counting measure 𝜇, the point processes on R𝑑 are defined
as follows.

6

Deﬁnition 1.2. Let (Ω, A, P) be a probability space, M(R𝑑 ) be the set ofcounting measures on (R𝑑 , B (R𝑑 )), and M (R𝑑 ) be the 𝜎-algebra on M(R𝑑 )generated by the mappings 𝜇 → 𝜇(𝐵). Then, a point process Φ is deﬁned asa measurable mapping Φ : Ω → M(R𝑑 ). The probability distribution of Φ isP Φ = P ◦ Φ −1 .
The point process Φ on R𝑑 could be considered as a random mapping
whose realization is a counting measure. Note that the law of Φ is specified
by its probability distribution PΦ .

1.2

Poisson point process

One important instance of the point process is a Poisson point process (PPP),
which is defined as follows.
Deﬁnition 1.3. Let Λ be a locally ﬁnite measure on R𝑑 and Φ be a point
process. Φ is called PPP with intensity measure Λ if for all pairwise disjoint
sets 𝐵1 , · · · , 𝐵 𝑘 ∈ B (R𝑑 ), the random variables Φ(𝐵1 ), · · · , Φ(𝐵 𝑘 ) are independent Poisson random variables with respective means Λ(𝐵1 ), · · · , Λ(𝐵 𝑘 );that is, for all 𝑛1 , · · · , 𝑛 𝑘 ∈ N,
P(Φ(𝐵1 ) = 𝑛1 , · · · , Φ(𝐵 𝑘 ) = 𝑛 𝑘 ) =𝑘Öe−Λ(𝐵𝑖 )𝑖=1

Λ(𝐵𝑖 ) 𝑛𝑖.
𝑛𝑖 !

If the intensity measure Λ is the Lebesgue measure on R𝑑 , then Λ(𝐵1 ) =𝜆|𝐵1 | for some 𝜆 > 0; that is, the intensity measure is described by theconstant 𝜆. In this case, Φ is called a homogeneous PPP of intensity 𝜆.
It is worth noting that a PPP is characterized by the properties describedin the following theorem.
Theorem 1.1 ( [1, Theorem 2.1.14.]). Let Φ be a point process on R𝑑 andsatisfy P(Φ({𝒙}) > 0) = 0 for any 𝒙 ∈ R𝑑 . Φ is a PPP if and only if Φ has
the following properties(i) Simplicity; P(Φ({𝒙}) ≤ 1 for all 𝒙 ∈ R𝑑 ) = 1.
(ii) Complete independence; the random variables Φ(𝐵1 ), · · · , Φ(𝐵 𝑘 ) are
independent for all pairwise disjoint sets 𝐵1 , · · · , 𝐵 𝑘 ∈ B (R𝑑 ).
Due to the above theorem, PPPs are often considered when one does notexpect any interactions between points. For example, PPPs on R are used
to model random events distributed in the time axis such as the independent7

arrival of customers at a store and the independent occurrence of earthquakes.
PPPs on R2 can represent the locations of scattered objects such as particles
colliding into a detector and trees in a forest [2]. Moreover, PPPs are often
used to model the locations of wireless nodes in cellular networks, e.g., basestations (BSs) and user equipments (UEs), although there should be somespatial correlations among those nodes [1, 3]. This is because the PPPs
are said to be the most analytically tractable among various sorts of point
processes due to the independence property [4].

1.3

Poisson-Poisson cluster process

Poisson-Poisson cluster processes (PPCPs) are another instance of point processes that can represent clustering point patterns. A stationary PPCP is
made by a parent process and daughter processes, where the parent processrepresents cluster centers allocated according to a homogeneous PPP, and thedaughter processes are independent, identical and finite PPPs located aroundeach point of the parent process. PPCPs are used to model a network if thenodes are clustered, e.g., users gathered around Wi-Fi hot spots [5, 6], andsmall BSs (SBSs) deployed organically to complement the capacity of thecellular networks at user hotspots or to patch their coverage dead-zones [7].
A PPCP is formally defined as follows.
Deﬁnition 1.4.

¯ on R𝑑 is(i) A PPCP Φ(𝜆 𝑝 , 𝑔, 𝑚)Ø¯ =Φ(𝜆 𝑝 , 𝑔, 𝑚)
𝒛 + C𝒛,

(1.1)

𝒛∈Ψ 𝑝

where Ψ𝑝 and C 𝒛 denote the parent point process and the oﬀspring
point process whose cluster center is at 𝒛 ∈ Ψ𝑝 , respectively. That is,
Ψ𝑝 is a homogeneous PPP on R𝑑 with the intensity 𝜆 𝑝 , and {𝒒 ∈ C 𝒛 } isan independently and identically distributed (i.i.d.) sequence of random
vectors with joint probability density function (PDF) 𝑔(𝒒). The number¯
of points in C 𝒛 is denoted by 𝑀 ∼ Poisson( 𝑚).
(ii) When the joint PDF 𝑔 of a PPCP Φ2 satisﬁes 𝑔(𝒒) = 𝑔𝑑 (∥𝒒∥), with a
PDF 𝑔𝑑 on R, then Φ2 is called an isotropic PPCP.
¯ consists of the offspring process C 𝒛 shifted by 𝒛 for
A PPCP Φ(𝜆 𝑝 , 𝑔, 𝑚)each 𝑧 ∈ Ψ𝑝 , and 𝒙 𝑖 ∈ 𝒛 + C 𝒛 , 𝑖 = 1, · · · , 𝑀, are conditionally i.i.d with the
PDF 𝑔(𝒙 𝑖 − 𝒛).
We next introduce some important properties regarding PPCPs
8

Proposition 1.1 (Conditional PPP properties of PPCPs [7]). If a PPCP
¯ is conditioned on the parent point process Ψ𝑝 , then Φ(𝜆 𝑝 , 𝑔, 𝑚)¯Φ(𝜆 𝑝 , 𝑔, 𝑚)Í¯ 𝒛∈Ψ 𝑝 𝑔(𝒙 − 𝒛).
is an inhomogeneous PPP with intensity 𝜆(𝒙) = 𝑚
Proposition 1.1 indicates a PPCP becomes a PPP under the condition
that its parent points are fixed. This property could be used to analyze
PPCPs since we can apply various analysis results regarding PPPs to PPCPs
by giving the condition. The following is the one of the results derived from
Proposition 1.1.
Lemma 1.1 (Contact distance distribution [7, Lemma 2]). If a PPCP Φ(𝜆 𝑝 ,¯ is conditioned on the parent point process Ψ𝑝 , then the PDF of the𝑔, 𝑚)¯ fromcontact distance, i.e., the distance of the nearest point of Φ(𝜆 𝑝 , 𝑔, 𝑚)
the origin, is given byÕÖ¯ 𝑑 (𝑟 |𝒛)¯
𝑓𝑐𝑑 (𝑟 | Ψ𝑝 ) = 𝑚
𝑓𝑑 (𝑟 | 𝒛)
e−𝑚𝐹
,
(1.2)
𝒛∈Ψ 𝑝

𝒛∈Ψ 𝑝

where 𝑓𝑑 (𝑟 | 𝒛) denotes the conditional PDF of the distance 𝑟 = ∥𝒙∥ of an¯ from the origin given its cluster center at
oﬀspring point 𝒙 ∈ Φ(𝜆 𝑝 , 𝑔, 𝑚)𝒛 ∈ Ψ𝑝 , and 𝐹𝑑 (𝑟 | 𝒛) denotes its cumulative distribution function (CDF).

1.4

Handover management problem in cellular networks

The development of fifth generation (5G) wireless cellular communication
systems is driven by the need to satisfy the ever-increasing capacity demand
resulting from the proliferation of mobile phones, tablets, and the other handheld mobile devices. One of the key features of the 5G evolution is the network densification through small cell deployment [8]. Densifying base stations(BSs) shrinks the service area of each BS, increases the spectral efficiency, andoffers more capacity, thereby enabling a significant increase in the quality ofservice. However, deploying more BSs promotes frequent handovers (HOs),
which increase the risk of disconnection and signaling overhead. Managing
and mitigating those frequent HOs is required upon conducting the network
densification.

1.4.1

Handover skipping

HO skipping is an approach to address the problem of frequent HOs [9–15],
where some HOs of a moving user equipment (UE) are skipped to reduce9excessive HOs. This approach is particularly effective when the UE moves
fast, or the network consists of densified small cells. The HO skipping enables
a UE to reduce the HO rate, whereas it may decrease the data reception
rate (data rate) because it demands the UE to maintain longer connection
duration with a BS. Thus, the HO skipping presents a trade-off relation
between the increased HO rate and the decreased data rate, and the tradeoff relation should be balanced to improve the performance for UEs.
A fundamental HO skipping scheme, which we call alternate HO skipping,
was introduced in [9]. In this scheme, a UE alternately performs HOs along
its trajectory, thereby achieving a 50% reduction in the HO rate. Moreover,they evaluated user throughput analytically for the UEs following the alternate HO skipping schemes. The work [9] was extended in [10] and [11]; [10]introduced the alternate HO skipping in two-tier networks and [11] introduced it in a BS cooperating network with CoMP transmission. Another HO
skipping scheme, called topology-aware HO skipping, was proposed in [12].
In this scheme, the HO skipping is triggered according to the UE’s distancefrom the target BS and the size of the cell; HOs are not performed when
the UE enters a cell whose area is smaller than a certain threshold, or acell whose dominant BS is farther from the UE than a certain threshold.
Thus, the topology-aware HO skipping can prevent the connection distancesfrom being unnecessarily long. The mathematical framework for evaluating
topology-aware HO skipping was provided in [13]. They derived the analytical expressions of the coverage probability and the expected data rate when
the topology-aware HO skipping is performed under the assumption that the
trajectory of a moving UE is a straight line.

1.4.2

Periodic handover skipping

However, both the alternate and the topology-aware HO skippings do notconsider the connection duration, i.e., the communication time connecting
to a BS until the next HO. Such connection duration is essential for the HO
management due to the problem of signaling overhead. More precisely, whena UE performs an HO, its data transmission is temporarily disrupted during
the signaling procedure of the HO. This sudden disruption causes unwanted
delay and may significantly degrade the performance of network applications. Thus, keeping connection for a certain duration is important for thecommunication quality of ultra-reliable applications, e.g., high-quality video
streaming services and vehicular safety applications. Owing to the above reason, an HO skipping scheme that can control the connection duration should
be developed.
We propose yet another HO skipping scheme, called periodic HO skip10

ping. Our proposed scheme considers a threshold of time, which we call theskipping period, to control frequency of HOs for a moving UE, that is, all HOs
attempted earlier than the threshold are skipped, and a UE is guaranteed tokeep connected during this threshold time. By adjusting the skipping period,
our proposed scheme enables to directly control the connection duration of
the UE with a BS.

1.5

Organization of this thesis

This thesis studies the performance analysis and its evaluation of the periodic
HO skipping scheme by using spatially stochastic point processes. Chapter 2
investigates the scheme on a basic model of a single-tier cellular network,where BSs are deployed according to a homogeneous PPP. This model is associated with the cellular network model where only one kind of BSs, often
called macro BSs (MBSs), are considered. The analytical framework for evaluating the transmission performance is provided in this chapter. Chapter 3
further develops the previous chapter and investigates optimal adjustmentof the system parameters introduced in the periodic HO skipping scheme.
The investigation in Chapter 4 extends the model of the previous chapter
to a two-tier cellular network consisting of MBSs and SBSs, which are allocated according to a homogeneous PPP and a stationary PPCP, respectively.
Finally, Chapter 5 concludes the thesis and presents directions for future research. The contents of this thesis have been published as follows. Chapter 2
is based primarily on [16] and [17]. Chapter 3 is based primarily on [17].
Chapter 4 is based primarily on [18].
Chapter 2 proposes the periodic HO skipping scheme and give a study ona single-tier cellular network based on a homogeneous PPP. We investigate
the performance of the proposed scheme on the basis of stochastic geometry.
Specifically, we derive analytical expressions of two performance metrics–
the HO rate and the expected downlink data rate– when a UE adopts theperiodic HO skipping. Numerical results based on the analysis demonstratethat the periodic HO skipping scenario can outperform the scenario without
any HO skipping in terms of a certain utility metric representing the tradeoff between the HO rate and the expected downlink data rate, in particularwhen the UE moves fast.
Chapter 3 studies the optimal adjustment of the skipping period, the system parameter introduced in the periodic HO skipping scheme proposed in
the previous section. Using the analytical results developed in the previous
chapter, we numerically evaluate a utility metric representing the transmission performance in the proposed scheme and show that there can exist an
11

optimal length of the skipping period, which locally maximizes the utility
metric, and approximately provide the optimal skipping period in a simple
form. Numerical comparison with some other HO skipping techniques is also
conducted.
Chapter 4 studies the periodic HO skipping scheme on a two-tier cellular
network based on a homogeneous PPP and a stationary PPCP. Inside the
network, we consider a random walk-based UE with the periodic HO skipping
technique. Based on the system model, we provide analytical results for boththe HO rate and the expected data rate, which are derived via the exact andthe approximate analyses, respectively. In addition, we conduct numericalexperiments to verify the efficiency of our proposed model.

12Chapter 2
Data Rate and Handover Rate
Analysis of Periodic Handover
Skipping in Homogeneous
Networks
2.1

Introduction

Theory of spatial point processes and stochastic geometry has become amajor tool for performance analysis of wireless communication networks andnumerous theoretical/practical results have been developed so far. Amongthem, many studies have analyzed the data reception rate (data rate forshort) for a user equipment (UE) in various settings (see, e.g., [19–21]) as
one of the primary performance metrics. However, most of them assume
that a UE is fixed at a certain position. On the other hand, there are studies
concerning mobility of UEs in the framework of stochastic geometry, which
are rather involved in the handover (HO) rate (see, e.g., [22–24]). In terms ofdata transmission, it would be better for a UE to be associated with a basestation (BS) offering higher data rate, which may encourage a mobile UE tohave frequent HOs. However, frequent HOs increase the risk of disconnection
and signaling overhead. In other words, there is a trade-off between the data
rate and the HO rate when a UE is moving.
To address the problem of frequent HOs, an approach of skipping some
opportunities of HOs has been considered, which is called HO skipping (see,e.g., [9–15]). While various HO skipping techniques have so far been proposedand studied in such a point of view, we propose yet another HO skipping
scheme, called periodic HO skipping. The proposed scheme prohibits the HOs
13

of a mobile UE for a certain period of time, referred to as skipping period,
thereby enabling flexible operation of the HO skipping by adjusting the lengthof the skipping period. In this chapter, we investigate the performance ofthe proposed scheme from the perspective of the trade-off between the HO
rate and the data rate.

2.1.1

Related work

A number of studies have so far analyzed the performance of cellular networks with mobile UEs and many of them have adopted stochastic geometry
as an analytical tool (see, e.g., recent tutorial articles [34, 35] and references
therein). In the stochastic geometry approach, the locations of wireless nodes(BSs and/or UEs) in a wireless network are modeled by stochastic point processes on the Euclidean plane, so that we can capture the spatial irregularityof wireless nodes and explore mathematical analysis of region-independent
network performance by virtue of the theory of point processes and stochastic
geometry. The first results along this line date back to the late 1990s [22,36],
where the cells associated with BSs in a cellular network are modeled as theVoronoi tessellation formed by a homogeneous Poisson point process (PPP)and some performance metrics concerning mobile UEs are discussed. Since
the 2010s, this stream has become more active. Lin et al. [23] propose amobility model of a UE on single-tier hexagonal/PPP networks and analyze
the HO rate and the expected sojourn time of a mobile UE staying in aparticular cell. The results in [23] are then extended in [37] to a two-tierheterogeneous network (HetNet). Bao and Liang [24] derive an analyticalexpression for the HO rate in a multi-tier HetNet modeled using overlaid
independent PPPs and provide a guideline for tier selection taking both the
HO rate and the expected downlink data rate into account. In addition, [29]develops a similar analysis to [24] for a single-tier network with BS cooperation. Sadr and Adve [31] analyze the HO rate in a PPP model of multi-tier
HetNets with orthogonal spectrum allocation among tiers and investigate
the negative impact of HOs on the coverage probability. Chattopadhyay etal. [30] evaluate the expected downlink data rate for a mobile UE taking into
account the data outage periods due to HOs in a two-tier HetNet and further
discuss the fraction of connecting BSs to reduce frequent HOs.
Several HO skipping techniques have also been proposed and analyzed using the stochastic geometry (see, e.g., [9–13]). Arshad et al. [9] introduce theso-called alternate HO skipping, where a mobile UE executes HOs alternately
along its trajectory, and quantify the average throughput representing thetrade-off between the HO rate and the expected downlink data rate when a
UE adopts the alternate HO skipping. The results in [9] are extended in [10]14

to a two-tier HetNet and in [11] to the incorporation with BS cooperation.
Furthermore, [12] proposes topologyaware HO skipping, where a UE skips
an HO when the target BS is far from the UE’s trajectory or the cell of thetarget BS is small, and evaluates its performance by Monte Carlo simulations. Demarchou et al. [13] then provide a mathematical analysis of the
topology-aware HO skipping. Compared with these sophisticated HO skipping techniques, our proposed scheme is simple and easy to implement since
it is enough for a mobile UE to observe the BS locations every fixed-length
period (as seen in the definition of the scheme in Sec. 2.2.2).

2.1.2

Contributions

The contributions of this work are summarized as follows.
1) We propose and advocate the periodic HO skipping, which prohibits the
HOs of a mobile UE during each cycle of the skipping period.
2) Applying the stochastic geometry approach, we derive the analytical expressions of the HO rate and the expected downlink data rate when amobile UE adopts the periodic HO skipping.
3) On the basis of the analytical results, we numerically demonstrate thatthe proposed scheme can outperform the conventional scenario without
any HO skipping, in particular when the UE moves fast.

2.1.3

Organization

The rest of this chapter is organized as follows. In the next section, we describe the network model and our proposed periodic HO skipping scheme.
We then define the user mobility model and the performance metrics; thatis, the expected downlink data rate and the HO rate. In Sec. 2.3, the performance of the proposed scheme is investigated, where for comparison, we
analyze the performance metrics not only for the proposed scheme but also
for the scenario without any HO skipping. Then, on the basis of the analytical results, the performances of the two scenarios are numerically comparedin terms of a certain utility metric representing the trade-off between thedata rate and the HO rate. Finally, this chapter is concluded in Sec. 2.5.

15

2.2

System model

2.2.1

Network model

In this chapter, we develop our proposed periodic HO skipping scheme implemented on the most basic spatially stochastic model of cellular networks;that is, a homogeneous PPP network with Rayleigh fading and power-law
path-loss (see, e.g., [19, 38]). Before defining the proposed scheme, we detailhere the network model.
BSs in a cellular network are deployed according to a homogeneous PPPÍΦ = 𝑖∈N 𝛿 𝑋𝑖 on the Euclidean plane R2 with intensity 𝜆 ∈ (0, ∞), whereN := {1, 2, · · · }, 𝛿 𝒙 denotes the Dirac measure with mass at 𝒙 ∈ R2 , and thepoints 𝑋1 , 𝑋2 , · · · of Φ are numbered in an arbitrary order. All the BSs transmit signals with the same power level, normalized to one, using a common
spectrum bandwidth. We suppose that the time is divided into discrete slots
and the downlink channels are affected by Rayleigh fading and power-law
path-loss. Moreover, we assume that shadowing effects are averaged out andnormalized to one. The reason is that the time variation is little compared to
the fading effects, and this assumption promotes tractability in our analysis.
Therefore, if a UE located at 𝒖 ∈ R2 at slot 𝑡 ∈ N0 := N ∪ {0} receives asignal from the BS at 𝑋𝑖 , 𝑖 ∈ N, the received signal power is represented by
𝐻𝑖,𝑡 ∥ 𝑋𝑖 −𝒖∥ −𝛽 , where ∥·∥ stands for the Euclidean norm, 𝐻𝑖,𝑡 , 𝑖 ∈ N, 𝑡 ∈ N0 , aremutually independent and exponentially distributed random variables withunit mean representing the fading effects, and 𝛽 > 2 denotes the path-loss
exponent.
We assume that, at any time slot, each BS has at least one UE in service
and transmits a signal to one of its UEs. Then, if a UE is located at 𝒖 ∈ R2 at
slot 𝑡 ∈ N0 and is served by the BS at 𝑋𝑖 , the downlink signal-to-interferenceplus-noise ratio (SINR) for this UE is represented by
SINR𝒖,𝑖 (𝑡) =𝐻𝑖,𝑡 ∥ 𝑋𝑖 − 𝒖∥ −𝛽,𝐼𝒖,𝑖 (𝑡) + 𝜎 2𝑖 ∈ N, 𝑡 ∈ N0 ,(2.1)

where 𝜎 2 denotes a nonnegative constant representing the noise power and
𝐼𝒖,𝑖 (𝑡) denotes the total interference power to this UE given byÕ𝐼𝒖,𝑖 (𝑡) =𝐻 𝑗,𝑡 ∥ 𝑋𝑖 − 𝒖∥ −𝛽 .
(2.2)
𝑗 ∈N\{𝑖}The instantaneous downlink data rate 𝜉 𝒖,𝑖 (𝑡) is then defined as𝜉 𝒖,𝑖 (𝑡) = log(1 + SINR𝒖,𝑖 (𝑡)),
16

(2.3)

where log stands for the natural logarithm for simplicity, but of course, it canbe converted into the conventional binary logarithm by multiplying the constant log2 e. Note that Shannon’s channel capacity is particularly consideredin the definition of the data rate. We further note that narrow bandwidthis mainly supposed for the utilized channel. The reason is that frequencyselective fading effects are mitigated in the narrow-band channel and therefore the channel capacity is likely to be achievable compared to the broadband
channel.

2.2.2

Periodic handover skipping

Suppose that a UE moves on R2 and is initially (at slot 0) served by its nearest BS, which offers the strongest signal power to the UE when the fading
effects are averaged out. In other words, the cells of respective BSs form a
Poisson-Voronoi tessellation (see [2, Sec. 9.7]). In our proposed periodic HO
skipping scheme, a UE is prohibited from executing HOs and retains the initial connection for 𝑠 time slots, referred to as skipping period, regardless of its
motion. After the skipping period of 𝑠 slots has passed, the UE reexamines
the connection and if the current connection is no longer with the nearest
BS due to its moving, the UE executes an HO and makes a new connection
with the nearest one. Afterward, this procedure is repeated in cycles of theskipping period. Namely, the UE reexamines its connection to a BS every
𝑠 time slots, during which it skips any chances of HOs even if it crosses theboundaries between cells. We assume that an HO, if it is done, is executed
instantly without any time loss. We should note that a UE is always connected to its nearest BS at the beginning of each cycle of the skipping period,
whereas it does not always execute an HO at the end of a cycle since thecurrent connection can still be with the nearest one.
One may claim that the skipping period described above is similar to
the time-to-trigger (TTT) in Long-Term Evolution (LTE) [39]. Indeed, theyboth suppress the number of HOs by prohibiting them for a certain period oftime. However, they are substantially different in that the TTT starts at the
instant that a UE crosses a boundary between two cells and it is in the orderof 100msec, which mainly aims to prevent ping-pong phenomena around theboundaries between cells. On the other hand, the skipping period repeats incycles and prohibits HOs during each cycle, which is in the order of seconds(though it may depend on the speed of the UE).
Clearly, the choice of the length of the skipping period is vital for our
proposed scheme. If the skipping period is too short, it results in frequentHOs, whereas the long skipping period may cause long-distance connections,which deteriorate the transmission performance. Therefore, we should decide17the length of the skipping period carefully.

2.2.3

Mobility model

Owing to the spatial stationarity of the network model, we can focus on
a UE that is supposed to be at the origin 0 = (0, 0) ∈ R2 at slot 0 and
we refer to this UE as the typical UE. Let 𝑆(𝑡) denote the location of thetypical UE at slot 𝑡 ∈ N0 . Since a UE is allowed to execute an HO every
cycle of the skipping period in our proposed scheme, it is enough to observe
the location of the typical UE every cycle and we model its motion as asimple and tractable random walk on R2 . Let 𝑌1 , 𝑌2 , · · · denote a sequenceof independent and identically distributed (i.i.d.) random variables on R2
representing the motions of the typical UE in respective cycles of the skipping
period. Then, the location of the typical UE just after 𝑛 cycles (that is, at
slot 𝑛𝑠) is provided as a random work;𝑛Õ𝑆(𝑛𝑠) =
𝑌𝑘 , 𝑛 ∈ N,
(2.4)
𝑘=1

with 𝑆(0) = 0. We assume that the typical UE moves along the straightline segment at a constant velocity during each cycle; that is, {𝑆(𝑡)}𝑡∈N0 is
piecewise deterministic and is given by𝑡 − 𝑛𝑠𝑌𝑛+1 , 𝑡 = 𝑛𝑠, 𝑛𝑠 + 1, · · · , (𝑛 + 1)𝑠, 𝑛 ∈ N0 .
(2.5)
𝑆(𝑡) = 𝑆(𝑛𝑠) +𝑠
An example of a path of the typical UE is illustrated in Fig. 2.1. Let
𝑌𝑛 = (𝐿 𝑛 , 𝜓𝑛 ), 𝑛 ∈ N, in the polar coordinates. Then, the moving speedof the typical UE during the 𝑛th cycle is equal to 𝑉𝑛 = 𝐿 𝑛 /𝑠. It is reasonable to suppose that the moving distance 𝐿 𝑛 in a cycle depends on the cycle
length 𝑠; that is, 𝐿 𝑛 is stochastically larger as 𝑠 is larger. Hence, we provide
the distribution of 𝑉𝑛 , instead of 𝐿 𝑛 , and that of 𝜓𝑛 for our mobility modeland assume that these distributions do not depend on the cycle length. The
distributions of 𝑉𝑛 and 𝜓𝑛 respectively represent changes in speed and direction of the typical UE over cycles of the skipping period, and the choice of
these distributions gives enough flexibility to our model to capture various
mobility patterns. For instance, P(𝑉𝑛 = 0) > 0 represents that the UE cantake a pause for 𝑠 time slots with a positive probability, and if 𝜓𝑛 takes aconstant, the UE always moves along a straight line.

2.2.4

Performance metrics

As discussed in Sec. 2.1, the HO skipping induces the trade-off between the
HO rate and the data rate. We thus evaluate the performance of our proposed18

Figure 2.1: A path of the typical UE in the random walk mobility model.
scheme in terms of the expected downlink data rate T and the HO rate H ,which are respectively defined as
"𝑚−1
#
Õ
1
T = lim E
𝜉 (𝑡) ,
(2.6)
𝑚→∞ 𝑚
𝑡=0
E[𝜁 (0, 𝑚)],𝑚→∞𝑚

H = lim(2.7)

where 𝜉 (𝑡) denotes the instantaneous downlink data rate for the typical UE
at slot 𝑡, specifically given in (2.3), and 𝜁 (𝑎, 𝑏) denotes the number of HOs
executed by the typical UE from slot 𝑎 to slot 𝑏. These performance metrics
are analyzed and evaluated in the following sections.

2.3

Performance analysis and evaluation

In this section, we investigate the performance of our proposed scheme introduced in the preceding section. For comparison, we analyze the performance
metrics defined in (2.6) and (2.7) not only in the scenario with the proposed
periodic HO skipping but also in the conventional scenario without any HO
skipping on the same network and mobility models described in Secs.2.2.1
and 2.2.3. In the scenario without HO skipping, the typical UE certainly
executes an HO whenever it crosses a boundary between two cells. We referto the scenario without HO skipping and that with the periodic HO skippingas Scenario 0 and Scenario 1, respectively, and distinguish elements in therespective scenarios by putting the superscript “(0)” or “(1)”; for example,
T (0) and H (0) respectively stand for the expected downlink data rate andthe HO rate in Scenario 0, whereas T (1) and H (1) are those in Scenario 1.

19

2.3.1

Expected downlink data rate analysis

Expected Downlink Data Rate in Scenario 0
In Scenario 0, the typical UE certainly executes an HO whenever it crosses
a boundary between two cells; that is, the typical UE is always connected toits nearest BS. The following proposition is directly derived from the existing
result in the literature.
Proposition 2.1. For the cellular network model described in Sec. 2.2, theexpected downlink data rate in Scenario 0 is given by∫ ∞∫ ∞
𝜌(𝑧, 𝑤)( 0)d𝑧 d𝑤,
(2.8)
T =
1+𝑧
0
0
where


𝜌(𝑧, 𝑤) = exp −𝜎 𝑧2 𝑤  𝛽/2𝜋𝜆

2𝑧 2/𝛽
−𝑤 1+
𝛽

∫ ∞

𝑣 2/𝛽−1d𝑣
1/𝑧 1 + 𝑣

,Proof. For 𝒖 ∈ R2 , let 𝐵(𝒖) denote the index of the nearest point of Φ =Í
𝑖∈N 𝛿 𝑋𝑖 to the location 𝒖; that is, 𝑋 𝐵(𝒖) − 𝒖 < ∥ 𝑋𝑖 − 𝒖∥ for 𝑖 ∈ N\{𝐵(𝒖)}.
Suppose that the typical UE is located at 𝑆(𝑡) = 𝒖 at slot 𝑡 ∈ N0 . SinceÍ∞
𝑖=1 𝛿 𝑋𝑖 −𝒖 is equal in distribution to Φ due to the stationarity and 𝐻𝑖,𝑡 , 𝑖 ∈N, 𝑡 ∈ N0 , are i.i.d., we have from (2.1) with (2.2) that SINR𝑢,𝐵(𝑢) (𝑡) is equal2
in distribution to SINR
 0,𝐵(0) (0) for any 𝒖 ∈ R . Thus, since {𝑆(𝑡)}𝑡∈N0 is
independent of Φ and 𝐻𝑖,𝑡 𝑖∈N,𝑡∈N , the definition of the expected downlink
0data rate in (2.6) leads to
T

( 0)

𝑚−1

1 Õ 
E 𝜉 𝑆(𝑡),𝐵(𝑆(𝑡)) (𝑡)
= lim
𝑚→∞ 𝑚
𝑡=0


= E 𝝃 0,𝐵(0) (0) ,which implies that the expected downlink data rate in Scenario 0 is equal
to that for a static UE. Hence, the existing result of the expected downlink
data rate for a static UE gives (2.8) (see, e.g., [19, Theorem 3]).
□
Remark 2.1. Proposition 2.1 implies that, if a moving UE is always connected to its nearest BS, the expected downlink data rate is identical to thatfor a static UE. Note that this fact is derived under the condition that Φ isstationary,𝐻𝑖,𝑡 , 𝑖 ∈ N, 𝑡 ∈ N0 , are i.i.d., and {𝑆(𝑡)}𝑡∈N0 is independent of Φand 𝐻𝑖,𝑡 𝑖∈N,𝑡∈N . In other words, this holds true even when the locations of0
BSs are according to a general stationary point process and the fading eﬀects
independently follow a general identical distribution. A similar discussion isfound in [30, Remark 2].
20

Expected Downlink Data Rate in Scenario 1
In Scenario 1, the typical UE is not always connected to its nearest BS but
remains connected to the BS that is the nearest at the beginning of eachcycle of the skipping period.
Theorem 2.1. For the cellular network model described in Sec. 2.2, theexpected downlink data rate in Scenario 1 with 𝑠 slots of the skipping period
is given by
𝑠−1 ∫
1Õ ∞
( 1)
𝜏(𝑡𝑣)d𝐹𝑉 (𝑣),
(2.9)
T =
𝑠 𝑡=0 0where 𝐹𝑉 denotes the distribution function of the moving speed 𝑉1 = ∥𝑌1 ∥ /𝑠of the typical UE in a cycle of 𝑠 slots and
∫ ∞


1
𝜏(𝑢) =
exp −𝜎 2 𝑧 − 𝜋𝜆𝐾 𝛽 𝑧2/𝛽 (𝜇(𝑧, 𝑢) − 1)d𝑧,(2.10)
𝑧
0
with
2𝜋
2𝜋
csc ,
𝛽
𝛽
∫ ∞
 

𝜇(𝑧, 𝑢) = 2𝜋𝜆𝑟 exp −𝜆 𝜋𝑟 2 − 𝐽 (𝑟, 𝑧, 𝑢) d𝑟,∫ 𝜋0 ∫ 𝑟𝑥
𝐽 (𝑟, 𝑧, 𝑢) = 2𝑧d𝑥 d𝜙,
𝛽
0
0 𝑧+𝑤
𝑥,𝑢,𝜙
𝐾𝛽 =

and 𝑤 𝑥,𝑢,𝜙 =(2.11)(2.12)(2.13)p
𝑥 2 + 𝑢 2 − 2𝑥𝑢 cos 𝜙.

The proof of Theorem 2.1 relies on the following lemma.
Lemma 2.1. Suppose that a UE is located at 𝒖 ∈ R2 with ∥𝒖∥ = 𝑢 at slot𝑡 ∈ {0, 1, . . . , 𝑠 − 1} and is served by the BS at 𝑋𝐵(0) , which is the nearest BS
to the origin. Then, the expected instantaneous downlink data rate for thisUE satisﬁes E 𝜉 𝒖,𝐵(0) (𝑡) = 𝜏(𝑢) given in (2.10).
□

Proof. See Section 2.4.1.

Proof of Theorem 2.1. As in the proof of Proposition 2.1, let 𝐵(𝒖) denoteÍthe index of the nearest point of Φ = 𝑖∈N 𝛿 𝑋𝑖 to 𝒖 ∈ R2 . In Scenario 1,
we see from (2.4) and (2.5) that the typical UE is connected to the BS at

21

𝑋𝐵(𝑆(𝑛𝑠)) at slot 𝑛𝑠 + 𝑡 for 𝑛 ∈ N0 and 𝑡 ∈ {0, 1, . . . , 𝑠 − 1}. Thus, the expected
downlink data rate in (2.6) is reduced to⌊𝑚/𝑠⌋ 𝑠−11 Õ Õ E 𝜉 𝑆(𝑛𝑠+𝑡),𝐵(𝑆(𝑛𝑠)) (𝑛𝑠 + 𝑡)
𝑚→∞ 𝑚
𝑛=0 𝑡=0

T (1) = lim

𝑠−1

1Õ 
=
E 𝜉 𝑆(𝑡),𝐵(0) (𝑡) ,𝑠 𝑡=0(2.14)

where the last equality follows from the distributional equivalence among
SINR𝑆(𝑛𝑠+𝑡),𝐵(𝑆(𝑛𝑠)) (𝑛𝑠 + 𝑡) and SINR𝑆(𝑡),𝐵(0) (𝑡) for 𝑡 ∈ {0, 1, . . . , 𝑠 − 1}, which
follows because Φ is stationary and isotropic, 𝐻𝑖,𝑡 , 𝑖 ∈ N, 𝑡 ∈ N
 0 , are i.i.d.,
and also 𝑌𝑘 , 𝑘 ∈ N, in (2.4) are i.i.d. and independent of Φ and 𝐻𝑖,𝑡 𝑖∈N,𝑡∈N .
0Hence, we obtain (2.9) since E 𝜉 𝑆(𝑡),𝐵(0) (𝑡) | 𝑆(𝑡) = 𝒖 = 𝜏(𝑡𝑣) when ∥𝒖∥ =𝑡𝑣 by Lemma 2.1 and ∥𝑆(𝑡)∥ = (𝑡/𝑠) ∥𝑌1 ∥ = 𝑡𝑉1 for 𝑡 ∈ {0, 1, . . . , 𝑠 − 1}by (2.5).
□
The expressions from (2.9) to (2.13) obtained in Theorem 2.1 are indeed
numerically computable. However, they include some nested integrals, which
may annoy us with a heavy computational load. In the rest of this subsection,
we discuss some ways of reducing the computational load.
Tips for Computational Load Reduction
We here introduce some tips to reduce the load of computing T (1) in Theorem 2.1 exactly or approximately. First, we find that a simple change ofvariables reduces the number of nested integrals.
Lemma 2.2. Function 𝐽 in (2.13) is equal to the following.
∫ 𝑢+𝑟𝑥
𝐽 (𝑟, 𝑧, 𝑢) = 2𝑧
𝐶 (𝑥, 𝑟, 𝑢)d𝑥,
𝑧 + 𝑥𝛽0with(
𝐶 (𝑥, 𝑟, 𝑢) =𝜋,arccos −1 ∨𝑥 2 +𝑢 2 −𝑟 2

2𝑥𝑢



∧1 ,

(2.15)

𝑢 = 0,
𝑢 > 0,

where 𝑎 ∨ 𝑏 = max{𝑎, 𝑏} and 𝑎 ∧ 𝑏 = min{𝑎, 𝑏} for 𝑎, 𝑏 ∈ R.
□

Proof. See Section 2.4.2.

We can observe through experiments that (2.15) reduces the computation
time of the expected instantaneous downlink data rate 𝜏(𝑢) by about 60%
22

𝑢( km)Use of eq. (2.13) (sec)Use of eq. (2.15) (sec)

0.1
1344
695

0.2
973
1118

0.3
10221
1196

0.4
7286
632

0.5
4754
974

0.6
1474
401

Table 2.1: Comparison of the computation time of 𝜏(𝑢) using (2.13)
and (2.15). The paramenter values are fixed at 𝜆 = 10 (units/km2 ), 𝛽 = 3and 𝜎 2 = 25.

on average compared to the use of (2.13) (see Table 2.1). Next, we give a
simple lower bound for 𝜏 in (2.10) under the interference-limited (noise-free)assumption.
Corollary 2.1. Suppose that 𝜎 2 = 0. Then, 𝜏 in (2.10) is bounded below asfollows.
∫𝛽 ∞𝑧 𝛽/2−12 𝑧 exp −𝜋𝜆𝑢𝜏(𝑢) ≥d𝑧, 𝑢 ≥ 0, (2.16)
2 0 (1 + 𝑧) 𝐾 𝛽/2 + 𝑧 𝛽/2
1+𝑧𝛽with 𝐾 𝛽 given in (2.11).
□

Proof. See Section 2.4.3.

Remark 2.2. As we can see in the proof, the lower bound in Corollary 2.1
is obtained by relaxing the condition that there must not be other BSs closer
than the nearest BS to the origin. Similar bounds/approximates are often
found in the literature (see, e.g., [40]).
The lower bound obtained in Corollary 2.1 is indeed of a simple form
(including a single integral), but as seen in Fig. 2.2, it causes non-negligible
gaps from the exact values, in particular when the moving distance 𝑢 is small,whereas the gaps decrease as 𝑢 increases. On the other hand, we know that𝜏(0) = T (0) since the expected downlink data rate in Scenario 0 is equal tothat for a static UE. Hence, we can approximate 𝜏 in (2.10) by interpolating
between T (0) and the lower bound in Corollary 2.1 as follows. Suppose
𝜎 2 = 0 as in Corollary 2.1 and let e𝜏 denote the lower bound of 𝜏 given on
the right-hand side of (2.16). Then, 𝜏 in (2.10) is approximated as
𝜏(𝑢) ≈ 𝜖 (𝑢)T (0) + (1 − 𝜖 (𝑢))e𝜏 (𝑢),𝑢 ≥ 0,(2.17)

where a function 𝜖 : [0, ∞) → [0, 1] is smooth and decreasing, and satisfies
𝜖 (0) = 1 and 𝜖 (𝑢) → 0 as 𝑢 → ∞; that is, it is chosen in such a way that theright-hand side of (2.17) is close to T (0) when 𝑢 is small, and it approaches
23ሺȀ Ȁ ሻǆƉĞĐƚĞĚŝŶƐƚĂŶƚĂŶĞŽƵƐĚĂƚĂƌĂƚĞDŽǀŝŶŐĚŝƐƚĂŶĐĞ‫ ݑ‬ሺሻ

Figure 2.2: Numerical comparison of 𝜏 in (2.10) with the lower bound (2.16),
the approximation (2.17), and the values from Monte Carlo simulation. The
BS intensity is fixed at 𝜆 = 1 (units/km2 ) and two patterns of 𝛽 = 3 and𝛽 = 4 are exhibited.
e𝜏 (𝑢) as 𝑢 becomes larger. Figure 2.2 compares the numerical results of 𝜏in (2.10) with its lower bound in (2.16) and the approximation in (2.17), as
well as with the values from Monte Carlo simulation, with respect to themoving distance 𝑢. In the approximation (2.17), the function 𝜖 is set as2
𝜖 (𝑢) = 𝑒 −10𝑢 , 𝑢 ≥ 0. The simulation results are computed as the mean of
10,000 independent samples. As stated above, the values of the lower boundhave some gaps from the exact values when 𝑢 is small, whereas these gaps
decrease as 𝑢 increases. This implies that the condition that other BSs neverexist closer than the nearest BS is nonnegligible when the typical UE is close
to the origin, but it is diminishing as the UE moves away from the origin.
On the other hand, the approximation (2.17) shows good agreement withthe exact values as expected. However, we should note that such agreement𝑏depends on a choice of the function 𝜖. An exponential function 𝜖 (𝑢) = 𝑒 −𝑎𝑢
as above seems an plausible choice as one with the desired properties (that is,smooth and decreasing with 𝜖 (0) = 1 and lim𝑢→∞ 𝜖 (𝑢) = 0 ), and statisticalfitting of 𝑎 and 𝑏 would lead to better results.

24

2.3.2

Handover rate analysis

We now proceed to the analysis of the HO rate. Similar to the proof of
Theorem 2.1, the HO rate in (2.7) is reduced to
⌊𝑚/𝑠⌋

1 Õ
H = limE[𝜁 (𝑛𝑠, (𝑛 + 1)𝑠)]
𝑚→∞ 𝑚
𝑛=0
=

E[𝜁 (0, 𝑠)],𝑠(2.18)

where the last equality follows since Φ is stationary and isotropic, and 𝑌𝑘 , 𝑘 ∈N, in (2.4) are i.i.d. and independent of Φ. By (2.18), it is enough to consider
the expected number of HOs in a cycle of 𝑠 slots, during which the typicalUE moves along a straight line segment, and we can use the existing resultsin both Scenarios 0 and 1.
HO Rate in Scenario 0
The HO rate in the scenario without any HO skipping has so far been studied
in the literature. The following is a direct consequence of it.
Proposition 2.2. For the cellular network model described in Sec. 2.2, theHO rate in Scenario 0 is given by
√
4 𝜆¯
𝑣
( 0)
H =
,
(2.19)
𝜋
where 𝑣¯ denotes the average moving speed of the typical UE in a cycle of 𝑠slots; that is, 𝑣¯ = E [𝑉1 ] with 𝑉1 = ∥𝑌1 ∥ /𝑠.
Proof.
 (0) Given 𝐿 1 = ∥𝑌1 ∥ = ℓ, the conditionally expected number of HOs
E 𝜁 (0, 𝑠) | 𝐿 1 = ℓ] in a cycle is equal to the expected number of intersections of a line segment of length ℓ with
√ Poisson (0) the boundaries
 of theVoronoi cells, and is well-known as E 𝜁 (0, 𝑠) | 𝐿 1 = ℓ = 4 𝜆ℓ/𝜋 (see,e.g., [23, 41, 42]). Applying this to (2.18) with 𝐿 1 = 𝑠𝑉1 derives (2.19) bytaking the expectation.
□
HO Rate in Scenario 1
The HO rate in a similar scenario to our Scenario 1 is studied in [31], whichhelps us to show the following.

25

Proposition 2.3. For the cellular network model described in Sec. 2.2, theHO rate in Scenario 1 with 𝑠 slots of the skipping period is given by∫ ∞∫ 𝜋∫ ∞1( 1)−𝜆𝜂(𝑟,𝑠𝑣,𝜙)H =1 − 2𝜆𝑟𝑒d𝑟 d𝜙d𝐹𝑉 (𝑣) ,(2.20)
𝑠
0
0
0
where


2
𝜂(𝑟, ℓ, 𝜙) = 𝑤 𝑟,ℓ,𝜙arccos𝑟 cos 𝜙 − ℓ+ 𝑟 2 (𝜋 − 𝜙) + 𝑟ℓ sin 𝜙,𝑤 𝑟,ℓ,𝜙(2.21)

pwith 𝑤 𝑟,ℓ,𝜙 = 𝑟 2 + ℓ 2 − 2𝑟ℓ cos 𝜙, and 𝐹𝑉 is (as in (2.9)) the distributionfunction of the moving speed 𝑉1 = ∥𝑌1 ∥ /𝑠 of the typical UE in a cycle of 𝑠slots.
Í
Proof. By the isotropy of Φ = 𝑖∈N 𝛿 𝑋𝑖 , we can assume without loss of generality that the typical UE moves in the positive direction along the horizontal
axis during a cycle of 𝑠 slots. Suppose that the typical UE initially connected to the BS at 𝑋𝐵(0) = 𝒙 = (𝑟, 𝜙) in the polar coordinates and moves
to 𝑌1 = 𝒚 = (ℓ, 0) in 𝑠 slots. Let 𝑏 𝒙 (𝑟) denote the disk centered at 𝒙 ∈ R2
with radius 𝑟 > 0. Since there are no BSs in 𝑏 0 (𝑟) and pthe distance to theinitial BS at 𝒙 from the location 𝑦 is equal to 𝑤 𝑟,ℓ,𝜙 = 𝑟 2 + ℓ 2 − 2𝑟ℓ cos 𝜙,
the typical UE executes an HO at the end of the cycle if and only if there
is at least one BS in the area 𝑏 𝑦 𝑤 𝑟,ℓ,𝜙 \𝑏 0 (𝑟). Hence, similar discussion
to [31] giveshiE 𝜁 (1) (0, 𝑠) | 𝑋𝐵(0) = (𝑟, 𝜙), 𝑌1 = (ℓ, 0)= 1 − 𝑒 −𝜆 | 𝑏 𝑦 ( 𝑤𝑟 ,ℓ, 𝜙 ) \𝑏0 (𝑟) |
 



𝑟 cos 𝜙 − ℓ22= 1 − exp −𝜆 𝑤 𝑟,ℓ,𝜙 arccos− 𝑟 𝜙 + 𝑟ℓ sin 𝜙 ,(2.22)
𝑤 𝑟,ℓ,𝜙
where | 𝐴| denotes the Lebesgue measure of 𝐴 ∈ B R2 . This can be uncon2
ditioned by integrating with respect to the density 𝑓0 (𝑟)d𝑟 = 2𝜋𝜆𝑟𝑒 −𝜋𝜆𝑟 d𝑟of 𝑋𝐵(0) over [0, ∞), d𝜙/𝜋 over [0, 𝜋), and d𝐹𝑉 (𝑣) over [0, ∞) with 𝑣 = ℓ/𝑠.
Finally, plugging it into (2.18), we have (2.20).
□

2.3.3

Numerical evaluation of performance metrics

We here numerically evaluate the expected downlink data rate and the HO
rate in the periodic HO skipping scheme, which are respectively obtainedin Theorem 2.1 (with Lemma 2.2) and Proposition 2.3. Throughout the26ǆƉĞĐƚĞĚĚĂƚĂƌĂƚĞሺȀ Ȁ ሻ

ʹ1000,ĂŶĚŽǀĞƌƌĂƚĞሺȀ ሻ^ŬŝƉƉŝŶŐƉĞƌŝŽĚ ‫ ݏ‬ሺ ሻ

ʹ1000^ŬŝƉƉŝŶŐƉĞƌŝŽĚ ‫ ݏ‬ሺ ሻ

Figure 2.3: The performance metrics as functions of the length 𝑠 of theskipping period for three patterns of the moving speed 𝑣 = 0.01, 0.02, and0.03 (km/sec).
experiments, we set as 1slot = 1msec, the intensity of the BSs, the path-loss
exponent, and the noise power are fixed at 𝜆 = 10 (units /km2 ), 𝛽 = 3,and 𝜎 2 = 25, respectively, and the moving speed 𝑉1 = ∥𝐿 1 ∥ /𝑠 of the typical
UE is given as a constant. Figure 2.3 shows the curves of T (1) and H (1)with respect to the length 𝑠 of the skipping period. For comparison, thevalues from Monte Carlo simulation are also plotted as the means of 1,000
Í𝑚−1 (1)independent samples of 𝑡=(𝑡)/𝑚 and 𝜁 (1) (0, 𝑚)/𝑚, respectively with0 𝜉
𝑚 = 1000 (cf. (2.6) and (2.7)). We find that the analytical results match wellwith the simulation results. Moreover, we can confirm the trade-off relation
between the HO rate and the expected downlink data rate; that is, bothare decreasing in the length of the skipping period. We further explore thistrade-off in the next subsection.

27

2.3.4

Utility metric

To discuss the trade-off between the expected downlink data rate T and the
HO rate H , we introduce a utility metric U as
U = T − 𝑐H ,(2.23)

where a utility constant 𝑐 > 0 is suitably chosen so as to convert the negative
impact of HOs into the loss of the downlink data rate. Note that similar
metrics are found in the literature (cf. [10, 24, 29, 35]) and are often referred
to as the user throughput accounting for the loss due to HOs. However, we
do not use the term ”throughput” in this chapter because U in (2.23) cantake negative values (see, e.g., Fig. 2.4 below).
Figure 2.4 compares the utility metrics U (0) and U (1) respectively for
Scenarios 0 and 1 with respect to the average speed 𝑣¯ = E [𝑉1 ] of the typical
UE. In the computation of U (1) , 𝜏 in (2.9) is replaced by its approximation
2(2.17) with the adjustment function 𝜖 (𝑢) = 𝑒 −10𝑢 . Four different distributions of the moving speed are experimented with the common average 𝑣¯;that is, exponential, second-order Erlang, second-order hyper-exponential,
and deterministic ones, where in the hyper-exponential distribution, two
exponential distributions with means 𝑣¯/2 and 3¯𝑣 /2 are mixed with equal
probability. The other parameters are fixed as 𝜆 = 1 (units /km2 ), 𝛽 = 3,
𝜎 2 = 0, 𝑠 = 50, 000(msec), and 𝑐 = 10 (nats/Hz). Note that only one line isexhibited for U (0) since it depends on the distribution of the moving speed
only through its average (as confirmed from (2.8) and (2.19)). From Fig. 2.4,
we observe that Scenario 0 shows better performance when the average moving speed is small (roughly 𝑣¯ ≤ 0.05( km/sec) ), whereas Scenario 1 becomes
better as the UE moves faster. This is thought to be because H (0) is linearly
increasing in 𝑣¯ (see (2.19)), whereas 𝜏(𝑢) in (2.10) is slowly decreasing in 𝑢(see Fig. 2.2). Moreover, we find an interesting property from the experimentthat the distribution of the moving speed has an impact on the utility metricin Scenario 1; that is, the utility metric takes larger values as the distributionof the moving speed is larger in variation. Exploration of this property willbe left for future work.

28hƚŝůŝƚǇŵĞƚƌŝĐሺȀ Ȁ ሻǀĞƌĂŐĞƐƉĞĞĚሺȀ ሻ

Figure 2.4: The values of U (0) and U (1) as functions of the average speed 𝑣¯of the moving UE with several distributions.

2.4

Proofs of Lemmas and Corollary

2.4.1

Proof of Lemma 2.1

Applying Hamdi’s Lemma [43, Lemma 1] to the expectation of (2.3) with
(2.1), (2.2) and 𝑖 = 𝐵(0), we have
"
−𝛽 !#


𝐻 𝐵(0),𝑡 𝑋𝐵(0) − 𝒖
E 𝜉 𝒖,𝐵(0) (𝑡) = E log 1 +𝐼𝒖,𝐵(0) (𝑡) + 𝜎 2∫ ∞ −𝜎2 𝑧  hih Í−𝛽 i 
𝑒−𝑧 𝑗 ∈N 𝐻 𝑗,𝑡 ∥ 𝑋 𝑗 −𝒖 ∥−𝑧𝐼𝒖,𝑩 ( 0 ) (𝑡)
d𝑧,
−E 𝑒
E 𝑒
=
𝑧
0
(2.24)For the second expectation in the last expression above, the Laplace transform of an exponential distribution and the generating functional of a PPP(e.g., [44, Example 9.4(c)]) yield"#! −1 
Ö
Ö


−𝛽
𝑧

E
𝑒 −𝑧𝐻 𝑗,𝑡 ∥ 𝑋 𝑗 −𝒖 ∥= E 1+𝛽 𝑗 ∈N
𝑋𝑗 − 𝒖
𝑗 ∈N
 

∫
1
= exp −𝜆𝑧d𝒙𝛽
R2 𝑧 + ∥𝒙∥= 𝑒 −𝜋𝜆𝐾 𝛽 𝑧2/𝛽,(2.25)

with 𝐾 𝛽 = (2𝜋/𝛽) csc(2𝜋/𝛽)∫ ∞ as given in (2.11), where we use the polar coordinate conversion and 0 𝑤 𝑎−1 /(1 + 𝑤)d𝑤 = 𝜋 csc 𝑎𝜋 for 𝑎 ∈ (0, 1) in the29

last equality. Next, we consider the first expectation in the last expressionof (2.24), which satisfieshi ∫ ∞ hi−𝑧𝐼𝒖,𝐵( 0 ) (𝑡)E 𝑒=
E 𝑒 −𝑧𝐼𝒖,𝐵(0) (𝑡) | 𝑋𝐵(0) = 𝑟 𝑓0 (𝑟)d𝑟,(2.26)
0where 𝑓0 (𝑟) = 2𝜋𝜆𝑟𝑒 −𝜋𝜆𝑟 gives the probability density function of 𝑋𝐵(0) .
Similar to obtaining (2.25), we havehiE 𝑒 −𝑧𝐼𝒖,𝐵(0) (𝑡) | 𝑋𝐵(0) = 𝑟
! −1
 Ö



𝑧


=E
1+
|
𝑋
=
𝑟
𝐵(
0
)

𝛽
 𝑗 ∈N\{𝐵(0)}
𝑋𝑗 − 𝒖
 


∫
1
d𝒙
= exp −𝜆𝑧𝛽∥𝒙∥>𝑟 𝑧 + ∥𝒙 − 𝒖∥


∫
1
2/𝛽
d𝒙 ,
(2.27)= exp −𝜋𝜆𝐾 𝛽 𝑧 + 𝜆𝑧
𝛽∥𝒙∥≤𝑟 𝑧 + ∥𝒙 − 𝒖∥2where the polar coordinate conversion gives∫
1d𝒙 = 𝐽 (𝑟, 𝑧, 𝑢),𝑧𝛽∥𝑥∥≤𝑟 𝑧 + ∥𝒙 − 𝒖∥(2.28)

with 𝐽 given in (2.13). Plugging (2.27) together with (2.28) into (2.26), we
have
h
i
2/𝛽
E 𝑒 −𝑧𝐼𝒖,𝐵(0) (𝑡) = 𝑒 −𝜋𝜆𝐾 𝛽 𝑧 𝜇(𝑧, 𝑢),(2.29)
with 𝜇 in (2.12). Finally, plugging (2.25) and (2.29) into (2.24) derives (2.10).

2.4.2

Proof of Lemma 2.2

It is immediate for the case of 𝑢 = 0 since 𝑤 𝑥,0,𝜙 = 𝑥 in (2.13). Suppose 𝑢 > 0.
On the left-hand side of (2.28), changing the variables as 𝒙 ′ = 𝒖 − 𝒙 leads to
∫
∫
1
1
d𝒙
=
d𝒙 ′,
𝛽
𝛽
′
𝑏𝒖 (𝑟) 𝑧 + ∥𝒙 ∥∥𝒙∥≤𝑟 𝑧 + ∥𝒙 − 𝒖∥where 𝑏 𝒖 (𝑟) denotes the disk centered at 𝒖 ∈ R2 with radius 𝑟 > 0. Recall
that ∥𝒖∥ = 𝑢 as in Lemma 2.1. When 𝑢 ≥ 𝑟, the polar coordinate conversion

30

gives (see Fig. 2.5a)
∫

1
d𝒙
𝛽
𝑏𝒖 (𝑟) 𝑧 + ∥𝒙∥
 2

∫ 𝑢+𝑟
𝑥 + 𝑢2 − 𝑟 2𝑥arccos=2d𝑥𝛽2𝑥𝑢
𝑢−𝑟 𝑧 + 𝑥
 2

∫ 𝑢+𝑟
𝑥 + 𝑢2 − 𝑟 2𝑥arccos∧ 1 d𝑥,=2(2.30)
2𝑥𝑢
𝑧 + 𝑥𝛽0where the last equality holds since 𝑓 (𝑥) = 𝑥 2 + 𝑢 2 − 𝑟 2 /(2𝑥𝑢) > 1 for 𝑥 ∈(0, 𝑢 − 𝑟) with 𝑓 (𝑢 − 𝑟) = 1 when 𝑢 ≥ 𝑟 > 0. On the other hand, when 𝑢 < 𝑟,
we have similarly (see Fig. 2.5b),∫1d𝒙𝛽𝑏 𝑢 (𝑟) 𝑧 + ∥𝒙∥
 2

∫ 𝑟−𝑢
∫ 𝑢+𝑟
𝑥
𝑥
𝑥 + 𝑢2 − 𝑟 2= 2𝜋d𝑥 + 2arccosd𝑥
𝛽2𝑥𝑢
𝑧 + 𝑥𝛽0𝑟−𝑢 𝑧 + 𝑥∫ 𝑢+𝑟
𝑥 2 + 𝑢2 − 𝑟 2𝑥arccos −1 ∨d𝑥,(2.31)=22𝑥𝑢
𝑧 + 𝑥𝛽0where the last equality holds since 𝑓 (𝑥) = 𝑥 2 + 𝑢 2 − 𝑟 2 /(2𝑥𝑢) < −1 for𝑥 ∈ (0, 𝑟 − 𝑢) with 𝑓 (𝑟 − 𝑢) = −1 when 0 < 𝑢 < 𝑟. Hence, unifying (2.30) and(2.31), we have (2.15) since 𝑥 2 + 𝑢 2 − 𝑟 2 /(2𝑥𝑢) ∈ [−1, 1] when |𝑢 − 𝑟 | ≤ 𝑥 ≤𝑢 + 𝑟.

2.4.3

Proof of Corollary 2.1−𝛽

In (2.24) with 𝜎 2 = 0, changing the variables as 𝑧′ = 𝑋𝐵(0) − 𝒖𝑧 leads toE 𝜉 𝒖,𝐵(0) (𝑡)∫ ∞i
1 h − ∥ 𝑋𝐵(0) −𝒖 ∥ 𝛽 𝑧′ 𝐼𝒖,𝑩 (0) (𝑡) −𝑧 ′ 𝑯 𝑩 ( 0 ),𝑡 )
E
𝑒
1
−
𝑒
d𝑧′
=
′
𝑧
∫0 ∞
h
i
𝛽
1
=
E 𝑒 − ∥ 𝑋𝑩 (0) −𝒖 ∥ 𝑧𝑰𝒖,𝑩 (0) (𝑡) d𝑧,(2.32)
1+𝑧0
where the second equality follows from E 𝑒 −𝑧𝐻 𝐵(0),𝑡 | 𝐵(0) = (1+𝑧) −1 because
𝐻𝑖,𝑡 , 𝑖 ∈ N, 𝑡 ∈ N0 , are mutually independent and exponentially distributedwith unit mean. Furthermore, since 𝑋𝐵(0) follows the probability density2function 𝑓0 (𝑟) = 2𝜋𝜆𝑟𝑒 −𝜋𝜆𝑟 and the angle between 𝑋𝐵(0) and 𝒖 is uniformly31

(a) Case of 𝑢 ≥ 𝑟, where 𝑥 varies from 𝑢 − 𝑟 to 𝑢 + 𝑟.

(b) Case of 𝑢 < 𝑟, where 𝑥 varies from 𝑟 − 𝑢 to 𝑢 + 𝑟.
Figure 2.5: Supplement
to the derivation of eqs. (2.30) and (2.31), where

2
2
2
−𝑟
𝜙 = arccos 𝑥 +𝑢
for each 𝑥 ∈ [|𝑢 − 𝑟 |, 𝑢 + 𝑟].
2𝑥𝑢
distributed on [0, 2𝜋), the expectation in the last expression of (2.32) satisfieshi𝛽
E 𝑒 − ∥ 𝑋𝐵(0) −𝒖 ∥ 𝑧𝐼𝒖,𝐵(0) (𝑡)∫ 2𝜋 ∫ ∞ hi𝛽1
E 𝑒 −𝑤𝑟 ,𝑢, 𝜙 𝑧𝐼𝒖,𝐵(0) (𝑡) | 𝑋𝐵(0) = 𝑟 𝑓0 (𝑟)d𝑟 d𝜙=2𝜋 00∫ 2𝜋 ∫ ∞


2
2 2/𝛽
≥𝜆
𝑟 exp −𝜋𝜆𝑟 − 𝜋𝜆𝐾 𝛽 𝑤 𝑟,𝑢,𝜙 𝑧
d𝑟 d𝜙
0
0
∫ 2𝜋 ∫ ∞hi−𝜋𝜆𝐾 𝛽 𝑢 2 𝑧2/𝛽
= 𝜆𝑒
𝑟 exp −𝜋𝜆 1 + 𝐾 𝛽 𝑧2/𝛽 𝑟 2 − 2𝐾 𝛽 𝑢𝑧 2/𝛽 𝑟 cos 𝜙 d𝑟 d𝜙
0
0
!
2/𝛽
𝐾
𝑧
1
𝛽
exp −𝜋𝜆𝑢 2,(2.33)=
1 + 𝐾 𝛽 𝑧2/𝛽
1 + 𝐾 𝛽 𝑧2/𝛽32pwhere 𝑤 𝑟,𝑢,𝜙 = 𝑟 2 + 𝑢 2 − 2𝑟𝑢 cos 𝜙 and the inequality follows from (2.27),
from which the nonnegative integral term is removed. In the last equality in(2.33), we apply the following; that is, for 𝑝 > 0 and 𝑞 ∈ R,
∫ 2𝜋 ∫ ∞0

𝑟𝑒−𝑝𝑟 2 +𝑞𝑟 cos 𝜙0∫ ∞∫ ∞𝑒 −𝑝 ( 𝑥 +𝑦 ) +𝑞𝑥 d𝑥 d𝑦−∞ −∞∫ ∞∫ ∞2 2
𝑞 2 /( 4 𝑝)
𝑒 −𝑝 ( 𝑥 +𝑦 ) d𝑥 d𝑦=𝑒d𝑟 d𝜙 =−∞

𝜋 2
= 𝑒 𝑞 /(4 𝑝) .
𝑝

2

2

−∞

Finally, plugging (2.33) into (2.32) and changing the variables as 𝑧′ = 𝐾 𝛽 𝑧2/𝛽 ,we obtain (2.16).

2.5

Conclusion

In this chapter, we have proposed a simple HO skipping scheme in cellular networks, called the periodic HO skipping, and have evaluated its performance analytically and numerically. Specifically, applying the stochastic
geometry approach, we have derived numerically computable expressions forthe expected downlink data rate and the HO rate when the UE adopts the
proposed scheme. Through the numerical experiments based on the analysis, we have confirmed that the proposed scheme can outperform the scenario
without any HO skipping in terms of a utility metric representing the tradeoff between the expected downlink data rate and the HO rate, in particularwhen the UE moves fast.

33

Chapter 3
Optimal Skipping Period
3.1

Introduction

The development of the fifth generation mobile communication systems (5G)
is driven by the ever-increasing demand for channel capacity due to the proliferation of mobile user equipments (UEs) such as mobile phones, tablets,and other handheld devices. One of the key solutions in the 5G evolution
is network densification through small cell deployments (see, e.g., [32, 33]).
Densifying base stations (BSs) offers more capacity, which improves the quality of service. On the other hand, it shrinks the service area of each BS andinduces frequent handovers (HOs), which may increase the signaling overhead
and the risk of disconnections.
HO skipping is an approach to address the problem of frequent HOs byskipping some opportunities of HOs (see, e.g., [9–15]). However, in turn, theHO skipping may decrease the data reception rate (data rate for short) since
it tends to force a UE to retain long-distance connection with a BS. In otherwords, the HO skipping induces a trade-off between the HO rate and thedata rate, and this trade-off should be balanced for the network densification
to work effectively.

3.1.1

Related work

To investigate the impact of HOs on transmission performance of UEs incellular networks, a number of studies have focused on analyzing the tradeoff relation between the HO rate and the data rate. For their theoreticalanalysis, the theory of spatial point processes and stochastic geometry havebeen adopted to model cellular networks (see, e.g., [34, 35] for surveys). In
the stochastic geometry approach, the deployment of nodes is modeled bya stochastic point process, which can capture its randomness in the real34

world. Moreover, applying the stochastic geometry approach enables us toobtain rigorous mathematical expressions of performance metrics as a function depending on various system parameters; thereby, we can grasp how
the system parameters are generally involved in those performance metrics
through the expressions. For example, the trade-off relation between the
HO rate and the data rate are analyzed for heterogeneous cellular networks(HetNets) in [24] and for a BS cooperated cellular network with coordinatedmulti-point (CoMP) transmission in [29]. Maximizing user throughput ina two-tier network is considered in [30], where they found the optimal proportion of macro and small BSs density and their respective transmit power
levels by balancing the HO rate and the data rate for multiple static andmoving UEs. Some related work considered a trade-off relation between the
HO rate and the coverage probability; the work [31] considered the HO rate
in multi-tier HetNets and analyzed the negative effect of performing HOs on
the coverage probability. Using the analysis results, they derived the optimal proportion of the BSs density in different tiers to maximize the coverage
probability of a typical UE.
The HO skipping is a user-centric approach for mitigating the effect of
frequent HOs [9–15]. This approach aims to balance the trade-off relation
between the HO rate and the data rate by controlling the frequency of HOs
directly for the UEs. In [9], an alternate HO skipping scheme is studied,
where a UE alternately performs HOs along its trajectory, thereby achieving
a 50% reduction in the HO rate. Considering the trade-off relation, they evaluated user throughput analytically for the UEs following the alternate HO
skipping schemes. The work [9] was extended in [10] and [11]; [10] introduced
the alternate HO skipping in two-tier networks and [11] introduced it in a BS
cooperating network with CoMP transmission. In [13], a topology-aware HO
skipping scheme is studied with its mathematical framework. In this scheme,
the HO skipping is triggered according to the UE’s distance from the target
BS and the size of the cell. They derived the analytical expressions of thecoverage probability and the expected data rate when the topology-aware HO
skipping is performed under the assumption that the trajectory of a moving
UE is a straight line. In [16], a periodic HO skipping scheme is studied. They
consider to control HOs of a mobile UE by a certain fixed period of time,called skipping period, and analyze the HO rate and the expected data rate
for the UE following the periodic HO skipping scheme. Moreover, they foundthe periodic HO skipping scheme could outperform the conventional non-HO
skipping scenario by comparing a utility metric representing the trade-off
between the HO rate and the data rate.

35

3.1.2Contribution

This work enhances [16], where the periodic HO skipping is already proposed,and considers maximization of the utility metric defined in [16] by adjusting
the skipping period of the periodic HO skipping scheme. Moreover we willfind later that, with the optimal adjustment of the skipping period, theperiodic HO skipping scheme can compete with the other sophisticated HO
skipping schemes such as ones in [9] and [13]. The contributions of this work
are summarized as follows.
1) We numerically observe that there can exist an optimal length of theskipping period and provide an approximate optimal skipping period ina simple computable form.
2) We numerically observe that the proposed scheme can compete with someother sophisticated HO skipping techniques.

3.1.3

Organization

The rest of this chapter is organized as follows. Section 3.2 is the preliminaries
regarding the utility metric and the results of analysis used in the following
sections. In Sec. 3.3, we discuss how to decide the length of the skipping
period, where we numerically observe that there exists an optimal lengthof the skipping period which locally maximizes the utility metric. We thenprovide a simple computable expression of an approximate optimal skipping
period. Some properties of the approximate optimal skipping period arealso revealed by numerical experiments. Numerical comparison with someother HO skipping techniques are made in Sec. 3.5. Finally, this chapter isconcluded in Section 3.7.

3.2

Preliminaries

Since we consider the same models regarding the network, the periodic HO
skipping scheme, and the mobility of UEs, we here omit that description (seeSec. 2.2 for the system model). We here describe again the definition of theutility metric and the results of analysis given in the previous chapter, which
are used in the following discussion of this chapter.

3.2.1

Utility metric

Let T and H denote the expected downlink data rate and the HO rate of
the typical UE in the scenario where the typical UE attempts the periodic
36

HO skipping scheme described in the previous chapter (see Sec. 2.2.2). Here,T and H are defined as follows
𝑚−1


1 Õ
1 
T = lim E
𝜉 (𝑡) , H = lim E 𝜁 (𝑚) ,
(3.1)
𝑚→∞ 𝑚
𝑚→∞ 𝑚
𝑡=0
where 𝜉 (𝑡) denotes the data rate at the time 𝑡 ∈ N0 , which is given in (2.3),
and 𝜁 (𝑚) denotes the number of experienced HOs from the time 0 to 𝑚 of
the typical UE in the scenario of the periodic HO skipping. Using the above
two metrics, the utility metric is defined as follows [24, 29]
U = T − 𝑐H ,(3.2)

where 𝑐 > 0 is the cost for an HO1 .
The intuitive background behind the definition in (3.2) is to indicate thetrade-off relation between the expected data rate and the HO rate, that is, theutility of data reception penalized by the cost of performing an HO. A related
performance evaluation function, called the mobility-aware throughput, isdescribed in [35], where they considered the trade-off relation between the
HO rate and the data rate to indicate the throughput of a moving UE.

3.2.2

Results of analysis

Theorem 3.1 (Theorem 2.1 and Lemma 2.2). The expected data rate T in
the periodic HO skipping scenario is1Õ
E[𝜏( 𝑡𝑉1 )],
𝑠 𝑡=0
𝑠−1

T =

(3.3)

where 𝑉1 is the moving speed of the typical UE described in Sec. 2.2.3, and𝜏(𝑢), 𝑢 ≥ 0, is given by
∫ ∞
∫ 2∫
𝜋 ∞
2
2 𝛽
𝜌(𝑧, 𝑟, 𝜃|𝑢)d𝑧d𝑟d𝜃,(3.4)
𝜏(𝑢) = 𝜆𝑟 e−𝜋𝜆𝑟 −𝜎 𝑤 𝑢,𝑟 , 𝜃 𝑧
1+𝑧
0
0 0
where



𝜌(𝑧, 𝑟, 𝜃|𝑢) = exp −2𝜋𝜆22𝑤 𝑢,𝑟,𝜃
𝐾𝛽 𝑧 𝛽 −

∫ 𝑢+𝑟

1

0

𝜑
e𝑢,𝑟,𝑥n

1  𝑥  𝛽 o −1
1+𝑥d𝑥
𝑧 𝑤 𝑢,𝑟,𝜃,

As described in Section 3.1, performing HOs has several negative impacts, such as thedisconnection and the signaling overhead. In our model, such negative impacts of HOs are
regarded as transformed to the amount of lost data by the constant 𝑐.

37

with
𝐾𝛽 =

𝜋
2𝜋
csc ,
𝛽
𝛽

(3.5)



𝑢2 − 𝑟 2 + 𝑥 2
𝜑e𝑢,𝑟,𝑥 = 𝜋 −1 cos−1 −1 ∨
∧1 ,
2𝑢𝑥
p
𝑤 𝑢,𝑟,𝜃 = 𝑢 2 + 𝑟 2 − 2𝑢𝑟 cos 𝜃.

(3.6)
(3.7)

Corollary 3.1 (Corollary 2.1). If 𝜎 2 = 0, that is, noise power is negligible,
then the expected data rate T in the periodic HO skipping scenario is givenin (3.3), where the lower bound of 𝜏(𝑢), 𝑢 ≥ 0, is given by
2𝜋𝜆𝑧 2∫ ∞exp−𝑢
1 +2 𝑧𝛽/2𝜏(𝑢) ≥ 𝜏′ (𝑢) =·𝑧−1 𝑑𝑧,
(3.8)
−
1
𝛽/
2
1
+
2𝑧
1 + (𝐾 𝛽 𝑧 )0with 𝐾 𝛽 which is given in (3.5).
Corollary 3.2 (eq.(2.17)). If 𝜎 2 = 0, then the expected data rate T in the
periodic HO skipping scenario is given in (3.3), where 𝜏(𝑢) is approximately
given by22
𝜏(𝑢) ≈ 𝜏′′ (𝑢, 𝜀) = e−𝜀𝑢 T0 + (1 − e−𝜀𝑢 )𝜏′ (𝑢),(3.9)
where T0 is given byT0 =∫ ∞∫ ∞
0

with



0



𝑤
𝜌0 (𝑧, 𝑤) = exp −𝜎 𝑧
𝜋𝜆
2

 𝛽/2

𝜌0 (𝑧, 𝑤)
d𝑧 d𝑤,1+𝑧

2𝑧 2/𝛽−𝑤 1+
𝛽(3.10)∫ ∞𝑣 2/𝛽−1d𝑣
1/𝑧 1 + 𝑣

.

and 𝜏′ (𝑢) are given in (3.10) and (3.8), respectively. Moreover, 𝜀 > 0 is aﬁtting parameter.
Proposition 3.1 (Proposition 2.3). The HO rate H in the periodic HO
skipping scenario is given as
1
H = E[𝜂(𝑠𝑉1 )],(3.11)𝑠where 𝜂(𝑙), 𝑙 > 0, is given by∫ 𝜋∫ ∞
2
𝜂(𝑙) = 1 − 2𝜆𝑟e−𝜆{𝜋𝑟 +𝜚(𝑟,𝜃|𝑙)} d𝑟d𝜃,(3.12)
0

0

with

h
 𝑙 sin 𝜃 i𝜚(𝑟, 𝜃|𝑙) = 𝑤 𝑙,𝑟,𝜃 2 𝜃 + sin−1− 𝑟 2 𝜃 + 𝑙𝑟 sin 𝜃,
𝑤 𝑙,𝑟,𝜃
and 𝑤 𝑙,𝑟,𝜃 is given in (3.7).
38(3.13)

3.3

Analysis of optimal skipping period

In this section, we consider to maximize the utility metric with respect tothe skipping period. First, we evaluate the utility metric U given in (3.2)
using some approximations. Next, we analyze the optimal skipping periodthat maximizes the approximated utility metric.

3.3.1

Approximation of utility metric

We assume that the moving speed of the typical UE is conditioned on 𝑉1 =𝑣 > 0. Then, by combining (3.3) and (3.11) with (3.2), the utility metric isgiven by1Õ𝑐
𝜏(𝑡𝑣) − 𝜂(𝑠𝑣),
𝑠 𝑡=0
𝑠
𝑠−1

U (𝑠) =

(3.14)

where 𝜏(𝑡𝑣) and 𝜂(𝑠𝑣) are given in (3.4) and (3.12), respectively. Although
the parameter 𝑠 in the function above is a positive integer, we modify thefunction to make 𝑠 continuous; we regard the sum on the right-hand side of(3.14) as a Riemann sum and approximate it by the corresponding integral,that is
∫
𝑠−1
Õ
1 𝑠𝑣
𝜏(𝑢) d𝑢.
𝜏(𝑡𝑣) ≈𝑣 0
𝑡=0
In addition, we replace 𝜏(𝑢) with 𝜏′ (𝑢) in (3.8) for the sake of simplicity. We
then obtain an approximate continuous function of U(s) as follows:
∫
1 𝑣 ′
𝑐
e
(3.15)
U (𝑠) ≈ U (𝑠) :=𝜏 (𝑠𝑤)d𝑤 − 𝜂(𝑠𝑣).
𝑣 0
𝑠

3.3.2

Analytical expression of optimal skipping periodee
We here consider to maximize the above utility metric U(s).
In fact, U(s)is expected to have a local maximum especially when the moving speed 𝑣 is
sufficiently small. We refer to the length 𝑠 of the skipping period which gives
the local maximum of the utility metric as the optimal skipping period. Here
we show this fact with the approximate expression of the optimal skipping
period.
Theorem 3.2. For the cellular network model described in Sec. 2.2, we suppose that the typical UE adopts the periodic HO skipping and moves at certain39

constant speed 𝑣. Then, an approximation of the optimal skipping period is
obtained as the closest integer of 𝑠∗ given by∫ −1
15 − 𝜋 2  ∞
1
1
∗
𝑠 ≈
𝑐
·
d𝑧
,
(3.16)−1 𝛽/2 (1 + 2𝑧) 2
4𝜋 2 𝛽0 1 + (𝐾 𝛽 𝑧 )where 𝐾 𝛽 is given in (3.5).
□

Proof. See Section 3.6.

Remark 3.1. Although this expression is obtained under the condition that 𝑣
is suﬃciently small, numerical studies suggest that it is applicable to generalcases of the moving speed 𝑣: see the discussion in Sec. 3.4.1.

3.4

Numerical evaluation

3.4.1

Accuracy of approximate utility metrice in (3.15) compared with
We first examine the approximation error of U(s)U (𝑠) in (3.14) in terms of their optimal skipping period. Fig. 3.1 shows thenumerical results of the utility metrics. From the figure, we see that boththe two utility metrics have similar shapes, although there are some errors
between them. In particular, both the metrics exhibit a local maximum. In
addition, the values of the optimal skipping period for the two metrics areobserved to be similar. Also, the optimal skipping period is almost invariantewith respect to 𝑣. Furthermore, since U(s) and U(s)converge to zero as𝑠 increases, we expect that when 𝑣 is sufficiently small, the local maximum
indicates the maximum uniquely at least on a finite value range of a practicalsetting. Since both the exact and approximate utility metrics have similar
optimal skipping period, we conclude that the approximate utility metrice has enough accuracy for evaluating the optimal skipping period.
U(s)

3.4.2

Numerical evaluation of 𝑠∗

We here numerically evaluate 𝑠∗ obtained in Theorem 3.2. Figure 3.2 shows
the numerical results of 𝑠∗ (the time unit = msec) with respect to the pathloss exponent 𝛽 and the utility constant 𝑐. The BS intensity and the moving
speed of the typical UE are fixed as 𝜆 = 1 (units /km2 ) and 𝑣 = 0.01(km/sec),
respectively. For comparison, the values of 𝑠, which locally maximize the ape in (3.15), are numerically searched and plotted
proximate utility metric U(s)in the figure. From Fig. 3.2, we observe that the values of 𝑠∗ agree well with40hƚŝůŝƚǇ&ƵŶĐƚŝŽŶሺȀ Ȁ ሻ,ĂŶĚŽǀĞƌ ^ŬŝƉƉŝŶŐdŝŵĞ  ሺ ሻe
Figure 3.1: Numerical samples of U(s) and U(s),which are plotted in
the red solid lines and the black dotted lines, respectively. Three patterns of the values of the moving speed 𝑣 are exhibited where 𝑣 =0.006, 0.012, 0.018 (km/sec), and the other system parameters are fixed as
𝜆 = 3 (units/km2 ), 𝑐 = 30 (nats/Hz), 𝜎 2 = 0, and 𝛽 = 3.
the values from the numerical search even for positive moving speed, in particular for large 𝛽 and small 𝑐, in spite that 𝑠∗ is derived under the conditionof sufficiently small moving speed. Figure 3.2a shows that 𝑠∗ is decreasing in
𝛽. This is thought to be because 𝜏′ (𝑢) decays more rapidly with respect to𝑢 when 𝛽 is larger (as confirmed in Fig. 2.2); that is, smaller∫ 𝑠 brings better
1 𝑣 ′eperformance when 𝛽 is larger since U(s) has the term 𝑣 0 𝜏 (𝑠𝑤)d𝑤. On
the other hand, Fig. 3.2b shows that 𝑠∗ is linearly increasing in 𝑐 (as is clear
e is linearly decreasing in 𝑐,from (3.16)). This is interpreted as because U(s)and thus better performance is given by larger 𝑠 which makes the HO rate
lower.
Figure 3.3 further compares the values of 𝑠∗ in (3.16) and the numericallysearched values of 𝑠 as above with respect to the constant moving speed 𝑣 of
the typical UE. Since 𝑠∗ does not depend on 𝑣 and 𝜆 (see (3.16)), its value
is given as a horizontal line for each pair of 𝛽 and 𝑐. Note that the values ofthe optimal skipping period obtained by the numerical search do not change
significantly with respect to the changes in 𝑣 and 𝜆, in particular for large 𝛽
and small 𝑐, which allows us to use 𝑠∗ in (3.16) as an approximation of the
optimal skipping period for any 𝑣 and 𝜆, in particular when 𝛽 is large and 𝑐is small.

41KƉƚŝŵĂůƐŬŝƉƉŝŶŐƉĞƌŝŽĚሺ ሻ

ʹ1000WĂƚŚͲůŽƐƐĞǆƉŽŶĞŶƚ (a) 𝑠∗ as a function of 𝛽.

KƉƚŝŵĂůƐŬŝƉƉŝŶŐƉĞƌŝŽĚሺ ሻ

ʹ1000hƚŝůŝƚǇĐŽŶƐƚĂŶƚ  ሺȀ ሻ(b) 𝑠∗ as a function of 𝑐.
Figure 3.2: The values of 𝑠∗ in (3.16) with respect to the path-loss exponent
𝛽 and the utility constant 𝑐 in (3.15).

3.5

Comparison with other handover skipping
schemes

We further investigate the optimal skipping period by conducting a comparison study with the two baseline schemes: the alternate HO skipping as
the first baseline [9], and the topology-aware HO skipping 2 as the second2

We particularly choose the criteria for the topology-aware HO skipping as the chord
length of a cell; HOs are not performed when the typical UE enters a cell whose chord
length made by the UE’s trajectory is less than the threshold 𝑇 > 0.

42

KƉƚŝŵĂůƐŬŝƉƉŝŶŐƉĞƌŝŽĚሺ ሻ

ʹ1000DŽǀŝŶŐƐƉĞĞĚ  ሺȀ ሻ∗(a) 𝑠 as a function of 𝑣 with several patterns of 𝜆 and 𝛽, where 𝑐 = 10

(nats /Hz ) is fixed.

KƉƚŝŵĂůƐŬŝƉƉŝŶŐƉĞƌŝŽĚሺ ሻ

ʹ1000DŽǀŝŶŐƐƉĞĞĚ  ሺȀ ሻ∗(b) 𝑠 as a function of 𝑣 with several patterns of 𝜆 and 𝑐, where 𝛽 = 5 isfixed.
Figure 3.3: The values of 𝑠∗ in (3.16) with respect to the moving speed 𝑣 of
the typical UE.
baseline [13]. We consider the utility metric defined in (3.2) for both thetwo baseline schemes and compare with the one of our proposed scheme,
where the expected data rate and the HO rate in those baseline schemes arecomputed via simulations. As for the utility metric of our proposed scheme,e
we consider U(s)given in (3.15) and regard 𝑠 as the closest integer to 𝑠∗computed by the expression in (3.16), where the time unit is msec.
The result of the comparison is shown in Fig. 3.4. We see that our proposed scheme outperforms the alternate HO skipping scheme for exhibited
values of 𝜆 and also outperforms the topology-aware HO skipping scheme particularly when 𝜆 is large. This is thought to be because our proposed scheme
does not cause excessive degradation of the data rate for any 𝜆, since the HO
rate 1/𝑠 is at least ensured and the UE can refresh its connection every time43hƚŝůŝƚǇŵĞƚƌŝĐሺȀ Ȁ ሻ^ŝŶƚĞŶƐŝƚǇ  ሺȀʹሻ

Figure 3.4: Comparison among the utility metrics of our proposed scheme(Periodic, in the graph legends) and the two baseline schemes (Alternate
and Topology-aware, in the graph legends). Simulation results are exhibited
discretely for the two baseline schemes, while numerical results are exhibitedfor the proposed scheme. The chord length threshold 𝑇 in the topology-aware
HO skipping scheme (the second baseline) is set as 𝑇 = 0.3 (km). The othersystem parameters are fixed as 𝑣 = 0.01 (km/sec), 𝑐 = 10 (nats/Hz), and𝛽 = 3.

interval of 𝑠. That is, the HO rate could be managed for preventing the data
rate degradation in our proposed scheme, which exhibits better performance
results compared with those baseline HO skipping schemes depending on thevalue of 𝜆.

3.6

Proof of Proposition 3.2

e in (3.15) with respect to 𝑠, we consider
To obtain the first derivative of U(s)the Taylor expansion around 𝑣 = 0, and ignore terms for which the order ofe (𝑠) yields
𝑣 is greater than 2. Then, the first term of U∫ 𝑣
1 d
𝜏′ (𝑠𝑤) d𝑤
𝑣 d𝑠 0∫ ∫2 2𝑣 ∞1 − 21+𝜋𝜆𝑧
𝛽 d2𝑧 𝑠 𝑤=
𝑧−1 d𝑧d𝑤 + 𝑜(𝑣 2 )𝑎𝑠 𝑣 → 0−1𝛽/22𝑣 d𝑠 0 0 {1 + (𝐾 𝛽 𝑧 ) }(1 + 2𝑧)
∫ ∞
2 2
1
1
≈ − 𝑠𝑣 𝜋𝜆𝛽·d𝑧.
(3.17)−1𝛽/2
3(1 + 2𝑧) 2
0 1 + (𝐾 𝛽 𝑧 )44

e (𝑠), we consider the asymptotical form ofRegarding the second term of U
𝜂(𝑠𝑣) given in (3.12) under the condition that 𝑣 ≈ 0. Since 𝑣 ≪ 𝑟 yields
sin 𝜃−1 𝑠𝑣 sin 𝜃
sin−1 𝑠𝑣
, 𝜚(𝑟, 𝜃|𝑠𝑣) in (3.13) is expressed as follows
𝑤 𝑠𝑣,𝑟 , 𝜃 ≈ sin𝑟
𝜚(𝑟, 𝜃|𝑠𝑣)h 𝑠𝑣 sin 𝜃 i
≈ 𝑤 𝑠𝑣,𝑟,𝜃 2 𝜃 + sin−1− 𝑟 2 𝜃 + 𝑠𝑣𝑟 sin 𝜃𝑟h𝑠𝑣 sin 𝜃 i− 𝑟 2 𝜃 + 𝑠𝑣𝑟 sin 𝜃 + 𝑜(𝑣 2 )= 𝑤 𝑠𝑣,𝑟,𝜃 2 𝜃 +𝑟= −2𝑠𝑣𝑟 (𝜃 cos 𝜃 − sin 𝜃) + 𝑠2 𝑣 2 (𝜃 − sin 2𝜃) + 𝑜(𝑣 2 )𝑎𝑠 𝑣 → 0,𝜃
where the Maclaurin expansion of sin−1 𝑠𝑣 sin
is applied in the first equality,𝑟and (3.7) is used in the last equality. Substituting the above expression into
(3.12) yields
𝜂(𝑠𝑣)

∫ ∫
𝜋 ∞
2
≈ 1 −2𝜆𝑟e−𝜆𝜋𝑟0 0
2𝜆𝑠𝑣𝑟 (𝜃 cos 𝜃−sin 𝜃)· e−𝜆𝑠 𝑣 (𝜃−sin 2𝜃) d𝑟d𝜃 + 𝑜(𝑣 2 )
∫ ∫
h
𝜋 ∞
−𝜆𝜋𝑟 2= − 2𝜆𝑟e
2𝜆𝑠𝑣𝑟 (𝜃 cos 𝜃 − sin 𝜃)
0 0
i
 2
2 2
2
+ 𝜆𝑠 𝑣 2𝜆𝑟 (𝜃 cos 𝜃 − sin 𝜃) − (𝜃 − sin 2𝜃) d𝑟d𝜃 + 𝑜(𝑣 2 )√15 − 𝜋 2 2 2
4 𝜆𝑠𝑣 −𝜆𝑠 𝑣 + 𝑜(𝑣 2 )𝑎𝑠 𝑣 → 0,=𝜋
6𝜋
where the Maclaurin expansion of the exponential form is applied in the firstequality. Therefore, we have
×e

2 2

d 𝑐
15 − 𝜋 2
𝜂(𝑠𝑣) ≈ −𝑐𝜆𝑣 2 .
(3.18)d𝑠 𝑠
6𝜋From (3.17) and (3.18), we derive the asymptotical form of the first-ordere asderivative of U(s)
∫ ∞
1
d e
2 2
15 − 𝜋 2 2
1·
U (𝑠) ≈ − 𝑠𝑣 𝜋𝜆𝛽
d𝑧
+
𝑐𝑣 𝜆.
−1 𝛽/2 (1 + 2𝑧) 2d𝑠
36𝜋
0 1 + (𝐾 𝛽 𝑧 )It follows from the expression above that the approximate form of the funce is a linear function
e is concave and has only one peak because d U(s)tion U(s)d𝑠e(𝑠) > 0 when 𝑠 is sufficiently small and d Uewith respect to 𝑠, and dd𝑠 U
d𝑠 (𝑠) < 0e(𝑠) corresponds towhen 𝑠 is sufficiently large. Thus, the extreme point of U
the global maximum with respect to 𝑠.
e(𝑠) = 0 for 𝑠 completes the proof.
Finally, solving dd𝑠 U
45

3.7

Conclusion

We have discussed how to decide the length of the skipping period and haveprovided a simple computable expression of the skipping period which approximately gives a local maximum of the utility metric. Numerical comparison with other related HO skipping techniques have also shown that the
proposed scheme is comparable to the others. Although we have consideredhere a simple mobility model on a homogeneous PPP network model, further development within more extended and generalized frameworks (e.g.,
HetNets with interference cancellation and/or the BS cooperation) would be
expected for future work and one direction of the extensions is found in [18].

46Chapter 4
An Extension to Heterogeneous
Networks Using
Poisson-Poisson Cluster
Process
4.1

Introduction

Network densification is regarded as a promising approach to cope with theever-growing traffic in the current fifth generation (5G) cellular networks [8].
In the densified cellular networks, miniaturized base stations (BSs), namely,
small BSs (SBSs), play a key role in complementing the capacity for the
traffic not sufficiently covered by macro BSs (MBSs) [45]. SBSs are often
considered to be deployed as they form clusters for intensive improvementof the capacity around the hot spots and the dead zones [46, 47]. On theother hand, the network densification could cause frequent handovers (HOs),
especially for mobile user equipments (UEs) passing through the area of theclustered SBSs.
During the process of an HO, a mobile UE exchanges signals with theserving BS and the target BS, which are connected via the core network.
If the HOs are processed frequently, enormous signals could cause huge signaling overheads among those networks, which diminishes performance for
the UE [11]. To mitigate those frequent HOs, HO skipping approaches havebeen proposed. HO skipping enables a UE to reduce the HO rate, whereas
it may decrease the data rate because it demands the UE to maintain longer
connection duration with a BS. Thus, the HO skipping presents a trade-off
relation between the increased HO rate and the decreased data rate, and the47

trade-off relation should be balanced to improve the performance for UEs.
Studies of performance analysis for user mobility in cellular networks havebeen extensively done in the literature. Among them, stochastic geometrybased approach have recently attracted researchers’ interest [19,24,31]. While
adopting cluster point processes for modeling the intensively deployed SBSs
has gained much attention owing to the aforementioned background, theexisting studies in this context are still few. In [48], they first analyzed the HO
rate in a single-tier network where BSs are deployed according to a PoissonPoisson cluster process (PPCP). Thereafter, they extend the result to a multitier heterogeneous network where either a Poisson point process (PPP) or a
PPCP is adopted for the deployment in each tier, and considered the optimal
tier selection for a moving UE by balancing the tradeoff between improving
the data rate and reducing the HO rate. In [49], they proposed a modified
random waypoint model based on the one proposed in [23]. Moreover, they
considered a heterogeneous network that consist of three tiers, in one of which
SBSs are deployed according to a PPCP, and derived the HO rate, the HO
failure rate, and the ping-pong rate of a moving UE that follows the modified
random waypoint model.
The technique of HO skipping has been investigated and analyzed bystochastic geometry approach, where the majority of the work consider tomodel the locations of BSs by a homogeneous PPP due to its simplicity andtractability [4]. Arshad et al. [9] proposed an HO skipping technique, in which
a UE skips HOs under certain conditions. More precisely speaking, in theirstudy, UE alternatively skips HOs along its moving trajectory. Moreover,
the authors evaluated the performance via quantifying throughput of the UE
in a PPP-based single-tier network. Thereafter, in [10], a velocity-aware HO
skipping technique was analyzed in a PPP-based two-tier network. In [12,13],
a topology-aware HO skipping technique was analyzed in a PPP-based singletier network. In [50], a user-centric cooperative transmissions-based handoverscheme, named the group-cell HO scheme, was proposed, and they developed
a new framework of HO skippings based on their proposed group-cell HO
scheme. In [16, 17], authors proposed a periodic HO skipping technique andevaluated it in a PPP-based single-tier network.
In our study, we consider a 2-tier heterogeneous cellular network, wheremacro and small BSs are respectively deployed by PPP and a PPCP. Realworld small BSs which tend to be deployed as clusters can be well represented
by the PPCP. For modeling moving UEs, inspired by [16, 17], we employ asimilar random walk-based model with a periodic HO skipping technique.
Note that we introduce two distinct skipping period parameters unlike [16,17]
since our network consists of two types of BSs. Based on this system model,
we provide two analytical results in Section 4.3: the exact expression of the48

HO rate and the approximated expression of the expected data rate. To thebest of our knowledge, no HO skipping technique has yet been analyzed basedon our network. Thus, these two analytical results are our main contributions. Moreover, based on the two analytical results, the efficiency of the HO
skipping technique in our network is evaluated by comparing it with non-HO
skipping counterpart in numerical experiments.

4.2

System model

4.2.1

Network model

As mentioned at the end of Section 4.1, our network consists of a macro BS
tier and a small BS tier. Macro BSs (resp. Small BSs) are referred to as
1st (resp. 2nd) tier. Macro BSs are assumed to be deployed on R2 as ahomogeneous PPP Φ1 with intensity 𝜆 1 . All the macro BSs transmit signal
power with the same level adopting a common spectrum bandwidth, andthe power level is denoted by 𝑃1 . Small BSs are deployed as a PPCP Φ2
by transmitting the same power level 𝑃2 , where we assume that Φ2 has the¯ is defined as in Definition 1.4
isotropy. The isotropic PPCP Φ2 = Φ(𝜆 𝑝 , 𝑔, 𝑚)
of Chapter 1. As described in (1.1), Φ2 consists of the offspring process C 𝒛
shifted by 𝒛 for each 𝒛 ∈ Ψ𝑝 , and 𝒙 𝑖 ∈ 𝒛 + C 𝒛 , 𝑖 = 1, · · · , 𝑀, are conditionallyi.i.d. with the probability density function (PDF) 𝑔(𝒙 𝑖 − 𝒛).
Let 𝒙 𝑘 ∈ R2 (𝑘 ∈ N) denote an element of Φ𝑖 (𝑖 = 1, 2), expressing theposition of either a macro or a small BS. Here, 𝑘 is an index of the BS. We
assume Rayleigh Fading and general power-law path-loss on downlinks while
ignoring shadowing effects. Thus, the UE at 𝒖 ∈ R2 receives the following
signal power at discrete time 𝑡 ∈ N0 from a BS at 𝒙 ∈ R2 : 𝑃 (𝒙) 𝐻𝒙,𝑡 ∥𝒙 − 𝒖∥ −𝛽 ,where 𝐻𝒙,𝑡 are i.i.d. random variables such that 𝐻𝒙,𝑡 ∼ exp(1), representing
the fading effect from the location 𝒙. As for 𝑃 (𝒙) , 𝑃 (𝒙) = 𝑃1 if 𝒙 ∈ Φ1 ,otherwise 𝑃 (𝒙) = 𝑃2 . The symbol 𝛽 (> 2) denotes the general path-loss
exponent for both tiers. We then define the downlink Signal-to-Interferenceplus-Noise Ratio (SINR) for UE at 𝒖 when it receives signal power at thetime 𝑡 from BS at 𝒙 ∈ Φ1 ∪ Φ2 as follows:
𝑃 (𝒙) 𝐻𝒙,𝑡 ∥𝒙 − 𝒖∥ −𝛽,
SINR𝒙,𝒖 (𝑡) =𝐼 (Φ1 ∪Φ2 )\𝒙,𝒖 (𝑡) + 𝜎 2(4.1)

where 𝐼 (Φ1 ∪Φ2 )\𝒙,𝒖 (𝑡) denotes the interference power at time 𝑡 ∈ N0 from all
the BSs except the one at 𝒙 when UE is at 𝒖, and 𝜎 2 > 0 denotes the noise49

power. The interference power is given byÕ𝐼 (Φ1 ∪Φ2 )\𝒙,𝒖 (𝑡) =𝑃 (𝒙 ′ ) 𝐻𝒙 ′ ,𝑡 ∥𝒙 ′ − 𝒖∥ −𝛽 .

(4.2)

𝒙 ′ ∈(Φ ∪Φ )\𝒙
1

2

Eventually, log 1 + SINR𝒙,𝒖 (𝑡) defines the instantaneous downlink data rate.

4.2.2

Mobility model for a moving user equipment

As for the network described in Section 4.2.1, a moving UE is modeled bya random walk with a periodic HO skipping technique explained as follows.
Initially, the UE is at 𝑆(0) ∈ R2 . For any 𝑗 ∈ N, at the beginning of the𝑗-th movement, the UE is at 𝑆( 𝑗 − 1) ∈ R2 . At the end of the movement,
UE is at 𝑆( 𝑗) ∈ R2 . During the 𝑗-th movement, the UE moves on straightly
with constant velocity 𝑉 𝑗 ∈ R2 , while it is always associated with the only
one same BS. The velocities 𝑉 𝑗 , 𝑗 ∈ N are assumed to be i.i.d. random
vectors. In addition, such associated BS in the movement is decided by thefollowing criterion: if a BS at 𝒙 𝑘 1∗ provides higher power to the UE at 𝑆( 𝑗 −1)
than a BS at 𝒙 𝑘 2∗ , UE will be associated with the BS at 𝒙 𝑘 1∗ . Otherwise the
BS at 𝒙 𝑘 2∗ . Here, 𝑘 𝑖∗ (𝑖 = 1, 2) denotes the index of BS which is the closest(with Euclidean metric) to the UE among 𝑖-th tier BSs. Moreover, if an 𝑖-thtier BS provides higher signal power to the UE at 𝑆( 𝑗 − 1), UE is supposedto spend a constant 𝑠𝑖 ∈ N units of time from 𝑆( 𝑗 − 1) to 𝑆( 𝑗), and then𝑆( 𝑗) = 𝑆( 𝑗 − 1) + 𝑠𝑖𝑉 𝑗 . Here, the periods of the time 𝑠1 and 𝑠2 are respectively
called the macro skipping period and the small skipping period.

4.2.3

Performance metrics

We furthermore define the HO rate H (𝑠1 , 𝑠2 , 𝑣) and the expected data rate
T (𝑠1 , 𝑠2 , 𝑣) for the UE attempting the periodic HO skipping scheme in our
2-tier network model. Following the previous chapters, H (𝑠1 , 𝑠2 , 𝑣) andT (𝑠1 , 𝑠2 , 𝑣) are defined as follows
E[𝜁 (0, 𝑚)]
,
𝑚→∞
𝑚
𝑚−1
1 Õ
E[log(1 + SINR𝒙,𝒖 (𝑡))],T (𝑠1 , 𝑠2 , 𝑣) = lim
𝑚→∞ 𝑚
𝑡=0

H (𝑠1 , 𝑠2 , 𝑣) = lim

where 𝜁 (0, 𝑚) denotes the number of HOs executed by the typical UE up to
the time 𝑚 ∈ N, and SINR𝒙,𝒖 (𝑡) is given in (4.1).

50

4.3

Analyses

We aim to analyze expectations of the HO rate and the expected data rate
defined in the preceding section. To do so, it suffices to analyze the two
expectations in the 1st movement with 𝑆(0) = 𝒐 and 𝑉1 = (𝑣, 0) ⊤ , where weassume 𝑣 ≥ 0 is fixed. The reason is due to the stationatiry of Φ1 , Ψ𝑝 and theisotropy of Φ1 , Ψ𝑝 , 𝑔; see the proof of Theorem 1 in [17]. For these analyses,
we use SINR𝒙,𝑢 (𝑡) instead of SINR𝒙,𝒖 (𝑡) in (4.1), where 𝒖 = (𝑢, 0) ⊤ and 𝑢 ≥ 0.
In addition, 𝐼·,𝑢 (𝑡) is used instead of the left term in (4.2). Moreover, while
we employ a polar coordinate system, a position 𝒙 ∈ R2 is expressed by (𝑥, 𝜃)
where 𝑥 = ∥𝒙∥ and 𝜃 ∈ [0, 2𝜋).

4.3.1

Handover rate analysis

Proposition 4.1. The HO rate H (𝑠1 , 𝑠2 , 𝑣) is given by (4.3),


∫ ∞
∫
1 𝜋
2𝜋𝜆 1
−𝜆1𝜋𝑟 2𝑟e
𝐴1 (𝑟) 1−
𝐷 1 (𝑟, 𝑠1𝑣 𝑗 , 𝜃)d𝜃 d𝑟
H (𝑠1 , 𝑠2 ,𝑣 𝑗 ) =𝜅(𝑠1 , 𝑠2 ) 0
𝜋 0


∫ ∞
∫
¯
𝑚
1 𝜋
−𝜆1𝜋 𝑃ˆ22,1𝑟 2+e
𝐴2 (𝑟) 𝐵(𝑟) −𝐶 (𝑟, 𝑠2 𝑣 𝑗 , 𝜃)𝐷 2 (𝑟, 𝑠2 𝑣 𝑗 , 𝜃)d𝜃 d𝑟, (4.3)
𝜅(𝑠1 , 𝑠2 ) 0
𝜋 0
where



∫ ∞



−𝑚𝐹
¯ 𝑑 ( 𝑃ˆ𝑖,2 𝑟 |𝑧)



𝑧 1−e
d𝑧 ,
𝐴𝑖 (𝑟) = exp −2𝜋𝜆 𝑝0∫ ∞¯ 𝑑 (𝑟 |𝑧)
𝐵(𝑟) = 2𝜋𝜆 𝑝𝑧 𝑓𝑑 (𝑟 | 𝑧) e−𝑚𝐹
d𝑧,
0
∫ ∞
¯ 𝑑 (𝑟 |𝑧)𝐶 (𝑟, ℓ, 𝜃) = 2𝜆 𝑝
𝑧 𝑓𝑑 (𝑟 | 𝑧) e−𝑚𝐹
∫ 𝜋0
∫


¯
×
exp −𝑚𝑔(𝒙 −𝒛)d𝒙 d𝜃d𝑧,
0D2,2 (𝑟,ℓ,𝜃)∫ ∞¯ 𝑑 ( 𝑃ˆ𝑖,2 𝑟 |𝑧)
𝐷 𝑖 (𝑟, ℓ, 𝜃) = exp −𝜆 1 | D𝑖,1 (𝑟, ℓ, 𝜃) | −2𝜆 𝑝𝑧 e−𝑚𝐹
0

∫ 𝜋n
∫

o
¯ 𝑔(𝒙 −𝒛)d𝒙 d𝜃d𝑧 ,×
1 − exp −𝑚
D𝑖, 2 (𝑟,ℓ,𝜃)0

and D𝑖, 𝑗 (𝑟, ℓ, 𝜃), 𝑖, 𝑗 = 1, 2, is a two dimensional region deﬁned by
D𝑖, 𝑗 (𝑟, ℓ, 𝜃) = 𝑏 (ℓ, 0), 𝑃ˆ𝑖, 𝑗 𝜔𝑟,ℓ,𝜃 ∩ 𝑏(𝒐, 𝑃ˆ𝑖, 𝑗 𝑟) 𝑐 ,

51

(4.4)
(4.5)

(4.6)

(4.7)

(4.8)

√
with 𝑃ˆ𝑖, 𝑗 = (𝑃 𝑗 /𝑃𝑖 ) 1/𝛽 and 𝜔𝑟,𝑢,𝜃 = 𝑟 2 + 𝑢 2 − 2𝑟𝑢 cos 𝜃. Moreover, 𝜅(𝑠1 , 𝑠2 )denotes the expected length of a skipping period, which is given by𝜅(𝑠1 , 𝑠2 ) = 𝑠1 (1 − P(S2 )) + 𝑠2 P(S2 ),
with

(4.9)

∫ ∞
∫ ∞


−𝜋𝜆 1 𝑃ˆ22,1 𝑟 2¯ 𝑝¯ 𝑑 (𝑟 | 𝑧) 𝑧d𝑧
P(S2 ) = 2𝜋 𝑚𝜆e𝑓𝑑 (𝑟 | 𝑧) exp −𝑚𝐹
0
0


∫ ∞


¯ 𝑑 (𝑟 | 𝑧) 𝑧d𝑧 d𝑟.
× exp −2𝜋𝜆𝑝1 − exp −𝑚𝐹
0Remark 4.1. We recall that 𝑓𝑑 (𝑟 | 𝒛) and 𝐹𝑑 (𝑟 | 𝒛) in the above expressions respectively denote the conditional PDF of the distance 𝑟 = ∥𝒙∥ of an¯ from the origin given its cluster center at
oﬀspring point 𝒙 ∈ Φ(𝜆 𝑝 , 𝑔, 𝑚)𝒛 ∈ Ψ𝑝 , and its CDF. Their speciﬁc expressions when the PPCP Φ2 is speciﬁed as Thomas Cluster Process (TCP) or Mattern Cluster Process (MCP)
are provided in [7].
Proof. For 𝑖 = 1, 2, let S𝑖 and 𝐻𝑖𝑐 respectively denote the events that the UE
is associated with the 𝑖-th tier in the 1st movement, and that an HO does
not occur at the end of the movement. Then, we have
E[𝑁 (𝑠1 , 𝑠2 , 𝑣)],
𝜅(𝑠1 , 𝑠2 )H (𝑠1 , 𝑠2 , 𝑣) =(4.10)

where 𝑁 (𝑠1 , 𝑠2 , 𝑣) denotes the number of HOs that occur in a certain skipping
period, and 𝜅(𝑠1 , 𝑠2 ) is given in (4.9). Here, E[𝑁 (𝑠1 , 𝑠2 , 𝑣)] is given byE[𝑁 (𝑠1 , 𝑠2 , 𝑣)] =

2
Õ
𝑖=1


1 − P(𝐻𝑖𝑐 | S𝑖 ) P(S𝑖 ).

(4.11)

Let 𝑟 = ∥𝒙 𝑘 𝑖∗ ∥ denote the magnitude of the UE’s associated BS in the event S𝑖 .
Conditioned on the parent PP Ψ𝑝 , the PDF of 𝑟 is obtained from Lemma 4of [7], that is
𝑓𝑐 (𝑟 | S𝑖 , Ψ𝑝 ) =

Ö
Λ(𝑖)
ˆ2 2
¯ 𝑑 ( 𝑃ˆ𝑖,2 𝑟 |𝑧)e−𝜆1 𝜋 𝑃𝑖,1 𝑟
e−𝑚𝐹P(𝑆𝑖 | Ψ𝑝 )
𝒛∈Ψ
𝑝



Í
¯ 𝒛∈Ψ 𝑝 𝑓𝑑 (𝑟 | 𝑧) , andwhere Λ(1), Λ(2) = 2𝜋𝜆1𝑟, 𝑚
∫ ∞
Ö
ˆ2 2
¯ 𝑑 ( 𝑃ˆ𝑖,2 𝑟 |𝑧)
P(𝑆𝑖 | Ψ𝑝 ) =Λ(𝑖) e−𝜆1 𝜋 𝑃𝑖,1 𝑟
e−𝑚𝐹
d𝑟.
0

𝒛∈Ψ 𝑝

52

(4.12)

Therefore, we haveP(S𝑖 )P(𝐻𝑖𝑐 | S𝑖 ) = E[P(S𝑖 | Ψ𝑝 )P(𝐻𝑖𝑐 | S𝑖 , Ψ𝑝 )]
∫ 2∫
𝜋 ∞
1
ˆ2 2
=
e−𝜆1 𝜋 𝑃𝑖,1 𝑟
2𝜋 0 0"#Ö¯ 𝑑 ( 𝑃ˆ𝑖,2 𝑟 |𝑧)
× E Λ(𝑖)e−𝑚𝐹
P(𝐻𝑖𝑐 | 𝑟, 𝜃, S𝑖 , Ψ𝑝 ) d𝑟d𝜃.

(4.13)𝒛∈Ψ 𝑝

We next let 𝐻𝑖,𝑐 𝑗 , 𝑖, 𝑗 = 1, 2, denote the event that there is no need toperform HOs from the 𝑖-th tier to the 𝑗-th tier at the end of the movement.
Note that under the condition of S𝑖 and 𝒙 𝑘 𝑖∗ = (𝑟, 𝜃), the event 𝐻𝑖,𝑐 𝑗 occurs if
and only if there are no BSs of the 𝑗-th tier offering stronger signal power
for the UE at the position (𝑠𝑖 𝑣, 0) than the BS at the polar coordinate (𝑟, 𝜃).
Also note that any BSs of the 𝑗-th tier could not be allocated inside thecircle region 𝑏(𝒐, 𝑃ˆ𝑖, 𝑗 𝑟) under the condition of S𝑖 and 𝒙 𝑘 𝑖∗ = (𝑟, 𝜃). Since Φ1and Φ2 conditioned on Ψ𝑝 are respectively PPPs with the intensities 𝜆 1 andÍÎ𝑐 |¯ 𝒛∈Ψ 𝑝 𝑓 (𝒙 | 𝒛) (see Proposition 1.1), P(𝐻𝑖𝑐 | 𝑟, 𝜃, S𝑖 , Ψ𝑝 ) = 𝑘=1,2 P(𝐻𝑖,𝑘𝑚𝑟, 𝜃, S𝑖 , Ψ𝑝 ) is given as follows
P(𝐻𝑖𝑐 | 𝑟, 𝜃, S𝑖 , Ψ𝑝 )=e

−𝜆 1 |D𝑖, 1 (𝑟,𝑠𝑖 𝑣,𝜃)|Ö¯
exp −𝑚

𝒛∈Ψ 𝑝



∫

𝑓 (𝒙 | 𝒛)d𝒙 ,D𝑖, 2 (𝑟,𝑠𝑖 𝑣,𝜃)(4.14)

where D𝑖, 𝑗 (𝑟, 𝑠𝑖 𝑣, 𝜃) is in (4.8) and the norm | · | denotes the area of a region.
Substituting (4.11), (4.12), (4.13), and (4.14) into (4.10), and applying theprobability generating functional (PGFL) and the sum-product functional(SPFL) (see Lemma 5 and 6 of [7]) to the parent PPP Ψ𝑝 , we obtain (4.3).
The rest part marely consists of algebraic manipulations, so it is omitted. □53

4.3.2

Data rate analysis

Proposition 4.2. The expected data rate T (𝑠1 , 𝑠2 , 𝑣) is approximately givenby𝑠 1 −1 ∫ ∞𝑤𝑦
2𝜆 1 Õ
−𝜆 𝜋𝑟 2 − 𝑟 ,𝑡𝑃𝑣, 𝜃 𝜎 2
1
𝑟e 1
T (𝑠1 , 𝑠2 , 𝑣) =𝐴1 (𝑟)
𝜅(𝑠1 , 𝑠2 ) 𝑡=0 0∫ 𝜋∫ ∞J1 (𝑦, 𝜃, 𝑟, 𝑡𝑣)K1 (𝑦, 𝜃, 𝑟, 𝑡𝑣)
×d𝑦d𝜃d𝑟1+𝑦0 0𝑠Õ2 −1 ∫ ∞𝑤𝑦¯𝑚−𝜆 𝜋 𝑃ˆ2 𝑟 2 − 𝑟 ,𝑡𝑃𝑣, 𝜃 𝜎 22
e 1 2, 1+𝐴2 (𝑟)𝐵(𝑟)𝜅(𝑠1 , 𝑠2 ) 𝑡=0 0∫ 𝜋∫ ∞J2 (𝑦, 𝜃, 𝑟, 𝑡𝑣)K2 (𝑦, 𝜃, 𝑟, 𝑡𝑣)
1
×d𝑦d𝜃d𝑟,
𝜋 0 01+𝑦(4.15)

where 𝐴𝑖 (𝑟) and 𝐵(𝑟) are respectively given in (4.4) and (4.5), and22J𝑖 (𝑦, 𝜃, 𝑟, 𝑢) = exp −2𝜆 1 𝐾 𝛽 𝑃ˆ𝑖,21 𝑤 𝑟,𝑢,𝜃
𝑦𝛽
!𝛽 


∫ 𝑃ˆ𝑖,1 𝑟+𝑢
−1
1
𝑥
ˆ
e
𝑥d𝑥 ,
−
𝑅(𝑥, 𝑃𝑖,1𝑟, 𝑢) 1 +𝑦 𝑃ˆ𝑖,1 𝑤 𝑟,𝑢,𝜃
0


∫ ∞ 
∫ ∞
¯
K𝑖 (𝑦, 𝜃, 𝑟, 𝑢) ≈ exp −2𝜋𝜆 𝑝𝑧 1 − exp −𝑚𝑓𝑑 (𝑥 | 𝑧)


0

1
𝑥
× 1+
𝑦 𝑃ˆ𝑖,2 𝑤 𝑟,𝑢,𝜃
with
𝜋 2 2𝜋𝐾𝛽 = csc ,𝛽𝛽(e ℎ, 𝑢) =
𝑅(𝑥,

𝜋,

!𝛽 

0

−1



d𝑥


d𝑧 ,





2
2 +𝑢 2cos−1 −1 ∨ 𝑥 −ℎ
∧1 ,
2𝑥𝑢

(𝑢 = 0)(𝑢 > 0)(4.16)

Proof. Similar to the proof of Proposition 4.1, we have
T (𝑠1 , 𝑠2 , 𝑣) =E[𝑇 (𝑠1 , 𝑠2 , 𝑣)],
𝜅(𝑠1 , 𝑠2 )(4.17)

where 𝑇 (𝑠1 , 𝑠2 , 𝑣) denotes the amount of received data in a certain period,

54and 𝜅(𝑠1 , 𝑠2 ) is given in (4.9). Here, E[𝑇 (𝑠1 , 𝑠2 , 𝑣)] is given by
E[𝑇 (𝑠1 , 𝑠2 , 𝑣)]
2
Õ=P(S𝑖 )E[𝑇 (𝑠1 , 𝑠2 , 𝑣) | S𝑖 ]𝑖=1"
𝑠 𝑖 −12 ∫ ∫Ö
1 Õ 2𝜋 ∞ −𝜆1 𝜋 𝑃¯12,𝑖 𝑟 2 ÕeE Λ(𝑖)
I2,𝑖 (𝑟 | 𝑧)=2𝜋 𝑖=1 0 0
𝑡=0
𝒛∈Ψ 𝑝
#
× E[log(1 + SINR𝒙 𝑘∗ ,𝑡𝑣 (𝑡)) | 𝑟, 𝜃, S𝑖 , Ψ𝑝 ] d𝑟d𝜃.
𝑖

(4.18)



We consider to analyze E[ln 1 + SINR𝒙 𝑘∗ ,𝑢 (𝑡) | 𝑟, 𝜃, S𝑖 , Ψ𝑝 ] for 𝑢 ≥ 0,𝑖𝑡 ∈ N, 𝑖 = 1, 2. Recall that 𝒙 𝑘 𝑖∗ = (𝑟, 𝜃) denotes the nearest point of Φ𝑖 fromthe origin, and let L·,𝑢 denote the Laplace transform of the interference 𝐼·,𝑢conditioned on Ψ𝑝 . Using (7) in [4], the expected data rate is expressed by
the Laplace transform, that is,h iE ln 1 + SINR𝒙 𝑘∗ ,𝑢 (𝑡) | 𝑟, 𝜃, S𝑖 , Ψ𝑝
𝑖
∫ ∞

2
= e−𝑐𝑖 (𝑦)𝜎 LΦ1 ∪Φ2\𝒙 𝑘∗ ,𝑢 𝑐𝑖 (𝑦) d𝑦
𝑖
∫ 0∞


2
= e−𝑐𝑖 (𝑦)𝜎 LΦ1 ∩𝑏(0,𝑃¯1,𝑖 𝑟) 𝑐 ,𝑢 𝑐𝑖 (𝑦) LΦ2 ∩𝑏(0,𝑃¯2,𝑖 𝑟) 𝑐 ,𝑢 𝑐𝑖 (𝑦) d𝑦,(4.19)
0where 𝑐𝑖 (𝑦) = 𝜔𝑟,𝑢,𝜃 (e𝑦 − 1)/𝑃𝑖 , and the last equality follows from the mutual
independence between Φ1 and Φ2 .
Regarding LΦ1 ∩𝑏(0,𝜈) 𝑐 ,𝑢 𝑐𝑖 (𝑦) , 𝜈 ≥ 0, since Φ1 is a homogeneous PPP
with the intensity 𝜆 1 , the analysis result is obtained in the proof of Lemma 1
in [17], which isLΦ1 ∩𝑏(0,𝜈) 𝑐 ,𝑢 (𝑐𝑖 (𝑦))


∫ 𝑢+𝜈
2/𝛽
e
−2𝜆 1 𝐾 𝛽 𝑃1 𝑐 𝑖 (𝑦) 2/𝛽 − 0 𝑅(𝑥,𝑣,𝑢)

=e



𝑥
d𝑥
1+𝑃 − 1 𝑐𝑖 ( 𝑦) − 1 𝑥 𝛽

1(4.20),

e 𝑣, 𝑢) are given in (4.16). As for LΦ2 ∩𝑏(0,𝜈) 𝑐 ,𝑢 𝑐𝑖 (𝑦) , 𝜈 ≥ 0,
where 𝐾 𝛽 and 𝑅(𝑥,since its analysis induces much complexity, we introduce an approximation
similar to Proposition 4.1: we regard interfering BSs as assigned over entire R2 by another independent PPCP Φ′2 whose parent process Ψ′𝑝 and theoffspring process are identical to ones of Φ2 . By this approximation, sinceΦ′2 conditioned on Ψ′𝑝 follows an inhomogeneous PPP with the intensity
55



Í
¯ 𝒛∈Ψ′𝑝 𝑓 (𝒙 | 𝒛) (see Proposition 1.1), LΦ2 ∩𝑏(0,𝜈) 𝑐 ,𝑢 𝑐𝑖 (𝑦) ≈ LΦ2′ ,𝑢 𝑐𝑖 (𝑦) is𝑚given as follows
LΦ2 ∩𝑏(0,𝜈) 𝑐 ,𝑢 𝑐𝑖 (𝑦)




Õ
−𝛽
≈ E exp − 𝑐𝑖 (𝑦)𝑃2 𝐻𝒙,𝑡 ∥𝒙 − 𝒖∥𝑟, 𝜃, S𝑖 , Ψ𝑝
𝒙∈Φ2′


1
𝑟, 𝜃, S𝑖 , Ψ𝑝
=E
−𝛽
1
+
𝑐
(𝑦)𝑃
∥𝒙∥
𝑖
2
′
𝒙∈Φ2


∫ ∞
Ö
𝑓𝑑 (𝑥 | 𝑧)¯d𝑥 ,=exp −𝑚−1 𝑃 −1 𝑥 𝛽
1
+
𝑐
(𝑦)
0
𝑖
′
2
𝑧∈Ψ
Ö

(4.21)

𝑝where the shift-invariance of the parent PP Ψ𝑝′ and the Laplace transformof 𝐻𝒙,𝑡 ∼ exp(1) are applied in the first equality, and the PGFL of an inhomogeneous PPP and Lemma 1 of [7] are applied in the last equality. By
substituting (4.18), (4.19), (4.20), and (4.21) into (4.17), applying the PGFL
and the SPFL to the mutually independent PPs Ψ𝑝 and Ψ𝑝′ , and doing somealgebraic manipulations like the preceding proposition, we obtain (4.15). □4.4

Evaluation of transmission performance

We conduct two numerical experiments referred to as Expt1 and Expt2. See
results of Expt1 in (a) and (b) of Fig.4.1, and results of Expt2 in (c) of
Fig.4.2. The purpose of Expt1 is to evaluate the exact result of H (𝑠1 , 𝑠2 , 𝑣) in(4.3) and the approximated result of T (𝑠1 , 𝑠2 , 𝑣) in (4.15) by comparing withsimulation. The purpose of Expt2 is to compare the two following scenarios.
In Scenario 0, we employ a random walk-based UE with non-HO skipping in
our network model: see Section 4.2.1. We note that the UE is, via performing
HOs, always associated with a BS that provides the highest signal power. In
Scenario 1, a UE model described in Section 4.2.2 is employed in the same
network as Scenario 1. For both Expt1 and Expt2, PPCP Φ2 is defined via
Matern cluster process (MCP) having the cluster radius 𝑟 𝑑 ; see Definition 4of [7], and we respectively set 1.0 (units/km2 ), 5, 0.1 (km), 1000 and 4 as 𝜆 1 ,¯ 𝑟 𝑑 , 𝑃1 /𝑃2 and 𝛽.
𝑚,
Additionally for Expt1, we respectively fix 𝑠1 and 𝑣 to 20 (sec) and 0.02
(km/sec), while regarding 𝑠2 and 𝜆 𝑝 as variables. As we can see in (a) and(b) of Fig.4.1, the horizontal axes represent values of 𝑠2 (sec). The verticalaxes in (a) and (b) respectively express H (𝑠1 , 𝑠2 , 𝑣) (/sec) and T (𝑠1 , 𝑠2 , 𝑣)
(nats/sec/Hz). All the twelve curves are functions of 𝑠2 and 𝜆 𝑝 , where 𝜆 𝑝
is fixed to 5, 10, or 15. In (a) (resp. (b)), the red curves represent the56

([SHFWHG+DQGRYHUUDWHሺȀ ሻ

6PDOOVNLSSLQJSHULRG ‫ݏ‬ଶ ሺ ሻ([SHFWHGGDWDUDWHሺȀ Ȁ ሻ(a) H (𝑠1 , 𝑠2 , 𝑣) as a function of 𝑠2 .

6PDOOVNLSSLQJSHULRG ‫ݏ‬ଶ ሺ ሻ(b) T (𝑠1 , 𝑠2 , 𝑣) as a function of 𝑠2 .
Figure 4.1: (a) and (b) correspond to results of Expt1. In (a) (resp. (b)), we
compare the analysis results of the HO rate (resp. the expected data rate)with the simulation results. The analysis results are expressed by red curvesin (a) and (b). We here fix 𝑠1 to 20.
analytical results of H (𝑠1 , 𝑠2 , 𝑣) (resp. T (𝑠1 , 𝑠2 , 𝑣)) defined via (4.3) (resp.
(4.15)), which are computed via the traditional trapezoidal integral method.
The black curves in (a) (resp. (b)) represent the values of H (𝑠1 , 𝑠2 , 𝑣) (resp.
T (𝑠1 , 𝑠2 , 𝑣)), which are obtained via Monte Carlo simulation with the averageof 10000 independent samples.
On the other hand, as for Expt2, we respectively fix 𝑠1 and 𝑠2 to 20 (sec)and 10 (sec), while regarding 𝑣 (km/sec) and 𝜆 𝑝 as variables. To evaluate
the efficiency of Scenario 𝑗 ∈ {0, 1}, following [17], we introduce the following
utility metric: U 𝑗 (𝑠1 , 𝑠2 , 𝑣) = T𝑗 (𝑠1 , 𝑠2 , 𝑣) − 𝑐H 𝑗 (𝑠1 , 𝑠2 , 𝑣). Here for 𝑗 = 1,
57

8WLOLW\PHWULFሺȀ Ȁ ሻ

9HORFLW\ ‫ ݒ‬ሺȀ ሻ

Figure 4.2: The result of Expt2. Based on the utility metric, we compare
Scenario 0 (non HO skipping) with Scenario 1 (periodic HO skipping). The
black and red curves respectively represent U0 (𝑠1 , 𝑠2 , 𝑣) and U1 (𝑠1 , 𝑠2 , 𝑣).
We also respectively set 20 and 10 as skipping periods 𝑠1 and 𝑠2 in Scenario
2.
T𝑗 (𝑠1 , 𝑠2 , 𝑣) (nats/sec/Hz) and H 𝑗 (𝑠1 , 𝑠2 , 𝑣) (/sec) are respectively defined by(4.15) and (4.3). For 𝑗 = 0, T𝑗 (𝑠1 , 𝑠2 , 𝑣) (resp. H 𝑗 (𝑠1 , 𝑠2 , 𝑣)) is the expectationof total downlink data rate (resp. the expectation of the number of HOs)
per second under Scenario 0. The symbol 𝑐 (nats/Hz) denotes the cost of
HO and is fixed to 5. In Fig.4.2, the horizontal axis represents the value of𝑣. All the six curves are functions of 𝑣 and 𝜆 𝑝 , where 𝜆 𝑝 is fixed to either
10, 20, or 30. The black curves represent the function U0 (𝑠1 , 𝑠2 , 𝑣), whose
values are obtained via the simulation. The red curves express the function
U1 (𝑠1 , 𝑠2 , 𝑣).
In (a) of Fig.4.1, we see that the analytical results of H (𝑠1 , 𝑠2 , 𝑣) well correspond to the simulation results only with some small gaps. The reason of
the gaps mainly come from the instability of the trapezoidal integral method
we applied in the computation. Moreover, we see that H (𝑠1 , 𝑠2 , 𝑣) is increasing in 𝑠2 when 𝑠2 is small. This is because E[𝑁 (𝑠1 , 𝑠2 , 𝑣)], the numerator in(4.10), increases in 𝑠2 , whereas the effect of the increase of 𝜅(𝑠1 , 𝑠2 ), the denominator in (4.10), is comparatively small particularly when 𝑠1 is large (see(4.9)). On the other hand, H (𝑠1 , 𝑠2 , 𝑣) is decreasing in 𝑠2 when 𝑠2 is large.
This is because E[𝑁 (𝑠1 , 𝑠2 , 𝑣)] converges to 1 as 𝑠2 increases, and the effectof the increase becomes comparatively small. In (b) of Fig.4.1, we observethat our analytical results correspond with the simulation results althoughsome gaps exist. The reason of the gaps mainly comes from the effect of the58

approximation introduced in the analysis of T (𝑠1 , 𝑠2 , 𝑣). Furthermore, similar to Fig.4.1(a), we see that T (𝑠1 , 𝑠2 , 𝑣) is increasing in 𝑠2 when 𝑠2 is small,and is decreasing in 𝑠2 when 𝑠2 is large. The reason is similar to the one for
H (𝑠1 , 𝑠2 , 𝑣); that is, the effect of the increase of E[𝑇 (𝑠1 , 𝑠2 , 𝑣)] in (4.17) islarge compared to 𝜅(𝑠1 , 𝑠2 ) when 𝑠2 is small, and the effect gradually becomeslittle.
As for Fig.4.2, we observe that the faster the velocity 𝑣 becomes, thebetter the periodic HO skipping technique outperforms non-HO skipping
counterpart. We furthermore would like to emphasize that there exists (𝑣, 𝜆 𝑝 )such that U1 (𝑠1 , 𝑠2 , 𝑣) > 0 and U1 (𝑠1 , 𝑠2 , 𝑣) > U0 (𝑠1 , 𝑠2 , 𝑣). For examples,(𝑣, 𝜆 𝑝 ) = (0.015, 30) and (0.020, 20). Therefore in some cases, in the PPCPbased network, our proposed HO skipping scheme outperforms the non-HO
skipping scheme for UE.
Remark 4.2. Currently in cellular networks, under a cloud radio access network umbrella, decoupling control plane and user plane is considered to reduce
the HO rate and control burden [51]. In this architecture, UEs associated witha small BS tier can receive data packets from a nearby small BS while being
controlled via a farther macro BS, which enables to mitigate handover delaysfor HOs among small BSs [52]. While the constant 𝑐 is considered for thecost of all kinds of HOs in our model, to consider diﬀerent costs for such
HOs in a small BS tier should be expected, which is left for future work.

4.5

Conclusion

In this study, we focus on a 2-tier heterogeneous cellular network. Here, the
1st tier BSs are deployed as a homogeneous PPP, while the 2nd tier BSs are
deployed as a PPCP. In this network, we consider a random walk-based UEwith periodic HO skipping technique. Then, we analyze the expected HO
rate and data rate by introducing approximations. The analytical results
are respectively shown in (4.3) and (4.15). As we can see in (a) and (b) of
Fig.4.1, the two obtained results approximate the corresponding exact values
well. Moreover, we evaluate the efficiency of the periodic HO skipping technique in the network by comparing it with the non-HO skipping counterpart.
The efficiency is quantized by the utility function. From the comparison result shown in Fig.4.2, we can have a deeper insight under what conditions
the periodic HO skipping scheme in the network outperforms the non-HO
skipping scheme, and also vice versa.

59Chapter 5
Conclusion
5.1

Summary

In this thesis, we have proposed yet another HO skipping scheme, periodic
HO skipping and investigated the performance in the proposed periodic HO
skipping scheme via stochastic geometric analysis.
The studies in Chapter 2 and 3 have been conducted under the modelof single-tier downlink cellular networks where BSs are allocated according
to homogeneous PPP. We derived theoretical expressions of the two metrics:
the expected data rate and the HO rate for a moving UE that performs theperiodic HO skipping. On the basis of these theoretical results, we havestudied transmission performance of a moving UE using the two metrics,
under the model of the periodic HO skipping. From our analysis, we havefound that the transmission performance in the periodic HO skipping scenario
can outperform the non-skipping scenario when the moving speed of a UE issufficiently large. Moreover, we have found that there is a local maximum of
the transmission performance with respect to the skipping period of a UE.
We have also investigated to derive a theoretical expression of the skipping
period that gives the maximum, and conducted the performance comparison
among other sophisticated HO skipping techniques.
The study in Chapter 4 has been conducted under the model of a two-tierheterogeneous cellular network, where the macro BSs and the small BSs are
assumed to be deployed as a homogeneous PPP and a PPCP, respectively.
Based on this network, we consider a random walk-based UE with periodic
HO skipping technique. Then, we separately analyze the expected HO rate
and the expected data rate by introducing certain approximations. Moreover,
we evaluate the efficiency of the periodic HO skipping scheme by comparing
it with the scenario of non-HO skipping, and successfully verify the efficiency

60

of the periodic HO skipping scheme in the two-tier network model.

5.2

Directions of future works

The studies in Chapter 2 and 3 have various points that need extension. For
example, we considered a deterministic length of the skipping period, which
could be enhanced to the random skipping period. Although we focused on
optimizing the performance of a single user, incorporating multiple users inthis model is essential for capacity management and load balancing in cellular
networks. Furthermore, in our experiment, we gave the utility constant 𝑐 > 0
as a hyperparameter, while it could be estimated from the implementationof cellular networks since it represents the cost of an HO.
In Chapter 4, studies of the transmission performance for the UE we
gave are very limited. Especially, the optimal skipping period is not studied.
Evaluation of the optimal skipping period in 2-tier cellular networks is ofprimal interest. Besides, in the experiment of the performance comparison,
we considered a common utility constant in all kinds of HOs occuring in the
2-tier network model. However, to consider distinct utility constants between
intra-tier HOs and inter-tier HOs should be expected. Moreover, the costs
of intra-tier HOs among MBSs and SBSs should also be different (see Remark 4.2). To introduce those different HO costs is expected for future work.
On the other hand, the given analytical results of the performance metrics
have complex forms, so that remedies for mitigating that complexity would
be expected, through such as approximate analysis or heuristic analysis.
For further study, we could consider incorporating the current wirelesstechnologies in 5G, such as the inter-cell interference coordination [53], into
the periodic HO skipping scheme. Moreover, this HO skipping scheme might
contribute to the HO management problem in the unmanned aerial vehicleassisted cellular networks [54] and the beam management problem in the
millimeter-wave cellular networks [55].

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