Deployment of K-tier Aerial Cellular Network
Item Type Article
Authors Wang, Ruibo;Kishk, Mustafa Abdelsalam;Alouini, Mohamed-Slim Citation Wang, R., Kishk, M. A., & Alouini, M.-S. (2023). Resident
Population Density-Inspired Deployment of K-tier Aerial Cellular Network. IEEE Transactions on Wireless Communications, 1–1.
https://doi.org/10.1109/twc.2023.3257222 Eprint version Post-print
DOI 10.1109/twc.2023.3257222
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Resident Population Density-Inspired Deployment of K-tier Aerial Cellular Network
Ruibo Wang, Mustafa A. Kishk, Member, IEEE and Mohamed-Slim Alouini,Fellow, IEEE
Abstract—Using Unmanned Aerial Vehicles (UAVs) to enhance network coverage has proven a variety of benefits compared to terrestrial counterparts. One of the commonly used mathematical tools to model the locations of the UAVs is stochastic geometry (SG). However, in the existing studies, both users and UAVs are often modeled as homogeneous point processes. In this paper, we consider an inhomogeneous Poisson point process (PPP)-based model for the locations of the users that captures the degradation in the density of active users as we move away from the town center. In addition, we propose the deployment of aerial vehicles following the same inhomogeneity of the users to maximize the performance. In addition, a multi-tier network model is also considered to make better use of the rich space resources. Then, the analytical expressions of the coverage probability for a typical user and the total coverage probability are derived. Finally, we optimize the coverage probability with limitations of the total number of UAVs and the minimum local coverage probability.
Finally we give the optimal UAV distribution parameters when the maximum overall coverage probability is reached.
Index Terms—Coverage probability, urban model, multi-tier UAV network, stochastic geometry, inhomogeneous Poisson point process.
I. INTRODUCTION
In the next generation mobile network (5G, beyond 5G), UAVs have many application scenarios [1]–[3], among which UAV-aided ubiquitous coverage becomes an important topic [4]. Because UAVs are easy to deploy, highly mobile, and have 3D deployment, they are often used to build temporary or dy- namic networks and provide ubiquitous coverage. Especially, UAV is widely used to relieve the pressure of large crowds gathering in small areas [4], [5]. They are proven to provide reliable system coverage in hot spots and provide additional system performance [6]. Due to the demand for high rate signals, multi-tier vertical heterogeneous networks (VHetNet) is proposed to make use of space resources in city centers [7], [8].
One of the main unanswered questions in the realm of UAV- enabled wireless networks is where and how high the UAVs should be deployed [9]. The common assumption in SG-based literature is that the user’s spatial distribution is homogeneous.
Consequently, existing literature typically assumes that the density of the UAVs is spatially invariant. However, according to recent studies on resident population densities, a more
Ruibo Wang and Mohamed-Slim Alouini are with King Abdullah Univer- sity of Science and Technology (KAUST), CEMSE division, Thuwal 23955- 6900, Saudi Arabia. Mustafa A. Kishk is with the Department of Electronic Engineering, Maynooth University, Maynooth, W23 F2H6, Ireland. (e-mail:
[email protected]; [email protected]; [email protected]).
proper assumption would be for the density of the users (and consequently the UAVs) drops as their distance from the town center increases [10]. Analyzing the influence of such a setup on the wireless network’s performance is the main objective of this paper. More details on the contributions of this paper are provided later in Sec. I-B.
A. Related Work
SG is a powerful mathematical method of analyzing com- munication networks with irregular topology [11]. Further- more, the SG framework is suitable for modeling and an- alyzing devices in motion, such as UAVs, cars [12], and LEO satellites [13], [14]. The SG-based analytical results of the network coverage probability can provide accurate approximations to the actual network [15], [16]. Next, the authors in [17] proposed an air-to-ground line-of-sight (LoS) probability model suitable for town centers. In this model, the probability of the UAV being blocked by the building decreases with the increase of the elevation angle of the UAV to the typical user. This model divides UAVs into LoS UAVs and non-line-of-sight (NLoS) UAVs. Since the model is related to density, area, and height of building [18], it is suitable for various scenarios. Based on the LoS probability model, there has been some literature on UAV networking in town centers [6], [19].
In the existing research, some resident population density models have been considered. Different user distributions in several urban environments are proposed in [20]. A disjoint clustered model for large resident population density is set up in [21]. A central model is provided in [10] and is adopted in this paper. In the central model, user density decreases with the distance from the user to the center. However, the above articles pay more attention to the modeling of users, while the UAVs are simply deployed. UAVs are deployed as a homogeneous PPP in [10], [21], while the locations of UAVs are determined by clustering in [20]. Therefore, the deployment of UAVs is also worth exploring. However, with regard to analyzing downlink network coverage performance, changing the distribution of UAVs brings much more difficulty in technical derivation than changing the distribution of users.
Given that UAVs form a homogeneous PPP, the downlink coverage performance of users at any location is the same.
Nevertheless, when the density of the UAV is not constant, the distributions of the distance between the serving UAV and the interfering UAV to the user are different for the users at different locations, which makes the analysis challenging.
To effectively utilize the deployable space of UAVs, devel- oping the vertical deployment mode of UAVs is also worth studying, in addition to designing the horizontal distribution of UAVs. Based on the SG framework, authors in [8], [9], [22]–[25] have put forward multi-tier VHetNets consisting of ground base stations (BSs), UAVs and high altitude platforms (HAPs) and low earth orbit (LEO)-satellites. The above re- searches all introduced the concept of association probability to describe the probability of users choosing a communication device in a specific tier (instead of other tiers) to provide services. Unfortunately, the above analytical framework is unsuitable for our study because the UAVs are not uniformly distributed in our paper. Designing a different method to obtain the association probability is another challenge.
B. Contribution
The contributions of this paper can be summarized as follows:
• We study a resident population density-inspired model of the urban area. The density of users decreases with the distance to the town center. UAVs follow a similar distribution to the distribution of users and are deployed at different altitudes with different densities.
• We derive the analytical result of coverage probability under the specific model and prove that it is consistent with the Monte-Carlo simulation. In addition, the existing coverage probability analysis framework is extended to data rate and energy efficiency.
• The coverage performance of multi-tier networks and single-tier networks are compared. We also compare the coverage performance of the population density-inspired distribution and the homogeneous distribution of UAV.
• By adjusting the distribution of UAVs in each tier, we optimize the coverage probability under different user dis- tributions. Furthermore, remarks on the parameter design criterion for UAV distribution are given.
II. SYSTEMMODEL
A. Network Model
User LoS UAV NLoS UAV
𝒙 𝒚
𝒛
ℎ! ℎ!"#
𝑇𝑖𝑒𝑟 𝑘 + 1
……
Typical user Tagged LoS UAV Tagged NLoS UAV
𝑇𝑖𝑒𝑟 𝑘
Fig. 1: Illustration of the system model.
As shown in Fig. 1, we consider a scenario in which ground users are distributed according to an inhomogeneous PPP,
which is inspired by the resident population distribution model proposed in [10]. We assume that the UAVs in the VHetNet are deployed based on the ground users density and, hence, their locations also follow an inhomogeneous PPP. Assuming that the center of the town is located at the origin, the densities of the users and the UAVs near the origin are relatively high, while the density goes down as we move away from the origin.
Assume the users are located at the ground, with horizontal distancezu to the origin, the density distribution of the users Λu(zu)can be represented as follows,
Λu(zu) =λue−βuzu, (1) where λu determines the total density of the plane, βu is a measure of homogeneity. When the value of βu is large, the users are spatially condensed at the origin. Whenβu= 0the process degenerates to a homogeneous PPP. Without loss of generality, we focus on a typical user located on the positive X-axis.
We assume that K tiers of UAVs are distributed at a set of some fixed heights hk independently. Their location distribution of each tier form a 2D inhomogeneous PPP, denoted byΦk=∆{xi,k} where xi,k refers to the 3D location of UAV i in tier k. We prefer polar coordinates (zi, θi, hk) to representxi,k, where zi is the horizontal distance between the UAV and the origin, θi is the angle between the X-axis and the line which connects the projection of the UAV and the origin. Because designers tend to place more UAVs in densely populated areas, it is reasonable to assume that they will be in the same distribution as the users. Thus, in tierk, the density distributionΛUAV,k can be described as,
ΛUAV,k(zi) =λke−βkzi, (2) where λk and βk are parameters of the UAVs in tier k, which have the same meaning as the users’ parameters in the density distribution. Furthermore, we assume that each tier of UAVs have the same transmitting powerρk. The quad Tk={hk, λk, βk, ρk}, k= 1,2, ..., K is used to represent the parameters of thekth tier.
B. Channel Model
To model the air-to-ground channel between a user and a UAV, we need to take into consideration the LoS and NLoS scenarios [17]. Considering a UAV in tierk, given the horizontal distance z between the UAV’s projection on the ground and the user, the probability of setting up an LoS link between the typical user and the UAV is [17], [26],
PkLoS(z) = 1
1 +aexp −b 180π tan−1 hzk
−a, (3) where a andb are environment-dependent parameters. From the perspective of the typical user, the inhomogeous PPP process corresponding to the K-tier UAVs can be split into two disjoint PPPs, that isΦk=ΦLoS,k∪ΦNLoS,k andΦLoS,k∩ ΦNLoS,k = ϕ, where ΦLoS,k and ΦNLoS,k denote the set of
UAVs which establish LoS and NLoS conditions for the typical user respectively,ϕis the empty set.
In this article, UAVs with an LoS link to the typical user are abbreviated as LoS UAVs, while the rest are abbreviated as NLoS UAVs. Therefore, the 3D VHetNet is split into 2K disjoint two-dimensional PPPs, with the density PkQΛUAV,k, where Q = {LoS,NLoS}. After LoS and NLoS states are defined, the channel fading model can be established, which is described by small-scale fading and large-scale fading.
For small scale fading, we denote channel fading power gains in terms of independent random variables GLoS and GNLoS, under LoS and NLoS conditions for the typical user, respectively. In order to represent several fading scenarios, Nakagami-m fading is experienced with shape parameters and scale parameters (mLoS,m1
LoS)and (mNLoS,m1
NLoS)for LoS and NLoS links, respectively. As a result, the probability density functions (PDF) of the power gains GQ is given by [27]
fGQ(g) =mQmQrmQ−1
Γ (mQ) e−mQg, (4) where Γ (mQ) = R∞
0 xmQ−1e−xdx is the Gamma function, Q={LoS,NLoS}.
For large scale fading,ηLoSandηNLoSare mean additional gain for LoS and NLoS transmissions [17], withηLoS> ηNLoS
satisfied. Combining small scale and large scale fading, the received power of the typical user, transmitted by a UAV in tier k, is given by,
Sk(r) =
ηLoSρkGLoSr−αLoS in case of LoS ηNLoSρkGNLoSr−αNLoS in case of NLoS . (5) where ρk is the transmission power of UAVs in tier k, αLoS andαNLoS are path-loss exponents for LoS and NLoS transmissions, withαLoS< αNLoSsatisfied,ris the Euclidean distance between the UAV and the typical user, which is computed by polar coordinates (zi, θi, hk) of the UAV and the horizontal distance zu from the user to the origin
rxi,k= q
(zicosθi−zu)2+ (zisinθi)2+h2k. (6) C. Interference
In each tier, the closest LoS and NLoS UAVs are called tagged UAVs. According to the strongest average received power association strategy [9], the typical user will associate with the UAV with the strongest average received power among these 2K tagged UAVs. Furthermore, denote the lo- cation of the associated UAV asxo. Note that the typical user may not associate with the closest UAV, since in different tiers, the transmitted power of the UAVs is different.
In urban areas where UAVs are densely distributed, it is necessary to consider interference between UAVs. Considering a worst-case scenario, except for the associated UAV, other LoS and NLoS UAVs interfere with the typical user, which we refer to as interfering UAVs. Given that the horizontal distance from the typical user to the origin is zu and the associated UAV is in tier j, the total interference can be expressed as a
function ofrxo, which is the Euclidean distance between the associated UAV and the typical user,
Ij(rxo, zu) =
K
X
k=1
X
xi,k∈ΦLoS,k\{xo}
ηLoSρkGLoSr−αx LoS
i,k
+ X
xi,k∈ΦNLoS,k\{xo}
ηNLoSρkGNLoSr−αx NLoS
i,k
.
(7)
As shown in the above formulation, the value of the total interference is related to zu, rxo and the tier where the associated UAV is located. rxo and the transmission power of tier j determine the received power from the associated UAV.
D. Performance Analysis
Assuming the associated UAV is located in tier j, the instantaneous signal-to-interference plus noise ratio (SINR) at the typical user is given by the following equation,
SINR =
ηLoSρjGLoSr−αxoLoS
Ij(rxo|zu)+σ2 xo∈ΦLoS,k
ηNLoSρjGNLoSrxo−αNLoS
Ij(rxo|zu)+σ2 xo∈ΦNLoS,k
. (8)
whereσ2is the additive white Gaussian noise (AWGN) power, andIj(rxo|zu)is the total interference power.
The reliability of the service provided can be evaluated by the average performance of SINR. In consequence, the coverage probability, which represents the probability that the system can provide reliable connections is defined as the probability that the SINR is greater than a predefined threshold γ:
PC=∆P[SINR> γ]. (9) III. PROBLEMFORMULATION
In this section, our objective is to obtain the analytical expression of coverage probability. From the definition of coverage probability in (9), we know that distinguishing the associated UAV and the interfering UAVs is a prerequisite for computing the coverage probability. Taking an LoS associated UAV for example, the following steps are used to obtain the analytic expression for the coverage probability: (i) derive the PDF of the distance distribution of tagged UAV in each tier k, denoted asfRLoS,k(r, zu), (ii) calculate the probability of the tagged UAV in tier k being associated with the typical user, defined as association probability PLoS,kA (r, zu), with fRLoS,k(r, zu)PLoS,kA (r, zu) being the PDF of the distance between the associated UAV in tier k and the typical user, (iii) for a specific distancerbetween the typical user and its associated UAV, denote the probability that the SINR is greater than the threshold γ as the conditional coverage probability PC(γ, zu|r). The average coverage probability is obtained by taking the expectation of the conditional coverage probability PC(γ, r, zu)with respect to the PDF ofrin step (ii). Steps (i) - (iii) will be explained in Sec. III-B Sec. III-C, and Sec. III-D, respectively.
A. Nearest Interfering UAVs
A clear understanding of the location range of interfering UAVs is necessary when analyzing tagged or associated drone distribution. Obviously, for tierk, the nearest interfering UAV should locate at a distance larger thanhk for the typical user.
Another possible lower bound of the distance for an interfering UAV is to ensure a lower average receiving power than the associated UAV. In the following lemmas, the distance between the typical user and the nearest interfering UAVs is given.
Lemma 1. Given that the typical user is associated with a LoS UAV located at distancerin tierj, the closest interfering LoS and NLoS UAV in tier k are at least at distances dLoS−LoS,j,k(r)and dLoS−NLoS,j,k(r), given by
dLoS−LoS,j,k(r) = max (
hk, ρk
ρj αLoS1
r )
, (10)
dLoS−NLoS,j,k(r)
= max (
hk,
ηNLoSρkE[GNLoS] ηLoSρjE[GLoS]
αNLoS1 r
αLoS αNLoS
) . (11) Proof. According to the received power in (5), the average received power of the associated UAV at distancerin tierj is Sj = ηLoSρjGLoSr−αLoS. The closest interfering LoS UAV in tier k is at least at distances dLoS−LoS,j,k, which can be obtained by solving the equality ηLoSρjGLoSr−αLoS = ηLoSρkGLoSd−αLoS−LoS,j,kLoS . Similarly, for NLoS interfering UAVs, dLoS−NLoS,j,k can be obtained by solving the equality ηLoSρjGLoSr−αLoS =ηNLoSρkGNLoSd−αLoS−NLoS,j,kNLoS .
Lemma 2. Given that the typical user is associated with a NLoS UAV located at distance r in tier j, the closest interfering LoS and NLoS UAV in tier k are at least at distancesdNLoS−LoS,j,k(r)anddNLoS−NLoS,j,k(r), given by
dNLoS−LoS,j,k(r)
= max (
hk,
ηLoSρkE[GLoS] ηNLoSρjE[GNLoS]
α1
LoSr
αNLoS αLoS
)
, (12)
dNLoS−NLoS,j,k(r) = max (
hk, ρk
ρj
αNLoS1 r
)
. (13) Proof. The proof is similar to that of lemma 1.
In the subsequent analysis, the horizontal distance is more practical than the Euclidean distance in this model. The horizontal distances zQ,j,k(r)corresponding to lemma 1 and lemma 2 are defined as
zQ,j,k(r) =q d2
Q,j,k(r)−h2k, (14) whereQ={LoS−LoS,LoS−NLoS,NLoS−LoS,NLoS− NLoS}.
B. Distance Distribution of Tagged UAV
Before deriving the PDF of the distance of associated UAV, obtaining the distance distributions of tagged UAVs is necessary. The distance distributions are given in the following lemmas.
Lemma 3. Given the distance between the typical user and the origin is zu, the CDF of the distance between the tagged LoS UAV in tierk and the typical user is given by,
FRLoS,k(r, zu) = 1
−exp − Z zu+√
r2−h2 k zu−√
r2−h2 k
Z φLoS−LoS(l,r,zu)
−φLoS−LoS(l,r,zu)
vkLoS(zu, l, θ)dθdl
! ,
(15) where
φQ,j,k(l, r, zu) = arccos l2+zu2−zQ,j,k2 (r) 2l zu
!
, (16) vkQ(zu, l, θ) =|l|ΛUAV,k(l)PkQ(du2U(zu, l, θ)), (17) whereQ={LoS−LoS,LoS−NLoS,NLoS−LoS,NLoS− NLoS}, the horizontal distanceszQ,j,k(r)are defined in (14), ΛUAV,k(r)andPkLoS(z)are given in (2) and (3), respectively.
The distance between the potential interfering UAV and the typical userdu2U(zu, l, θ)in (17) is given by,
du2U(zu, l, θ) = q
(zu−lcosθ)2+ (lsinθ)2. (18) Proof. See Appendix A.
Lemma 4. Given the distance between the typical user and the origin iszu, the CDF of distance between the tagged NLoS UAV in tierkand the typical user is given by
FRNLoS,k(r, zu) = 1−
exp −
Zzu+√
r2−h2k
zu−√
r2−h2 k
Z φNLoS−NLoS(l,r,zu)
−φNLoS−NLoS(l,r,zu)
vNLoSk (zu, l, θ)dθdl
! , (19) wherevQk (zu, l, θ)andφQ,j,k(l, r, zu)are given in (17) and (16), respectively.
Proof. The proof is similar to that of Lemma 3, therefore omitted here.
Lemma 5. Given the distance between the typical user and the origin is zu, the PDF of distance between the tagged Q
UAV in tierk and the typical user is given by, fRQ,k(r, zu)
= exp − Z zu+√
r2−h2 k zu−√
r2−h2 k
Z φQ−Q(l,r,zu)
−φQ−Q(l,r,zu)
vQk (zu, l, θ)dθdl
!
×
Z zu+√
r2−h2 k zu−√
r2−h2 k
− 4r1(r > hk)vQk (zu, l, φQ−Q) q
4l2z2u−(l2+zu2−r2+h2k)2 dl
+
Z φQ−Q zu+√
r2−h2k,r,zu
−φQ−Q zu+√
r2−h2k,r,zu
r vQk
zu, zu+p
r2−h2k, θ pr2−h2k dθ
+
Z φQ−Q zu−√
r2−h2k,r,zu
−φQ−Q zu−√
r2−h2k,r,zu
r vQk
zu, zu−p
r2−h2k, θ pr2−h2k dθ
! ,
(20) where 1(r > hk) is an indicator function, its value is 1 when r > hk is satisfied, otherwise 0, vQk (zu, l, θ) and φQ,j,k(l, r, zu) are given in (17) and (16), respectively. For a LoS tagged UAV, Q in (20) is replaced with LoS, while Q is replaced with NLoS for an NLoS tagged UAV.
Proof. See Appendix B.
C. Association Probabilities
Association probability is used to describe the probability that a tagged UAV will eventually be selected as the associated UAV. For the tagged LoS UAV in tier k, there will be no LoS UAVs providing stronger power in tierkthan the tagged UAV, while the NLoS UAVs in tier k may provide stronger average received power, and in other tiers, both LoS and NLoS UAVs may provide stronger power. As a result, the association probabilities are given in the following lemmas.
Lemma 6. Given the distance between the typical user and the origin is zu, for the LoS tagged UAV from tierj at Euclidean distancerfrom the typical user, the probability that the typical user is associated with this specific UAV is given by
PLoS,jA (r, zu)
=
K
Y
k=1,j̸=k
exp −
Z zu+zLoS−LoS,j,k(r) zu−zLoS−LoS,j,k(r)
Z φLoS−LoS(l,r,zu)
−φLoS−LoS(l,r,zu)
vLoSk (zu, l, θ)dθdl
!
×
K
Y
k=1
exp −
Z zu+zLoS−NLoS,j,k(r) zu−zLoS−NLoS,j,k(r)
Z φLoS−NLoS(l,r,zu)
−φLoS−NLoS(l,r,zu)
vNLoSk (zu, l, θ)dθdl
! ,
(21)
where vkQ(zu, l, θ) andφQ,j,k(l, r, zu) are given in (17) and (16), respectively.
Proof. See Appendix C.
Lemma 7. Given the distance between the typical user and the origin iszu, for the NLoS tagged UAV from tierjat Euclidean distancerfrom the typical user, the probability that the typical user is associated with this specific UAV is given by
PNLoS,jA (r, zu)
=
K
Y
k=1
exp −
Z zu+zNLoS−LoS,j,k(r) zu−zNLoS−LoS,j,k(r)
Z φNLoS−LoS(l,r,zu)
−φNLoS−LoS(l,r,zu)
vLoSk (zu, l, θ)dθdl
!
×
K
Y
k=1,j̸=k
exp −
Z zu+zNLoS−NLoS,j,k(r) zu−zNLoS−NLoS,j,k(r)
Z φNLoS−NLoS(l,r,zu)
−φNLoS−NLoS(l,r,zu)
vkNLoS(zu, l, θ)dθdl
! ,
(22)
wherevQk (zu, l, θ)and φQ,j,k(l, r, zu)are given in (17) and (16), respectively.
Proof. The proof is similar to that of Lemma 6, therefore omitted here.
D. Coverage Probability
As an indispensable intermediate result to enable computing coverage probability, the Laplace Transform of interference is given in the following lemma.
Lemma 8. Given that the distance between the typical user and the origin iszu, the Laplace transform of the interference power conditioned on the associated UAV in tier j with Euclidean distancerfrom the typical user is given by,
LIQ1,j(s, r, zu)
=
K
Y
k=1
hLIQ
1−LoS,j,k(s, r, zu)× LIQ
1−NLoS,j,k(s, r, zu)i , (23) whereQ1is replaced with LoS when the typical user is associ- ated with LoS UAV,Q1is replaced with NLoS when NLoS UAV is associated, and LIQ1−Q2,j,k(s, r, zu),Q2 = {LoS,NLoS}
is given by (24) at the top of next page, where wQ2,k(s, r) = 1
−
mQ2
mQ2+sηQ2ρkGQ2(d2u2U(zu, l, θ) +h2k)
−αQ2 2
mQ2 2
,
(25) vkQ(zu, l, θ)andφQ,j,k(l, r, zu)are given in (17) and (16), respectively. lemma that Q1 represents the type of associated UAV, while Q2 represents the type of interfering UAVs.
Proof. See Appendix D.
As the distance distributions of tagged UAVs and the associ- ation probabilities have been derived, we are ready to calculate
LIQ1−Q2,j,k(s, r, zu)
= exp −
Z max{0,zu−zQ1−Q2,j,k(r)}
0
Z π
−π
vQk2(zu, l, θ)wQ2,k(s, zu, l, θ)dθdl
!
×exp − Z +∞
zu+zQ1−Q2,j,k(r)
Z π
−π
vQk2(zu, l, θ)wQ2,k(s, zu, l, θ)dθdl
!
×exp −2
Z zu+zQ1−Q2,j,k(r) zu−zQ1−Q2,j,k(r)
Z π
φQ1−Q2,j,k(l,r,zu)
vQk2(zu, l, θ)wQ2,k(s, zu, l, θ)dθdl
! ,
(24)
the local coverage probability. The definition and derivation of the local coverage probability are given as follows.
Definition 1(Local coverage probability). The local coverage probability PC(zu, γ)is the probability that the SINR of the typical user at distance zu from the origin is greater than threshold γ.
Theorem 1. The exact coverage probability PC(zu, γ) for the typical user is given by,
PC(zu, γ) =
K
X
k=1
Z +∞
hk
fRLoS,k(r, zu)PLoS,kA (r, zu)
×
mLoS−1
X
n=0
(−s)n n!
∂n
∂snLULoS,k(s, r, zu)
s=µLoS,k(r,γ)
dr
+
K
X
k=1
Z +∞
hk
fRNLoS,k(r, zu)PNLoS,kA (r, zu)
×
mNLoS−1
X
n=0
(−s)n n!
∂n
∂snLUNLoS,k(s, r, zu)
s=µNLoS(r,γ)
dr, (26)
LUQ,k(s, r, zu) = exp −σ2s
LIQ,k(s, r, zu), (27) µQ,k(r) =mQγηQ−1ρ−1k rαQ, (28) Q = {LoS,NLoS} in (27) and (28), fRQ,k(r, zu), PLoS,kA (r, zu)andPNLoS,kA (r, zu)are defined in (20), (21) and (22).
Proof. See Appendix E.
Based on the local coverage probability, the definition and derivation of the overall coverage probability are given as follows.
Definition 2 (Overall coverage probability). The overall cov- erage probability is the average coverage probability of all users.
From the definition, the overall coverage probability for the typical user is the normalized expectation of the local coverage probability with regard to zu.
Corollary 1. The overall exact coverage probability with the SINR thresholdγ is given by,
POverallC (γ) = R+∞
0 Λu(zu)PC(zu, γ)zudzu R+∞
0 Λu(zu)zudzu
. (29)
As is shown in (26), higher-order derivatives of the Laplace transform are needed while deriving the exact coverage proba- bility. Because the computational complexity increases rapidly as the order of the derivative increases, the amount of com- putation is not acceptable under large shape parametersmLoS
andmNLoS. Therefore, we provide an approximate evaluation of the coverage probability using the upper bound of the CDF of the Gamma distribution [28].
Theorem 2. The approximate coverage probabilityPeC(zu, γ) for the typical user is given by,
PeC(zu, γ) =
K
X
k=1
Z +∞
hk
fRLoS,k(r, zu)PLoS,kA (r, zu)
×
mLoS
X
n=1
mLoS
n
!
(−1)n+1LULoS,k(n ωLoSµLoS,k(r, γ), r, zu)dr
+
K
X
k=1
Z +∞
hk
fRNLoS,k(r, zu)PNLoS,kA (r, zu)
mNLoS
X
n=1
mNLoS
n
!
(−1)n+1LUNLoS,k(n ωNLoSµNLoS(r, γ),r,zu)dr, (30) where
ωQ = (mQ!)−
1
mQ , Q={LoS,NLoS}, (31) fRQ,k(r, zu), PLoS,kA (r, zu), PNLoS,kA (r, zu) and µQ,k are defined in (20), (21), (22) and (28).
Proof. See Appendix F.
The same as the overall exact coverage probability, the over- all approximate coverage probability is given in the following corollary.
Corollary 2. The overall approximate coverage probability is given by,
PeOverallC (γ) = R+∞
0 Λu(zu)PeC(zu, γ)zudzu R+∞
0 Λu(zu)zudzu
. (32)
IV. NUMERICALRESULTS
In this section, we compare the coverage performance of different systems and optimize the overall coverage probability by changing the distribution of UAVs in different tiers. Refer- ring to [9], [17], [29], we assume the channel parameters as follows: the LoS and NLoS path-loss exponents areαLoS= 2 andαNLoS= 3, the mean additional gains for LoS and NLoS transmissions are ηLoS = 0dB andηNLoS =−20dB, m pa- rameters of Nakagami-m fading for LoS and NLoS UAVs are mLoS= 2 andmNLoS= 1, the noise power isσ2= 10−7W, the parameters for the probability of establishing an LoS link in (3) are a = 4.88 and b = 0.429. The deterministic parameters of users’ distribution in (1) are λu = 10−3m−2 and βu = 5 × 10−3. As a non-homogeneous PPP, the distribution of users can be realized by thinning property [11]
of homogeneous PPP. Finally, we assume three tiers of UAVs are deployed in a small town center square with sides of 5km, at 50,100 and 150 meters height, with the corresponding transmission power 2, 7, and 12 dBm, respectively, and the same value ofλ1=λ2=λ3= 4×10−5m−2.
In Fig. 2, a curve of local coverage probability for the typi- cal user as a function of the distance between the origin and the typical user is plotted. We compare the coverage performance of population density-inspired UAV systems (one-tier and three-tier) with that of the uniformly distributed UAV system.
All of the above systems have the same UAV deployment density on average as 25 UAVs/km2 (i.e., λh = 10−6m−2 and βh = 0 for uniform distribution). The distribution pa- rameters β of the other three systems are shown in the table, λ= 4×10−5m−2 as mentioned above.
0 200 400 600 800 1000 1200 1400 1600
Distance between the Typical User and the Origin [m]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Coverage Probability for Typical User One-tier, 50m
One-tier, 100m One-tier, 150m Three-tier Uniform Distribution
Fig. 2: Local Coverage Probability for the Typical User.
As shown in Table I, different β are chosen to keep the density of the UAVs as1 UAV/km2. Assume the threshold of coverage probability is γ = −15dB. It can be seen that the results of the Monte-Carlo simulation (lines) coincide well with the results of the theoretical analysis (points). With the
same number of UAVs, no matter how far away the typical user is from the origin, the performance of the three-tier network is always better than that of the one-tier networks. As shown in Table I, the three-tier network has significant advantages in terms of overall coverage. At the edge, the advantage of the three-tier network is further expanded, indicating that the lower limit of network coverage or SINR can be guaranteed.
In addition, the coverage performance of all three systems is declining from center to edge due to the reduced density of UAV deployment. Compared with the proposed distribution, the uniform distribution system only has a slight advantage in the edge area, but the overall performance is far inferior to the other four systems. The clustering effect brings the non- uniform distribution advantages over the uniform distribution.
According to Fig. 2, when the user is close to the center, the coverage probability of the UAV network under resident popu- lation density-inspired distribution is significantly greater than that under the uniform distribution. Most users are clustered in the area close to the center, so the proposed distribution has significant advantages in the overall coverage probability.
We study the following optimization problems and record the results in Fig. 3 and Fig. 4. We want to maximize the overall coverage probability by changing the distribution of UAVs, under the premise that the total number of UAVs is limited (the first constraint) and the coverage of users in any location is guaranteed (the second constraint). Therefore, the mathematical representation of the optimization problem is as follows,
arg max
β1,β2
POverallC (γ1)
s.t.
K
X
k=1
Z +∞
0
ΛUAV,k(z) dz≤Nmax, P[SINR≥γ2]≥0.95, ∀zu≥0,
(33)
where the threshold of overall coverage probability is γ1 =
−8dB and the threshold of local coverage for the typical user is γ2 = −20dB. The total number of UAVs is limited to Nmax = 1000, that is, the maximum density of UAV is 40 UAVs/km2. We deploy UAVs at the first the second tiers (h= 50,100 m).
A paradoxical but interesting conclusion in Fig. 3 is, the UAV distribution preferred by the system is a non-uniform one influenced by the resident density distribution, but the optimal distribution is not closely related to the residents numerically.
The optimal overall coverage probability POverallC∗ and the corresponding β1∗ and β2∗ value are marked in the figure. It can be seen that there is a considerable difference among β1∗, β∗2 and βu. For a large β, the first constraint cannot be satisfied due to the high density of UAVs. Under this condition, the coverage probability is set to 0, so the dark blue area at the bottom left appears. When β increases to 10−3, there is a large amount of interference near the centre because the system still tends to be uniformly distributed and the density is relatively high. With the increase ofβ, the overall coverage
TABLE I: Table of Parameters
hk [m] β POverallC Density of UAVs
One-tier 50 3.2×10−3 0.9026 1 UAV/km2
One-tier 100 3.2×10−3 0.9203 1 UAV/km2
One-tier 150 3.2×10−3 0.9367 1 UAV/km2
Three-tier (50,100,150) (4.5,5.8,7.6) ×10−3 0.9713 1 UAV/km2
Uniform Distribution 100 0 0.3845 1 UAV/km2
𝜷𝟏∗= 𝟏. 𝟔×𝟏𝟎#𝟐 𝜷𝟐∗= 𝟏×𝟏𝟎#𝟑 𝑷𝑶𝒗𝒆𝒓𝒂𝒍𝒍𝑪∗ = 𝟎. 𝟖𝟓𝟎𝟑
Fig. 3: Overall coverage probability under different UAV distributions.
probability is improved rapidly. Near the optimal area, the coverage performance of the system is no longer very sensitive to both β1 and β2. This is interesting because in such a tolerant system, we do not have to select an accurate set of optimal parameters to determine the distribution of UAVs, but only to estimate a range. For a large β, the small number of drones are almost all concentrated in the central area, making it difficult for users in the edge area to maintain good communication conditions. Therefore, the second constraint cannot be satisfied, and the dark blue area appears at the top right of the image. Finally, with the same number of UAVs as the optimal distribution, the overall coverage probability of the uniform distribution in the same condition (h= 50,100 m, λ1=λ2= 4×10−6m−2) is only 0.2883, which is much lower than 0.8503.
Although the optimization problem (33) has been carefully studied in Fig. 3, it is still necessary to study the behavior of the system hidden in the dark blue area. Fig. 4 shows the influence of the UAV distribution in a single tier on the overall coverage probability. We observe the special case of β2 = 10−2 in Fig. 3 and broaden the range ofβ1.
First, it is easy to find that the overall coverage probability increases at the beginning and then decreases with the increase of the value of β1. This can simply be explained by the fact that too many UAVs will cause too much interference in the
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Overall Coverage Probability
Doesn't satisfy the second
constraint Doesn't
satisfy the first constraint
Feasible Region
Fig. 4: The influence of the UAV distribution in single tier on the overall coverage probability.
central area, while having too few UAVs will make it difficult for the user to find a close UAV to establish an LoS link. When β1≤10−5, the number of UAVs in the first tier is much larger than that in the second tier, so the overall coverage probability is stable and tends to be similar to that of uniform distribution in the first tier. Whenβ1≥5.6×10−2, the number of UAVs in the first tier is rapidly decreasing, which means there are no available LoS UAVs nearby for some users. It is not hard to predict that the final result will converge to the scenario where only the second tier of UAVs are providing the service.
V. FURTHERREMARKS
A. Analytic Framework Extension
This subsection presents how to extend the existing analysis framework to other scenarios and network models. Enhancing the coverage is one of the application scenarios for UAV networks. UAV networks can also relieve the pressure of insufficient channel capacity in town centers.
Remark 1. Based on Shannon’s theorem and the definition of coverage probability in (9), the channel capacity can be expressed as [30],
P[Blog2(1 + SINR)>R]
=P h
SINR>2RB −1i
=P[SINR>eγ], (34) whereeγ= 2RB −1 is the rate threshold. By replacingeγ into SINR thresholdγ, the local probability and overall probability
that the channel capacity is greater than the rate threshold can be obtained byPeC(zu,γ)e given by (30) andPeOverallC (eγ)given by (32), respectively.
Next, we study green communications in a small hot spot area centered on a base station. Energy efficiency, the number of bits that can be transmitted per unit of energy consumed, is used as a performance metric for green communication.
For convenience, we calculate the energy efficiency as the ratio of the number of bits transmitted per unit time (channel capacity) to the energy consumed per unit time (transmis- sion power). The introduction of the UAV network allows the central base station to reduce its coverage area, thereby reducing transmission power and enhancing energy efficiency.
The following remark illustrates how the coverage probability analytic framework can be applied to the above scenario.
Remark 2. From the definition of energy efficiency, it can be calculated by the ratio of channel capacity to transmission power,
P B
ρk log2(1 + SINR)>E
=P h
SINR>2EρBk −1i
=P[SINR>bγ],
(35)
By replacing γb= 2EBρk −1 into SINR threshold γ, the local probability and overall probability that the energy efficiency is greater than the rate threshold can be obtained byPeC(zu,bγ) given by (30) and PeOverallC (bγ)given by (32), respectively.
Furthermore, there is no need for dense deployment of UAVs near the base station in this case. Fortunately, the network model can be easily extended to the above scenario by adjusting the density distribution of UAVs in (2). Under the premise that the density (whether of the user or the UAV) is only related to the distance to the town center, the analytical framework of this paper is applicable to any distribution model.
B. Distribution Parameter Design
According to the above theorems, it can be seen that the relationship between the coverage probability and the spatial distribution of UAVs is not straightforward, and obtaining the optimal parameters by optimization tools is challenging.
Therefore, the following qualitative criteria for parameters about UAVs’ vertical and horizontal distributions are given.
Notice that all of the remarks have been verified by simulation.
Remark 3. Remarks on altitudesh1, h2, . . . , hK are given as follow.
• In most cases, UAVs have an optimal altitude, and it is better to deploy the UAVs near the optimal altitude.
• While facing a low communication quality, the UAVs are suggested to be distributed at a low altitude so that UAVs are closer to users and users in the LoS region can be covered.
• In a good communication environment, by increasing the deployment altitude, more users can establish LoS links with UAVs, therefore, increasing the coverage probability.
Remark 4. Remarks on the number of tiers K are given as follows.
• With the increase of tiers, more parameters can be optimized so that the coverage performance can be im- proved to some extent. A network with fewer tiers can be considered a special case of a network with more tiers.
However, the improvement in coverage performance is limited when more than three tiers are applied.
• Consider a more general case where the receivers (users) can be divided into M classes according to different gain and demodulation capabilities. Multi- tier distribution has significant advantages over single- tier distribution, and the number of deployment tiers K is recommended to be larger than M. Assuming that the optimal UAV deployment altitude for the re- ceiver of the m-th class is h∗m, the height of UAVs is suggested to satisfy min{h∗1, h∗2, . . . , h∗M} < hk <
max{h∗1, h∗2, . . . , h∗M},∀k.
Remark 5. Remarks on homogeneityβ are given as follows.
• The value of β is related to the strength of interference power relative to noise. When the interference power is significantly stronger than the environmental noise, it is suggested to choose a smallerβ to make the distribution of UAVs more homogeneous and vice versa. Considering that the value of βk will affect the average number of UAVs in hot areas whenβk is changed,λk is adjusted to keep the average number of UAVs unchanged.
• We use the exhaustive search to solve the optimization problem about β given in (33), which results in the calculation complexity increases exponentially with K.
An improved alternate maximization method can be a substitution for the exhaustive search as described in [31]. The complexity of this method isO N K2
, where N is the preset maximum number of rounds. The set of suboptimal parameters is obtained by optimizing from β1, β2, . . . toβK in order. When optimizingβk, theβk is repeatedly reduced by the predefined step size for at most Ntimes, and one of theβin the set{β1, β2, . . . , βk−1}is increased, so that the coverage probability is maximized when the UAV density is unchanged. The optimization of βk ends when the coverage probability no longer increases.
The example in Table II provides further explanation for the above remarks. In Table II, we compare the coverage performance under different numbers of tiers K. The total density and the density of UAVs in each tier are fixed as 1 UAV/km2 and λk = 4×10−5. The set of homogene- ity {β1, β2, . . . , βK} is obtained by alternate maximization method. Overall, increasing the number of tiers allows more parameters to be optimized, thus achieving better coverage
TABLE II: Optimization ofK andβ.
1 UAV/km2 One-tier Two-tier Three-tier Five-tier h1= 50m N/A N/A β1= 4.5×10−3 β1= 5.4×10−3
h2= 75m N/A N/A N/A β2= 6.5×10−3
h3= 100m N/A β3= 4.2×10−3 β3= 5.8×10−3 β3= 7.8×10−3
h4= 125m N/A N/A N/A β4= 8.2×10−3
h5= 150m β5= 3.2×10−3 β5= 5.4×10−3 β5= 7.6×10−3 β5= 9.8×10−3
POverallC 0.9367 0.9557 0.9713 0.9786
performance. The UAV deployment in the one-iter network (β5 = 3.2×10−3) can be regarded as a special case of that of a two-tier network ({β3, β5}={+∞,3.2×10−3}), but it is not optimal. Finally, the gain in coverage probability from deploying more than three tiers of UAV networks is limited.
VI. CONCLUSION ANDFUTUREWORK
In this paper, We studied the coverage performance of multi- tier UAV networks in a centralized urban model. We first de- rived the distance distribution of tagged UAVs and association probability for the selected typical user. Based on this, the analytical expression of downlink coverage probability is given and proved to be consistent with the Monte-Carlo simulation results. As a result, the coverage probability for the typical user and intermediate products are all related to the distance zu. Both the local and total coverage performance are significantly improved by increasing the number of UAV network tiers. The urban population density-inspired model has a huge advantage over the uniform distribution performs. However, too much concentration of UAVs in the central area will bring more noise to the town center and fail to maintain communication for users at the edge. Therefore, how to design the distribution of each tier of UAVs is crucial.
One future research direction is introducing interference and noise mitigation technologies into the framework based on the proposed resident population density-inspired model. In urban areas, the relatively dense deployment of UAVs may cause strong interference. Strong environmental noise in town centers is also one factor limiting the performance of wireless communication. Under the SG framework, orthogonal channel [32] and directional antenna gain [33] can be introduced into the system model respectively to reduce interference and noise power. In addition, we model the users as a PPP, which means that the user’s movement is undirected and random.
Considering that there is a directional flow of people in the town [34], analyzing the coverage probability of the urban system based on SG will be challenging and application- oriented.
APPENDIXA PROOF OFLEMMA3
When the distance between the typical user and the origin is fixed, given that the distance between the tagged LoS UAV
in tierkand userRLoS,k is a random valuable, the Cumulative Distribution Function (CDF) ofRLoS,k is given by
FRLoS,k(r, zu) =P[RLoS,k< r]
= 1−P[RLoS,k> r] = 1−P[N(Ak(r)) = 0]
(a)= 1−exp − Z
Ak(r)
ΛUAV,k(l)ldldθ
! ,
(36)
whereΛUAV,k(l)is defined in (2),N(Ak(r))in (36) counts the number of the UAVs in region Ak(r), which is a circle at the height of hk centered directly above the typical user with radiusp
r2−h2k, and(a)is given by the property of the general PPP [35],
P[N(Ak(r)) =n] = exp − Z
Ak(r)
ΛUAV,k(l)ldldθ
!
× exp
−R
Ak(r)ΛUAV,k(l)ldldθn
n! .
(37)
𝑥 𝑦
𝑧
𝜽𝝋 𝒍
𝒅u2U
𝒛𝑸𝟏%𝑸𝟐,𝒋,𝒌 𝒛𝒖
Fig. 5: Vertical Viewed System Schematic Figure.
wherezu is the horizontal distance from the typical user to the origin,λudetermines the total density of the plane,βu is a measure of homogeneity.
To integrate formulation in (36) over the region ofAk, the area is divided into infinite concentric circular arcs centered at the point which is directly above the origin at the height of