Performance Evaluation of RF-powered IoT in Rural Areas: The Wireless Power Digital Divide
arXiv:2607.25817v1 [cs.NI] 28 Jul 2026
Hao Lin, Student Member, IEEE, Mustafa A. Kishk, Member, IEEE and Mohamed-Slim Alouini, Fellow, IEEE
Abstract—Bridging the digital divide is one of the goals of mobile networks in the future, and further building IoT networks in rural areas is a feasible solution. This paper studies the downlink performance of rural wireless networks, where IoT devices we consider are battery-less and powered only by ambient radio-frequency (RF) signals. We model a rural area as a finite area that is far from the city center. The base stations (BSs) in the whole city and the access points (APs) in the finite network both act as sources of wireless RF signals harvested by IoT devices. We assume that BSs follow an inhomogeneous Poisson Point Process (PPP) with a 2D-Gaussian density, and a fixed number of APs are uniformly distributed inside the finite area following a Binomial Point Process (BPP). The IoT devices we consider can harvest energy and receive downlink signals in each time slot, which is divided into two parts: (1) a charging sub-slot, where the RF signals from BSs and APs are harvested by IoT devices, and (2) a transmission sub-slot, where each IoT device uses the harvested energy to receive and process downlink signals. We consider two main system requirements: minimum energy requirement and signal-to-interference-plus-noise ratio (SINR). Using these two parameters, we investigate the overall coverage probability (OCP) related to them. We first study the effect of remoteness in rural areas on energy harvesting performance. Then we analyze the influence of IoT device’s location and the number of APs on coverage probability when the effect of BSs can be ignored. This paper shows that the IoT devices located inside the rural area can obtain about twice the ECP and OCP of IoT devices located near the edge. For the average downlink performance in rural areas with radii less than 100 m, more than 80% of the RF-powered IoT devices can be supported when there are 100 APs deployed. Index Terms—Stochastic geometry, finite wireless network, energy harvesting, Binomial Point Process, overall coverage probability.
I. I NTRODUCTION Although 5G cellular networks are gradually being implemented and 6G key technologies are also being studied, most people in rural areas still face challenges when using the Internet. Due to the limitation of population density and economic benefits, the communication infrastructure deployed in rural areas around the world has lagged behind that in urban Hao Lin is with the Electrical and Computer Engineering Program, Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia (e-mail: [email protected]). Mustafa A. Kishk is with the Department of Electronic Engineering Program, Maynooth University, Maynooth, W23 F2H6 Ireland (e-mail: [email protected]). Mohamed-Slim Alouini is with the CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia (e-mail: [email protected]).
areas [1], which has caused the digital divide. The digital divide is mainly seen as a challenge, but also as an opportunity for further development of wireless networks. Many methods for understanding and eliminating the digital divide have been proposed, including data-driven policy-making [2], high altitude platform stations [3], mega configurations of low earth orbit (LEO) [4], wind-turbine-mounted base stations (WTBS) [5], and cellular-connected unmanned aerial vehicles (UAVs) toward 6G [6]. It is worth noting that, it is a feasible choice to use Internet of Things (IoT) devices to help build modern agricultural industry technology systems and bridge the digital divide in the whole world. The core concept of the Internet of Things is to equip everyday objects with the functions of identification, sensing, networking, and processing so that people can communicate or interact with each other through the Internet to achieve the goals they expect [7]. The application scenarios of IoT devices include modern medical care, urban transportation, smart home, smart agriculture, and other scenarios with development needs and potential [8]. The effective use of IoT also contributes to the achievement of the United Nations Sustainable Development Goals (SDGs). Unlike the devices that people carry with them every day like mobile phones and tablets, IoT devices may be deployed in some areas hard to be directly reached, such as inside instruments or equipment and remote locations like forests or farms [9], [10]. Their spatial location makes wired charging and battery replacement difficult while harvesting radio frequency (RF) signals becomes a viable option [11], [12]. Authors in [13] summarized contributions of energy harvesting wireless communications (EHWC), including offline/online energy management, joint wireless energy and information transfer, and the energy consumption models. Communication networks with simultaneous wireless information and power transfer (SWIPT) is an important concept where the signal sources also work as energy stations of RF-powered devices. A general receiver operation has been proposed in literature [14], which can split the received signals for energy charging and information decoding, to study the trade-off between the energy demand and the data rate. Stochastic geometry framework is widely used in the performance evaluation of wireless networks. Poisson Point Process (PPP) and Binomial Point Process (BPP) are both used to model the distribution of RF sources and IoT devices, which are suitable in infinite space and finite space, respectively [15]. In this paper, we investigate the performance of RF-powered IoT devices in rural and remote areas while capturing the
influence of the lack of infrastructure in such regions. More details on the contributions are provided in Sec. I-B. A. Related Work Stochastic geometry is an important mathematical tool, which can help us tractably analyze various types of wireless networks without losing much accuracy. Energy harvesting (EH), especially from ambient wireless signals, can become particularly necessary in rural and remote areas when it is impractical to rely on frequent battery replacement. In this article, we mainly study the effect of relying on energy harvesting for IoT on the communication performance in rural areas. Therefore, we categorize relevant literature into i) stochastic geometry for energy harvesting and ii) communications in rural areas. Stochastic geometry for energy harvesting: Energy harvesting wireless networks have many benefits, including increasing the lifetime of devices and reducing reliance on traditional energy sources [16]. For different energy harvesting models, authors in [17] characterized the harvested energy and average achievable downlink rate for SWIPT in cell-free massive multiple-input multiple-output (MIMO). Authors in [18] investigated the application of SWIPT in networks with cooperative non-orthogonal multiple access (CNOMA) and derived the expression of outage probability and throughput based on PPP assumption. In literature [19], joint energy and signal-tointerference-plus-noise ratio (SINR) coverage probability was proven useful for evaluating the performance of RF-powered devices with a time-slotted architecture. In addition, authors in [20] studied the end-to-end outage probability in EH-enabled device-to-device (D2D) cellular networks, and the network performance for cache- and EH-enabled D2D enabled cellular networks was investigated in literature [21]. To maximize coverage and EH probabilities, authors in [22] optimized the altitude of RF-powered aerial base stations serving terrestrial devices based on a stochastic geometry analysis. In addition, the work in [23]–[25] characterized the outage probability and self-sustainability of RF-powered D2D IoT networks. Communications in rural areas: Several concepts and theories about communications in rural areas have been introduced in [26], including information systems and communication networks for agriculture and people in rural areas. The economic, social, educational, and knowledge inequalities between those with information and communication technology (ICT) and those without ICT were defined as digital divide, and many rural communities in the world are seeking solutions to bridge the digital divide [27], [28]. After the 5G standard was proposed, the challenges and opportunities of future rural wireless communications have been discussed, where novel low-cost network enablers (including non-terrestrial networks and drone base stations) were proved feasible to support future applications in rural areas such as smart agriculture [1]. Authors in [29] analyzed the feasibility, architecture, and cost considerations of using TV white spaces (TVWS) to enhance rural Internet access in 5G cellular networks. The work on [30] presented the advantages and challenges of employing TVWS spectrum with 5G enabled high altitude platforms
(HAPs). In [31], renewable energy charging stations were demonstrated which address the limited battery of EH-UAVs. A new stochastic geometry framework was presented in [32] to evaluate the coverage probability in a region including urban and rural areas, and it introduced aerial base stations (ABSs) to compensate for the deficiency in such a UAV-assisted cellular network. Other studies such as [5] proved that it is feasible to use wind turbine-mounted base stations (WTBSs) to enhance rural connectivity. Authors in [33] suggested that artificial intelligence (AI) technologies, especially reinforcement learning (RL) and federated learning (FL), are expected to become the key to space, air, and ground optimization and integration for rural connectivity. However, existing works lack a unified mathematical framework for communication networks in a large-scale city with urban areas and rural areas and the overall performance of RF-powered IoT devices. There are few discussions of the constraints for RF-powered IoT networks in rural areas. B. Contributions The contributions of this paper can be summarized as follows: • System modeling and problem statement: We design a large-scale city model to describe a novel terminology, which is the wireless power digital divide, that refers to the unbalanced capabilities to harvest wireless energy between IoT devices in urban and rural areas. For that purpose, we propose a system model composed of a cellular network with BSs distributed throughout the city according to an inhomogeneous PPP, to capture the reduction in BS density in rural areas. The proposed system also involves a fixed number of APs deployed in a finite area and modeled as a BPP, while the rural area is modeled as a finite area far away from the city center. The proposed system enables mathematically capturing the unfairness in access to wireless power between urban and rural users. • Technical and mathematical challenges: This paper is the first one to compute the distribution of wireless harvested energy with the locations of RF sources distributed as a superposition of an inhomogeneous PPP and a BPP. Given that RF sources are distributed according to different point processes, a level of complexity arises in both energy coverage analysis as well as interference analysis, which are both handled carefully in this paper. In addition, as a result of the assumption of both RF-powered IoT devices and local RF sources to be distributed in a finite area, few novel distance distributions are derived in this paper and properly utilized for the performance analysis of the considered system. • Main findings and revealed insights: In this paper, we first derive the expression of harvested energy, SINR and coverage probabilities. Then, we show how the remoteness of finite areas affects energy harvesting performance, which causes the wireless power digital divide. We also describe the effect of the number of APs and the distance between the IoT device and the rural center on energy harvesting
performance and overall performance. Finally, we prove the existence of APs’ optimal distribution when the IoT devices and APs inside rural areas are further limited in a special case. II. S YSTEM M ODEL When we study the energy harvesting performance and overall performance of each IoT device with energy harvesting, we need to consider these factors: the location or distribution of IoT devices, the remoteness of the finite area, and the distribution of BSs in the whole city and APs in the finite area. In the large city model we introduce in this section, a rural area is considered a finite area far from the city center. We describe the basic system model in Fig.1. A. IoT devices Modeling As introduced in [15], we can define a finite area as A = b(od , rd ) ⊂ R2 which means a circular area centered at od with radius rd . The BSs and APs both act as sources of wireless RF signals harvested by IoT devices. Next, we model the location of the typical IoT device and distribution of IoT devices (working as receivers) using different coordinate systems in two different cases: 1) considering both BSs and APs and 2) only considering APs. The second case is particularly relevant in remote areas with no presence of cellular networks, which is investigated in a more detailed manner. 1) Considering both BSs and APs: If the RF signals from BSs can not be ignored, we assume that the rural center od is on x − axis, i.e. od = (rc , 0) where rc is the distance between od and oc . In the finite area, we assume that x′ − axis is the line that passes through the typical IoT device located at x0 and the rural center od . As shown in Fig.2, the angle between x′ − axis and x − axis is ψ, and the coordinates of the typical IoT device location is x0 = (ζ0 , γ0 ), where p 0 sin ψ . ζ0 = rc2 + v02 + 2rc v0 cos ψ and γ0 = arctan rcv+v cos ψ 0 ′ In general, using x − axis and x − axis, we can obtain the positional relationship between the typical IoT device, rural center and city center when rc , v0 and ψ are known. When the finite area is not far away from the city center, the performance of each IoT device is affected by the cellular network in the whole city (BSs) and the local network in the finite area (APs). IoT devices in different positions have different energy harvesting performance and overall coverage performance, which are relevant to the value of rc , v0 and ψ. 2) Only considering APs: When the finite area is far away from the city center, the effect of BSs on IoT devices can be ignored. Therefore, we only need to consider APs uniformly distributed in the finite area A. The performance of the typical IoT device only depends on the distance to the center of A because A is a circular area and APs are uniformly distributed in it. Such a finite area A far away from the city center is used to model a rural area, whose center is called ‘rural center’ od . If the typical IoT device location is fixed, we still assume that it is on the x′ − axis and x0 = (v0 , 0), where v0 = ∥x0 − od ∥ ∈ [0, rd ] is the distance to the rural center. The CDF of v0 is the
ratio of πv02 and πrd2 , and the PDF is the derivative which can be written as: fV (v0 ) =
2v0 , 0 < v0 < rd . rd2
(1)
In the actual situation, some places in rural areas are not suitable to set up IoT devices or APs, such as rivers or muddy areas. The center and border areas of rural areas are generally more likely to be used for networking, such as residential areas or transportation tracks. So we propose a new distribution called ‘center-edge distribution’, in which the IoT devices are uniformly and randomly distributed in these two areas: i) A1 = b(od , R1 ), a circular area centered at od with radius R1 , and ii) A\A2 = b(od , rd )\b(od , R2 ), a ring area centered at the od with inner radius R2 and outer radius rd , where R2 > R1 . By setting the values of R1 and R2 , we can model finite areas based on different natural environments and planning. The new PDF of the device-to-center distance v0 in this case is expressed as: 2v0 , 0 < v0 < R1 or R2 < v0 < rd , rd2 − R22 + R12 (2) and we use an ordered pair (R1 , R2 ) to represent the range of the IoT device’s distribution in this case. fVCE (v0 ) =
B. Base Stations Modeling In a large city, we model the rural area as a limited space where the IoT devices can not only receive RF signals from APs located in this space but also be affected by signals from BSs throughout the whole city. We use ζj = ∥wj − oc ∥ to represent the distance between the city center oc and a BS located at wj = (ζj , γj ), where ∥ · ∥ means the Euclidean norm. The distance between a BS at wj and the typical IoT device at x0 is q ξj = ζj2 + ζ02 − 2ζj ζ0 cos(γj − γ0 ). (3) As introduced in [32], in order to capture the imbalance of ICTs between urban and rural areas, BSs can be modeled using an inhomogeneous PPP ΦBS ≡ {wj } ⊂ R2 with a ep (ζ) = G(ζ)λp , where 2D-Gaussian density λ ζ2 1 (4) G(ζ) = √ exp − 2 . 2σp σp 2π C. Access Points Modeling In the considered system, we assume that a fixed number of APs form a local network and serve the RF-powered IoT devices. If the wireless network is built in an infinite area, we always assume that all APs follow a PPP which is a basic point process hypothesis. But the rural area is usually finite and the funding to deploy ICTs is limited. Therefore, it is better to model the distribution of APs using a BPP [15], [34], where the APs are i.i.d. in the finite region A and the number of APs is N t . The APs are all active and their locations are denoted
The Gaussian Density of BSs
Urban Area City Center ��
Base Stations (BSs)
Access Points (APs)
Typical IoT Device
Finite Area
Finite Area (Faraway, Rural Area)
Fig. 1. Illustration of the system model. There is a finite area in a large city. A typical IoT device (receiver) is located inside this finite area. A finite area ep (ζ) that is far from the city center is used to model a rural area. The BSs in the whole city follow an inhomogeneous PPP with a 2D-Gaussian density λ and the APs inside the finite area are distributed according to a BPP.
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A Large City with Cellular Networks (BSs) A Finite Area with Local Networks (APs) Fig. 2. Illustration of the mathematical model of networks in the large city and the considered finite area inside it. In the polar coordinate system, od and rd are the center and radius of the finite area, and oc is the center of the whole city. The location of any of BSs is wj = (ζj , γj ) and the location of the typical q IoT device (receiver) is x0 = (ζ0 , γ0 ). The distance between them is ξj = ζj2 + ζ02 − 2ζj ζ0 cos(γj − γ0 ).
as a set ΦAP ≡ {yi } ⊂ R2 . Thus, the PDF of any AP located at yi ∈ ΦAP is 1 , ∥y − o ∥ ≤ r , d d i f (yi ) = (5) πr2 0,d otherwise. For calculations of performance evaluation, we define the distance between AP at yi and the typical IoT device located at x0 as Di .
D. Time-slotted Architecture of IoT Devices In this system, we assume that the RF signals are the only energy source of IoT devices. The APs and IoT devices are assumed to be fully-synchronized, which can be realized through network time protocol (NTP) or mesh time-synchronization protocol (MTP) [35], [36]. These synchronization methods can make the clock accuracy at the millisecond (ms) level, and help the devices switch between charging and transmission modes. All RF signals are required for communication by IoT devices in each given time slot with T seconds. The energy is harvested and provided in the same time slot as introduced in [11], [37]. As shown in Fig.3, we assume that each IoT device adopts a special architecture called the ‘time-slotted architecture’. In this architecture, each time slot is divided into a charging sub-slot where Tch = τ T and a transmission sub-slot where Ttr = (1 − τ )T . The antenna can be used for energy charging or downlink information receiving in their sub-slots, respectively. This system can help us to evaluate the performance of energy harvesting wireless networks in rural areas using two necessary conditions described next. 1) Charging sub-slot: In the charging sub-slot, all BSs and APs act as RF energy sources for IoT devices. The energy harvested in this sub-slot EH should be greater than the minimum energy demand Emin to make the transmission succeed, i.e. EH ≥ Emin , where EH can be expressed as X X EH = τ T η( pAP Gyi Di−α + pBS Gwj ξj−α ), yi ∈ΦAP
wj ∈ΦBS
(6) where η < 1 is the efficiency of the RF-to-DC conversion. The power of signals received by the IoT device from any of APs or BSs can be represented as pAP Gyi Di−α and pBS Gwj ξj−α , where Gyi , Gwj ∼ exp(1) model the Rayleigh fading gains, and Di−α , ξi−α model the standard power law path-loss with
Charging Sub-slot
Transmission Sub-slot
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downlink transmission, the SINR should be larger than the threshold β, i.e. SINR ≥ β. Similar to the harvested energy, when the effect of BSs can be ignored, the interference I1 in (9) is simplified as shown in (10): X pAP Hyi Di−α . (10) I1 =
Downlink Time Slot �
yi ∈ΦAP \y1
Closest AP RF energy
Using the events EH ≥ Emin and SINR ≥ β, we define the coverage probabilities that can help us study the performance of IoT devices. [35], [36], [38]
Data
III. D ISTANCE D ISTRIBUTION = IoT Device (Receiver)
= Access Points (APs)
= Base Stations (BSs)
Fig. 3. Illustration of the IoT devices’ time-slotted architecture. In the charging sub-slot, each IoT device harvests all RF signals from APs and BSs. In the transmission sub-slot, the signal from the closest AP is considered as the information source while signals from other APs and BSs are considered as interference.
In the considered system model, BSs and APs are both the RF energy sources of IoT devices. The harvested energy and interference are both related to the distance between the IoT device and BSs or APs. Because we select the closest AP to be the serving one, it is necessary to analyze the distribution of distances between APs and the typical IoT device. A. Distance Distribution in a Finite Area
exponent α > 2. We assume that the distributions and fading gains of APs and BSs are all independent when the location of the typical IoT device is fixed. Especially, when the finite area is far away from the city center, i.e. rc is much larger than σp , we can ignore the influence from BSs. Therefore, the expression of harvested energy EH can be simplified as X EH = τ T η pAP Gyi Di−α . (7) yi ∈ΦAP
2) Transmission sub-slot: In the transmission sub-slot, we assume that the IoT devices receive the information from their associated serving AP which is chosen from all APs in the finite area A = b(od , rd ), where yl is the location of the chosen AP. Considering both the interference and thermal noise, we achieve the expression of the signal-to-interferenceplus-noise ratio (SINR) from the serving AP located at yl to the IoT device located at x0 : SINR =
pAP Hyl Dl−α . Il + σ 2
(8)
In (8), the interference Il is the summation of signals from APs in ΦAP \yl and all BSs in ΦBS , where ΦAP \yl contains all APs except the chosen one located at yl . Especially, in our assumptions, APs inside the finite area form a local network. In order to obtain effective information in the local network and the largest possible SINR, we assume that the typical IoT receives the information from the closest AP located at y1 and substitute l = 1 into (8) to represent the SINR in this case. Therefore, the interference can be given as follows: X X I1 = pAP Hyi Di−α + pBS Hwj ξj−α , (9) yi ∈ΦAP \y1
wj ∈ΦBS
in which Hyi , Hwj ∼ exp(1) model the Rayleigh fading in this sub-slot and σ 2 represents the thermal noise power in each IoT device’s circuit. If we want to have a successful
When we analyze the distribution of APs, it is not convenient to use the distances between the rural center od and APs directly. To calculate the harvested energy and SINR, we have introduced Di which is the distance between the typical IoT device located at x0 and AP located at yi . In the considered setup, all elements in {Di : Di = ∥yi − x0 ∥} are correlated since they are all functions of the random variable x0 . However, when we condition on x0 , and given that the locations of APs {yi } are i.i.d. , the elements in {Di } can be considered i.i.d. , which helps us simplify the calculations. Because APs follow a BPP, we can calculate the CDF of Di using the intersection of two circular regions: A = b(od , rd ) and a circular area centered at x0 with radius di , which is defined as Bi = b(x0 , di ). The CDF of Di is derived by E [1(yi ∈ A)1(yi ∈ Bi )] . E [1(yi ∈ A)] (11) There are two cases: i) 0 ≤ di ≤ rd −v0 where Bi ⊂ A, and ii) rd − v0 < di ≤ rd + v0 where Bi \A ̸= ∅. Based on this conditional probability, we introduce the distance distribution in a finite area in Lemma 1. FDi (di ) = P(yi ∈ Bi |yi ∈ A) =
Lemma 1. The CDF of the distance between AP located at yi and the typical IoT device located at x0 is FDi,1 (di ), 0 ≤ di ≤ rd − v0 FDi (di ) = , (12) FDi,2 (di ), rd − v0 < di ≤ rd + v0 d2
d2
d
d 2
with FDi,1 (di ) = r2i and FDi,2 (di ) = πri2 (θ∗ − 12 sin 2θ∗ ) + 2
2
di +v0 −rd 1 1 ∗ ∗ ∗ ∗ π (ϕ − 2 sin 2ϕ ), where θ = arccos ( 2v0 di ) and ϕ = v02 +rd2 −d2i arccos ( 2v0 rd ). By taking the derivative of FDi (di ) and using the basic
algebraic manipulations, the PDF fDi (di ) can be expressed as: fDi,1 (di ), 0 ≤ di ≤ rd − v0 fDi (di ) = , (13) fDi,2 (di ), rd − v0 < di ≤ rd + v0
d2 +v 2 −r 2
2di i i 0 d with fDi,1 (di ) = 2d and fDi,2 (di ) = πr 2 arccos ( 2v0 di ). r2 d
d
Similar to the typical IoT device, APs are also constrained by spatial conditions. Therefore, we adopt the ‘center-edge’ distribution to model the distribution of APs and introduce the distance distribution in Corollary 1. Corollary 1. We also adopt the ‘center-edge distribution’ to further limit the distribution of APs. The APs are distributed uniformly and randomly in these two areas: i) A3 = b(od , R3 ), a ring area centered at od with radius R3 , and ii) A\A4 = b(od , rd )\b(od , R4 ), a circular area centered at od with inner radius R4 and outer radius rd , where R4 > R3 . We use another ordered pair (R3 , R4 ) to represent the distribution range of APs. We use D to simply represent the independent variable Di . The new PDF of Di is shown as: rd2 A4 A3 (f A (d) − fD (d) + fD (d)), (14) rd2 − R42 + R32 D and the new CDF of D is: CE fD (d) =
CE FD (d) =
rd2 A4 A3 (F A (d)−FD (d)+FD (d)), (15) rd2 − R42 + R32 D
AK AK (d) next in Lemma (d) and FD where we introduce the fD 2.
After adopting the ‘center-edge’ distribution of APs, the spatial area is divided into several parts. In Lemma 2, we simplify the expressions of updated PDF and CDF of Di in different domains. Lemma 2. Considering the positional relationship between AK = b(od , RK )(K = 3, 4) and B = b(x0 , d), we divide the entire domain of rural area into six parts shown in Fig.4: 0 < d < v0 − RK IK : IIK : v0 − RK < d < v0 + RK , if 0 < RK < v0 < rd , IIIK : v0 + RK < d < v0 + rd (16) and 0 < d < RK − v 0 IVK : VK : RK − v0 < d < RK + v0 , if 0 < v0 < RK < rd . VIK : RK + v0 < d < rd + v0 (17) AK The FD (d) can be expressed as: IK : 0 II , V : F(d, RK ) K K 2 RK AK IIIK , VIK : , (18) FD (d) = πrd2 2 d IVK : πrd2 where d2 1 R2 1 F(d, RK ) = 2 (θK − sin 2θK ) + K2 (ϕK − sin 2ϕK ), πrd 2 πrd 2 (19) and 2 2 d + v02 − RK θK = arccos 2 2v02d . (20) v + RK − d 2 ϕK = arccos 0 2v0 RK
Fig. 4. Illustration of distance distribution introduced in Lemma 2.
AK (d) is: The fD
IK , IIIK , VIK : IIK , VK : AK fD (d) = IVK :
0 2d θK . πrd2 2d rd2
(21)
A A Remark 1. The expressions of FD (d) and fD (d) are the same as (12) and (13). Especially, F(d, rd ) = FDi,2 (d) in Lemma 1.
B. Conditional Distance Distribution in a Finite Area When we analyze the coverage performance, we define the distance between the closest AP to the typical IoT device as a random variable R. If the APs’ distribution and location of the typical IoT device are known, R = r = ∥x0 − y1 ∥, where y1 is the location of the closest AP. In a finite area, we have introduced the BPP to model the distribution of APs, and some studies like [15] have introduced the distribution of the k th closest AP for a typical IoT device. The PDF of the distance between the k th closest AP and the typical IoT device can be expressed as: (k)
t
t
N ! k−1 fDi (r)(1 − FDi (r))N −k . fR (r)= (k−1)!(N t −k)! FDi (r) (22)
To achieve the distribution of R, we substitute k = 1 into (22) and achieve the distribution of the distance between the closest AP and the typical IoT device in Lemma 3. Lemma 3. The PDF of the distance between the typical IoT device located at x0 and its closest AP located at y1 is: t
fR (r) = N t fDi (r)(1 − FDi (r))N −1 ,
(23)
where fDi (di ) and FDi (di ) are given in (13) and (12), respectively.
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where d− = rd − v0 and d+ = rd + v0 . FDi (d) and fDi (d) are given in (12) and (13). Proof: Because the condition is R = r, we can use the Bayesian Probability Formula to achieve the conditional probability easily. Similar to the concept in [15], we consider the situation that all APs are outside the circle centered at x0 with radius r because we choose the closest AP for downlink transmission. Similar to Corollary 1 and Corollary 2, the Corollary 3 describes the expression of new PDF of Di conditioned on R = r. Corollary 3. Following the assumptions in Corollary 1, 2 and Lemma 4, the APs are distributed following the ‘center-edge distribution’. In this case, fU (u) can be expressed as:
�
fUCE (u) =
��
(26)
CE CE (d) are given in (14) and (15). (d) and FD where fD
IV. C OVERAGE P ROBABILITY A NALYSIS
Typical IoT Device (Receiver) The Closest AP (Serving AP) Other APs (Interfering APs) Fig. 5. Illustration of the BPP model and the distance distribution. Because we assume that the typical IoT device can receive the signal from the closest AP, other APs are regarded as interfering APs. Lemma 3 introduces the distribution of the distance between the typical IoT device and the closest AP, and Lemma 4 introduces the distribution of Di conditioned on R = r.
We also update the PDF of R when adopting the ‘centeredge’ distribution in Corollary 2. Corollary 2. When we adopt the assumption in Corollary 1, the new PDF of the distance between the typical IoT device located at x0 and its closest AP located at y1 is t
CE CE fRCE (r) = N t fD (r)(1 − FD (r))N −1 ,
(24)
CE CE where the fD (d) and FD (d) are shown in (14) and (15).
To simplify the calculation in coverage probability analysis, we need to define a new distribution of Di conditioned on R = r using a new variant U . The conditional PDF of Di is expressed as fU (u) in Lemma 4.
In our considered system, we assume that APs follow a BPP ΦAP and BSs follow an inhomogeneous PPP ΦBS with a 2D-Gaussian density. There are two necessary conditions to achieve device coverage: energy supply and SINR requirement. In this section, we first define the energy coverage probability (ECP) and transmission coverage probability (TCP). After that, we derive the expression for the overall coverage probability (OCP) Pcov using the conditional energy and transmission coverage probabilities.
A. Energy Coverage Probability To activate its circuit, the typical IoT device needs to harvest enough energy during the charging sub-slot. In order to analyze the energy harvesting performance of IoT devices, we introduce the energy coverage probability in Theorem 1. Theorem 1 (Energy Coverage Probability, ECP). Conditioned on the location of the typical IoT device and R = r, the probability of the event that EH ≥ Emin is: P(EH ≥Emin |r, v0 , ψ)
Lemma 4 (Conditional Distance Distribution in a Finite Area). The PDF of Di conditioned on R = r is fDi (u) fU (u) = 1 − FDi (r) fDi,1 (u) , 1 − FDi,1 (r) fDi,2 (u) , = 1 − FDi,1 (r) fDi,2 (u) , 1 − FDi,2 (r)
CE fD (d) , CE (r) 1 − FD
= exp − rα [C(τ ) − ΩAP (r) −
pBS ΩBS ]+ , pAP
where −
0<r<d ,
r<u<d
−
ΩAP (r) = (N t − 1) 0 < r < d− ,
d− < u < d+ ,
d− < r < d+ ,
r < u < d+
Z rd +v0
u−α fU (u)du,
r
and Z 2π Z ∞ (25)
ΩBS = 0
0
ep (ζ)ξ(ζ, γ)−α ζdζdγ. λ
(27)
TABLE I TABLE OF N OTATIONS
Notation ΦAP ; yi ; N t ΦBS ; wj λp ; σ p x0 ; y1 od ; oc rd ; rc Gyi ; Gwj Hyi ; Hwj pAP ; pBS τ; T; η σ 2 ; α; β R 1 ; R2 R 3 ; R4
Description BPP modeling the locations of APs; location of any AP; the number of all APs Inhomogeneous PPP modeling the locations of BSs; location of any BS Parameters in the Gaussian density of the inhomogeneous PPP ΦBS Location of the typical IoT device; location of the closest AP to the typical IoT device Location of the center of the finite area; location of the city center Radius of the finite area; distance between od and oc Rayleigh fading gains in the charging sub-slot Rayleigh fading gains in the transmission sub-slot Uniform power of APs; uniform power of BSs Time slot division parameter for charging sub-slot; length of each time slot; efficiency of the RF-to-DC conversion Power of thermal noise; path loss exponent (α > 2); SINR threshold for a successful transmission Parameters used to limit the distribution of IoT devices in rural areas Parameters used to limit the distribution of APs in rural areas
min and fU (u) is shown in In addition, C(τ ) = τ TEηp AP (25). The expression of energy coverage probability is the expectation of conditional ECP over r.
P(EH ≥ Emin |v0 , ψ) = ER [P(EH ≥ Emin |R, v0 , ψ)] i h pBS α + = ER exp −R [C(τ ) − ΩAP (R) − ΩBS ] pAP Z S = fR (r)dr Z 0rd +v0 pBS ΩBS ] fR (r)dr. + exp − rα [C(τ ) − ΩAP (r) − pAP S (28) where fR (r) is shown in (23), S is the threshold satisfying pBS C(τ ) − ΩAP (S) − ΩBS = 0. pAP Proof: See Appendix A. When the remoteness of the finite area increases, the BS density decreases and such a finite area can be used to model a rural area. For most rural areas far from the city center, it is an acceptable assumption to ignore the effect of BSs. In this case, the energy coverage probability can be simplified as shown in Corollary 4. Corollary 4. If the RF signals from BSs can be ignored, the conditional ECP can be written as: P(EH ≥ Emin |r, v0 ) = exp −rα [C(τ ) − ΩAP (r)]+ , (29) and the ECP can be written as: P(EH ≥ Emin |v0 )= ER [P(EH ≥ Emin |R, v0 )].
(30)
Remark 2. When the location of the typical IoT device is fixed, the energy harvesting performance can be analyzed using P(EH ≥ Emin |v0 ). But when it is uniformly and randomly distributed in the rural area, the energy harvesting performance should be evaluated using the expectation EV [P(EH ≥ Emin |V )].
ability can be expressed using the Laplace transform of the interference of APs and BSs as shown in Lemma 5. Lemma 5 (Transmission Coverage Probability, TCP). Conditioned on R = r, the probability of the event SINR ≥ β is: P(SINR ≥ β|r, v0 , ψ) βr α σ 2 pBS α α = exp − pAP LAP (βr )LBS pAP βr ,
(31)
where Z 2π Z ∞ e LBS (s) = exp − λ(ζ)(1 − 0
0
1 )ζdζdγ 1 + sξ(ζ, γ)−α (32)
and Z rd +v0 LAP (s) = 0
N t −1 1 f (u)du , U 1 + su−α
(33)
where fU (u) is given in (25) and fR (r) is given in (23). Proof: See Appendix B. Transmission coverage probability is a vital tool to calculate the overall coverage probability. We also simplify the expression of TCP in Corollary 5 when the interference of BSs can be ignored. Corollary 5. If the effect of BSs can be ignored, the interference can be expressed as shown in (10). In this situation, the conditional TCP can be written as: βrα σ 2 P(SINR ≥ β|r, v0 ) = exp − LAP (βrα ). (34) pAP
C. Overall Coverage Probability
B. Transmission Coverage Probability
We have obtained the expressions of ECP and TCP conditioned on R = r, but our final goal is to achieve the expression of overall coverage probability. Therefore, we introduce the overall coverage probability in Theorem 2.
Conditioned on the distance between the typical IoT device and its closest AP R = r, the transmission coverage prob-
Theorem 2 (Overall Coverage Probability, OCP). The overall coverage probability can be derived using the conditional
energy coverage probability and transmission coverage probability. It can be expressed as: P cov (β|v0 , ψ) Z rd +v0 pBS α + exp −r [C(τ ) − ΩAP (r) − = ΩBS ] pAP 0 α 2 βr σ pBS α × exp − LAP (βrα )LBS βr fR (r)dr, pAP pAP (35) + min , [x] = max{0, x}. The Ω (r), Ω where C(τ ) = τ TEηp AP BS AP and LAP (s), LBS (s) are defined in Theorem 1 and Lemma 5, while fR (r) and fU (u) are given in (23) and (25). Proof: See Appendix C. As the combination of ECP and TCP, the expression of overall coverage probability can be simplified when ignoring RF signals from BSs in Corollary 6. Corollary 6. If the RF signals from BSs can be ignored, the overall coverage probability can be written as: Z rd +v0 Pcov (β|v0 )= exp −rα [C(τ ) − ΩAP (r)]+ 0 (36) βrα σ 2 LAP (βrα )fR (r)dr. × exp − pAP Remark 3. When the location of the typical IoT device is fixed, the overall performance can be analyzed using Pcov (β|v0 ). But when the IoT devices are uniformly and randomly distributed inside the finite area, the overall coverage performance should be evaluated using EV [Pcov (β|V )]. V. S IMULATION R ESULTS AND D ISCUSSION We first consider both the signals from APs and BSs, and prove that the remoteness of the finite area can affect the energy harvesting performance of IoT devices inside the finite wireless network. We set the radius of the finite area as rc = 2 km and there are 2 × 104 APs in it. The parameters of the Gaussian BS density are set as λp = 2 BSs/km2 and σp = 2 km. Other parameters in the simulation are set as follows: α = 3, τ = 0.2, T = 1 s, pAP = 1 W, pBS = 10 W(1 dBW), η = 0.5, and Emin = 10−4 J. The parameters we choose have been verified in the literature [19], [32], [37], [39], [40], respectively. As shown in Fig.6, if rc is much less than σp , the IoT devices inside the finite area have a uniform and high energy supply. When rc is close to σp , the IoT devices closer to the city center can have better energy coverage than those farther from the city center. As the finite area moves away from the city center, the energy coverage probability decreases, When rc is much larger than σp , the IoT devices also have a uniform energy supply but lower than in the previous case. Such a finite area can be used to model a communication network in rural area. Firstly, we analyze the performance of a typical IoT device inside a rural area. We consider a small rural area and reset the radius of rural area rd = 100 m, while other parameters about transmission are set as: β = 0.1(−10 dB) and σ 2 = 10−9 . We simulate the energy coverage probability and overall coverage probability to observe their relation to v0 and N t . Fig.7(a) and
Fig.7(b) show the theoretical analysis and simulation results of ECP and OCP when we choose different values of N t . When the typical IoT device is located inside the finite space (v0 < 80 m), ECP and OCP are both generally stable. This shows that APs beyond a certain distance only have a small impact on IoT devices. When the AP density in the area near IoT devices is uniform, their performance becomes more stable. But when it is close to the edge of the finite space, the ECP is lower. Both ECP and OCP generally decrease when the distance between the IoT device and the rural center increases, which is because the IoT device near the edge can not harvest enough energy to support its operation. Secondly, we evaluate the average performance of IoT devices inside a rural area. In Fig.8(a) and Fig.8(b), we plot the theoretical analysis curves of the expectation of ECP and OCP over v0 , which are mentioned in Remark 2 and Remark 3. We compare them to the simulation results. When we set up enough APs to provide energy to IoT devices, the ECP approaches 1. When N t is small, the OCP has a similar trend to the ECP, but it does not approach 1 when N t is large due to high interference. Since the number of APs N t has different effects on ECP and TCP, when we use OCP as the evaluation standard, we can choose the optimal N t to achieve the tradeoff. For example, when rd = 20 m, the optimal number of APs is 10. But when rd = 50 m, the optimal N t is 60. As introduced in Corollary 1, we adopt the ‘center-edge distribution’ when we limit the distribution area of IoT devices and APs. When we change the distribution range of the IoT devices, the optimal location of (R3 , R4 ) is changed at the same time. Fig.9 is an example when (R1 , R2 ) = (30, 60). Fig.10 shows the theoretical analysis and simulation result of the optimal edge of APs’ distribution in this case. The optimal value of R3 is less than R1 , and the optimal value of R4 is larger than R2 . That means the optimal distribution range of APs satisfies A3 ⊂ A1 and A\A4 ⊂ A\A2 in our simulation. In Fig.11, the grid color corresponding to the better ordered pair (R3 , R4 ) is closer to dark red. When we fix the value of R1 and increase the value of R2 , the optimal R3 is not greatly affected, but the value of the optimal R4 increases. Similarly, the optimal R4 is larger when R2 increases. In the considered system, all APs and IoT devices are synchronized. However, in realistic applications, the synchronization issues may result in loss of wireless energy, signal power, and interference, which complicates the performance analysis. Therefore, synchronization challenges and their impact on rural wireless networks are expected to be further analyzed. In addition, the real-world constraints of IoT devices and APs are complex, which makes the performance analysis more difficult. Electromagnetic environment simulation platforms can help analyze the coverage performance. However, the cost of building a virtual environment for each different rural area is high. Therefore, although our proposed system is an idealistic model, it is still a tractable and convenient solution to analyze the coverage analysis of RF-powered IoT networks in rural areas before deployment. VI. C ONCLUSION In this paper, we build a mathematical model to analyze
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Fig. 6. The figure shows the effect of remoteness of finite areas on energy harvesting performance. When rc = 0 km, the energy harvesting performance is the highest. When rc is close to σp , the IoT devices closer to the city center have better energy harvesting performance than the devices further away from the city center. When rc ≫ σp , the energy harvesting performance is the lowest.
(a) Energy coverage probability derived in Theorem 1.
(a) The expectation of energy coverage probability over v0 mentioned in Remark 2.
(b) Overall coverage probability derived in Theorem 2. Fig. 7. Theoretical analysis and simulation results of ECP and OCP conditioned on v0 .
the downlink performance of RF-powered IoT devices in rural areas. We use an inhomogeneous PPP to describe the difference in ICT density between the urban area and rural area in a large-scale city. At the same time, we use a BPP to model the distribution of a fixed number of APs inside a considered finite area. We focus on the performance of battery-less IoT devices inside the finite area, which
(b) The expectation of overall coverage probability over v0 mentioned in Remark 3. Fig. 8. Theoretical analysis of ECP and OCP compared to the simulation results conditioned on N t .
harvest and process RF signals in different sub-slots. The energy coverage, transmission coverage, and overall coverage probability are defined to analyze the performance of IoT devices. We show that due to the effect of BSs in the whole city, IoT devices closer to the city center have higher energy
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Fig. 11. The optimal choice for (R3 , R4 ) when (R1 , R2 ) have different choices and N t = 10 (simulation).
shown in (6), we can derive the conditional energy coverage probability:
P(EH≥ Emin |r, v0 , ψ) X = P τ T η( pAP Gyi Di−α yi ∈ΦAP
X
+
pBS Gwj ξj−α ) ≥ Emin r, v0 , ψ
wj ∈ΦBS X −α = P Gy1 D1 +
Gyi Di−α
yi ∈ΦAP \y1
pBS + pAP Fig. 10. The theoretical analysis and simulation result of the optimal edge of APs’ distribution when (R1 , R2 ) = (30, 60) and N t = 10.
harvesting performance than those far away from the city center. This means that unbalanced BS density causes the wireless power digital divide. Because rural areas can be modeled as a finite area far from the city center, the effect (including energy supply and interference) from BSs can be ignored. We prove that the IoT devices closer to the center of the finite area can work better than those towards the edge. We propose the ‘center-edge distribution’ to model the deployment constraints of IoT devices and APs in rural areas. Based on this, we prove that an optimal APs’ distribution exists when the IoT devices’ distribution is constrained.
X
Gwj ξj−α ≥ C(τ ) r, v0 , ψ
wj ∈ΦBS
pBS ΩBS ≥ C(τ ) r, v0 , ψ , ≈ P Gy1 r + ΩAP (r) + pAP (37) min where C(τ ) = τ TEηp . In step (a), we define Ω (r) as AP AP a conditional expectation of fading gains of APs except the closest one, and it can be derived as: (a)
−α
X
ΩAP (r) = E
Gyi Di−α r, v0 , ψ
y ∈ΦAP \y1
i (b)
= E
(c)
X
Di−α |r, v0
yi ∈ΦAP \y1 Z rd +v0 t
= (N − 1)
A PPENDIX A P ROOF OF T HEOREM 1 In the considered system, we assume that r = ∥x0 − y1 ∥ ∈ (0, rd + v0 ), which is the distance between the closest AP and the typical IoT device. Fig.2 shows the illustration of BSs’ distribution. Using the expression for harvested energy EH
r
(38)
1 fU (u)du, uα
where step (b) comes from Gyi ∼ exp(1), and step (c) follows the assumption that the APs are i.i.d. in our system. Because BSs follow an inhomogeneous PPP with a 2D-Gaussian denep (ζ) and Gw ∼ exp(1) models the Rayleigh fading, sity λ j
ΩBS can be calculated as: X ΩBS = E Gwj ξj−α |r, v0 , ψ ∈ΦBS wjX =E ξj−α |v0 , ψ
h i+ . (44) P(EH ≥ Emin |r, v0 ) = exp −rα C(τ )−ΩAP (r) (39)
w ∈Φ
Z 2π jZ ∞BS ep (ζ)ξ(ζ, γ)−α ζdζdγ. λ = 0
0
Because Gy1 ∼ exp(1), the expression of the energy coverage probability can be derived as follows: P(EH≥ Emin |r, v0 ) h i pBS α = P Gy1 ≥ r C(τ ) − ΩAP (r) − ΩBS r, v0 pAP h i+ pBS (d) α = exp − r C(τ ) − ΩAP (r) − ΩBS , pAP
A PPENDIX B P ROOF OF L EMMA 5 The overall coverage probability, based on the distribution of the closest distance to APs, is conditioned on the harvested energy and SINR. To analyze it, we need to derive the expression of the transmission coverage probability conditioned on R = r, which is written as P(SINR ≥ β|r, v0 ). With this condition, the power of the signal from the closest AP received by the typical IoT device is pAP Hy1 r−α , and the summation of those from other APs and BSs can be written as (9). The conditional transmission coverage probability can be derived as follows: pAP Hy1 r−α ≥ β|r, v , ψ P (SINR ≥ β|r, v0 , ψ) = P 0 I1 + σ 2 α 2 X βr σ = P Hy1 ≥ Hyi Di−α + βrα pAP yi ∈ΦAP \y1 pBS α X −α + βr Hwj ξj r, v0 , ψ pAP wj ∈ΦBS βrα σ 2 = exp − p AP X ×E exp − βrα Hyi Di−α r, v0 , ψ
(40) where step (d) comes from the CCDF of exponential distribution of Gy1 , and [x]+ = max{0, x} in this step. Since ΩAP (r) is a monotonically decreasing and continuous function of r where ΩAP (r) ∈ (0, +∞), there is only one solution r = S for the equation C(τ ) − ΩAP (r) − ΩBS = 0. Thus, BS ΩBS ]+ can be written as: [C(τ ) − ΩAP (r) − ppAP h i+ pBS C(τ ) − ΩAP (r) − ΩBS pAP ( pBS ΩBS , S < r < rd + v0 C(τ ) − ΩAP (r) − pAP = . yi ∈ΦAP \y1 p 0, 0<r≤S X BS −α α (41) ×E exp − βr Hwj ξi r, v0 , ψ . pAP wj ∈ΦBS We can calculate the expectation of energy coverage prob(45) ability using fR (r): We define the Laplace transform of fading gains from other APs conditioned on R = r as: P(EH ≥ Emin |v0 , ψ) = ER P(EH ≥ Emin |R, v0 , ψ) h i+ pBS X = ER exp − Rα C(τ ) − ΩAP (R) − ΩBS −α r, v0 , ψ LAP (s) = E exp − s Hyi Di pAP Z rd +v0 i+ h y ∈Φ \y p AP i 1 BS ΩBS = exp −rα C(τ ) − ΩAP (r) − fR (r)dr Y −α p AP 0 = E r, v exp − sH D H 0 yi i Z S AP \y1 = fR (r)dr yi ∈Φ Y 1 (f ) Z 0rd +v0 i h = E r, v0 pBS α 1 + sDi−α ΩBS fR (r)dr. + exp − r C(τ ) − ΩAP (r) − yi ∈ΦAP \y1 pAP Z rd +v0 N t −1 S (42) 1 (g) = fU (u)du , If the rural area is far from the city center, the effect of BSs 1 + su−α 0 (46) can be ignored, which means the summation of BSs’ fading gains ΩBS approach 0. In this case, the conditional ECP can be where step (f) comes from Hyi ∼ exp(1), and step (g) follows simplified as P(EH ≥ Emin |r, v0 ) using the harvested energy from the conditional PDF of Di . Differently, the Laplace transform of relative fading gains from BSs is: in (7): X −α P(EH≥ Emin |r, v0 ) LBS (s) = E exp − s Hwj ξi r, v0 , ψ P = P τTη pAP Gyi Di−α ≥ Emin Y wj ∈ΦBS yi ∈ΦAP = EH exp − sHwj ξi−α v0 , ψ X −α −α (43) = P Gy1 D1 + Gyi Di ≥ C(τ ) j ∈ΦBS wY 1 y ∈Φ \y AP i 1 v , ψ = E exp 0 (e) 1 + sξi−α ≈ P Gy1 r−α + ΩAP (r) ≥ C(τ ) , wj ∈ΦBS Z 2π Z ∞ 1 (h) e = exp − λ(ζ) 1 − ζdζdγ , where step (e) is similar to step (a). In this case, the conditional 1 + sξ(ζ, γ)−α 0 0 ECP is: (47)
where step (h) is based on the PGFL of the inhomogeneous PPP. Thus, the conditional TCP is P(SINR≥ β|r, v0 , ψ) p βrα σ 2 BS = exp − LAP (βrα )LBS βrα . pAP pAP
(48)
When the BSs’ interference can be ignored, the interference can be simplified as (10), and the conditional TCP is simplified as follows:
Similarly, when the effect of BSs can be ignored, the OCP is simplified as: Pcov (β|v0 ) Z rd +v0 = P(SINR ≥ β|r, v0 )P(EH ≥ Emin |r, v0 )fR (r)dr Z0 rd +v0 = exp −rα [C(τ ) − ΩAP (r)]+ 0 βrα σ 2 × exp − LAP (βrα )fR (r)dr. pAP (52)
R EFERENCES pAP Hy1 r−α ≥ β r, v0 , ψ P (SINR ≥ β|r, v0 ) = P [1] Y. Zhang, D. J. Love, J. V. Krogmeier, C. R. Anderson, R. W. Heath, I1 + σ 2 and D. R. Buckmaster, “Challenges and opportunities of future rural α 2 X wireless communications,” IEEE Communications Magazine, vol. 59, βr σ −α α no. 12, pp. 16–22, 2021. = P Hy1 ≥ + βr Hyi Di r, v0 pAP [2] J. Parekh, C. Parekh, and A. Ghasemi, “Enabling universal connectivity y ∈Φ \y AP i 1 via data-driven policymaking: A north American case study,” IEEE α 2 X Communications Magazine, vol. 59, no. 12, pp. 23–29, 2021. βr σ −α α Hyi Di r, v0 − βr = E exp− [3] S. Euler, X. Lin, E. Tejedor, and E. Obregon, “High-altitude platform pAP stations as international mobile telecommunications base stations: A yi ∈ΦAP \y1 X primer on HIBS,” IEEE Vehicular Technology Magazine, vol. 17, no. 4, α 2 βr σ pp. 92–100, 2022. = exp − Hyi Di−α r, v0 E exp − βrα [4] X. Lin, S. Cioni, G. Charbit, N. Chuberre, S. Hellsten, and J. F. pAP yi ∈ΦAP \y1 Boutillon, “On the path to 6G: Embracing the next wave of low earth βrα σ 2 orbit satellite access,” IEEE Communications Magazine, vol. 59, no. 12, α = exp − LAP (βr ). pp. 36–42, 2021. pAP [5] M. Matracia, M. A. Kishk, and M.-S. Alouini, “Exploiting wind(49) turbine-mounted base stations to enhance rural connectivity,” IEEE
A PPENDIX C P ROOF OF T HEOREM 2 Considering the two main system requirements EH and SINR, the overall coverage probability can be derived as Pcov (β|v 0 , ψ) = E[1(SINR ≥ β)1(EH ≥ Emin )] = ER EH [1(SINR ≥ β)|R, v0 , ψ]
(i)
×EG [1(EH ≥ Emin )|R, v0 , ψ] = ER P(SINR ≥ β|R, v0 , ψ) × P(EH ≥ Emin |R, v0 , ψ) , (50) where step (i) is the result of the assumption that fading gains during two sub-slots are independent. The two conditional probability has been proved in Theorem 1 and Lemma 5. Substituting these two conditional probabilities in (50), the overall coverage probability can be written as: Pcov (β|v0 , ψ) Z rd +v0 = P(SINR ≥ β|r, v0 , ψ) 0
×P(EH ≥ Emin |r, v0 , ψ)fR (r)dr p BS ΩBS ]+ = exp −rα [C(τ ) − ΩAP (r) − pAP 0 βrα σ 2 pBS α × exp − LAP (βrα )LBS ( βr )fR (r)dr. pAP pAP (51) Z rd +v0
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