A Coverage Area-Based CoMP Technique for SWIPT-Enabled Mobile Networks

The limited power resources for battery-operated mobile devices and the highly dense network deployment in the sixth generation networks make the simultaneous wireless information and power transfer (SWIPT) and coordinated multi-point transmission (CoMP) technologies ever more attractive. However, the CoMP suffers from the heavy handover signalling of high-velocity mobile users (MUs). In this paper, we propose a novel coverage area-based CoMP (CA-CoMP) scheme for SWIPT-enabled mobile networks, aiming at reducing the handover rate and enhancing the SWIPT performance. In particular, by taking into consideration the heterogeneity of base stations’ (BSs’) footprints, the CA-CoMP scheme enables a MU to select and communicate with multiple BSs, whose coverage areas are greater than a pre-defined threshold. By leveraging stochastic geometry tools, we study the CA-CoMP scheme in terms of several key performance metrics, i.e., inter- and intra-cell handover rate, success probabilities, average spectral efficiency, average harvested energy as well as energy efficiency. Our results reveal that the proposed CA-CoMP scheme achieves a much lower handover rate and offers a better SWIPT performance, compared to conventional CoMP schemes, e.g., a 41% improvement of the spectral efficiency can be achieved for MUs with a velocity of 25 m/s; while the optimal area threshold for achieving the best SWIPT performance is numerically evaluated.


I. INTRODUCTION
D RIVEN by the explosive growth of smart devices, ultra- reliable and green communications technologies will be accommodated in the sixth generation (6G) communication networks, which are envisioned to provide unlimited and ubiquitous wireless connections with extremely low latency [1].The concept of massive Internet-of-Things (mIoT) is expected to be a crucial component of 6G wireless networks, accommodating numerous static and mobile devices, such as remote sensors, unmanned aerial vehicles, autonomous cars, etc, which leads to an unprecedented increase of multi-user interference.
The demand of providing enhanced quality of service (QoS) for the diverse range of end-user devices, highlights the paramount importance of advanced downlink transmission and interference mitigation technologies.Moreover, the limited energy of battery-operated devices in mIoT communications restricts the performance of the communication systems, triggering the urgent need for novel energy harvesting (EH) technologies.Hence, an energy efficient downlink transmission framework that takes into consideration the mobility of the end-user devices is significant on the way to 6G success.
By aiming to prolong the lifetime of the network nodes in 6G networks, the simultaneous wireless information and power transfer (SWIPT) technology is proposed to provide the ability for static and/or mobile users (MUs) to decode information and harvest energy simultaneously from the received radiofrequency (RF) signals, where the harvested energy can be used for data processing or stored in a battery for future use [2].In practice, SWIPT is achieved by separating the received RF signals into two parts; one part is used for information decoding (ID) and the other part is used for EH.The division of the RF signals can be performed in the time, power or space domain, by employing time switching (TS), power splitting (PS) or antenna switching (AS) schemes, respectively [2], [3], [4].Throughout the literature, the potential benefits of SWIPT have been widely acknowledged and the performance of SWIPT-enabled large-scale communications has been extensively investigated.The authors in [3] investigate the performance of a SWIPT-enabled multipleinput multiple-output (MIMO) wireless broadcast system in the context of several receiver architectures for TS, PS and AS schemes, revealing the fundamental trade-off between information and energy transfer.Moreover, motivated by the dense and random topologies of the next-generation networks, the SWIPT technology is also investigated from a macroscopic point-of-view [4].Specifically, the authors in [4] propose a stochastic geometry (SG)-based framework to investigate a SWIPT-enabled MIMO system, while the associated optimal TS and PS ratios that jointly achieve the best ID and EH performance are demonstrated.Even though SWIPT technology is well-investigated in the literature, most of the exiting works assume a simplistic linear EH model, i.e., the conversion efficiency of RF to direct current (DC) is assumed to be constant [2], [3], [4].Nevertheless, in practice, the conversion efficiency is a highly non-linear function and its impact on the network performance is not well studied [5], [6], [7].In particular, by taking into account the non-linear nature of the rectification process, the authors in [5] propose a non-linear EH model capturing the dynamics of the RF energy conversion efficiency for different input power levels, where the amount of the harvested energy is modelled based on a parametric logistic function.Furthermore, an alternative non-linear EH model is proposed in [6], [7], where the random noise in the detection and conversion of the actual harvested energy is considered, offering tractability for system-level analysis in wireless-powered networks.
Due to the immense deployment of mIoT applications in 6G networks, the unprecedented increment of the interference becomes a crucial factor, jeopardizing the ID performance of the MUs.Motivated by this, the coordinated multi-point transmission (CoMP) technique is proposed to enhance the link reliability of the mobile networks, by improving the intended received signal strength and by mitigating the multiuser interference [8], [9], [10].On the one hand, it has been shown that the CoMP technique can effectively enhance the system performance.In particular, the authors in [8] demonstrate the aforementioned behaviour by investigating the CoMP technique in a downlink heterogeneous cellular network and revealing the significant gain achieved in terms of coverage probability.On the other hand, the multi-connection of a MU with multiple base stations (BSs) generally leads to more frequent handovers, alleviating the network performance due to the heavy signalling cost [9], [10].Specifically, the authors in [9] study the CoMP technique in the context of user-centric cooperation networks, where analytical expressions for the handover rate are derived, revealing that the handover rate is greatly increased by the CoMP operation.By aiming to decrease the handover rate, the authors in [10] propose a movement-aware CoMP handover scheme that exploits the trajectory and the cell dwell time of the MUs, to ensure a lower inter-cell handover rate.Nevertheless, the severe signalling overhead required by the cooperative techniques makes such approaches impractical for MUs with scarce power resources, thus motivating the investigation of energy-efficient low-complexity techniques.
To unlock the full potential of mIoT communications, multiple-antenna techniques such as transmit beamforming, are essential solutions to enhance the link reliability and improve the data rate.Despite their many advantages, these techniques pose several new challenges for 6G mobile networks, such as mobility management and heavy handover signalling cost [11].Specifically, in addition to the inter-cell handover discussed in the aforementioned works, another type of handover, namely beam reselection or intra-cell handover should be taken into account [11], [12].More specifically, the authors in [11], [12] study the beam management based on SG, where a closed-form expression for the beam reselection rate is analytically derived and the optimal number of beams that ensures maximum spectral efficiency is demonstrated.By taking into consideration both the inter-and intra-cell handover, the authors in [13] propose an evolutionary game theory-based approach to solve the problem of access model selection, aiming at improving the spectral efficiency in sub-6 GHz/millimetre-wave cellular networks.Moreover, the authors in [14] propose a velocity-based cell association technique in the context of multi-tier cellular networks, aiming to reduce the beam reselection overheads and improve the data rate, by associating MUs with different network tiers according to their velocities [14].
Although several CoMP techniques are developed for the mobile networks, which induce a higher network management complexity due to the multi-tier network topologies, lowcomplexity mobility management and energy-efficient CoMP techniques are missing from the literature.Moreover, the intracell handover analysis under a CoMP transmission scenario is overlooked from the current studies.Hence, the aim of this work is to fill these gaps by providing a novel CoMP scheme for SWIPT-enabled single-tier mobile networks, aiming at reducing the inter-and intra-cell handover rate and enhancing the SWIPT performance.Specifically, the main contributions of this paper are summarized as follows.
• We develop an analytical framework based on SG, which comprises the co-design of SWIPT and CoMP techniques, shedding light on the analysis of SWIPT-enabled large-scale mobile networks.In particular, the developed framework takes into account the ability of MUs to jointly communicate with multiple BSs in a non-coherent manner and perform non-linear EH, where all MUs have arbitrary velocity and trajectory within the considered network area.• We propose a novel low-complexity CoMP scheme, namely the coverage area-based CoMP (CA-CoMP) scheme, aiming to reduce the handover rate of the high mobility MUs and to enhance their SWIPT performance.More specifically, the CA-CoMP scheme enables the high mobility MUs to jointly communicate with multiple serving BSs, whose coverage areas are greater than a pre-defined threshold; thus frequent handovers associated with small-coverage area BSs are avoided.• By leveraging tools from SG, we derive closed-form expressions for both the inter-and intra-cell handover rate for the proposed CA-CoMP scheme.Analytical expressions for the ID and EH success probabilities, the average spectral efficiency, the harvested energy, as well as the energy efficiency are derived.Furthermore, under specific practical scenarios, closed-form expressions for the aforementioned metrics are derived.These closed-form expressions provide a quick and convenient method to evaluate the network performance and obtain insights into how key parameters affect the network.• Our results reveal that with the employment of the proposed CA-CoMP scheme, MUs experience significantly fewer inter-and intra-cell handover processes compared to that observed with the employment of conventional CoMP schemes.In addition, by properly selecting the area threshold parameter, the numerical results exhibit considerable spectral efficiency increase by 41% compared to that achieved with the conventional schemes; the optimal area threshold that offers the highest SWIPT performance is also illustrated.Finally, we demonstrate that a higher average harvested energy and energy efficiency can be achieved by the proposed CA-CoMP compared to conventional CoMP.
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TABLE I SUMMARY OF NOTATIONS
The rest of the paper is organised as follows: Section II introduces the considered system model.Section III-A presents our proposed CA-CoMP scheme and the associated handover analysis.Section IV provides the achieved ID and EH performance of the CA-CoMP scheme.Finally, numerical results are presented in Section V, followed by our conclusions in Section VI.
Notation: R d denotes the d dimensional Euclidean space; x denotes the Euclidean norm of x ∈ R d ; P{X } denotes the probability of the event X and E{X } represents the expected value of X; Γ(•) and Γ(•, •) denote the complete and the upper incomplete functions, respectively; 2 F 1 (•, •; •; •) is the Gauss hypergeometric function; and G[a, b] denotes the Gamma distribution with a shape and a scale parameter a and b, respectively; U (•, •) denotes the uniform distribution.

II. SYSTEM MODEL
In this section, we provide details for the considered system model.The network is studied from a macroscopic point-ofview based on SG.The main mathematical notations related to the system model are summarized in Table I.

A. Network and Channel Model
We consider a single tier cellular network as illustrated in Fig. 1, where the BSs are spatially distributed according to a homogeneous Poisson point process (PPP), i.e., Φ b = {x i ∈ R 2 }, with spatial density λ b , where x i denotes the spatial coordinate of the i-th BS within the considered bi-dimensional network deployment.Moreover, we assume that the spatial locations of the MUs follow an independent homogeneous PPP Φ u with a density λ u λ b [4], [15].In addition, we assume that all BSs transmit signals with power P t (dBm) and each BS serves only one MU at a time per resource block [11], [16].In the context of the proposed CA-CoMP scheme (detailed description in Section III-A), each MU jointly communicates with N cooperating BSs in order to enhance the received signal power and mitigate the inter-cell interference.Based on the Slivnyak's theorem and without loss of generality, we perform our analysis for the typical MU, which is initially located at the origin, and our results are applicable to all MUs within the network area [15], [16], [17].
We assume that all wireless signals experience both largescale path-loss effects and small-scale fading.Specifically, for the mathematical tractability, the small-scale fading of the wireless channels is modelled as Rayleigh fading, where different links are assumed to be independent and identically distributed (i.i.d.).Hence, the power of the channel fading h is modelled by an exponential random variable with unit mean, i.e., h ∼ exp(1) [4], [17].Regarding the large-scale path-loss effects, we assume a non-singular path-loss model [18].In particular, the path-loss between the typical MU and the i-th BS located at x i is given by (r i ) = 1 1+r α i , where r i = x i is the Euclidean distance from the typical MU to the i-th BS, and α > 2 is the path-loss exponent [18].

B. Directionality Model
By aiming to compensate the path-loss effects for the long distance transmission, all BSs employ high-gain directional antennas [11], [19].Specifically, we assume that all BSs are equipped with multiple antennas to generate directionality towards to the MUs, while all MUs are equipped with a single omnidirectional antenna.We consider that each BS has a codebook of M beamforming vectors with M = 2 m for m ∈ N, where the patterns of these beamforming vectors have non-overlapping main lobes and cover the full angular range, i.e., [0, 2π) [11].For simplicity, the beamwidth is considered as the central angle of a sector, i.e., θ = 2π M [4], [11].Moreover, the main lobe is assumed to be restricted to the beamwidth and thus, the main lobe and the side lobe antenna gain are given by and G s = ξG m , respectively, where ξ ∈ (0, 1) is the loss coefficient of the antenna directivity [19].This antenna model approximates the actual beam pattern with sufficient accuracy and captures the directivity loss effect, while also providing tractability for the analytical process.Therefore, the antenna gain of the link between the typical MU and the i-th closest interfering BS, denoted as G I,i , is given by G I,i = {G m , G s }, with the corresponding probabilities p G = { θ 2π , 2π−θ 2π }; while the antenna gain of the link between the typical MU and its i-th closest serving BS, denoted as G S,i , is evaluated in Section III-A.

C. Mobility Model
We consider a general mobility model, where all MUs move freely and randomly within the network area without any restrictions.In particular, the mobility of the MUs is modelled by a widely-adopted random waypoint mobility model, which is specified as following [20].Initially, all MUs are uniformly located according to the PPP Φ u ∈ R 2 .Afterwards, each MU moves with certain velocity towards a randomly selected destination point (which also refers to a waypoint), within the network area.In addition, motivated by the 3GPP mobility model [21], we assume that each MU moves along a straight line with a uniformly random speed v ∼ U(0, v max ) and direction φ ∼ U(0, 2π).Moreover, the mobility of different MUs is considered to be independent of each other, and thus, all MUs have identical stochastic mobility properties.During the motion of the MUs, both inter-and intra-cell handovers may occur.Specifically, the inter-cell handover refers to the handover process associated with different BSs along MUs' trajectories, while the intra-cell handover refers to the beam reselection Section III-A).

D. Information and Power Transfer Model
We assume that the BSs have continuous power supply, while the MUs are battery-operated.Specifically, we assume that all MUs have SWIPT capabilities, and thus, they are able to decode information and harvest energy simultaneously.Since the MUs are equipped with a single antenna, we assume the employment of either the TS or the PS scheme for SWIPT [4].In particular, by adopting the TS scheme, a MU allocates a fraction τ ∈ [0, 1] of the time slot for communication with its serving BSs, while it harvests energy for the remaining time slot [4].Analogously, for a PS-based receiver, an extra power splitting circuit is integrated, such that the received power is divided into two parts with a splitting ratio ρ ∈ [0, 1], where a fraction ρ of the received power is used for ID purpose, while the remaining power is directed to the rectenna for EH [2].It is important to mention that, the ratios τ and ρ for the TS and PS schemes, respectively, are design parameters that can be adjusted accordingly, aiming to satisfy the specific requirements for various mIoT applications [2], [3], [4].
Regarding the downlink data transmission, we adopt a non-coherent joint transmission policy.Specifically, the noncoherent joint transmission allows multiple cooperative serving BSs transmit the same signal to their associated MU without prior phase-alignment and tight synchronization to that MU, which is thus suitable for the scenario where MUs exhibit high mobility [9], [22].Then the MU combines the signals from multiple BSs using non-coherent combining techniques.In addition, a scheduling mechanism is employed in which each MU is scheduled for communicating with its serving BSs at different time-frequency resources.Therefore, no intra-cell interference exists since intra-cell users are served within orthogonal time-frequency resources, while only intercell interference is taken into consideration [23], [24].Hence, the signal to interference plus noise ratio (SINR) observed at the typical MU can be formulated as where S represent the set of serving BSs (Detailed in Section III-A), σ 2 n is the thermal noise, σ 2 c = σ 2 cov and σ 2 c = σ 2 cov /ρ for the TS and the PS scheme, respectively, σ 2 cov is a constant that accounts for the noise induced by the signal conversion from RF to baseband [4].
Each MU is equipped with a rectifier circuit that is capable of converting a portion of the received RF signals into DC power to either charge its battery or power its circuits, while the RF signals may include the intended signal from the coordinated serving BSs, as well as interfering signals from other BSs.Due to the non-ideal RF-to-DC circuitry of the practical end-user devices, we adopt a non-linear EH model, which captures the randomness in the detection and conversion of the actual harvested energy [6], [7].More specifically, the amount of the instantaneous harvested energy of a MU is quantified as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where F is an exponential random variable 1 with mean and η is a constant representing the RF-to-DC conversion efficiency.Note that (1) and ( 2) hold for both the TS and the PS scheme, i.e., ρ = 1 and 0 ≤ τ ≤ 1 are adopted for the TS scheme, while τ = 1 and 0 ≤ ρ ≤ 1 are adopted for the PS scheme.

III. COVERAGE AREA-BASED COMP SCHEME
AND HANDOVER ANALYSIS In this section, we introduce the proposed CA-CoMP technique in the context of SWIPT-enabled cooperative cellular networks.Our technique exploits the cooperation among randomly located BSs, that are selected according to their coverage areas, aiming at both enhancing the network performance and reducing the handover rate.In addition, we develop a tractable analytical framework to compute the handover rate associated with the CA-CoMP scheme, while closed-form expressions for the inter-and intra-cell handover rates and the misalignment probability are derived, which will be useful for evaluating the achieved ID and EH performance in Section IV.

A. Coverage Area-Based CoMP Scheme
The proposed CA-CoMP scheme is based on a two-stage procedure.In the first stage, a set of candidate BSs is selected based on their coverage areas.More specifically, due to the irregular shape of the cells, different BSs generally have various coverage areas, while the MUs only communicate with the BSs that have relative large coverage areas, named as candidate BSs.Hence, at the first stage the candidate BSs are determined, which consists of the BSs that their coverage areas are greater than a pre-defined area threshold A (m 2 ), i.e., Φb = {x i |∀x i ∈ Φ b , A i > A}, where A i is the coverage area of the i-th BS located at x i .In the second stage, the set of serving BSs with which the MUs communicates, i.e., S, is selected among the set of candidate BSs.In particular, in order to ensure high intended signal power, each MU communicates with its N closest BSs among the set of candidate BSs.Hence, the final set of the serving BSs is formulated as, It is worth mentioning that, the proposed CA-CoMP scheme provides flexibility for the design of large-scale SWIPTenabled communication networks, since according to the MUs' mobility within the network, the area threshold can be appropriately adjusted.Particularly, for high-velocity MUs, i.e., v → ∞, the handover overhead is the dominant factor that jeopardizes the network performance.Therefore, a large area threshold is beneficial to be adopted in order to reduce the MUs' handover rate.On the other hand, for MUs with low velocity or static, i.e., v → 0, the handover overhead is negligible, and hence, a small area threshold ensures that the MUs are associated with the N closest BSs in order to achieve the highest received signal strength.Moreover, it is worth emphasising that, the proposed CA-CoMP has low-complexity; more 1 Such coefficient in the energy transfer phase is used to capture the random noise in the detection and conversion of the actual harvested energy [6].
specifically, it only requires information about the coverage area of each BS, which is initially deterministic for a given network deployment and could be easily disseminated to MUs via the central network.In addition, compared to some other techniques, e.g., heterogeneous networks-based CoMP schemes [9], [10], that assign MUs with different network tiers based on their velocity for combating high handover rates, the proposed CA-CoMP is operated on a single-tier network topology with much lower complexity of the network structure and management.

B. Inter-Cell Handover Analysis
We explore the inter-cell handover process of the typical MU, by considering that the typical MU is moving with an arbitrary trajectory.Along this trajectory, connections between the typical MU and its serving BSs change according to its location, such that the typical MU maintains the connectivity with the network.In addition, since each MU jointly communicates with N serving BSs according to the CA-CoMP scheme, the average inter-cell handover rate is defined as the total number of the triggered handovers with respect to the N serving BSs per unit time.More specifically, since the serving BSs of the typical MU are the N closest BSs among the set of candidate BSs, the inter-cell handover is triggered when the typical MU crosses the boundaries of an N-th order Voronoi cell [9], [22].The inter-cell handover rate experienced by a MU in the considered network topology is evaluated in the following proposition.
Proposition 1: Based on the CA-CoMP scheme, the intercell handover rate of the typical MU moving with velocity v, is given by where λb = and K = 3.5.Proof: See Appendix A. Although (3) can be easily evaluated by using numerical tools, intuitions on how key system parameters affect the intercell handover rate are difficult to derive.In the following corollary, we simplify the analysis by considering an extreme scenario, where the MUs are able to jointly communicate with a large number of serving BSs, i.e., N 0. Corollary 1: For the special case with N 0, the inter-cell handover rate, μ c , is given by Proof: According to [25], for x 0, we have 2 , b = 0 and x = N, the above expression is derived.
From (4) we can easily observe that, for a certain density of candidate BSs, i.e., λb , when the number of cooperative serving BSs becomes large, i.e., N 0, the inter-cell handover rate significantly increases.This is expected since, MUs experience more frequent handovers with multiple serving Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
BSs than the scenario with a single serving BS.Moreover, we can observe that the inter-cell handover rate of the MUs increases with their velocity.This is based on the fact that a MU with higher velocity travels a longer distance along its trajectory, compared to a lower-velocity MU, resulting in a higher probability of crossing a cell boundary.We can also observe that the CA-CoMP scheme with λb ≤ λ b , achieves a lower inter-cell handover rate than the conventional CoMP scheme, corresponding to the special case of our proposed CA-CoMP scheme with A = 0 [9].The impact of the pre-defined area threshold, i.e., A, on the inter-cell handover rate is evaluated in the following corollary.
Corollary 2: By increasing the coverage area threshold of the CA-CoMP scheme, i.e., A → ∞, the inter-cell handover rate of the MUs reduces, i.e., μ c → 0.
Proof: By using the expression in [26, 8.356.4], the first order derivative of μ c with respect to A can be derived, i.e., where It is worth mentioning that the CA-CoMP enables MUs to ignore some nearby BSs, which have small coverage area, to mitigate the frequent handover; on the other hand, these nearby BSs could provide strong signal due to short propagation distances.Hence, we obtain a trade-off between the handover overhead and the received signal quality; the optimal area threshold for achieving the best network performance is discussed in Section V.

C. Intra-Cell Handover Analysis
We now focus our attention on the intra-cell handover process, where both geometry-and measurement-based handover procedures are considered.In particular, a geometry-based intra-cell handover is triggered when a MU crosses the beam boundaries of the serving BSs, where a new beam is reselected for the downlink transmission.Nevertheless, there are various types of intra-cell handover in modern wireless communication systems, such as channel handover and sector handover; in our work, we mainly focus on the intra-cell handover caused by the beam alteration [11].Similarly to the inter-cell handover scenario, the geometry-based intra-cell handover rate equals to the average number of beam reselections with respect to the sum of N serving BSs per unit time.The following proposition evaluates the geometry-based intra-cell handover rate experienced by the typical MU with respect to its i-th closest serving BS, with 1 ≤ i ≤ N .
Proposition 2: Based on the CA-CoMP scheme, the geometry-based intra-cell handover rate of the typical MU with respect to its i-th closest serving BS, i.e., μ b,i , is given by Proof: See Appendix B.
From the expression in ( 6), we can easily observe that the intra-cell handover rate of the typical MU with respect to its i-th closest serving BS, i.e., μ b,i , is directly proportional to the number of beams and the velocity of the typical MU, i.e., M and v, respectively.Therefore, the downlink performance of high-velocity MUs is diminished by the frequent intra-cell handovers, especially for scenarios where large number of beams are employed at the BSs, e.g., 6G cellular networks.Note that the geometry-based intra-cell handover rate decreases with the increase of the distance from the typical MU to its serving BSs.This result can be theoretically justified by showing the ratio of μ b,i with μ b,i+1 , i.e., In practice, a MU that is moving with a shorter distance to the serving BSs, is also close to the beam boundaries of these BSs, thereby resulting in a higher probability of crossing beam boundaries.Hence, although the closest serving BSs could provide the strongest intended signals due to the shortest propagation distances, the most frequent intra-cell handover occurs with the closest serving BSs, which jeopardizes the SWIPT performance of the MUs.
Note that the beam misalignment of the link between serving BSs and the typical MU may occur by taking into account the measurement-based intra-cell handover.More specifically, according to [11], [27], the measurement-based intra-cell handover is operated based on measurements of the synchronization signal block (SSB) burst, which is transmitted periodically with a period T SSB (ms).The beam misalignment occurs when a MU moves inside the side lobe area of the serving BSs before receiving the SSB burst, which results in a weak QoS at the end-user devices.Hence, in the following proposition, we characterize the misalignment probability, which is useful for evaluating the distribution of the antenna gain and the corresponding ID and EH performance of the MUs.
Proposition 3: The misalignment probability of the link between the typical MU and its i-th closest serving BS is given by Proof: See Appendix C. Based on the above proposition, the misalignment probability depends on the MUs' velocity, the intra-cell handover rate as well as the period of the SSB burst, where high velocity of MUs results in a high probability of misalignment.Note that, for scenarios with extremely small or equal to zero period of the SSB burst, i.e., T SSB → 0, we have P (i) mis → 0, which indicates that, the geometry-and measurement-based intra-cell handovers are triggered at exact the same location, i.e., achieving perfect beam alignment [4].Hence, the antenna gain of the link between the typical MU and its i-th closest serving BS can be modelled as a discrete random variable, i.e., Moreover, the joint probability mass function (PMF) of G S,i is given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where P S,i (G m ) = 1 − P (i) mis and P S,i (G s ) = P (i) mis .Then, based on the above discussion, the total intra-cell handover rate of the typical MU with respect to N serving BSs per unit time is evaluated in the following theorem.
Theorem 1: The total average intra-cell handover rate of the typical MU, for the CA-CoMP scheme is given by Proof: See Appendix D.
From (10) we can observe that, as the number of beams increases, i.e., M → ∞, the intra-cell handover rate increases, i.e., μ b → ∞.This is based on the fact that, by increasing the number of beams, the spatial density of the beam boundaries increases, thus resulting in a higher probability of beam boundary crosses by the MUs.Moreover, since the density of the candidate BSs is smaller than the complete BSs set, i.e., λb ≤ λ b , the CA-CoMP achieves a lower intra-cell handover rate compared to conventional CoMP schemes.Finally, it is easily to observe that the intra-cell handover rate is generally greater than the inter-cell handover rate, and is directly proportional to the inter-cell handover rate, i.e., μ b = M 4 μ c .

IV. SWIPT PERFORMANCE WITH CA-COMP SCHEME
We study the information and energy transfer performance of the SWIPT-enabled mobile cellular networks achieved by the proposed CA-CoMP scheme.We start by evaluating the information transfer performance via computing the ID success probability as well as the average spectral efficiency.Subsequently, we assess the EH ability of the MUs in terms of the EH success probability and the average harvested energy.Finally, the energy efficiency of the SWIPT-enabled MUs is evaluated for the CA-CoMP scheme.The analytical expressions for the aforementioned performance metrics are presented.

A. Interference Characterization
We first characterize the interference observed at the typical MU, i.e., I = x i ∈Φ \S b P t G I,i h i (r i ), by calculating the Laplace transform, i.e., L I (s) = E{exp(−sI)}, where a closed-form expression is derived in the following lemma.
Lemma 1: The Laplace transform of the received interference at the typical MU is given by (11), shown at the bottom of the page.
Proof: See Appendix E. It can be observed from (11) that the Laplace transform of the interference consists of two terms inside the exponential function.The first term is related to the interference generated by the nearby interfering BSs, that their distance to the typical MU is less than r N (i.e., the distance from the N-th serving BS to the typical MU).The second term refers to the interference from other distant interfering BSs.For simplicity and due to the high directionality of the transmitter's antennas, the second term could be neglected [28].

B. Information Transfer Analysis
By applying the Laplace transform of the interference, we now evaluate the ID success probability, which is defined as the probability that a MU is able to achieve a certain SINR threshold β (dB), i.e., P ID (β) = P{SINR ≥ β}.The analytical expression for the ID success probability is provided in the following theorem.
Theorem 2: The ID success probability of the typical MU for the CA-CoMP scheme, i.e., P ID (β) is given by where , and f r (r 1 , . . ., r N ) is the joint probability density function (PDF) of the distance from the typical MU to the serving BSs, which is given by [29] f r (r 1 , . . ., r N ) = 2 λb π N exp − λb πr 2 N N i=1 r i .(13) Proof: See Appendix F. Remark 1: In the interference-limited region, i.e., σ n = σ c = 0, the TS and the PS schemes achieve the same ID success probability.
It should be noted that the ID success probability only provides the statistics of the instantaneous SINR observed at the typical MU.Therefore, in order to show the impact of the handover overhead on the information transfer performance, we further investigate the average spectral efficiency, denoted as η SE , achieved by the proposed CA-CoMP scheme.In particular, the average spectral efficiency is defined as the ergodic Shannon rate achieved by the typical MU per unit bandwidth, i.e., η SE = E{T ID eff log(1 + SINR)}, which is evaluated in the following proposition.
Proposition 4: The average spectral efficiency achieved at the typical MU for the CA-CoMP scheme, is given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where T ID eff (μ c , μ b ) is the average effective time allocated for ID purpose per unit time with and T c and T b are the signalling overhead delay of the interand intra-cell handovers, respectively.Proof: The proof is directly from the definition of the ergodic Shannon rate and the average spectral efficiency [11].
Based on the expression derived in Proposition 4, we can observe that the handover overhead degrades the average spectral efficiency.In particular, the effective time of the information transfer depends on the handover rate, i.e., a higher handover rate results in a shorter time interval for information transfer.Note that due to the heavy signalling overhead, a large handover rate may result in an outage for the information transfer, i.e., η SE = 0, which is discussed in the following two corollaries.
Corollary 3: An information transfer outage (i.e., η SE = 0) occurs, when the velocity of a MU exceeds v = Finally, by solving the above equation in respect of v , the final result is derived.Note that, for the high mobility MUs, a relatively large coverage area threshold should be selected to ensure the communication of the MUs.The minimum area threshold for the CA-CoMP scheme to avoid the information transfer outage is derived in the following corollary.
Corollary 4: In order to avoid an information transfer outage, when a MU is moving with a velocity v, the minimum area threshold is given by where ) is the inverse regularized incomplete Gamma function and Proof: The proof follows a similar methodology with Corollary 3. By substituting the expression of λb , which is given in Proposition 1, and by solving equation (15) in respect of A, the final expression is derived.
The results provided in the above corollaries can be easily utilized for the network designing under the CA-CoMP scheme, to avoid the potential information transfer outage and provide stable downlink services for the MUs.In addition, these results hold for the energy transfer process, which will be discussed in the next subsection.

C. Energy Transfer Analysis
We now focus our attention on the achieved energy transfer performance of the CA-CoMP scheme.We evaluate the EH success probability, which is defined as the probability that the harvested energy of the typical MU is higher than the EH threshold (dBm), i.e., P{Q ≥ }.The analytical expression for the EH success probability is presented in the following theorem.
Theorem 3: The EH success probability of the typical MU for the CA-CoMP scheme is given by where is the Laplace transform of the interference, which is given in Lemma 1.
Proof: See Appendix G.
From (17) we can observe that the expression of the EH success probability consists of two main terms, i.e., the intended signal part and the interference signal part.Motivated by the high directionality of the antennas, the interference power can be ignored.Moreover, for the MUs moving with low-velocity, the misalignment probability approaches zero, and thus a perfect beam alignment is achieved, i.e., G(G m , . . ., G m ) = 1.Therefore, based on the aforementioned observations, we provide an approximated expression for the EH success probability in the following remark.
Remark 2: By ignoring the interference power, the EH success probability of the low-velocity MU is given by Note that, the EH success probability reveals the statistics of the instantaneous harvested energy, while by taking into account the handover signalling cost, we can assess the EH performance in terms of the average harvested energy.More specifically, the average harvested energy is defined as the average amount of energy harvested by the typical MU per unit time; the analytical expression is provided in the following theorem.
Theorem 4: The average harvested energy per unit time of the typical MU for the CA-CoMP scheme is given by Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE II SIMULATION PARAMETERS
where QS and QI are given by ( 19) and ( 20), shown at the bottom of the page, respectively; B = 2πηP t (1 − ρ)T EH eff and T EH eff is the effective time for the EH procedure, with Proof: See Appendix H.The above theorem reveals the negative impact of the handover process on the average harvested energy, i.e., a higher handover rate leads to less harvested energy per unit time.Moreover, for scenarios with high velocities, energy transfer outage may occur, i.e., Q → 0. In order to avoid such energy transfer outage, the area threshold of the CA-CoMP scheme should be greater than A min , which is provided in Corollary 4. Finally, by adjusting the PS or TS parameters, i.e., ρ or τ , the proposed CA-CoMP scheme can satisfy different requirements for various mIoT applications.

D. Energy Efficiency Analysis
In order to provide a comprehensive evaluation for the performance of CA-CoMP scheme, we investigate another equally important metric, namely, the energy efficiency, which refers to the ability of a MU to receive and process data with the minimum possible energy.Since a SWIPT-enabled MU is capable of converting part of the received RF signal into DC power, which is then used to power its circuitry, the total energy consumed by a MU is calculated by subtracting the amount of harvested energy from the total energy used for data receiving and processing.Hence, the energy efficiency of a SWIPT-enabled MU is defined as the ratio of the average downlink data rate to the amount of energy consumed at MU's battery [30].For the sake of simplicity, we only consider communication-related energy consumption at the MUs.In particular, for a certain input data rate, i.e., R, the energy consumed at the MU for processing the downlink input signal is given by [31] where R = B w η SE is the downlink rate, B w is the bandwidth, K BB is the logic operations per bit in the baseband processor, F 0 is the fanout, i.e., the number of loading logic gates, a is the activity factor of transistors for the chip in MUs' devices, k B is the Boltzmann constant, and T env is the temperature of the environment, G a ≈ 454.2 is the gap between the switch energy consumption for the transistor and the Landauer limit, and C cir represents the constant power consumption of the baseband processor.Then, based on the ID and EH performance metrics derived in the previous section, we provide the energy efficiency of the typical MU in the following proposition.Proposition 5: The energy efficiency of the typical MU with the CoMP scheme is given by where Q is the average harvested energy by the typical MU.

V. NUMERICAL AND SIMULATION RESULTS
We present analytical and simulation results to validate the accuracy of our model and illustrate the performance of the proposed CA-CoMP scheme.Unless otherwise stated, we use the parameters given in Table II.It is worth mentioning that, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the proposed analytical framework is generic, and the selection for these parameter values is for the purpose of illustration.

A. Handover Performance
Fig. 2 demonstrates the impact of the number of serving BSs and the MUs' velocity on the handover rate achieved by the proposed CA-CoMP scheme.In particular, Fig. 2(a) plots the inter-cell handover rate, i.e., μ c , versus the number of serving BSs, i.e., N, for MUs that move with different velocities, i.e., v ∈ {5, 15, 30} m/s.For comparison purposes, we also present the inter-cell handover rate obtained based on the conventional CoMP scheme [9], where each MU jointly communicates with N closest BSs, denoted as "CoMP".Firstly, we can observe that the proposed CA-CoMP scheme achieves a much lower inter-cell handover rate compared to the conventional CoMP.This was expected since, by employing the proposed CA-CoMP scheme, the communication of MUs with small area cells that induce frequent handover processes is avoided, thereby achieving a significantly lower inter-cell handover rate.Hence, the proposed scheme alleviates the signalling overhead and thus is promising for practical and low-complexity implementations.Moreover, we can observe that, the inter-cell handover rate increases with the increase of the number of serving BSs as well as the velocity of the MUs.This can be justified by the fact that, for the joint communication of a MU with multiple BSs as well as for the communication of a high-velocity MU, frequent handover operations are required to ensure high QoS.Moreover, another interesting observation is that a higher gain is achieved by the CA-CoMP scheme over the conventional CoMP, when the MU is connecting with more serving BSs or moving with a higher velocity.This result is in line with the expression in (3), where for a certain given area threshold, the achieved gain by the CA-CoMP scheme is directly proportional to the number of serving BSs and the velocity of the MUs.
Similar results can be observed in Fig. 2(b), which plots the intra-cell handover rate versus the number of serving BSs.Initially, the comparison of inter-and intra-handover rates that are depicted in Fig. 1(a) and Fig. 1(b), respectively, reveals that the intra-handover rate is the dominant limiting factor compared to the inter-handover rate.This was expected since, each BS has multiple beams (hundreds of beams in 6G networks), which could result in a much more frequent intra-cell handover compared with the inter-cell handover.Finally, the agreement between the theoretical curves (solid and dash lines) and the simulation result (markers) validates our theoretical analysis.i.e., N = {1, 3}.Firstly, it can be easily observed that, for a fixed area threshold, the ID success probability increases with the increase of the number of serving BSs.This could be explained by the fact that the increased number of serving BSs leads to an improved power of the intended received signal and to the reduction of the number of interfering BSs, and thus the overall observed interference is reduced, which results in an enhanced SINR.We can further observe that, by increasing the area threshold A, the success probability drops.This is based on the fact that, the density of the BSs that satisfy the coverage (i.e., BSs) decreases with the increase of the area threshold, resulting in a longer distance between the MU and its serving BSs, thereby reducing the received signal strength.Note that, the interference experienced by a MU is significantly stronger compared to the noise power, and thus, the ID success probability achieved with the employment of the PS scheme is approximately identical to the performance achieved with the TS scheme.Hence, due to space limitation, only the TS scheme is presented in Fig. 3. Finally, we can clearly observe from Fig. 3 that the CA-CoMP scheme achieves a higher coverage probability than the MACH scheme [10] for any number of serving BSs.It was expected since the employment of the MACH scheme associates the MU with multiple BSs which are in the direction of MU's movement, while several nearby BSs that do not belong in the set of cooperative BSs cause severe inter-cell interference, compromising the performance of the MU.Fig. 4 reveals the impact of the area threshold on the downlink spectral efficiency for the proposed CA-CoMP scheme.In particular, Fig. 4 plots the spectral efficiency, i.e., η SE , versus the area threshold, i.e., A, where the MUs employ either the PS or the TS scheme.We can observe from the figure that the network performance is enhanced for low area threshold values by increasing the area threshold.However, by increasing the area threshold beyond a critical point, i.e., the optimal spectral efficiency, the spectral efficiency decreases.This observation is based on the fact that at low area threshold constants, the MUs experience less intra-and inter-cell handover operations, while the MUs are still able to communicate with their serving BSs.In contrast, for large area threshold values, the distances between a MU and its serving BSs increase, and thus the spectral efficiency significantly decreases.Furthermore, it can be observed that, the spectral efficiency achieved with the employment of the PS scheme outperforms the TS scheme, independently of the MUs' velocity.This can be explained by the fact that, by employing the TS scheme, the MUs assign a fraction of the time slot for ID, while the PS scheme enables the MUs to allocate more effective time for ID, resulting in a higher achieved data rate.Furthermore, as expected, for the MUs moving with a high velocity, the information transfer outage occurs (i.e., η SE = 0) due to severe handover overhead, which could be avoided with the employment of the CA-CoMP scheme by increasing the area threshold.Finally, the performance achieved with the conventional CoMP is also illustrated in Fig. 4 for comparison purposes.Note that CoMP does not depend on the area threshold, thus the achieved spectral efficiency remains constant.We can easily observe that the CA-CoMP scheme outperforms the CoMP for both the PS and TS schemes.In particular, by selecting the area threshold equal to A = 1 λ b , a 41% gain is achieved by the CA-CoMP scheme over the conventional CoMP for a velocity of v = 25 m/s.}.It can be observed that, the EH success probability increases with the decrease of the area threshold.This is based on the fact that, a smaller area threshold enables the communication of the MUs with their closest BSs, even if their coverage areas are small, enhancing the harvesting power, and thus improving the EH success probability.Moreover, Fig. 5 presents the approximated EH success probability (provided in Remark 2), which is represented by the dash lines.We can observe that, by ignoring the interfering signal power, i.e., only the intended signals are harvested by the MUs, it achieves a tight lower bound for the actual performance.This was expected since the BSs employ high gain directional antennas that can transmit strong signals to specific directions to enhance the intended signal power at the MUs, while the observed interference signal power at the MUs is much lower and thus it can be neglected.Fig. 6 shows the effect of the area on the MUs' EH terms of the average harvested energy.In particular, Fig. 6 plots the average harvested energy of MUs versus the area threshold A, for different velocities of MUs, i.e., v ∈ {25, 30, 40} m/s.In correspondence to the observations obtained in Fig. 4, Fig. 6 illustrates that the amount of the average harvested energy initially increases and then decreases by increasing the area threshold.Hence, for a MU moving with a certain velocity, we can find an optimal area threshold to achieve the highest EH performance for the CA-CoMP scheme.Moreover, it can be observed that the MUs a lower velocity are able to achieve a better energy harvesting performance, since smaller handover rates provide more effective time for EH process.Finally, we observe that our proposed CA-CoMP scheme enhances the EH performance of MUs for different velocities, which achieves a better EH performance compared to conventional CoMP scheme.This was expected since the CA-CoMP scheme reduces the handover rate, and thus MUs can dedicate more time for EH.Fig. 7 presents the achieved energy efficiency of the CA-CoMP scheme.We can easily observe that the mobility has a negative impact on the energy efficiency, i.e., MUs with lower velocity can achieve a higher energy efficiency.This is based on the fact that, MUs moving with higher velocity need to allocate more time to ensure connection with multiple BSs, which results in less effective time for data and energy transfer for the network.We can also observe that, by employing the CA-CoMP scheme, there is an optimal area threshold to achieve the highest energy efficiency.Moreover, it can be observed that the optimal area threshold increases with the increase of MUs' velocities.This was expected since, the frequent handover becomes a dominant factor that reduces the ID and EH performance, and thus the area threshold for the CA-CoMP scheme should be increased to mitigate the handover rate.Finally, based on the results illustrated in Fig. 4, 6 and 7, we can observe that the CA-CoMP scheme can satisfy different requirements of ID, EH or energy efficiency for diverse applications at MUs by properly selecting the area threshold.

VI. CONCLUSION
For the purpose of addressing the new challenges of 6G mobile networks, in this paper, we have studied a novel CA-CoMP scheme in the context of SWIPT-enabled mobile networks.By exploring the different cell areas caused by the irregular network deployment, our proposed CA-CoMP scheme enables high-mobility MUs to jointly communicate with multiple nearby serving BSs, whose coverage areas satisfy a pre-defined threshold.In order to provide a comprehensive interpretation of the mobility effect on the network performance, we have investigated both the inter-and intra-cell handover operations, where the handover rates were derived in closed form.Moreover, by using tools from SG, we have studied the ID and EH performance achieved with the employment of CA-CoMP scheme and the performance metrics are derived in analytical expressions.Our results have shown that the proposed CA-CoMP can greatly reduce the handover rate, and by properly selecting the area threshold, the CA-CoMP scheme can avoid the information and power transfer outage for the high-velocity MUs.Furthermore, we have shown that compared to conventional CoMP schemes, our proposed technique offers a much better SWIPT performance in terms of the spectral efficiency and the average harvested energy, while the optimal area threshold of the CA-CoMP scheme to achieve the highest SWIPT performance has been numerically demonstrated.Finally, our results have revealed that the PS scheme outperforms the TS scheme in terms of SWIPT performance for high-velocity MUs under the CoMP scenario.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

APPENDIX A PROOF OF PROPOSITION 1
Since each MU only communicates and handovers with the BSs from the candidate BSs set, i.e., Φb , we characterize the spatial distribution of Φb .According to [29], [32], the area of an arbitrary Voronoi cell created by a homogeneous PPP is a random variable, of which the distribution could be accurately approximated by the Gamma distribution with a shape and a scale parameter K and K λ b , respectively, i.e., A i ∼ G[K , K λ b ] with K = 3.5.Let P A denote the probability that the coverage area a BS A i ) is larger than pre-defined threshold A. Hence, P A can be derived as follows Moreover, since the original spatial distribution of the BSs follows a homogeneous PPP, and based on the thinning property, the distribution of the candidate BSs is still uniform, with a density λb = λ P A [16].Therefore, by associating a MU with its N closest BSs from the set of the candidate BSs Φb , the Euclidean plane R 2 is separated into regions, forming an N-th order Voronoi tessellation with PPP Φb and density λb [9].Then, by using the results in [9, Th. 3], the inter-cell handover rate under the proposed CA-CoMP scheme can be derived.

APPENDIX B PROOF OF PROPOSITION 2
The proof follows a similar approach as in [11].Without loss of generality, we consider the movement of a typical MU at the origin, with trajectory from (0, 0) to (1, 0) along the x-axis.As shown in Fig. 8, the triangles denote the location of i-th and (i − 1)-th closest serving BSs of the typical MU, i.e., BS i and BS i−1 , the red circle ω i denotes the location where the typical MU conducts the intra-cell handover with respect to BS i , and θ i is the angle of a beam boundary of BS i with respect to the direction of the movement of the typical MU.Let Ψ i denote the point process of intra-cell handover with BS i .Hence, the average intra-cell handover rate with respect to BS i is equivalent to the intensity of Ψ i .
In order to compute the intensity of Ψ i , we start considering the case where there is at most one intra-cell handover corresponding to BS i , i.e., there are two beams of the same size for a BS.Thus, the event of the intra-cell handover corresponding to BS i occurs when the following two events occur simultaneously, i.e., 1) the point of the intra-cell handover lies in the N-th order Voronoi cell of the N serving BSs, i.e., , where BS i is the i-th closest serving BSs of the typical MU.
2) the point of the intra-cell handover lies on the unit line connecting (0, 0) and (1, 0), i.e., ω i ∈ [0, 1].Firstly, conditioning on θ i , the location of BS i should be located on the strip between the two lines passing through the origin and the point (1, 0).Moreover, the distance from the typical MU to the BS i should not be less than the distance to BS i−1 , i.e., r i ≥ r i−1 , where r i = Hence, by conditioning on both θ i and r i−1 , the possible locations of BS i are displayed as the shaded area in the Fig. 8, and the average number of intra-cell handovers in [0, 1] can be formulated as where erf(z ) = 2 π z 0 e −t 2 dt is the Gauss error function.Then, by averaging over θ i , which is uniformly distributed in [0, π], we have Subsequently, by averaging over r i−1 , we can derive the linear intensity of Ψ i for the case of two beams as following , where is the PDF of distance from the typical MU to the i-th closest BS [29].It should be noted that the above expressions also holds for the case with i = 1, where the proof is same as in [11] and hence is omitted.Then, for the case where each BS has M beams, there are M 2 lines passing each BS to formate M beams.The intensity of the intersection points, i.e., ω i , is the summation of the intensity with respect to M 2 lines, i.e., there are M 2 possibilities of intra-cell handovers corresponding to the i-th closest serving BS.Finally, by also multiplying the velocity of the MU, i.e., v, the result in Proposition 2 is proven.

APPENDIX C PROOF OF PROPOSITION 3
The beam misalignment occurs during the intra-cell handover process.Similar to the Proof of Proposition 2 (Appendix B), we consider that the MU is moving from the original point (where a SSB burst is just received) to the point (vT SSB , 0) (where the next SSB burst will be arrived).Although the beam misalignment could be avoided when the intersection between the MUs' trajectory and beam boundary (i.e., ω i ) is coincident with point (vT SSB , 0), the probability is mathematically equal to zero.Moreover, during one period of SSB burst, the typical MU may cross more than one beam boundaries, e.g., MU moving with high velocity or close to the serving BSs, hence, we derive the probability of a nonmisalignment.The linear intensity of the intersections between the typical MU's trajectory and the beam boundary of its i-th closest serving BS is derived in Appendix B, i.e., μ b,i /v .By considering the process of intersections as a one-dimensional Poisson point process, the void probability of the intersections, i.e., non-misalignment probability, can be evaluated as [11] P(i) ms = P[non-intersections within(0, 0) to (vτ, 0)] = exp −T SSB μ b,i .
Then, the misalignment is derived as P (i) mis = 1 − P(i) ms .

APPENDIX D PROOF OF THEOREM 1
Since each MU is jointly served by N serving BSs, where each BS has M beams, the total average intra-cell handover rate of the typical MU is the summation of the intra-cell handover rate with respect to N serving BSs, i.e., Mv λb Γ(i − 0.5) .
Let X = Mv √ λb π 3/2 , we have where (a) is based on the property of Gamma function that is, Γ( 1 2 + i ) = i − 1 2 i i !√ π; (b) follows the Hockey-stick identity [33]; (c) is based on the identity of the binomial coefficients, and the final step follows the inverse process of step (a).Hence, the final expression for the intra-cell handover rate in Theorem 1 is proven.

APPENDIX E PROOF OF LEMMA 1
The proof directly follows from the definition of the Laplace transform, i.e., where the second step is based on the fact that the channel power gain h i is an i.i.d.exponential random variable, and the last step follows from the probability generating functional (PGFL) of the PPP and the moment generating function (MGF) of an exponential random variable.Then by using [26, 3.194.5],the above integrals in could be solved.

APPENDIX F PROOF OF THEOREM 2
The ID success probability can be re-written as S,i h S,i h 1 2 i 1 2 (r i )| 2 is an exponential random variable with mean x i ∈S P t G S,i (r i ).Then, by evaluating the expectation over distance r 1 , . . ., r N , we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Finally, by evaluating the expectations over G S,i for 1 ≤ i ≤ N , where the joint PMF is given by ( 9), the final results in Theorem 2 are derived.

APPENDIX G PROOF OF THEOREM 3
Since F is an exponential random variable with mean ζ, i.e., F ∼ exp(ζ), the EH success probability can be re-written as where S and I are the intended and interfering signals, respectively; the last step is derived based on the cumulative distribution function (CDF) of an exponential random variable and by ignoring the term exp(ζ), which approaches to one for small ζ [6].Note that the second expectation in (25) includes the Laplace transform of the interference, i.e., By averaging over the distance from the typical MU to its N-th closest serving BS, i.e., r N , of which the PDF is given by (24), the expectation in (25) can be evaluated.Then, let ψ = νη(1−ρ)ζ , the first term of (25) can be solved as following Then, by evaluating the expectations over G S,i , of which the joint PMF is given by ( 9), the final results in Theorem 3 are proven.

APPENDIX H PROOF OF THEOREM 4
The average harvested energy per unit time can be calculated by averaging the instantaneous harvested energy over the random channel, path loss components and the antenna gains, i.e., where QS and QI represent the average harvested energy from intended and interfering signals, respectively.We evaluate QS as where B = T EH eff (1 − ρ)ηP t , and (a) follows from the fact that the channel power gain h i are i.i.d.exponential random variables with mean one.Then, the above expectation can be evaluated by averaging over the antenna gains and distance from the typical MU to the serving BSs, of which the joint PMF and the joint PDF are given by ( 9) and ( 13), respectively.In addition, QI can be calculated based on the Campbell's Theorem [16], i.e., QI = where the integrals can be easily evaluated based on the resulting expression [26, 3.194.5].Finally, by evaluating the expectation over r N , of which the PDF is given by ( 24), the results in Theorem 4 could be derived.

Fig. 1 .
Fig. 1.The Voronoi tessellation of a single-tier cellular network, where the BSs and the waypoints of a MU are represented by triangles and points, respectively.The candidate BSs are represented by solid triangles.The solid and dash lines represent the cell and beam boundaries, respectively, while the trajectory of the typical MU is illustrated by the dotted line.

2 √
λb Γ(0.5+N )(MT b +4Tc ) , i.e., v ≥ v .Proof: According to the definition of the information transfer outage, i.e., η SE = 0, we have 1 − T b μ b − T c μ c = 0. Then by applying the result μ b = M 4 μ c and by substituting the expression of μ c given in Proposition 1, we have

Fig. 3
Fig. 3 illustrates the effect of the area threshold and the number of serving BSs on the ID success probability.In particular, Fig. 3 plots the ID success probability (Theorem 2) with respect to the SINR threshold for different area threshold, i.e., A = {0, 1 λ b }, and different number of serving BSs,

Fig. 5
Fig. 5 depicts the impact of the area threshold on the EH success probability achieved by the employment of the proposed CA-CoMP scheme.More specifically, Fig. 5 plots the achieved EH success probability (Theorem 3) versus the EH threshold, for A ∈ {0, 1 λ b , 2 λ b

Fig. 7 .
Fig. 7. Energy efficiency (E) versus the area threshold (A) for MUs with different velocities.