Impact of Wireless-Powered Communications in Coexisting Mobile Networks

In this letter, we investigate the impact of wireless-powered communications when energy is harvested from multiple static and/or mobile wireless coexisting networks. In a first step, we characterize the aggregate power received by a harvester node when it harnesses the energy generated by the coexisting wireless networks. Considering that the harvester node acts as a transmitter after the harvesting duration, we derive the outage probability for such coexisting scenario. In addition, the throughput achieved by the harvester node is also characterized, and the optimal harvesting duration is identified taking into account the mobility of the coexisting networks, the features of the static networks, the energy harvesting process, as well as the communication performance between the harvester node and the receiver. This letter shows that the distribution of the power received by the harvester from the coexisting networks can be accurately approximated by an $\alpha -\mu $ distribution. Moreover, the mobility also impacts on the optimal throughput of the wireless-powered communications, which is accurately confirmed by the proposed analysis and extensive simulations.


Impact of Wireless-Powered Communications in Coexisting Mobile Networks
in a fixed point of the network is investigated, when the interferers move according to the Random Waypoint (RWP) mobility model. The RWP model was also adopted in [6], which proposed a spatial circular model (SCM) to derive the distribution of the received power caused by mobile nodes at a given spatial position. Most of the existing works do not address large scale WPC networks due to the challenges associated with the characterization of the harvested RF energy in the presence of dominant transmitters. An exception to this is the work in [7], which uses the probability generating functional of a Poisson point process (PPP), to characterize the distribution of the harvested energy.
To the best of the authors' knowledge, the characterization of WPC systems that harvest energy from multiple coexisting networks, including static and/or mobile ones, has not yet been addressed in the literature. In order to fill partly this gap that exists in the literature, this letter aims to first study the distribution of the harvested energy from multiple energy sources belonging to different coexisting networks. Admitting that the harvester node acts as a transmitter in a different band after the energy harvesting period, the main contributions of this letter are: the characterization of the power distribution received from the coexisting nodes; the derivation the outage probability during the transmission period; the identification of the optimal harvesting duration. We study the throughput achieved by the harvester node, identifying the optimal energy harvesting time allocation having into account the mobility of the mobile networks, the features of the static networks, the energy harvesting process, as well as the communication performance between the harvester node and the receiver. 1

II. SYSTEM MODEL A. Coexisting Wireless Networks
In this letter, we consider the scenario illustrated in Fig. 1, where multiple Υ η networks, with η = 1, . . . , υ, coexist in the same RF band, and same spatial region with area X max × Y max . The υ networks can be static or mobile. The nodes of the static networks are deployed according to a homogeneous PPP. The nodes of the mobile networks move according to the RWP mobility model [9], where each node is initially placed in a random position (x, y) sampled from the uniform distributions represented by x ∈ [0, X max ] and y ∈ [0, Y max ], and move to a random ending point with velocity uniformly sampled from [V min , V max ]. The nodes stop at the ending point for pause time T p . After reaching the ending 1 Notations and functions: f X (.) and P(X = x) represent the probability density function (PDF) and the probability of a random variable (RV) X, respectively. Γ(.) represents the complete Gamma function [8, eq. (8. , which was studied in [9]. For modeling purposes, we adopt the SCM, where the analysis of the energy received by the harvester node (node N Tx in Fig. 1) from the network Υ η is derived by considering the nodes located in the L η annuli centered on the harvester node. The radius of the larger and smaller circles of the annulus l ∈ {1, . . . , L η }, are represented by R η,l+1 = (R η,l + l ρ) and R η,l , respectively, where ρ denotes the annulus' width. The nodes of a given network Υ η are thus located in a circular region with area The number of transmitters of the network Υ η located in a particular annulus l ∈ {1, . . . , L η } is represented by the RV X η,l . For both static and mobile networks we consider that X η,l is distributed according to a truncated Poisson distribution given by where λ η,l is the node's spatial density, n η is the total number of nodes of the network Υ η , and τ η is the individual transmission probability. We highlight that for static networks λ η,l is equal for all L η annulus. However, for RWP mobile networks, the spatial distribution of the nodes is approximated by an inhomogeneous PPP. Consequently, for mobile RWP networks, λ η,l takes a different value for each annulus l. In this letter, we consider that the density parameter λ η,l adopted to model the mobile RWP networks is computed as proposed in [5, eq. (8)], which takes into account the annuli's geometry (ρ; L η ; R η,1 ), and mobility parameters (V min ; V max ; X max ; Y max ; T p ).

B. Propagation Effects
The power received by the harvester node (N Tx ) from the n η,l transmitters of the network Υ η located in the annulus l is denoted by I η,l = n η,l i=1 I η,l,i . I η,l,i represents the power received from the i-th transmitter, which can be written as where P Tx η is the transmitted power, ψ i is the instantaneous value of the fading channel and shadowing gain, r η,l is the distance between the i-th transmitter and N Tx , and m denotes the path-loss coefficient. The values ψ i and r η,l represent instantaneous values of the RVs Ψ i and R η,l , respectively. Ψ i represents small-scale fading and shadowing effects. Small-scale fading amplitude is assumed to be Rayleigh distributed with mean power 2σ 2 ζ = 1. Lognormal shadowing is also assumed, with mean and standard deviation of the RV's natural logarithm given by μ ξ = −σ 2 ξ /2 and σ ξ > 0, respectively. To simplify the composite fading model, we consider that the power of the Rayleigh and Lognormal effects can be jointly approximated by a Gamma distribution [10] with scale and shape parameters given by

C. Wireless-Powered Communications
We consider a WPC network with a time-switching protocol. In particular, wireless energy transfer is assumed in the downlink (DL) band, where the node N Tx accumulates energy from the transmitters of the υ different coexisting wireless networks (Fig. 1). We highlight that N Tx operates in two different bands: first it harvests energy during the time interval cT from the DL RF band, and then uses the harvested energy to transmit data to N Rx over the uplink (UL) band. Distinct DL and UL bands are assumed to avoid the adoption of interference mitigation techniques problem at the receiver node N Rx . The transmission lasts (1 − c)T, where T is the total duration of a time-switching cycle and c represents the time splitting factor. We consider an unitary cycle duration, i.e., T = 1.
A Rayleigh fading channel between the nodes N Tx and N Rx is considered, and the distance between the nodes is denoted as d 1 . The transmission power for information transfer depends on the energy harvested in the DL band and is denoted by P N Tx . Consequently, the signal received by N Rx can be written as where h 1 is the channel coefficient from the transmitter N Tx to the receiver N Rx , x c is the normalized information signal transmitted by N Tx , and n d is the zero-mean additive white Gaussian noise (AWGN) at the receiver.
III. HARVESTED ENERGY In this section, we derive the energy harvested by the node N Tx from the υ coexisting networks. Specifically, we describe how the aggregate power received by N Tx from all transmitters of the υ coexisting networks can be approximated by an α-μ distribution. Then, the aggregate power is used to derive the energy harvested during the harvesting period cT.
The work in [11] has proved that the aggregate power received from multiple nodes located over a single circle of a homogenous Poisson network is distributed according to a Gamma distribution. More recently, the work in [5] showed that the aggregate power received at the center of a SCM from the nodes located in an annulus l can be also approximated by a Gamma distribution, when path-loss, fast fading and shadowing effects are considered. In this letter, it is assumed the SCM described in [5]. Consequently, using the method of the moments, the shape and the scale parameters of the Gamma distribution that characterize the aggregate power received by N TX are given by where E[I η,l ] and Var[I η,l ] are the expectation and variance of the power received from the transmitters of the annulus l of the network η, which are respectively given by [5] To derive the aggregate power received from all nodes of a given network, Υ η , the summation of the power received from the L η annuli must be considered. Let {Z η,l } Lη l=1 be independent non-identically distributed (i.n.i.d.) Gamma RVs with parameters k η,l and θ η,l . The aggregate power received from the network Υ η can be written as I η = Lη l=1 Z η,l . In the same way, the aggregate power received by all coexisting network is written as I agg = υ η=1 I η , and represents the aggregate power caused by the nodes located within the L N annuli of the υ coexisting networks, with L N = υL η .
Let {Z j } L N j =1 be i.n.i.d. Gamma RVs with parameters k j and θ j , and W j ∼ Nakagami(m j , Ω j ). The aggregate power can be written as with k j = m j and θ j = Ω j /m j . According to [12], the sum of i.n.i.d. Nakagami-m RVs can be accurately approximated by an α-μ distribution. Consequently, the PDF of the aggregate power received from all transmitters of the coexisting networks (I agg ) can be approximated by an α-μ distribution as follows Var[Y α ] . α and μ parameters can be numerically computed with available software packages using the following moment-based estimators [12] and where the Nakagami-m moments are given as Using the parameters α and μ, the parameterr can be estimated byr Therefore, the harvested energy at the node N Tx , E h , is written as where 0 < ς < 1 represents the energy conversion efficiency, and the RV I agg follows an α-μ distribution with α, μ andr computed from (5), (6), and (7).
IV. THROUGHPUT ANALYSIS After having harvested energy during the harvesting period cT, the node N Tx transmits data in the UL band with P N Tx power, represented by Using (1), the signal-to-noise ratio (SNR) at the receiver node can be defined as where σ 2 n d is the variance of the zero-mean AWGN. Considering a Rayleigh channel with mean power 2σ 2 h between the nodes N Tx and N Rx , |h 1 | 2 is exponentially distributed with parameter 1/(2σ 2 h ). Given a SNR threshold γ 0 , the outage probability of the transmission can be written as Using (10), and considering the PDFs of I agg and |h 1 | 2 , (11) can be rewritten as in (12), as shown at the bottom of the page, where is the scaling value of the product of the RVs I agg and |h 1 | 2 . Since the integral in (12) can only be numerically solved, we propose to approximate the SNR at the receiver by an α-μ distribution, thus P out (γ 0 ) is given as follows where α p , μ p andr p can be obtained by solving the system of equations formed by (5), (6) and (7), substituting the symbols α, μ,r , and E[I agg n ] by α p , μ p ,r p , and respectively. As will be seen, (13) shows to be a very tight approximation, being evaluated instantaneously. (14) denotes the n-th moment of the product of the RVs I agg and |h 1 | 2 . Assuming a communication rate R (in bits/T) and the transmission duration (1 − c)T, the throughput of the communication channel between N Tx and N Rx can be written as From (15), one can notice that (1 − P out (γ 0 )) increases with c. However, the transmission duration decreases with c. The optimal time allocation ratio is defined as where c * can be numerically computed through the roots of ∂Rτ (c) ∂c , which are represented in the condition (17), as shown at the bottom of the page.

V. MODEL VALIDATION AND DISCUSSIONS
In this section, we validate the methodology described in Sections III and IV, by comparing the numerical results with simulations. We consider two coexisting networks (υ = 2): a static network (Υ 1 ), and a network (Υ 2 ) where the nodes move according to the RWP. Three different mobility scenarios are analyzed for Υ 2 , considering the case where nodes are static (E[V] = 0 m/s), or mobile with different average velocities (E[V] = {10.82, 1.50} m/s). The parameters adopted in the validation are presented in Table I, which are divided in the parameters related with the "Propagation Effects" described in Section II-B, the parameters of the network Υ 1 ("Static Network"), the parameters of the network Υ 2 ("Mobile Network"), and other parameters adopted in the WPC model (Section II-C), SCM, and simulations.
The assessment of the model is carried out by comparing Monte Carlo simulation results with numerical results Fig. 2.
In Fig. 2, we compare the cumulative distribution function (CDF) of the aggregate power (I agg ) generated by the coexisting networks (Υ 1 and Υ 2 ) for the different mobility scenarios considered in network Υ 2 (E[V] = {10.82 m/s, 1.50 m/s, 0 m/s}). Table II presents the α-μ distribution parameters adopted in (4) to approximate I agg for the different average velocities E[V]. As can be seen, the numerical results (represented by the "Model" curves) are close to the results obtained through simulation. This indicates that the α-μ distribution in (4) can effectively approximate the distribution of I agg with high accuracy, meaning that the aggregate power of the coexisting networks is accurately modeled when the network Υ 2 is static (0 m/s) or mobile (1.5 m/s or 10.82 m/s). Moreover, we observe that the aggregate power increases with the mobility of the network Υ 2 , which is due to the higher density of nodes closer located to N Tx as the node's mobility increase [9]. Next, we characterize the throughput (R τ ) achieved by the WPC system in the same coexisting scenarios adopted in Fig. 2. To this purpose, we have considered the optimal energy conversion (ς = 1), the range d 1 = 5 m, and the SNR threshold γ 0 = 5 dB at the receiver. Moreover, it is assumed that N Tx uses all the harvested energy to transmit the information. In Fig. 3, we present different curves of the throughput as a function of the time splitting ratio c. The numerical results (represented by the "Model" curves) have considered the approximation proposed in (13) to compute the outage probability. The parameters used in (13) were previously computed as described in Section IV, and their values for the different mobility scenarios are presented in Table III. The throughput, computed with (15), is close to the throughput obtained in the Monte Carlo simulations, as illustrated in Fig. 3. This fact validates the throughput's analytical derivation. The results also show that there is an optimal c value, clearly identifying an upward where extending the harvesting period increases the transmission power, and a downward zone, where the extension of the harvesting period shortens the transmission period. Finally, higher throughput values are achieved for higher mobility scenarios of the RWP network Υ 2 because the amount of harvest energy increases with the velocity of the nodes of the network Υ 2 , as justified by the aggregate power results in Fig. 2.
Finally, we evaluate the accuracy of the optimal time allocation ratio (c * ) proposed in (17). We consider the higher mobility scenario (E[V] = 10.82 m/s), and multiple SNR thresholds (γ 0 = {−10, −5, 0, 5, 10, 15} dB). The simulation results of the throughput and the numerical results of the optimal time allocation ratio (c * ) are illustrated in Fig. 4 (a marker "o" was adopted to indicate the c * value numerically computed with (17)). The throughput inversely increases with γ 0 , as expected. We also observe that γ 0 also impacts on the shape of the throughput curves. As can be seen, the optimal time allocation ratio is accurately approximated by (17) for all γ 0 values, as depicted in the figure. This validates the proposed analysis because to compute c * both harvested energy at the DL and UL communication models are considered.

VI. CONCLUSION
We proposed a novel model to characterize the power received by a harvester node. We consider the case where mobile and/or static networks may coexist together in the same band. The harvester node first collects energy in the band of the coexisting networks and uses the harvested energy to transmit in a different dedicated band. In this letter we have derived exact expressions for the outage probability and throughput achieved by the harvester node, when the harvester node acts as a transmitter after the energy harvesting period. We also derived the optimal harvesting duration, which takes into account the mobility level of the coexisting networks, as well as the radio propagation aspects not only during the harvesting period but also during the communication period. Several numerical and Monte Carlo simulation results are compared to assess the accuracy of the proposed analysis.