Throughput Analysis in a Wireless Powered Mobile Communication System

In this paper, important statistical characteristics of the ratio of double generalized-gamma random variables have been analytically evaluated. The derived results have been applied in a wireless powered mobile-to-mobile communication scenario, which operates in the presence of interference. The performance is investigated using the throughput criterion under two different scenarios, namely delay limited and delay tolerant transmissions. Moreover, since the links’ reliability depends on the mobile nodes locations, the $2D$ Poisson point process is employed in order to more realistically model the physical layer behavior. The analytical results provided generalize previously reported ones, while the numerical results presented reveal the impact of various system’s and channel’s parameters, e.g., distance among nodes, interference, the non-linearity of the wireless medium.


I. INTRODUCTION
In mobile communication scenarios, in which both the transmitter (Tx) and the receiver (Rx) are in motion, a realistic model for the received amplitude behavior includes the product of fading amplitudes, as a result of the doublebouncing propagation mechanism [1]. Depending upon the propagation environment characteristics, various cascaded models have been investigated in the past, e.g., double-Nakagami [2] and the double-Weibull [3]. In addition to the physical justification, the cascaded fading models offer a good fit to experimental data related to mobile-to-mobile (M2M) communications [4].
Another parameter that affects the reliability of these systems is the locations of the Tx, Rx, and the interfering nodes. In this context, the stochastic geometry has been employed to faithfully model the locations of the wireless nodes as random point process, e.g., Poisson line process [5], Poisson point process (PPP) [6], double stochastic Poisson Line Cox Point (PLC) process [7]. For example in [7], the outage The associate editor coordinating the review of this manuscript and approving it for publication was Abderrahmane Lakas . probability (OP) has been analyzed in a non-orthogonal multiple access (NOMA) vehicular communication system based on the PLC process. The outcome from these studies is that stochastic geometry is a useful tool to realistically model the geometrical patterns of the vehicles positions.
In practical scenarios, the M2M communication channel is also subject to co-channel interference due to the hidden terminal problem in contention-based accessing schemes, e.g., [8]- [11]. For example, in [10], assuming Nakagami-m fading channels, novel expressions for the OP of the signalto-interference-plus-noise ratio (SINR) have been provided in a vehicular cooperative communication scenario. Moreover, in [9], the outage probability has been investigated in vehicular links assuming high signal-to-noise ratio (SNR) and weak interference scenarios. On the other hand, a promising technique that has gained the interest of the researchers and engineers is the energy harvesting (EH) from the radio frequency signals in an effort to increase the power of the (energy constrained) nodes [12]- [14]. A common approach in these papers is that the product and/or the ratio of random variables is required to be statistically investigated. More specifically, in [13], the statistical characterization of the VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ product of two κ − µ shadowed random variables (RVs) has been performed and applied to analyze the performance of a wireless powered communication (WPC) system. In [14], an EH mobile cooperative communication scenario has been investigated, in which the double-Rayleigh distribution has been adopted, and the symbol error rate has been evaluated. In this paper, by generalizing previously reported results, the throughput of a wireless powered M2M communication (WPMC) system has been analytically investigated. To this aim, the double generalized-gamma (dGG) distribution has been employed, which is able to model a plethora of different mobile propagation conditions [15]. As a result, by exploiting the generic form of this distribution, various models are included as special cases. Moreover, the PPP is also adopted to model the mobile nodes' random locations. The main contributions are as follows: • Novel expressions for important statistical characteristics of the ratio of independent but not identically distributed dGG RVs are provided. More specifically, the probability density function (PDF), the cumulative distribution function (CDF), the moment generating function (MGF), and the mean are derived; • Novel expressions for the PDF and the CDF of the ratio of random distances in a PPP model are also evaluated; • As an application example, a WPMC scenario has been considered, in which the throughput of delay limited transmission (DLT) and delay tolerant transmission (DTT) scenarios are analytically studied, by assuming deterministic locations for the mobile nodes; • Finally, an approximate expression is also provided for the throughput, in a scenario, in which the locations of the mobile nodes are modeled as PPP. The remainder of the paper is organized as follows. In Section II, important statistical metrics of the ratio of double GG RVs have been provided. In Section III, the statistics of the ratio of PPP have been analytical studied. In Section IV, the performance of WPMC scenario has been investigated in terms of the throughput, while in Section V several numerical evaluated results have been provided. Finally, Section VI concludes this paper.

II. STATISTICS OF THE RATIO OF DOUBLE GENERALIZED-GAMMA RVs
Let X i denote a dGG RV with PDF given as [15] (1), m i , β i are the distribution's shaping parameters, reflecting the clustering and non-linearity of the wireless medium, respectively, whilē γ i denotes distribution's scaling parameter, related to the average SNR, x = m x m x+1 γ xγx+1 , and θ ± x = m x+1 ±m x . It is noted that for m i = m i+1 = 1, (1) simplifies to the double-Weibull PDF, for β i = 2, it simplifies to the double-Nakagami, while for m i = m i+1 = 1 and β i = 2, it simplifies to the double-Rayleigh. Assuming integer values for m i+1 , 1 the corresponding CDF is given by where µ ± i = k ± m i . Theorem 1: Let Z denote the ratio of two independent dGG RVs, with PDF given in (1), i.e., The CDF of Z can be evaluated as 2 where x , . . . , y+x−1 x , and G m,n p,q [·|·] denotes the Meijer's G-function [16, eq. (9.301)], which is a built-in function in many mathematical software packages, e.g., Mathematica. It is noted that for m i , m j → ∞, (4) approximately simplifies to a previous reported result for the ratio of GG RVs [19].
1 This is a reasonable assumption since in practice, the measurement accuracy of the channel is sometimes only of integer order (for the fading parameter) of m i+1 . On the other hand, these results can be considered as upper/lower bounds for scenarios with non-integers values [17]. 2 An alternative representation could be obtained by using the results presented in [18].
Corollary 1.2: The MGF of Z is given by .

III. ON THE STATISTICS OF THE RATIO OF RANDOM DISTANCES IN A PPP
In this section, assuming a PPP model, important statistical characteristics of the ratio of random distances are analytically investigated. Let R i denoting the distance between a point and its nth nearest neighbour in a PPP in R m . The PDF of R i is [23] where λ denotes the node's spatial density, while The corresponding CDF expression is given by where γ (·, ·) denotes the lower incomplete gamma function [16, eq. (8.350/1)]. Proposition 1: Let Y = R i R j denote a RV defined as the ratio of two independent random distances each of them following the PDF given in (12). The CDF of Y is given by (2n)/n (1 + r m ) 2n 2 F 1 1, 2n; n + 1; where 2 F 1 (·) denotes the Gauss hypergeometric function [16, eq. (9.100)]. Proof: By substituting (12) and (13) in the definition of the ratio of two RVs given in (5) and using [16, eq. (6.455/2)], (14) is extracted and this completes the proof. Based on (14), the corresponding PDF expression can be directly evaluated as The mean value of Y can be evaluated by substituting (15) inȲ = E {r} and using [16, eq. (3.251/11)], resulting tō It is worth-noting that the node spatial density λ does not influence the expectation of the ratio of the two random distances in a PPP model.

IV. WIRELESS POWERED M2M COMMUNICATION SYSTEMS APPLICATION EXAMPLE
The previous presented analytical framework can be applied in various wireless communication scenarios, in which the ratio of dGG RVs is observed. Here, as an application example, a throughput analysis of a WPMC system is performed. In this context, the system model under investigation includes a mobile-source (MS) that communicates with the mobiledestination (MD) in the presence of a dominant mobileinterferer (I ), 3 all equipped with single antennas. The system model under consideration is given in Fig. 1(a). In this figure, the subscripts s, h, i denote the parameters related to MS-MD, I -MS, I -MD, respectively. Moreover, the MS is assumed to be energy-constrained and thus it harvests the transmitted energy from I (or other transmitting source). Assuming T is the duration of a block, in the first phase of transmission, MS harvests energy in an interval τ T , where 0 < τ < 1 [12]. Under this scenario, the harvested energy at the MS is given by where 0 < η < 1 is the energy conversion coefficient, P h is the transmission power of the nearest interferer at this phase I , g h follows the PDF given in (1), and d h denotes the distance between I and MS. In the second phase of communication, the information is transmitted in the rest of the block, i.e., (1 − τ )T , while the MD is also experiencing interference (from the same with the first phase node or other nearby). The time-switching operation between EH and information processing procedures is depicted in Fig. 1(b). Moreover, it is assumed that the level of interference is such that the effect of thermal noise on system's performance can be ignored. Certainly, such a scenario is less generic; however, it has clear practical interest and importance and this is the reason why it has been studied many times in the past, e.g., [12]. Therefore, in this interference limited scenario, the received signal-to-interference ratio (SIR) at the MD is given by [12] γ = ητ P h g h g s d a In (18), g s , g i denote the square amplitude coefficients of the channel for the MS − MD and I − MD links, respectively, both following (1), d s , d i denote the distances between MS − MD and I − MD, respectively, and P i denotes the transmit power for I at the second phase. Under the assumption of a DLT scenario and a constant transmission rate R, the average throughput can be expressed as with P out denoting the OP of the system under consideration. Regarding DTT scenario, the throughput of the system is given by In (20), (α) is due to the Jensen's inequality. For evaluating the throughput for both DLT and DTT scenarios, expressions for the OP and the moments should, respectively, be extracted.

A. DETERMINISTIC DISTANCES
In this subsection the performance of the system under consideration has been investigated under the assumption of deterministic distances among the nodes. For evaluating the OP, the following proposition will be exploited Proposition 2: Let W = Z · X z , with Z , X z are RVs with CDF and PDF given in (6) and (1), respectively, assuming equal values for β. The CDF of W can be derived as .
The P out can be evaluated as follows where Pr{·} denotes probability, γ th = 2 R − 1, and A th = analytical expression for the OP is obtained The mean value of γ in (18) is defined asγ = ∞ 0 γ f γ (γ )dγ . By employing the integration by parts in this and after some mathematical simplifications, it results to the following CDF-based definition for the expectation of γ Substituting (23) in this definition, using [20, eq. (21)], and after some mathematical manipulations, finally yields to the following closed-form expression forγ

B. RANDOM DISTANCES
In this section, it is assumed that the location of both the node that transfers power, d h , and the interferer, d i , are modeled as independent homogeneous 2 − D PPP of intensity λ. 5 Under this scenario, the CDF expression in (23) is conditioned on the ratio d h /d i which defines a new RV with PDF given in (15). In order to remove this conditioning, (23) and (15) should be included in an integral of the form given in (5). However, by making this substitution, an exact closed-form solution to this integral cannot be obtained. Therefore, for numerically evaluating the OP, the Gauss-Laguerre quadrature is employed as follows [26] where the abscissas and weights, x i , w i , respectively, can be found in [26,Table 25.9]. After numerically approximating the P out , the throughput for the DLT scenario can be directly 5 The distance between the MS and MD is considered to be constant modeling for example a scenario in which MS, MD are members of a vehicleto-vehicle platoon group [25].  evaluated using (26) in (19). It is noted that the result presented in (26) could be also considered as an upper bound for the OP in a scenario, where dependency exists between the distances d i , d h in (18). Moreover, it is noteworthy that the numerical evaluation of the OP in (26) can only be performed by exploiting the results given in (23) and (15).

V. NUMERICAL RESULTS
In this section, the theoretical results are verified through Monte Carlo simulations, which have been obtained by averaging 10 6 independent trials. Unless otherwise stated, it is assumed R = 1 bps/Hz, i.e., γ th = 1, β = 2.5, d s = 10m, and α = 1.8. 6 In all the figures, the throughput has been evaluated as a function of the energy harvesting time τ . In Fig. 2, a DLT scenario has been assumed and the following parameters were used in (23): m i = m h = 2, η = 0.7, d h = 5m, P i = 7dBW , P h = 12dBW . As it is shown in this figure, the performance improves as the channel conditions of the MS − MD link get better, i.e., m s increases. However, the improvement decreases as m s increases. Moreover, the difference among the performances increases for larger values of the interfering node distance d i . It is also interesting to be noted that variations on d i result to different EH intervals τ that offer the best throughput values. In Fig. 3, also DLT scenario has been assumed and the following parameters were used in (23): m s = 1, d h = 5m, d i = 40m, P i = 7dBW , P h = 12dBW . As it is shown, the best performance is obtained by assuming mild fading conditions for the EH link and severe for the interfering, while the worst for the opposite scenario. It is noted that the performance is better in scenarios with bad channel conditions in both links as compared to the ones with good channel conditions, which reveals the increased impact of the interference to the system's performance. This difference is more clearly observed at the low energy coefficient values.
In Fig. 4, a DTT scenario has been assumed and the following parameters were used in (20) and (25): As it is shown, a considerable reduction on the performance is obtained as the interferer's transmit power P i increases. It is important to mention that the upper bound becomes more tight for higher values of P i , while the gap among the exact results and the ones based on the bound is due to Jensen's property. Finally, in Fig. 5, focusing on the stochastic distances case, assuming a DLT scenario, and with the aid of (26), the following parameters were set: m s = m i = m h = 1, R = 0.5 bps/Hz, i.e., γ th = 0.4, d s = 5m, P i = 0dBW , P h = 15dBW , while n = 50. As it is shown in this figure, the performance decreases as α increases, while as the non-linearity of the medium becomes less severe, i.e., assuming higher values of β, the performance gets better. It is interesting to be noted that the analytical approximated results are quite close to the simulated ones. Moreover, in this figure, the corresponding performance of a scenario with deterministic distances is also included, with d i = d h = 20m. As it is seen, for similar channel conditions and average distances, the deterministic scenario provides a pessimistic result of the throughput and thus can be considered as a lower bound to the random one. In all cases, the simulation results that have been included conform to the analytical ones.

VI. CONCLUSION
In this paper, a throughput analysis of a wireless powered M2M communication scenario has been performed. Towards this objective, important statistical metrics of the ratio of dGG RVs have been evaluated, such as the PDF, CDF, and the MGF. The presented analysis generalizes previously reported results, while it also takes into account the stochastic behavior of the distance among the nodes. It is revealed the important impact of i) the interference, ii) the non-linear behavior of the wireless medium, and iii) the impact of the position randomness, on the system's performance.