Energy efficiency maximization of two-time-slot and three-time-slot two-way relay-assisted device-to-device underlaying cellular networks

energies Article Energy Efficiency Maximization of Two-Time-Slot and Three-Time-Slot Two-Way Relay-Assisted Device-to-Device Underlaying Cellular Networks Van-Van Huynh 1, Nguyen Tan-Loc 2,∗, Ma Quoc-Phu 3,4, Lukas Sevcik 3, Hoang-Sy Nguyen 3,4 and Miroslav Voznak 3 1Modeling Evolutionary Algorithms Simulation and Artificial Intelligence, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam; [email protected] 2Faculty of Technology and Engineering, Thu Dau Mot University, Thu Dau Mot City, Binh Duong Province, Vietnam 3Faculty of Electrical Engineering and Computer Science, Technical University of Ostrava, 17. Listopadu 2172/15, 708 33 Ostrava-Poruba, Czech Republic; [email protected] (L.S.); miroslav[email protected] (M.V.) 4Faculty of Information Technology, Robotics and Artificial Intelligence, Binh Duong University, Thu Dau Mot City, Vietnam; [email protected] (M.Q.-P.); [email protected] (H.-S.N.) *Correspondence: [email protected] Received: 28 May 2020; Accepted: 29 June 2020; Published: 2 July 2020   Abstract: The continuous development of fifth generation (5G) communication and Internet of Thing (IoT) inevitably necessitates more advanced systems that can satisfy the growing wireless data rate demand of future equipment. Device-to-Device (D2D) communication, whose performance is evaluated in terms of the overall throughput, energy efficiency (EE) and spectral efficiency (SE), is considered a promising solution for the aforementioned problem. Thereby, this paper aims at improving the performance of the D2D communication underlaying cellular networks operating on multiple bands by maximizing the EE in its uplink. Thanks to the stochastic geometry theory, it is possible to derive the closed-form expressions for the successful transmission probability (STP), the total average transmission rate (TATR), and the total average energy efficiency (TAEE) of cellular and D2D users in different time slot setting. Particularly investigated and compared in this study, there are one-hop, direct, D2D communication in two time slots (2TS), and multi-hop, indirect, D2D communication in three time slots (3TS) with an additional D2D user acting as a two-way relay to assist the communication. Moreover, an optimization problem is formulated to calculate the maximum TAEE of D2D users and the optimum transmission power of both the cellular and D2D users. Herein this optimization study, which is proven to be non-convex, the Quality of Service (QoS) is ensured as the STP on every link is considered. The herein approach is referred to as relay-assisted D2D communication which is capable of delivering a notably better QoS and lower transmission power for communication among distant D2D users. Keywords: device-to-device (D2D) communication; two-way relay; successful transmission probability (STP); stochastic geometry 1. Introduction The more the world is progressing toward building more IoT systems and smart cities, the more important the development of the D2D communication becomes. Because it is the foundation for the 5G wireless networks upon which the aforementioned structures are built [ 1 ]. Focuses of D2D Energies 2020,13, 3422; doi:10.3390/en13133422 www.mdpi.com/journal/energies Energies 2020,13, 3422 2 of 23 communication underlaying cellular networks infrastructure studies are paid on the cellular capacity increase, user throughput improvement, and battery lifespan extension of the user equipment units (UEs) by deploying the physical proximity between devices in the communication, along with reusing the spectrum resources [2–4]. Proximate devices can communicate with each other, and thanks to this, D2D communication can deliver higher data rate, SE and EE with noticeably lower latency and power consumption to the services [ 5 – 8 ]. Nonetheless, as the spectrum is shared between the cellular users in D2D communication, it inevitably produces problems that require knowledge about interference management. Besides, it should be mentioned that D2D communication can operate on a licensed band (in-band), or an unlicensed band (out-band). Specifically, in-band communication is classified into overlay and underlay, in which there are dedicated spectrum and shared spectrum, respectively. Paper [ 9 ] studied the power controlling in D2D communications aiming at minimizing the total downlink transmission power, which is under the influence of the QoS from UEs. For better scalability of D2D communications, a certain number of different resource allocation methods were investigated in [ 10 ]. Moreover, interference management for better reliability was discussed in [ 11 ]. Accordingly, there are state-of-the-art mechanisms utilized for interference management in D2D networks which are investigated in [ 12 , 13 ]. Paper [ 14 ] described the D2D communications under influence of several users in Multiple-Input and Multiple-Output (MIMO) channels. Last but not least, the author group in [ 15 ] proposed a dynamic graph optimization framework specifically developed for multi-hop D2D communication underlaying cellular networks. To solve the aforementioned interference problem, there is a need for an effective power allocation method. One of the solutions for this is to have a relay user placed in between the D2D pair [ 16 ]. This is referred to as relay-assisted D2D communication and was proven to be an efficient approach which provides better QoS and lower transmission power for communications among distant D2D users. In a relay-assisted D2D communication network, the relay works as a middle contact point which receives and then forwards the signal from a source to a destination. This technique is well-known since it is capable of delivering notably high SE and EE for cellular networks [ 17 ]. A relay-assisted network is categorized depending on how the signal is transmitted in the network. There are three types of network relay: one-way, two-way, and multi-way. A two-way relaying network (TWRN) provides higher SE in comparison with the one-way one as it permits the relay user to conduct concurrent communication with two D2D end users. Recent studies have proven that by utilizing two-way relays to assist a D2D communication, network designers can significantly improve the system performance (indicated by the improvement of outage probability (OP) and sum rate) [18,19]. The throughput performance of two-way relay-assisted D2D communication networks studies were presented in [ 20 – 22 ]. Specifically, in [ 20 ], the ergodic capacity, the finite signal-to-noise ratio (SNR), and the OP of simultaneous wireless information and power transfer (SWIPT) protocol for an amplify-and-forward (AF) TWRN were studied. The author group in [ 21 ] proposed three time switching based energy harvesting policies-namely, dual-source (DS) power transfer, single-fixed-source (SFS) power transfer, and single-best-source (SBS) power transfer. Last but not least, there are two relaying protocols designed in SWIPT, which were presented in [ 22 ]—namely, power time splitting-based two-slot (PTSTW), and power time splitting-based three-slot (PTSTH). Moreover, a so-called time switching-based network coding relaying (TSNCR) protocol was utilized for two-way energy harvesting and presented in [ 23 ]. Notwithstanding, the time slot processing problems have to be taken into account to balance the information transmission and energy harvesting process. Otherwise, the two-way transmission performance can not be improved. Besides, in [ 24 ] proposed a framework to compromise among D2D network parameters, specifically, the density of D2D users and the transmit power, to optimize the ergodic capacity while ensuring minimization of the overall power consumption. Recent study such as [ 25 ] even considered applying a potential game-based power adjustment in in-band multi-hop relay-assisted D2D communication to improve the EE and throughput in the downlink resulting in better coverage quality. Energies 2020,13, 3422 3 of 23 1.1. Related Work There are several pieces of research conducted on the EE of the D2D communications underlaying cellular networks. Before discussing these studies further, it should be noted that the EE metric is important for such wireless networks because it indicates how efficiently the energy is utilized. Thereby, for instance, to improve the system performance, [ 26 ] studied the EE of D2D communications whereas the channel reuse was removed. The D2D communications deployed in heterogeneous networks results in better EE in comparison with the full small-cell deployment, hence, a greener solution for cellular network deployment. Likewise, the EE of D2D systems with cellular network deployment was analyzed and presented in [ 17 , 27 ]. Specifically in [ 27 ], the authors studied the compromise between the EE and the delay in D2D systems under stochastic traffic arrivals and time-varying channel conditions. Moreover, in [ 28 ], a D2D system whereas every user is self-seeking and individually attempts to maximize its EE was investigated. Therein, the EE and SE were studied provided that the system was constrained by the SE and the maximum transmission power limit. In [ 29 ], the extended radio resource management algorithms were integrated into one-way relay-assisted D2D communications aiming to balance off the SE and EE under constraints of mode selection and resource allocation. In the context of one-way D2D communications, the authors in [ 16 , 30 ] studied the system’s maximum achievable transmission capacity. The OP for both cellular and D2D link was guaranteed during this study. In [ 31 ], network users were modelled based on the stochastic geometry on multiple bands and the D2D users’ EE was maximized. Nevertheless, therein the study, the D2D communication was investigated in one way only, given that the influence of the transmission power of the cellular users was neglected. Differing itself from the above papers, this study focuses on the maximization and comparison study of the EE, which is henceforward addressed interchangeably as the TAEE, of the two-way D2D communication underlaying cellular networks in 2TS and 3TS scenario. 1.2. Main Contributions As aforementioned, based on the stochastic geometry theory, an effective solution to maximize the EE of the cellular network in two-way D2D multi-hop communication is investigated. Accordingly, the optimum power of multiple bands is derived. In particular, the users are spatially and randomly distributed within the network following the homogeneous Poisson point process (PPP). It should be noted that the D2D and the cellular users in each band are distributed with different density levels. Additionally, the number of users participating in the transmission process is different. Because of the resource fluctuation and the developing network conditions of 5G, studies conducted in multi-hop scenario are considered more practical than one-hop ones. Besides, the dynamic behaviour of the system in those studies can as well be observed. Moreover, to manage the variation of the wireless channel parameters so as to enhance the network EE, multi-hop D2D communication as 2TS and 3TS mode with each possessing its adjustable power transmission is studied. Last but not least, as there are multiple bands for multiple hops, the D2D users, cellular users and other devices do not interfere with each other. This reduces considerably the complication of the interference management and improves the D2D communication performance. Listed below are the main contributions of this paper: • Firstly, the closed-form expressions for the STP, the TATR, and the TAEE for the cellular users and D2D users in 2TS and 3TS mode of the two-way D2D communication network are derived. • Secondly, there is a proposed optimization problem aiming at maximizing the EE of D2D users while ensuring the QoS of the cellular and D2D users. This problem is solved thanks to a DB algorithm formulated from the fact that the multi-hop D2D users are subject to transmission power and OP constraints. This problem is proven to be non-convex and separated into two sub-problems. Solving the first sub-problem provides the maximized EE of the D2D users along with the optimum transmission power of the direct (one-hop) D2D user in 2TS mode. In the second sub-problem, an objective function is formulated to calculate the maximized EE and the Energies 2020,13, 3422 4 of 23 optimum transmission power of the D2D users in 3TS mode with a two-way relay assisting the communication (multi-hop). It is noteworthy that the objective function is a sum of a number of sub-functions. When all of the sub-functions are maximized, by summing them up, it is possible to obtain the optimal result for the second sub-problem. • Finally, a comparison between this study and the study in [ 31 , 32 ] given similar system model and assumptions is done. It can be observed from the simulation results that the DB algorithm provides a near-optimal solution in 2TS and an outstanding performance in 3TS comparing to the conventional Branch and Bound (BB) algorithm [33]. Ones can find in Section 2the scenario and system model descriptions. Section 3describes the analytical studies and the problem formulation on multiple bands, especially the EE optimization problem. Section 4describes in details the proposed DB algorithm utilized to solve the aforementioned optimization problem. Accordingly, Section 5presents the simulation results. Section 6concludes this paper. Notations : E {.} denotes expectation operation, Pr {.} is OP function. L(x) is Laplace transform of x.Γ(.)is the gamma function with Γ(z) = R∞ 0tz−1e−tdt. 2. System Model A. Scenario Description: In this study, a D2D communication underlaying a general cellular network is considered. It should be noted that the cellular network’s uplink frequency resources are shared with the D2D communication. The resource allocation of the whole network is handled by a base station (BS). Moreover, there are network and channel model depicted. Dissimilar to [ 34 ] where the authors conducted a power allocation problem study on a single band, the power allocation on multiple bands is considered in this paper. As its name suggests, the cellular network operates in a spectrum that is split into K number of bands, whereas each K has a subscript i indicating the ith band, given that i= 1, 2, .., K . Figure 1depicts the two transmission modes focused in this study. The two are based on a cellular network with cellular links between cellular users and the BS receiver. As D2D users exchange information with one another in multiple bands scenario, they operate in two modes: •Two-time-slot (2TS) mode: direct, one-hop communication. • Three-time-slot (3TS) mode: indirect, via an in-between D2D user working as a two-way relay to assist the signal transmission, multi-hop communication. Figure 1. System Models. Energies 2020,13, 3422 5 of 23 B. Network Models: There are four assumptions that are made based on the stochastic geometry theory in this study: Assumption 1. There is a two-dimensional plane < symbolized by Φc,i . Herein, by utilizing the homogeneous PPP, the random spatial distribution of the cellular users in the ith band can be carried out given that the density is δc,i . To denote the cellular users’ transmission power in the ith band, Pc,i is used. The total transmission power of all the cellular users, according to [35], is calculated as Pc=K ∑ i=1 Pc,i. Assumption 2. On < , in the same manner with modelling cellular users, D2D users are distributed with homogeneous PPP Φd,i , and density δd,i . Every transmitter D2D user is coupled with a receiver D2D user which is a distance Rd,i away from each other, and assigned with a Rayleigh fading coefficient fd,i . Similarly, there is a denotation for transmission power of D2D users in the ith band, Pd,i, and Pd=K ∑ i=1 Pd,ito sum them up. Assumption 3. The statistics of the PPP are not influenced by the presence of any typical receiver at the origin, as stated in Palm theory, [ 36 ]. Thus, at the < ’s origin, a typical receiver is placed. It plays the role of a BS typically utilized for cellular uplink transmitting task and a typical D2D receiver assisting the D2D communication. This BS is studied in combination with the D2D users. Moreover, there are presumed reciprocal, invariant uplink and downlink channels founded on consecutive equal time slots model. Assumption 4. If a pair of D2D users, D2D a and D2D b , are considerably distanced from each other, there is a need for another D2D user placed in the middle of the two to assist their signal transmission. This device works as a two-way relay which is denoted as D2D r , given that the transmission power of the relay nodes (RNs) equals to the D2D power in the ith band, Pr,i=Pd,i . Communication between D2D a and D2D b is realized utilizing decode-and-forward (DF) protocol with the assistance of D2D r . The three D2D users are distributed on < following the stationary PPP Φr,i , with density δr,i . Furthermore, the distances from D2D r to D2D a and D2D b , denoted respectively as Rar d,i and Rrb d,i , are constrained so that they can not be equal to or greater than Rd,i , being the distance between D2D a and D2D b . To formulate the relation between the distances, α fraction is deployed so that Rar d,i=Rd,i1 1+α , and Rrb d,i=Rd,iα 1+α . Moreover, for fading channels, the fd,i,lk is used to indicate the the Rayleigh fading coefficient of l-k link (l,k=a,b,r) which satisfies fd,i,lk =fd,i,kl. C. Channel Models: The whole network is a combination of a cellular system and D2D communication. In this context, from [ 36 ], the path loss and Rayleigh fading are placed in a propagation channel model and their effects on the network are studied. Accordingly, a parameter namely received power for either cellular users or D2D users, Prx, is proposed and its expression is given as follows Prx =Ptx f R−m, (1) whereas Ptx is the transmit power. f is the Rayleigh fading coefficient. It should be noted that f follows an independent exponential distribution, and is conditioned so that the unit mean is assigned for every communication link in the system. R denotes the distance between the transmitter and the receiver. mstands for the path-loss exponent given that it is an even number and m≥2. Remark 1. Upon reusing the cellular resources, the D2D communication will cause interference to the typical receiver. This interference is generated by the cellular users, the D2D users, and last but not least, the RNs. 3. Problem Formulation This section presents the work of maximizing the network’s EE in two modes. Firstly, the signal-to-interference-plus-noise ratio (SINR) and the STP of typical receivers are considered. Then, following the STP formula, the closed-form expressions for the TATR and TAEE of D2D Energies 2020,13, 3422 6 of 23 communication of multiple bands are calculated. Eventually, the EE maximization, treated as an optimization problem, is formulated. 3.1. The Signal to Noise Plus Interference Ratio The performance analysis of the cellular transmission and D2D communication in two-way relay-assisted setup is conducted and presented in this subsection. In particular, as the D2D pair and two-way relay user share common cellular spectrum resources, the D2D communication in two modes, 2TS and 3TS, are investigated. A. Two Time Slot Communications Mode A.1. D2D Pair between the Direct Links and Cellular Links In this mode, the D2D pair formed between the direct and cellular links, UED a and UED b , is investigated. They are in charge of exchanging information via direct one-hop D2D link in maximum two time slots without the two-way relay assisting the D2D transmission in the overlapping region. According to [ 37 ], the received signal at the D2D receiver can be calculated in the ith band utilizing the following formula y2TS d,i=Id,00,i+Iid,c0,i+Iid,d0,i+N0, (2) whereas the signal from D2D user is Id,00,i=qPd,iR−m d,00,ifd,00 . Herein, there are fd,00 being the Rayleigh fading channel coefficient, and Rd,00,i being the distance from the D2D transmitter to its corresponding D2D receiver in the ith band. Similarly, the interference caused by the cellular users is formulated as Iid,c0,i=∑ j∈Φc,iqPc,iR−m c,j0,ifc,j0 , in which fc,j0 is the Rayleigh fading channel coefficient, and Rc,j0,i is the distance from the jth cellular user to the typical D2D in the ith band. Moreover, there is the interference caused by the D2D users Iid,d0,i=∑ l∈Φd,iqPd,iR−m d,l0,ifd,l0 . It is constructed of fd,l0 being the Rayleigh fading channel coefficient and Rd,l0,i being the distance from the lth D2D user as the transmitter to its corresponding receiver in the ith band. Last but not least, there is the thermal noise term N0. Due to the nature of the wireless broadcasting process, a typical receiver has to suffer the interference caused by the cellular and D2D communication in the same network. As aforementioned, the D2D communication operates utilizing the same uplink frequency resources of the cellular system. This, indeed, makes the interference an important parameter in this study. The whole network now is interference limited with neglect-able thermal noise. It should be noted that, in this scenario, the D2D pair communicates in 2TS mode. Accordingly, SINR is rewritten as the signal-to-interference ratio (SIR) of the typical D2D receiver in the ith band and expressed as γ2TS d,i=fd,00R−m d,00,i ∑ l∈Φd,i fd,l0R−m d,l0,i+∑ j∈Φc,i Pc,i Pd,ifc,j0R−m c,j0,i . (3) A.2. Cellular Transmissions of the Typical BS As aforementioned, the cellular and D2D transmission process cause interference to the typical receiver. Thus, to calculate the signal receiver that a typical BS, and, accordingly, its cellular user in the ith band, ones can utilize the following formula y2TS c,i=Ic,00,i+Iic,c0,i+Iic,d0,i+N0, (4) Energies 2020,13, 3422 7 of 23 whereas Ic,00,i=qPc,iR−m c,00,ifc,00 is the cellular signal of the typical BS. As in (2), Iic,c0,i= ∑ j∈Φc,iqPc,iR−m c,j0,ifc,j0 , and Iic,d0,i=∑ l∈Φd,iqPd,iR−m d,l0,ifd,l0 are the interference from the cellular users to the typical BS, and the interference from D2D users to the typical BS, respectively, in the ith band. The SIR of the typical cellular receiver in the ith band can be obtained in the same manner as in (3). Thus, it is written as γc,i=fc,00R−m c,00,i ∑ l∈Φd,i Pd,i Pc,ifd,l0R−m d,l0,i+∑ j∈Φc,i fc,j0R−m c,j0,i . (5) B. Three Time Slot Communications Mode This session presents the study conducted on 3TS communication whereas a D2D communication in stochastic geometry, specifically in this case, a relay-assisted D2D link, aids a pair of other D2D users in information exchange with physical-layer network coding scheme [ 38 ]. The main focus is paid on the EE of the 3TS transmission given that the UER is embedded with DF protocol. In two-way 3TS communication mode, a two-way relay D2D user is placed in between the two D2D users to assist the data transmission. In the first and second time slots, information is respectively transmitted from the two D2D users, D2D a and D2D b , to the two-way relay, D2D r , with power Pd,i . Then, within the third time slot, D2D r decodes the information sent from the two aforementioned sources. These two information flows are mixed utilizing some network coding technique, for example, the XOR coding operation. As a result, the information is network-coded and broadcasted back to the D2D a and D2D b . The assisting task of the D2D r is accomplished so far as the information transmission is finished. The two D2D users then can extract the information they need from the returned mixed flow of information. The signals received by the D2Druser in the first and second time slot are expressed as follows yar,i=Iar,00,i+Iir,c0,i+Iir,d0,i+Iir,r0,i+N0, (6) and ybr,i=Ibr,00,i+Iir,c0,i+Iir,d0,i+Iir,r0,i+N0, (7) whereas there are formulas for calculating the signal from the two typical D2D users Iar,00,i= rPd,iRd,00,i 1+α−mfar,00 , and Ibr,00,i=rPd,iαRd,00,i 1+α−mfbr,00 . The far,00 and fbr,00 are the Rayleigh fading channel coefficient between the two D2D users and the two-way relay which assists the D2D user communication in the ith band. Moreover, there are the received interference power from the cellular user, Iir,c0,i=∑ j∈Φc,i Pc,ifc,j0R−m c,j0,i , from the D2D user Iir,d0,i=∑ l∈Φd,i Pd,ifd,l0R−m d,l0,i , and from the two-way relay assisting the D2D users to the typical receiver Iir,r0,i=∑ k∈Φr,i Pd,ifr,k0R−m r,k0,i. Within the third time slot, a portion β , (0 <β< 1), of the transmission power at the two-way relay D2D user, Pd,i , is utilized by the D2D r to broadcast the mixed flow of the signal to the two D2D users. Particularly, βPd,i is allocated for transmission from D2D r to D2D a , and (1−β)Pd,i for D2D r to D2D b . Hence, at D2D a and D2D b , the resulting received signals are calculated respectively as follows yra,i=Ira,00,i+Iir,c0,i+Iir,d0,i+Iir,r0,i+N0, (8) and yrb,i=Irb,00,i+Iir,c0,i+Iir,d0,i+Iir,r0,i+N0, (9) whereas Ira,00,i=rβPd,iRd,00,i 1+α−mfra,00, and Irb,00,i=r(1−β)Pd,iαRd,00,i 1+α−mfrb,00. Energies 2020,13, 3422 8 of 23 Similar to what is previously done, the SIR in the ith band of the typical D2D receivers, D2D a and D2Dbuser, via D2Druser transmission with DF protocol is obtained from γ3TS a,i=min (γar,i,γbr,i,γra,i), (10) and γ3TS b,i=min (γar,i,γbr,i,γrb,i), (11) whereas with regard to the Assumption 3, the instantaneous SIR in the first, second and third time slot can be calculated as γar,i=Pd,ifar,00Ja, (12) γbr,i=Pd,ifbr,00Jb, (13) γra,i=βPd,ifra,00Ja, (14) γrb,i= (1−β)Pd,ifrb,00Jb, (15) Ja=Rd,00,i 1+α−m ∑ j∈Φc,i Pc,ifc,j0R−m c,j0,i+∑ l∈Φd,i Pd,ifd,l0R−m d,l0,i+∑ k∈Φr,i Pd,ifr,k0R−m d,k0,i , (16) Jb=αRd,00,i 1+α−m ∑ j∈Φc,i Pc,ifc,j0R−m c,j0,i+∑ l∈Φd,i Pd,ifd,l0R−m d,l0,i+∑ k∈Φr,i Pd,ifr,k0R−m d,k0,i . (17) Ones can find the derivation of the STP of the typical D2D receiver in the next subsection. 3.2. The Successful Transmission Probability of Typical Receivers Herein this subsection, the STP of typical receivers is derived. Signal transmission is considered successful as long as the transmitter can send the signal packet to its receivers within the third time slot given that the SIR recorded at the receivers is not lower than the whole D2D communication’s threshold value. On the other hand, negative feedback is sent to the D2D communication and the failed-to-be-sent packet is put back on top of the queue waiting for another transmission round. A. Two Time Slot Communications Mode A.1. D2D Pair Communication The Proposition 1 below defines the STP of a typical D2D user receiver when it accomplishes one round of communication in the duration of two time slots. Proposition 1. The SIR threshold of the D2D communication in the ith band is denoted as ζd,i . From (2), the STP of the typical D2D pair receivers in the ith band has to satisfy Pr γ2TS d,i≥ζd,i=e −Θ2TS d,i δc,iPc.i Pd,i2 m+δd,i , (18) whereas Θ2TS d,i=πζ 2 m d,iR2 d,00,iΓ(1+2 m)Γ(1−2 m). Energies 2020,13, 3422 9 of 23 Proof. As aforementioned in the network model section, fd,00 is independently and exponentially distributed with unit mean. In combination with the (2), the STP of the typical D2D receiver in the ith band is written as follows Pr nγ2TS d,i≥ζd,io=Pr      fd,00R−m d,00,i ∑ l∈Φd,i fd,l0R−m d,l0,i+∑ j∈Φc,i Pc,i Pd,ifc,j0R−m c,j0,i ≥ζd,i     .(19) As the study is conducted on 2D planes and γ2TS d,i follows the independent exponential distribution of D2D users and cellular channel gains, with regard to the (19), the STP can be rewritten as Pr nγ2TS d,i≥ζd,io=Pr (fd,00 ≥ζd,iRm d,00,i ∑ l∈Φd,i fd,l0R−m d,l0,i+∑ j∈Φc,i Pc,i Pd,ifc,j0R−m c,j0,i!) =1−Rx=∞ 0xζd,iRm d,00,id"Pr ∑ l∈Φd,i fd,l0R−m d,l0,i+∑ j∈Φc,i Pc,i Pd,ifc,j0R−m c,j0,i≤x!# =E"∏ l∈Φd,i e−ζd,iRm d,00,ifd,l0R−m d,l0,i#×E ∏ j∈Φc,i e−ζd,iRm d,00,i Pc,i Pd,ifc,j0R−m c,j0,i  =Lfd,l0ζd,iRm d,00,i× Lfc,j0ζd,iRm d,00,i . (20) From Laplace transform definition and stochastic geometry theory in [ 35 ], below equations are obtained Lfd,l0ζd,iRm d,00,i=e−δd,iRx=∞ 0E(fd,l0)1−e−ζd,iRm d,00,ix−mdx =e−δd,iπR2 d,00,iζ2 m d,iΓ(1+2 m)Γ(1−2 m), (21) whereas Γ(x)=R∞ 0e−ttx−1dt. Similarly, the function Laplace transformation of Iid,c0,ias Lfc,j0ζd,iRm d,00,i=e −δc,iRx=∞ 0(fc,j0) 1−e−ζd,iRm d,00,i Pc,i Pd,ix−m dx  =e −δc,iπζ 2 m d,iR2 d,00,iPc,i Pd,i2 mΓ(1+2 m)Γ(1−2 m)  . (22) By substituting (21) and (22) into (20), the result in (18) can be obtained. The proof for Proposition 1 ends here. Remark 2. Proposition 1 discloses the relation between a certain number of the network’s key parameters and the typical D2D receiver’s STP. In particular, as the threshold value in the ith band, ζd,i , increases, the γd,i≥ζd,i condition becomes more difficult to be satisfied leading to the decrease of STP. Moreover, as Rd,00,i increases, the STP decreases. This is because the increase in the distance makes the channel fading more serious. On the other hand, the lower the density of the D2D users δd,i or cellular users δc,i , the higher the STP. This can be utilized for alleviating the network interference resulted from different users. Moreover, Pd,i decrease is associated with the STP decrease. Last but not least, ζd,i , Rd,00,i , δc,i , δd,i , and Pc,i increase will lead to the STP decrease. The parameters mentioned in the two previous sentences characterize the cellular transmission power, all of which, after being modified as stated, will introduce interference to the D2D communication, thus, causing the decline of the STP. Energies 2020,13, 3422 16 of 23 Provided that Pd,i is adjusted from Pd,i,index to Pd,i,index −f , then, EEd is reduced by an amount of fi(Pd,i,index)−fi(Pd,i,index −f) . The reduced value is determined predominantly by f0 i(Pd,i,index) as stated in (45). Thus, the f0 i(Pd,i,index) is calculated for i= 1, 2, ..., K and Pd,j is adjusted from Pd,j,index to Pd,j,index −f , whereas j satisfies the condition of f0 jPd,j,index≤f0 i(Pd,i,index) , ∀i= 1, 2, ..., K . This process is replicated for a minimum of n times until which the condition K ∑ i=1 Pd,i=Pd is satisfied, then, the near-optimal solution to (40) is obtained. Based on the aforementioned analysis, the DB algorithm is formulated in the following Algorithm 1. Key steps are presented in details as follows Algorithm 1: The DB algorithm for 2TS Data input: K;Wi;m;δc,i;θc,i;ζc,i;Rc,00,i;Pc,i;δd,i;θd,i;ζd,i;Rd,00,i;Pd;P2TS d,i,down,P2TS d,i,up. Result output: TAEE2TS din (40) Initialize the tolerance ε←10−3that controls the loop; fis the adjustment step of Pd,i, and nis the parameter that controls f. Calculate Pd,i,max based on Theorem 1; Set Pd,i=Pd,i,max; Let d=K ∑ i=1 Pd,i−Pd; and f=d n; Calculate deri=f0 i(Pd,i); while  K ∑ i=1 Pd,i−Pd≥εdo j=arg (min {deri}); if Pd,j−f>P2TS d,j,up or Pd,j−f<P2TS d,j,down then Set derj= +∞; else Pd,j=Pd,j−f; Update derj=f0 jPd,j; end end return TAEE2TS d=K ∑ i=1 fi(Pd,i) B. Three Time Slot Communication Being approached as the same manner as in Algorithm 1, the DB algorithm for solving the non-convex TAEE optimization problem of the two-way relay-assisted D2D communication with DF Energies 2020,13, 3422 17 of 23 protocol in 3TS mode in Kbands, as mentioned in (41), is shown in the following Algorithm 2. Algorithm 2: The DB algorithm for 3TS Initialize the tolerance ε←10−3that controls the loop; Calculate Pd,i,max based on Theorem 1 for fa,i(Pd,i)and fb,i(Pd,i); Set Pd,i=Pd,i,max; Let d=K ∑ i=1 Pd,i−Pdand f=d n; Calculate dera,i=f0 a,i(Pd,i), and derb,i=f0 b,i(Pd,i); while  K ∑ i=1 Pd,i−Pd≥εdo Set j=arg (min {dera,i}), and l=arg min derb,i; if dera,j≤derb,lthen if Pd,j−f>P3TS d,j,up or Pd,j−f<P3TS d,j,down then dera,i= +∞; else Pd,j=Pd,j−f; Update dera,j=f0 a,jPd,j; end else if Pd,l−f>P3TS d,l,up or Pd,l−f<P3TS d,l,down then derb,l= +∞; else Pd,l=Pd,l−f; derb,l=f0 b,l(Pd,l); end end end return TAEE3TS d=K ∑ i=1 fa,i(Pd,i)+K ∑ i=1 fb,i(Pd,i). 6. Numerical Results Herein this section, the TAEE performance of the D2D communication in the 2TS and 3TS case is studied with the help of the DB algorithm in the K bands. The Table 1below lists out the primary simulation parameters [32]. Figures 2and 3plot the TAEE in all bands versus the density of D2D users, along with the transmission power of the cellular users calculated from the DB algorithm, in respectively 2TS and 3TS mode. There is the density setting of [δd,1,δd,2,· · · ,δd,5]=δd, ref ×[ 10, 1, 10, 10, 10 ] . It can be observed that the TAEE obtained from the DB algorithm is nearly coincident with the optimal solution. On the other hand, there exists a considerable gap between the performance plot of the DB algorithm and the conventional BB algorithm [ 33 ]. It can be concluded that the DB algorithm yields near-optimal results and significantly outperforms the ones from conventional BB algorithm. The two figures under consideration share the same common pattern being that after rising to a certain peak, the TAEE gradually declines as the δd,re f continues to increase further. This is because as the density of the users is small, the interference they cause through spectrum sharing is insignificant. This remains true up to a certain checkpoint. Hence, from the beginning, as δd,re f rises together with the ATR, the rise of the interference stays neglect-able resulting in higher and higher TAEE. Nevertheless, as δd,re f reaches its peak and continues to grow, the interference turns to be significant as there is a Energies 2020,13, 3422 18 of 23 higher demand for energy consumption to coordinate the interference which causes the decline of the TAEE. To the direction of the black arrow indicating the Pc,i , the lowest curve corresponds to the highest Pc,i and the highest curve corresponds to the lowest Pc,i . Accordingly, it can be observed that the TAEE curve tends to shift downward as the transmission power of the cellular users Pc,i increases. In fact, the rise in Pc,i results in more severe interference caused by the cellular transmission to the D2D communication. This rises as well the power demand for coordinating the interference thus causing the TAEE in overall to decrease. Table 1. Simulation parameters. Primary Parameters Description Values KNumber of bands 5 WiBandwidth ith 20 MHz Pc,iCellular transmission power 125 mW Pd,iD2D user transmission power 60 mW Pd,i,up D2D user transmission power threshold 20 mW mPath loss coefficient 4 αDistances fraction between relay and D2D user 0.5 βPower allocation at relay D2D user 0.5 εc,iCellular OP threshold 0.05 εd,iD2D user OP threshold 0.05 ζc,iCellular SIR threshold 0 dB ζd,iD2D user SIR threshold 0 dB [Rc,00,1,Rc,00,2,· · · ,Rc,00,K]Cellular link distance [50, 60, 70, 80, 90] [Rd,00,1,Rd,00,2,· · · ,Rd,00,K]D2D link distance [10, 20, 30, 20, 10] [δc,1,δc,2,· · · ,δc,K]Cellular user density [10, 1, 10, 10, 10]×10−5 [δd,1,δd,2,· · · ,δd,K]D2D user density [10, 1, 10, 10, 10]×10−4 [δr,1,δr,2,· · · ,δd,K]Potential two-way relay D2D user density [10, 1, 10, 10, 10]×10−4 12345678910 10-4 10 15 20 25 30 35 40 45 5.2 5.4 10-4 35 40 45 Pc,i = 425; 375; 325 mW Figure 2. The TAEE of D2D communication in 2TS versus δd,re f . Energies 2020,13, 3422 19 of 23 1 2 3 4 5 6 7 8 9 10 10-4 10 15 20 25 30 35 40 45 50 55 Pc,i = 425; 375; 325 mW Figure 3. The TAEE of D2D communication in 3TS versus δd,re f . It can be observed that TAEE performance in 3TS mode is remarkably better than in 2TS mode. This is because, in 2TS mode, the D2D direct link is utilized for signal transmission without two-way relay D2D users assisting the transmission with distance extension, Algorithms 1and 2, resulting in lower performance. On the other hand, in two-way relay-assisted case, with small δd,re f , the TAEE performance of the D2D network is highly boosted with the two-way relay’s assistance. In particular, the TAEE rises sharply as δd,re f increases from 1 to 3. The TAEE continues to increase with a lower rate in association with the increase of the D2D user density. The TAEE in 2TS and 3TS reaches its maximum value at δd,re f equals respectively to 5 or 6. Then, as the D2D user density increases further causing excessive interference to the D2D network, the TAEE performance gradually shrinks. Figures 4and 5plot the TAEE versus the cellular user density δc,re f in 2TS and 3TS mode. The higher the cellular user density, the lower the TAEE. This is because more cellular users introduce higher interference on D2D users. The D2D users, thus, have to consume more power to maintain the QoS leading to the exponential decline of the TAEE. It can be observed that the DB algorithm performs better than the conventional BB one. Nevertheless, as a very high-density level, there is not much difference between them since the TAEE at this level can not be enhanced further. Moreover, the TAEE is plotted for different Rd,re f being the distance between two D2D users. As the Rd,re f becomes higher, the TAEE tends to shift further downward since maintaining the QoS over longer distance requires higher power usage. Moving to the next figure, it can be seen that the TAEE performance in 3TS mode is similar to the 2TS mode but with an overall higher level. This further emphasizes the role of the communication mode in optimizing the transmission power of the cellular users. Besides, the two-way TAEE in 2TS and 3TS mode is compared with the one-way direct link TAEE in [ 31 ]. The TAEE is calculated by first solving the transmission power of the D2D users as in (40) and (41). Accordingly, after putting the transmission power of the cellular users, which is assumed to be constant, Pc,i= 325mW, and the power of devices into the DB algorithm, the TAEE is obtained. Similar to the TAEE in 2TS, the δc,re f increase is associated with the increase of the cellular users’ interference causing the TAEE to decline exponentially. The fact that higher overall TAEE is achieved in 3TS mode helps as well to enhance the D2D network in all bands. The TAEE, in the beginning, is considerably higher calculated from Algorithm 2because they operate at the optimum transmission power. Energies 2020,13, 3422 20 of 23 0.5 1 1.5 The reference density of cellular users c,ref (user/m2)10-5 0 2 4 6 8 10 12 14 16 18 20 Total Average Energy Efficiency TAEEd (Kbps/J) Figure 4. The TAEE of D2D communication in 2TS versus δc,re f . 0.5 1 1.5 The reference density of cellular users c,ref (user/m2)10-5 0 5 10 15 20 25 30 35 40 45 Total Average Energy Efficiency TAEEd (Kbps/J) Figure 5. The TAEE of D2D communication in 3TS versus δc,re f . 7. Conclusions This paper presents the maximization study of the TAEE of the D2D communication underlaying cellular networks in multi-hop 2TS and 3TS communication mode. First of all, by utilizing the stochastic geometry theory, the closed-form expressions for the STP and the TATR of both cellular and D2D users on multiple bands are derived. Accordingly, the optimization problem of maximizing the TAEE is formulated having considered the STP as the QoS of the cellular and D2D users in multi-hop two-way relay-assisted D2D network. Having proven that the TAEE optimization problem is non-convex, by taking advantage of the property of the objective function to be a sum of several sub-functions, a DB algorithm is proposed to collectively optimize the sub-functions to achieve the near-optimal solution Energies 2020,13, 3422 21 of 23 for the objective one. The simulation results reveal that the performance of the authors’ proposed scheme is better than the currently known schemes in the literature. Besides, there is a remarkable EE enhancement accomplished by joint optimization of the transmission power of the cellular and D2D users. Developers can consider applying this method to the D2D communication of the 5G wireless networks in the future. Furthermore, it can be observed from the simulation results that the D2D direct link distance in 2TS mode; the one-way communication of cellular and D2D pair; and the D2D two-way relay user density in 3TS mode are the three factors that cause interference to the overall D2D network with different intensity. For the future study, band selection and power allocation can be considered for cross-tier interference minimization between the cellular and D2D users. Author Contributions: Conceptualization, H.-S.N., M.V., methodology, T.-L.N., H.-S.N., experimental analysis and set-ups, V.-V.H., T.-L.N.; writing-original draft preparation, L.S., M.Q.-P. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The research leading to these results received funding from the Czech Ministry of Education, Youth and Sports under grant No. SP2020/65 conducted at VSB - Technical University of Ostrava. This work was supported also by The Czech Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPS II) project "IT4Innovations excellence in science - LQ1602" and computational time was specifically provided by the projects OPEN-19-38 and OPEN-16-32. Finally, the acknowledgement is given also to the Thu Dau Mot University for supporting this research under grant number DT.20.2-018. Conflicts of Interest: The authors declare that they have no conflict of interest. References 1. Tehrani, M.N.; Uysal, M.; Yanikomeroglu, H. Device-to-device communication in 5G cellular networks: Challenges, solutions, and future directions. IEEE Commun. Mag. 2014,52, 86–92. [CrossRef] 2. Doppler, K.; Rinne, M.; Wijting, C.; Ribeiro, C.; Hugl, K. Device-to-device communications as an underlay to LTE-advanced networks. IEEE Commun. Mag. 2009,47, 42–49. [CrossRef] 3. Lei, L.; Zhong, Z.; Lin, C.; Shen, X. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).