Full-duplex Relayng for D2D Communcaton n mmwave based 5G Networks Boang Ma Hamed Shah-Mansour Member IEEE and Vncent W.S. Wong Fellow IEEE Abstract Devce-to-devce D2D communcaton whch can offload data from base statons by drect transmsson between moble devces s a promsng technology for the ffth generaton 5G wreless networks. However the lmted battery capacty of moble devces s a barrer to fully explot the benefts of D2D communcaton. Meanwhle hgh data rate D2D communcaton s requred to support the ncreasng traffc demand of emergng applcatons. In ths paper we study relay-asssted D2D communcaton n mllmeter wave mmwave based 5G networks to address these ssues. Multple D2D user pars are asssted by fullduplex relays that are equpped wth drectonal antennas. To desgn an effcent relay selecton and power allocaton scheme we formulate a mult-obectve combnatoral optmzaton problem whch balances the trade-off between total transmt power and system throughput. The problem s transformed nto a weghted bpartte matchng problem. We then propose a centralzed relay selecton and power allocaton algorthm and prove that t can acheve a Pareto optmal soluton n polynomal tme. We further propose a dstrbuted algorthm based on stable matchng. Smulaton results show that our proposed algorthms substantally reduce the total transmt power and mprove the system throughput compared to two exstng algorthms n the lterature. Index Terms Full-duplex relayng D2D communcaton mult-obectve optmzaton matchng theory. I. INTRODUCTION Devce-to-devce D2D communcaton whch s regarded as a promsng technology for the ffth generaton 5G wreless networks allows moble devces to communcate wth each other drectly. D2D communcaton can provde hgh data rate transmsson and offload data traffc from cellular base statons [1]. Relays n D2D networks can further reduce the energy consumpton of moble devces enhance the qualty of data transmsson assst connecton establshment among devces and ncrease the range of D2D communcaton [2]. The recently developed full-duplex technques allow relays to smultaneously transmt and receve sgnals by enablng Manuscrpt receved on October 19 216; revsed on Aprl 4 217 September 2 217 and February 7 218; accepted March 25 218. Ths work was supported by the Natural Scences and Engneerng Research Councl NSERC of Canada. The revew of ths paper was coordnated by Prof. Tark Taleb and Prof. Guolang Xue. Part of ths paper was presented at the IEEE Internatonal Conference on Communcatons ICC Kuala Lumpur Malaysa May 216. B. Ma s wth Kwantlen Polytechnc Unversty Surrey BC Canada V3W 2M8 e-mal: boang.ma@kpu.ca. H. Shah-Mansour s wth Vancosys Data Securty Inc Vancouver Canada V6B 2Y9 emal: hamed@vancosys.com. V. W.S. Wong s wth the Department of Electrcal and Computer Engneerng The Unversty of Brtsh Columba Vancouver BC Canada V6T 1Z4 e-mal: vncentw@ece.ubc.ca. loop-nterference cancellaton. By usng full-duplex relayng n D2D communcaton where relays operate n full-duplex mode the spectrum effcency can be mproved over tradtonal half-duplex relayng systems [3] [4]. Resource allocaton n D2D networks has been wdely studed n the lterature [5] [11]. L et al. n [5] nvestgated uplnk resource allocaton for D2D networks. They proposed a scheme for transmsson mode selecton and resource sharng by formulatng a coalton formaton game. In [6] Wang et al. studed a ont channel and power allocaton problem for D2D communcaton and proposed an teratve algorthm to optmze the energy effcency. Qao et al. n [7] proposed a resource sharng scheme to enable concurrent transmssons for D2D communcaton n mllmeter wave mmwave wreless networks. In [8] Nguyen et al. proposed far schedulng polces to explot spatal and frequency reuse n D2D-enabled cellular systems. Xng et al. n [9] proposed resource management algorthms to mprove the spectrum effcency of D2D communcaton n cellular networks. In [1] Sheng et al. proposed an teratve algorthm based on fractonal programmng and Lyapunov optmzaton to balance the trade-off between energy effcency and delay n D2D communcaton. Zhang et al. n [11] studed the channel allocaton problem by usng hypergraph theory where D2D users are allowed to share the uplnk channels wth cellular users. Several exstng studes show that relayng can enhance the performance of D2D communcaton [12] [16]. In [12] Sh et al. studed the energy-effcent spectrum sharng problem n D2D wreless networks. They proposed a mechansm for relayasssted networks to optmze energy consumpton and channel utlzaton. Hasan et al. studed a mult-user relay-asssted D2D network n [13] and formulated a robust optmzaton problem by consderng the channel uncertantes. They showed that relay-asssted communcaton can mprove the aggregate data rate. Wang et al. n [14] studed the feasblty of enablng full-duplex capablty to D2D communcaton n heterogeneous networks. They provded solutons to address the nterference mtgaton ssue n full-duplex D2D networks. In [15] Zhang et al. proposed a power allocaton scheme whch ams to maxmze the data rate of D2D users n a relay-asssted D2D network underlayng the cellular system. Al-Houran et al. n [16] derved a closed-form expresson for energy savng geometrcal zone where relayng s effcent. The aforementoned exstng works reveal the benefts of usng relays n D2D communcaton. Nevertheless none of them ontly consder the mportance of energy savng for batteryoperated devces and system throughput for hgh data rate applcatons.
Smlar resource allocaton problems are studed n cogntve rado networks CRNs. Gharehshran et al. n [17] consdered subchannel allocaton for orthogonal frequencydvson multplexng OFDM based CRNs. Xu et al. n [18] proposed flexble cooperaton schemes to coordnate the prmary and secondary users n orthogonal frequency-dvson multple access OFDMA based CRNs. In [19] Dadallage et al. formulated a ont beamformng power and channel allocaton problem n CRNs. However the aforementoned resource allocaton problems n CRNs do not consder the characterstcs of full-duplex relay-asssted D2D communcaton wth drectonal transmsson [17] [19]. Therefore t s essental to utlze the dstnct features of full-duplex relayasssted communcatons to mprove the system throughput and energy savng for D2D communcaton. D2D communcaton whch can offload data from the base staton and utlze possble drect nter-devce connectons s promsng for 5G systems. However the lmted battery capacty of moble devces and occasonal poor lnk qualty may affect the aforementoned benefts of D2D communcaton. Full-duplex relays wth mmwave technology and drectonal transmsson can save the power of moble devces and mprove the lnk qualty and data rate. Furthermore the relay selecton schemes proposed for half-duplex relays may not be effcent n full-duplex relay-asssted D2D communcaton snce loop-nterference does not exst n half-duplex relayng systems. Utlzng an effcent relay selecton scheme desgned for full-duplex systems s essental to reduce the transmt power of moble devces extend ther battery lfetme and ncrease the data rate. Nonetheless most of the exstng works e.g. [12] [19] do not consder the features of mmwave based full-duplex relayng n D2D communcaton. In ths paper we study relay-asssted D2D communcaton for 5G wreless networks where moble devces transmt data usng drectonal antennas n mmwave frequency band. We consder that the relays have full-duplex capablty whle D2D users transmt n half-duplex mode. The relays can assst devce dscovery connecton establshment and data transmsson for the potental D2D communcaton. We also consder that loop-nterference cannot be fully elmnated due to mperfect self channel estmaton and hardware constrants. The contrbutons of ths paper are as follows: Full-duplex relayng n D2D communcaton: We desgn a ont relay selecton and power allocaton scheme for fullduplex relay-asssted D2D communcaton n mmwave based wreless networks. We formulate a mult-obectve combnatoral optmzaton problem whch consders the mpact of loop-nterference n full-duplex relayng systems. The formulated problem ams to reduce the total transmt power and mprove the system throughput whle satsfyng certan qualty of servce QoS requrements and physcal constrants. Low complexty centralzed and dstrbuted algorthms based on matchng theory: We frst transform the problem nto a one-to-one weghted bpartte matchng problem. We then propose a centralzed algorthm that solves the matchng problem optmally n polynomal tme. We prove that the centralzed algorthm acheves a Pareto optmal soluton of the mult-obectve optmzaton problem. Based on stable matchng we further propose a dstrbuted algorthm whch has a lower nformaton exchange overhead than the centralzed approach. Enhanced performance: Through extensve smulatons we evaluate the performance of our proposed algorthms under dfferent network settngs and compare them wth exstng algorthms proposed n the lterature. Smulaton results llustrate the trade-off between total transmt power and system throughput. Results also show that our proposed algorthms mprove the total transmt power and system throughput substantally compared to the algorthms proposed n [2] and [21]. We extend the work n [22] from several aspects. We mprove the channel modelng by consderng the possble nonlne-of-sght NLOS communcatons. In [22] the obectve s to mnmze the power consumpton. In ths paper we also consder the system throughput as an mportant desgn goal. Dfferent from the sngle-obectve problem formulaton n [22] our mult-obectve optmzaton problem balances the trade-off between total transmt power and system throughput. Our proposed centralzed algorthm s proved to acheve a Pareto optmal soluton n polynomal tme. We further propose a dstrbuted algorthm to reduce the nformaton exchange overhead compared to the centralzed algorthm. Ths paper s organzed as follows. In Secton II we present the system model. In Secton III we formulate the ont relay selecton and power allocaton problem and transform the problem nto a one-to-one weghted matchng problem. We also propose the centralzed and dstrbuted algorthms n Secton III. Smulaton results are presented n Secton IV. Concluson s gven n Secton V. The key notatons and varables used n ths paper are lsted n Table I. II. SYSTEM MODEL We consder an mmwave cellular network wth full-duplex relay-asssted D2D communcaton as shown n Fg. 1. Fullduplex relays are deployed to assst D2D users 1 who ether suffer from poor drect lnk qualty or requre an extended communcaton range. The base staton sends control messages through the control channel to D2D users and relays to coordnate the resource allocaton process. We denote the set of relays as R. The sets of source and destnaton devces whch are asssted by relays are denoted as S and D respectvely where S D =. We further denote the set of sourcedestnaton D2D user pars as L. The th source devce s S and the th destnaton devce d D form a source-destnaton user par l = s d L. We assume that the base staton D2D users and relays are wth the same servce provder. The relays whch assst D2D user pars share parts of the channel resources that the servce provder owns whle those users that do not requre assstance from relays use dfferent spectral resources. A relay can assst multple D2D communcaton pars usng dfferent channels. Each relay s equpped wth two sets of antennas that 1 In the remanng parts of ths paper we use the terms devce and user nterchangeably.
TABLE I LIST OF KEY NOTATIONS AND VARIABLES. Symbols Meanng B Bandwdth of each channel C l r Throughput of D2D user par l asssted by relay r Cl mn Mnmum throughput requrement for D2D user par l D Set of destnaton devces E Set of edges F A matchng G Bpartte graph h Channel gan between transmtter and recever h LI Loop-nterference channel gan L Set of source-destnaton user pars Lz Path loss functon N Background nose power N r Number of channels that relay r can use P s r transmt power of source devce s to relay r P r d transmt power of relay r to destnaton devce d Pr max Maxmum transmt power of each relay Ps max Maxmum transmt power of a source devce P s transmt power matrx of source devces R Set of relays R v Set of vrtual relays R v l Set of vrtual relays that D2D user par l requests S Set of source devces w l r Weght of edge l r W Weghtng matrx of the edges W l Weghtng vector of the feasble edges of D2D user par l x l r Relay selecton ndcator for user par l and relay r X Relay selecton matrx λ 1 Non-negatve coeffcent for total transmt power Non-negatve coeffcent for system throughput λ 2 enable full-duplex operaton. Decode-and-forward protocol s employed by the relays. As dscussed n 3GPP specfcaton [23 pp. 1] and related studes [24] [26] each D2D user par s assgned a non-overlappng orthogonal channel n dedcated mode where nterference among users s avoded. In ths paper the D2D user pars communcate n dedcated mode and are allocated non-overlappng orthogonal channels. A. Channel Model Recently over 1 GHz of spectrum above 24 GHz has been made avalable by the Federal Communcatons Commsson FCC for 5G wreless communcatons [27]. One of the mmwave frequency bands s the 38 GHz band [27] [29]. We consder a relay-asssted D2D communcaton system that operates on the frequency of 38 GHz. Ths band s selected based on ts sgnal propagaton feature and lcensng ssue. The channel model of mmwave communcaton s dfferent from the current cellular channel model. One mportant dfference s that mmwave communcatons requre drectonal antennas. We adopt the channel model ntroduced n [3] and consder the mpact of blockage and reflectons for both lne-of-sght LOS and NLOS cases. If the LOS s blocked we consder potental NLOS communcatons due to reflectons [31] [32]. Let z denote the dstance between the transmtter and recever. Fg. 1. An mmwave based D2D network wth full-duplex relays and drectonal transmssons. D2D user pars can choose whether or not to use full-duplex relays. The base staton coordnates the resource allocaton n the network. The path loss functon Lz n db s L LOS z + 1α LOS logz + Z σlos f LOS exsts Lz = L NLOS z + 1α NLOS logz + Z σnlos f LOS s blocked where L LOS z and L NLOS z are the free-space path loss at reference dstance z for LOS and NLOS sgnals respectvely. Moreover α LOS and α NLOS are the path loss exponents for LOS and NLOS cases and Z σlos and Z σnlos are zero-mean Gaussan random varables wth standard devatons σ LOS and σ NLOS that model the shadowng effects of LOS and NLOS envronments respectvely. Channel estmaton s used to determne the aforementoned parameters. The transmtter and recever perform channel estmaton perodcally va transmttng and analyzng orthogonal plot sequences [33]. In mmwave technology drectonal antenna s used to mprove the antenna gan. A sectored drectonal transmttng antenna model s proposed n [34]. We use ths sectored antenna model where the antennas acheve a constant hgh gan n the man lobe and a constant low gan n the sde lobe. Let Θ t represent the angle of departure of sgnals. The transmttng antenna gan s gven as follows: { G t Θ t M t Θ t Θ t HPBW = m t Θ t HPBW < Θt 18 2 where M t m t and Θ t HPBW are the man lobe gan sde lobe gan and half power beamwdth for the transmttng antenna respectvely. Smlarly let Θ r represent the angle of arrval of sgnals. The recevng antenna gan s gven as follows: { G r Θ r M r Θ r Θ r HPBW = m r Θ r HPBW < Θr 18 3 where M r m r and Θ r HPBW are the man lobe gan sde lobe gan and half power beamwdth for the recevng antenna respectvely. The antenna gan between devces and s denoted as G = G t Θ t Gr Θ r where Θt s the angle of departure of sgnal from transmtter to recever and Θ r s the angle of arrval of sgnal n recever sent from transmtter. If and belong to a par of communcatng 1
devces Θ t and Θr are both o snce we assume the transmttng and recevng antennas are accurately algned. The total gan ncludng channel and antenna gans between devces and can be represented as h = G /Lz where z s the dstance between the correspondng devces. B. Throughput of a User Par To obtan the throughput of a user par we need to determne the sgnal-to-nterference plus nose rato SINR n each hop of the relayed communcaton. For user par l = s d whch s asssted by relay r we denote as the transmt power of source devce s to relay r. We denote the transmt power of relay r to destnaton devce d as P rd. The SINR from source s to relay r s SINR sr = h s r h LI P rd + N 4 where h LI P rd represents the loop-nterference receved by full-duplex relay r and N s the nose power. Smlar to [15] [35] [36] we use the loop-nterference channel gan h LI to determne the loop-nterference power receved by the fullduplex relay. The channel gan h LI s defned as the rato between the receved loop-nterference power and transmt power of the full-duplex relay. It characterzes the result of power leakage from the transmtter of the full-duplex relay to ts recever due to mperfect loop-nterference cancellaton. The mutual nterference among dfferent user pars s avoded snce each user par s allocated an orthogonal channel. The SINR from relay r to destnaton devce d s SINR rd = h rd P rd + N 5 where s the nterference nduced by source devce s. Note that both the source-to-relay SINR and relay-todestnaton SINR have ncluded the effects of drectonal transmsson and possble NLOS due to reflecton n mmwave communcatons. In a full-duplex relayng system usng decodeand-forward protocol the throughput of user par l L asssted by relay r R can be obtaned from [35] B mn log 2 1 + SINR sr log 2 1 + SINR rd 6 where B s the bandwdth of each channel. III. PROBLEM FORMULATION In ths secton we formulate a ont relay selecton and power allocaton problem by takng both the total transmt power and the system throughput nto consderaton. Ths s motvated from the fact that the lmted battery capacty of moble devces requres a low transmt power whle applcatons such as vdeo sharng requre a hgh throughput data transmsson. Our obectve s to mnmze the total transmt power and smultaneously maxmze the system throughput of the D2D network. We formulate the problem as a mult-obectve optmzaton problem to consder both of the aforementoned factors. We consder the mpact of loop-nterference n fullduplex relayng systems and derve a closed-form expresson of the throughput of a user par. Note that for full-duplex relays a hgher transmt power of a relay ncreases the receved power n the destnaton devce of D2D user pars. However t also nduces a hgher loop-nterference at the same tme. Ths means that transmttng wth full power does not necessarly ncrease the throughput. Accordng to 6 the throughput of a user par asssted by a relay s the mnmum throughput of the source-to-relay and relay-to-destnaton lnks. Thus the transmt power of the source devce s wasted f the sourceto-relay throughput s greater than the relay-to-destnaton throughput. The same stuaton holds regardng the transmt power of the relays. In other words the power of a full-duplex relay can be adusted accordng to the allocated transmt power of the source devces. The source-to-relay throughput should be equal to the relay-to-destnaton throughput n order to save transmt power [36] [37]. Ths helps us to obtan a closedform expresson for the source-to-destnaton throughput. Consder D2D user par l = s d L and relay r R the above condton mples that SINR sr = SINR rd. In ths case from 4 and 5 P rd can be expressed as a functon of as follows: f rd + N 2h2 r d N h rd P rd = 2h LI h rd where f rd = 4h sr h LI h rd + 4N h sr h LI h rd. Therefore when we substtute 5 and 7 nto 6 the throughput of user par l L asssted by relay r R can be expressed as follows: C lr = B log 2 1 + h r d P rd. 9 + N To ntroduce our obectves we consder the transmt power of moble devces. Snce the relays are plugged nto the power source and have suffcent power supply the obectve s to mnmze the total transmt power of the transmttng D2D devces by selectng relays effcently. We denote P s = s Sr R as the transmt power matrx. The total transmt power can be represented as: f 1 P s =. 1 s S r R Another mportant desgn obectve s to maxmze the system throughput. Maxmzng the system throughput s equvalent to mnmzng the followng functon: f 2 P s = C lr. 11 l L r R We should note that dfferent power consumpton models wll result n dfferent trade-off between transmt power and throughput. However our proposed mechansm s applcable to general power consumpton models. We wll later address the trade-off between transmt power and throughput when formulatng the mult-obectve optmzaton problem. Both desgn obectves n 1 and 11 are mportant. A sngle-obectve formulaton that solely consders ether 1 7 8
or 11 s nsuffcent to capture the complete desgn goals of the system. However these obectves are conflctng n the sense that a hgher throughput may result n a hgher transmt power. Ths fact motvates us to formulate a mult-obectve problem whch consders both obectves smultaneously. To formulate the mult-obectve optmzaton problem we ntroduce the QoS requrements for dfferent applcatons as well as the physcal constrants of the devces and relays. We denote Cl mn as the mnmum throughput requrement for D2D user par l. To guarantee that the mnmum data rate requrement s satsfed for each D2D user par we ntroduce the followng constrant: C lr Cl mn l = s d L. 12 r R We defne matrx X = x lr l Lr R to ndcate the relay selecton for the user pars where bnary varable x lr = 1 f user par l selects relay r. Otherwse x lr =. Then the followng constrant ensures that each user par can be asssted by only one relay. x lr = 1 l L. 13 r R Furthermore the number of D2D user pars asssted by relay r should be less than or equal to the number of channels that r can use. Let N r denote the number of channels n relay r. We have x lr N r r R. 14 l L We denote Ps max as the maxmum transmt power of each moble devce and Pr max as the maxmum transmt power of each relay. To ensure the transmt powers of moble devces and relays do not exceed the maxmum transmt power allowed we ntroduce the followng constrants: x lr P max s l = s d L r R 15 P rd x lr Pr max l = s d L r R. 16 Notce that snce P rd s always non-negatve. Let P 1 r d denote the nverse of functon P rd. Snce P rd gven n 7 s strctly ncreasng the nverse functon always exsts. Thus from 16 we have P 1 r d x lr Pr max l = s d L r R. 17 When x lr = 1 P 1 r d x lr Pr max = P 1 r d Pr max whle = when x lr =. Therefore nequalty 17 can be rewrtten as: x lr P 1 r d P max r l = s d L r R. 18 Consequently constrant 16 s equvalent to the followng constrant: x lr P max s l = s d L r R 19 where constant P max s By combnng 15 and 19 we have x lr mn = P 1 r d P max r. 2 P max s l = s d L r R. max P s 21 Then the mult-obectve relay selecton and power allocaton problem can be formulated as: mnmze F P s = f 1 P s f 2 P s T 22a XP s subect to x lr { 1} l L r R 22b constrants 12 14 and 21. Problem 22 s a mult-obectve combnatoral optmzaton problem. The weghted sum approach s commonly used to transform a mult-obectve optmzaton problem nto a scalar optmzaton problem [38]. Note that n problem 22 we take the mpact of the loop-nterference channel gan nto consderaton when ntroducng constrants 12 and 21. In addton the drectonal antenna gan and channel gan for mmwave communcatons are ncorporated n the data rate n constrant 12. A. Reformulaton Usng the Weghted Sum Method In ths subsecton we reformulate problem 22 usng the weghted sum method. The weghted sum approach consders a lnear combnaton of all desgn obectves and s commonly used n solvng mult-obectve optmzaton problems [39]. Pareto optmalty s an mportant soluton concept for the mult-obectve optmzaton problems. An outcome s Pareto optmal when a sngle desgn obectve e.g f 1 P s cannot be mproved wthout degradng the other obectve e.g f 2 P s. Problem 22 whch consders both total transmt power and system throughput can be reformulated as a weghted sum problem. The Pareto optmal soluton of problem 22 can be obtaned by solvng the followng problem: mnmze XP s λ 1 f 1 P s + λ 2 f 2 P s 23 subect to constrants 12 14 21 and 22b where λ 1 and λ 2 are non-negatve coeffcents to adust the weghts of obectves f 1 and f 2. For example f λ 1 = 1 λ 2 = problem 23 mnmzes the total transmt power. By changng the value of λ 1 and λ 2 dfferent Pareto optma can be obtaned. Problem 23 can be solved usng methods such as branch-and-bound generalzed Benders decomposton or outer approxmaton. However none of the aforementoned methods can guarantee to obtan the soluton n polynomal tme. In the next subsecton we transform the mult-obectve combnatoral optmzaton problem nto a matchng problem [4] to obtan the Pareto optmal soluton. We then propose a Pareto optmal relay selecton and power allocaton algorthm. B. Bpartte Graph Constructon Matchng theory can provde tractable solutons for combnatoral problems. For resource allocaton n wreless net-
Fg. 2. a A bpartte graph wth four D2D user pars and three relays. b A matchng example wth four edges l 1 r 2 l 2 r 1 l 3 r 2 and l 4 r 3. c The one-to-one matchng wth vrtual relays n dashed frames. works matchng theory can address how resources can be allocated to users [4]. Users and resources are consdered as vertces n dsont sets that wll be matched to each other. In ths paper we regard vertex sets as the set of D2D user pars L and the set of relays R. We consder all possble relay selectons as dfferent matchngs. The goal s to fnd the best matchng.e. relay selecton between D2D user pars and relays whch results n the optmal soluton of problem 23 and s also the Pareto optmal soluton of problem 22. To acheve ths goal we construct a bpartte graph whch conssts two dsont vertex sets and edges as shown n Fg. 2a. A matchng s represented by a set of dstnct edges. We use tuple l r to denote the edge that connects D2D user par l wth relay r. For nstance as shown n Fg. 2b the graph wth four sold edges l 1 r 2 l 2 r 1 l 3 r 2 and l 4 r 3 corresponds to a matchng example. A weght s allocated to each edge of the graph. The purpose of constructng the weghted graph s to transform problem 23 nto an equvalent matchng problem. Thus we wll determne the weght of each edge n order to acheve ths goal. We also defne the mnmum weghted matchng as a matchng where the sum of the weghts of those edges selected n the matchng has the mnmum value. We then obtan the mnmum weghted matchng from whch we can determne the optmal soluton of problem 23. To determne the weght of each edge n the graph we frst ntroduce the matchng rules. These rules guarantee that the optmal matchng s wthn the feasble regon of problem 23. By consderng constrant 13 only a sngle edge can be connected wth a D2D user par n the matchng. Meanwhle we allow at most N r edges to be connected wth relay r R n order to satsfy constrant 14. Constrants 13 and 14 ndcate that the equvalent matchng problem s a many-toone matchng. We further consder 12 and 21 whch are related to the transmt power varables. Accordng to 13 each D2D user par can only be asssted by one relay. We assume that D2D user par l s asssted by relay r.e. x lr = 1 x lr k = r k R \ {r }. If there exsts a transmt power that satsfes both of the followng nequaltes: C lr C mn l 24 and mn P max s max P s 25 then s n the feasble regon determned by constrants 12 and 21 and r s a feasble relay for D2D user par l. Otherwse we regard relay r as an nfeasble relay. That s D2D user par l wll not use relay r. In ths case nfeasble relay r s excluded from the consderaton of D2D user par l and edge l r wll not be selected n the matchng. By dong so the optmal matchng wll be n the feasble regon of problem 23 and satsfes all of ts constrants. To fnd the feasble relays for a D2D user par we frst consder 24 and 25. By substtutng 9 nto 24 we have B log 2 1 + h r d P rd Cl mn + N 26 whch can be rewrtten as h rd P rd 2 C mn l /B 1. 27 + N Then by substtutng 7 nto 27 we obtan f rd + N 2h2 r d N h rd 2 Cmn l /B 1. 2h LI + N 28 The left-hand sde of 28 s an ncreasng functon of. When relay r s selected to assst user par l we can fnd the mnmum transmt power of source devce s. To acheve the mnmum data rate requrement Cl mn the mnmum transmt power of source devce s wth assstance of relay r denoted by Ps mn r can be obtaned from 28. We have P mn s r h LI N 2 Cmn l /B 1 2 + N h rd 2 Cmn l /B 1 2. h sr h rd h LI 2 Cmn l /B 1 29 From 25 we can determne the maxmum transmt power of source s to relay r denoted by Ps max r where Ps max r mn Ps max max P s. 3
If Ps mn r > Ps max r there s no feasble transmt power that satsfes both 24 and 25. In ths case relay r s an nfeasble relay for D2D user par l and edge l r s an nfeasble edge n the graph. In order to exclude such nfeasble edge we set ts weght to + so that the edge wll not be consdered when usng the mnmum weghted matchng method. After excludng all nfeasble edges from consderaton the remanng edges are the feasble relays for the correspondng D2D user pars. We now determne the weght of each feasble edge n the graph. We assume r s a feasble relay for D2D user par l.e Ps mn r Ps max r. We denote the weght of edge l r by wl r where l = s d L r R. To determne wl r we assume that only D2D user par l and ts feasble relay r exst n the network. In ths case the bpartte graph only has one edge.e. edge l r. Ths assumpton mples that x lr = 1 x lmr n = and P smr n = for all l m r n l r l m = s m d m L r n R. We can obtan the optmal transmt power of source devce s by solvng the followng problem: mnmze P s r λ 1 λ 2 C lr 31a subect to P mn s r P max s r. 31b For a partcular edge l r the weght wl r s determned by the optmal value of problem 31. We wll later show that solvng problem 23 s equvalent to fndng the mnmum weghted matchng where the sum of the weghts of the edges selected n the matchng has the mnmum value. Let P s r denote the optmal soluton of problem 31 that unquely exsts due to the followng theorem. Theorem 1. Problem 31 s a strctly convex optmzaton problem whch has a unque optmal soluton. The proof of Theorem 1 s gven n the Appendx. Thus the optmal soluton Ps r whch represents the optmal power of D2D user par l = s d asssted by relay r can be obtaned by usng technques such as nterorpont method. Once Ps r has been obtaned the optmal throughput denoted by Cl r = C lr Ps r can be found as well. We can then compute the weght of edge l r when we assume that user par l s asssted by relay r. The weght of edge l r s set to g lr Ps r = λ 1 Ps r λ 2 Cl r. Gven a matchng the resource allocaton matrx X s fxed and the constrants of problem 23 are all satsfed. For example n the matchng shown n Fg. 2b that ncludes edges l 1 r 2 l 2 r 1 l 3 r 2 and l 4 r 3 we have x l1r 2 = x l2r 1 = x l3r 2 = x l4r 1 = 1. Let F denote a matchng and E denote the set of all edges of the bpartte graph. For matchng F E we show that the mnmum weghted sum of transmt power and throughput s equvalent to the summaton of the weghts of all edges n the matchng. To do so we formulate the followng problem whch mnmzes the weghted sum of transmt power and throughput for matchng F: mnmze λ 1 λ 2 C lr P s l r F l r F 32a subect to P mn s r P max s r l r F 32b where l = s d. Snce λ 1 and λ 2 are constants the obectve functon s equal to λ1 λ 2 C lr l r F = l r F g lr. 33 The obectve of problem 32 s to mnmze the summaton of all user pars weghted sum for a gven matchng. Accordng to 32 and 33 the optmal transmt power of each source devce s ndependent from others when we know the matchng. Therefore gven matchng F for each edge l r F we can obtan the optmal transmt power of source devce s denoted by Ps r by solvng problem 31. The optmal value of problem 31 for edge l r s g lr Ps r. Thus gven matchng F the optmal value of problem 32 s g lr Ps r. 34 l r F Notce that w lr = g lr Ps r l r F. Thus the optmal value of problem 32 can also be represented as l w r F l r whch s the summaton of the weghts of all edges n matchng F. Up to now gven a matchng we have shown that the mnmum weghted sum of transmt power and throughput equals to the summaton of the weghts of all edges n the matchng. We now focus on fndng the optmal matchng.e. mnmum weghted matchng. We construct bpartte graph G = L R E W where D2D user par set L and relay set R are the sets of vertces and W = w lr s l Lr R the weghtng matrx of the edges. The mnmum weghted matchng can be obtaned as follows: F = arg mn F E l r F w lr. 35 By constructng the bpartte graph we have transformed the combnatoral relay selecton and power allocaton problem nto a many-to-one matchng problem. However t s stll dffcult to obtan the optmal soluton for ths many-to-one matchng problem n an effcent manner. Although each relay can assst multple D2D user pars each user par s allocated a non-overlappng channel. By utlzng ths feature we can further transform the many-to-one matchng problem nto a one-to-one matchng problem whch can be solved optmally n polynomal tme. Snce each relay r has N r channels we replace r wth N r vrtual relays whch are located at the same locaton. Each vrtual relay s assgned a non-overlappng channel as shown n Fg. 2c. We use r k to represent the vrtual relay that operates on the k th channel of relay r. We denote the set of vrtual relays as R v = {r 11 r 1Nr1 r R 1 r R Nr R } whch represents a new vertex set. For ease of exposton we denote r v as the th vrtual relay n set R v. We denote the new set of edges as E v and the new weghtng matrx as W v. By dong so we transform the many-to-one matchng problem
nto a one-to-one matchng problem denoted by a new bpartte graph G v = L R v E v W v. Note that one-to-one matchng problems can be solved n polynomal tme [41 Ch. 3]. C. Pareto Optmal Relay Selecton Up to now we have constructed the bpartte graph to obtan a soluton of problem 23 usng a one-to-one matchng. In the followng we prove that the obtaned soluton s a Pareto optmal resource allocaton n terms of total transmt power and system throughput. We frst formally defne the Pareto optmal resource allocaton n Defnton 2 wth the help of Defnton 1 [42 Ch. 1]: Defnton 1. In a mnmzaton problem gven resource allocaton decson matrces A and A the allocaton outcome f 1 A f 2 A s domnated by f 1 A f 2 A f f 1 A f 1 A f 2 A f 2 A and f 1 A f 2 A f1 A f 2 A. Defnton 2. A resource allocaton decson A s Pareto optmal f and only f there does not exst another resource allocaton A domnatng A. Based on the defnton of Pareto optmal resource allocaton we have the followng theorem. Theorem 2. The soluton of problem 23 denoted by X Ps s a Pareto optmal resource allocaton. Proof. Assume X P s s the soluton of problem 23 and s not Pareto optmal. Then there exsts an allocaton X P s such that f 1 P s f 1 P s and f 2 P s f 2 P s. Gven λ 1 and λ 2 we have λ 1 f 1 P s + λ 2 f 2 P s λ 1 f 1 P s + λ 2 f 2 P s. Ths means that X P s s not the optmal soluton of problem 23. Here we have the contradcton whch completes the proof. D. Algorthm Desgn In ths subsecton we develop the centralzed and dstrbuted algorthms. 1 Centralzed Relay Selecton and Power Allocaton Algorthm: We frst propose a centralzed relay selecton and power allocaton algorthm as shown n Algorthm 1 based on the Hungaran method [43] to solve the one-to-one matchng problem. As proven n Theorem 2 our proposed algorthm acheves a Pareto optmal soluton of problem 23. We defne X v as the optmal vrtual relay selecton matrx. The algorthm contans two man steps. As shown n Algorthm 1 we frst create the weghted bpartte graph by computng the weghts of all edges n the graph Lnes 2 to 16. By computng the mnmum value of the weghted sum of transmt power and mnus throughput we obtan the weght of each edge Lne 7. If the mnmum data rate requrement cannot be satsfed we set the correspondng weght to + Lne 9. Snce the Hungaran algorthm used n ths algorthm s to fnd the mnmum sum of all weghts those edges wth weght + wll not be consdered n the Hungaran algorthm.e. Algorthm 2. Thus we exclude the vrtual relays whch would result n an nfeasble soluton for the correspondng D2D user Algorthm 1: Centralzed relay selecton and power allocaton algorthm 1 nput L S D R v Cl mn h s r h r d h s d 2 for l L do 3 for r v R v do 4 Calculate P mn s r v 5 f P mn s r v l L Ps max Pr max λ 1 λ 2 h LI l = s d L r R P max s r v max and Ps r v then usng 29 and 3 6 Solve problem 31 to obtan Ps r v 7 wl v r v := λ1p s r v λ2c l r v P s r v 8 else 9 wl v r v := + 1 end 11 end 12 f Ps mn r v > P s max r v rv R v then 13 Return nfeasble 14 end 15 end 16 W v := w l r v l Lr v Rv 17 X v := HungaranW v 18 output: Vrtual relay selecton decson X v and transmt power matrx Ps. pars. If the mnmum data rate requrement for a user par cannot be satsfed regardless of whch vrtual relay the user par uses Lne 12 ths user par wll have to communcate usng the cellular base staton nstead of D2D mode and operate under the resource allocaton rules for regular cellular users. If the mnmum data rate requrement can be satsfed then the Hungaran method s used to obtan the optmal vrtual relay selecton matrx X v Lne 17. The optmal matchng that mnmzes the summaton of the weghts can be obtaned by the Hungaran method [43]. For the sake of completeness we present the Hungaran method n Algorthm 2. We use z = zl 1 zl zl L to ndcate the current matchng result. For example zl 3 = r5 v means that user par l 3 s matched wth vrtual relay r5. v We defne m = m lr v as a markng ndcator of the l Lr v Rv edges of the bpartte graph.e. the elements of W v. Ths ndcator s used to mark the element that has the mnmum weght. The edge correspondng to the marked element s regarded as a potental edge n the optmal matchng. Element w lr v s marked when m lr v s set to 1. If m lr v = element w lr v s not marked. We defne cr = cr l and l L cc = cc r v as the row cover ndcator and column r v R v cover ndcator respectvely. The purpose of coverng a row or a column s to exclude the elements of that row or column from further operatons. For example row of the weghtng matrx W v s covered and excluded from operatons f cr l = 1. The man steps of the algorthm are as follows. Frst the mnmum value of each row of W v s subtracted from all elements n ts row Lnes 3 to 5. Then zeros n the modfed matrx W v are found and ther ndces are recorded n z Lnes 6 to 1. Next the algorthm verfes whether all user pars and vrtual relays are matched by checkng the number of zeros n z Lnes 15 to 17. If there s no zero n z the matchng s obtaned. Otherwse the weghtng
Algorthm 2: Hungaran method 1 nput Weghtng matrx W v 2 ntalze z = z l l L := {} 1 L X v = x l r v l Lr v Rv := {} L R v m = m l r v := {} l Lr v Rv L R v A := L and B := R v 3 for l L do 4 Set w l r v := w l r v mn r k v Rv{w l r k v } rv k R v 5 end 6 whle A = do 7 Set zl := r v l A r v B such that w l r v = 8 Set A := A \ {l } 9 Set B := B \ {r v } 1 end 11 whle true do 12 Set cr = cr l l L := {} 1 L 13 Set cc = cc r v r v R v := {} 1 R v 14 Set cr l := 1 l L such that zl > 15 f zl l L then 16 Break 17 end 18 whle true do 19 Set m l r v := 1 l L rv R v such that cc r v = and w l r v = 2 f l r v l L r v R v such that m l r v = 1 cr l = and w l r v = then 21 Break 22 else 23 Set cr l := 1 cc r v := l L r v R v such that m l r v = 1 cr l = and w l r v = 24 end 25 Set J := {l r v L R v cr l = cc r v = } 26 Set a := mn l r v J l r v } 27 Update w l r v := w l r v + a l L such that cr l = 28 Update w l r v := w l r v a rv R v such that cc r v = 29 end 3 Set zl := r v l L r v R v such that w l r v = m l r v = 1 and cr l = cc r v = 31 end 32 for l L do 33 Set x l r v := 1 rv R v such that zl = r v 34 end 35 output: Vrtual relay selecton decson X v matrx s updated accordng to the mark ndcator m and cover ndcators cr and cc Lnes 18 to 29. Specfcally the mnmum uncovered element s subtracted from each covered row and then added to each uncovered column. By dong so new zeros wll be resulted. The algorthm then fnds and records the ndces of the new zeros Lne 3 and runs the whle loop Lnes 11 and 31 agan. If z has no zeros the recorded ndces wll be returned as the matchng results of the algorthm Lnes 32 to 34. The convergence of the Hungaran method s proven n [43]. We now dscuss the computatonal complexty of Algorthm 1. The computatonal complexty of the Hungaran method s O L 3 [41]. The complexty of computng the weghts of Algorthm 3: Dstrbuted relay selecton and power allocaton algorthm 1 nput L S D R v Cl mn R v l l L Ps max Pr max λ 1 λ 2 h LI h s d l = s d L 2 for l L do 3 D2D user par l sends relay request. Relays that receve the request on set R v l. 4 for r v R v l do l = s d L. 5 Relay r v estmates h s r v h r vd 6 end 7 end 8 for l L do 9 for r v R v l do 1 Relay r v calculates Ps mn r v 3 11 f P mn s r v P max s r v then max and Ps r v 12 Solve problem 31 to obtan Ps r v 13 wl v r v := λ1p s r v λ2c l r v P s r v 14 else 15 R v l := R v l \ {r v } 16 end 17 end 18 f R v l = then 19 Return nfeasble 2 end 21 Wl v := w l r v r v Rv l 22 end 23 X v = x l r v := {} l Lr v L R v Rv usng 29 and 24 whle rankx v L do 25 Fnd the smallest b such that w br v k = rk v R v and set l := b 26 a := r v such that w l r k v = mn r k v Rv{w l r k v } rv k R v l 27 for l k L \ {l } do 28 f There exsts x l k r v = a then l k L\{l }r v Rv 29 f wl v r v < wv l k r v then 3 x l r v := rv 31 x l k r v := 32 else 33 x l r v := 34 R v l := R v l \ {r v } 35 end 36 else 37 x l r v := rv 38 end 39 end 4 end 41 output: Vrtual relay selecton decson X v and transmt power matrx Ps. all edges Lnes 2 to 16 s O L R v whch s polynomal. Thus Algorthm 1 can obtan a Pareto optmal soluton n polynomal tme. However the centralzed algorthm needs to exchange the control messages between the relays and base statons. Each relay needs to send the channel state nformaton CSI of ts lnks to the base staton. In return the base staton sends the resource allocaton decson to all relays. 2 Dstrbuted Relay Selecton and Power Allocaton Algorthm: We now propose a suboptmal dstrbuted relay selecton and power allocaton algorthm to reduce the commu-
ncaton overhead mposed by exchangng control messages. In ths algorthm D2D user pars and relays act n a dstrbuted manner. As shown n Algorthm 3 D2D user pars frst broadcast ther requests for potental relays locally. We denote the set of relays whch receve the request of D2D user par l as R v l Lne 3. Each relay n set R v l estmates the CSI of the correspondng source-relay and relay-destnaton lnks Lnes 4 to 6. Note that the CSI of all lnks are not shared globally. The next step s to determne the feasble relays for each D2D user par Lnes 8 to 22. We exclude the vrtual relays whch would result n an nfeasble soluton for the correspondng D2D user par Lne 15. After excludng the nfeasble relays each D2D user par can determne the set of all feasble relays. The weght of the feasble edges for D2D user par l s stored n a vector denoted by Wl v Lne 21. Then the dstrbuted relay selecton s executed based on the dea of stable matchng Lnes 24 to 4 [44]. The relay allocaton s completed when all D2D user pars are allocated one vrtual relay Lne 24. Any user par that has not been allocated a vrtual relay wll send a request to ts preferred vrtual relay Lne 26. That vrtual relay wll accept the request f no other user par s allocated to the relay. If the vrtual relay e.g. r v has been allocated to another D2D user par Lne 28 the vrtual relay wll autonomously compare the weghts of two edges. These weghts are wl v k r.e. the v weght of the edge from r v to ts currently allocated D2D user par l k and wl v r.e. the weght of the edge from r v v to D2D user par l whch s requestng to be asssted by r v Lne 29. Vrtual relay r v wll choose to assst the D2D user par wth a smaller weght Lnes 29 to 35. By repeatng Lnes 25 to 39 untl convergence each user par wll be allocated a vrtual relay and a stable relay selecton wll be acheved [44]. We now dscuss the computatonal complexty of Algorthm 3. The complexty of computng the weghts of all edges Lnes 2 to 22 s O L R v. The complexty of stable matchng s O L 2 [44]. The dstrbuted algorthm does not requre each relay to send or receve any CSI to or from the coordnator. The centralzed algorthm requres nformaton exchange to allocate the resources and ncurs at least 2 L more message exchange compared to the dstrbuted algorthm. IV. PERFORMANCE EVALUATION In ths secton we evaluate the performance of our proposed algorthms and compare them wth two recently proposed relay selecton algorthms namely all relays selecton ARS algorthm [2] and dstrbuted relay selecton DRS algorthm [21]. Each relay can use four channels and communcate wth up to four D2D user pars smultaneously.e. N r = 4 r R [45]. The bandwdth of each channel B = 1 MHz [1] and the nose power spectral densty s 174 dbm/hz. We also consder a cell radus of 5 m [34]. Other smulaton parameters are as follows [3] [35]: Ps max = 2 Watt Pr max = 1 Watt Cl mn = 4 Mbps l L M t = M r = 1 db m t = m r = 5 db Θ t HPBW = Θr HPBW = 15o α = 2 and σ = 1.5. We use Monte Carlo smulatons and calculate the average value of the total transmt power and the system throughput over dfferent network settngs. Total transmt power mw 1.4 1.2 1.8.6.4.2 Exponental Dstrbuton Unform Dstrbuton Webull Dstrbuton 1 2 3 4 5 6 7 8 9 1 11 12 13 Number of D2D user pars L Fg. 3. Total transmt power versus number of D2D user pars L wth R = 4 λ 1 = 1 and λ 2 =. System throughput Gbps 14 12 1 8 6 4 2 Webull Dstrbuton Unform Dstrbuton Exponental Dstrbuton 1 2 3 4 5 6 7 8 9 1 11 12 13 Number of D2D user pars L Fg. 4. System throughput versus number of D2D user pars L wth R = 4 λ 1 = and λ 2 = 1. We frst nvestgate the performance of our proposed centralzed algorthm for dfferent dstrbutons of the dstance between the relays and base staton. We consder Webull exponental and unform dstrbutons. For the Webull dstrbuton the cumulatve dstrbuton functon cdf s F d rb ; k µ = 1 e d rb µ k where d rb s the dstance between a relay and the base staton and k and µ are constants. We set k = 7 and µ = 4. For the exponental dstrbuton the cdf s F d rb ; γ = 1 e d rb γ. For far comparson we set the mean parameter γ = 374 whch results n the same average dstance as Webull dstrbuton. For the unform dstrbuton relays are unformly dstrbuted n the cell area. The total transmt power and the system throughput are shown n Fgs. 3 and 4 respectvely. As can be observed dfferent dstrbutons result n dfferent total transmt power and system throughput. However our proposed algorthm s n general applcable for any dstrbutons. We now choose the Webull dstrbuton as an example to evaluate the performance of our proposed algorthms. We assume that the dstance between relays and the base staton d rb follows a Webull dstrbuton wth the aforementoned parameters. We frst set coeffcents λ 1 = 1 and λ 2 = to study the sngle-obectve optmzaton problem for mnmz-
Total transmt power mw 2.5 2 1.5 1.5 DRS Proposed Dstrbuted Algorthm ARS Proposed Centralzed Algorthm 1 2 3 4 5 6 7 8 9 1 11 12 13 Number of D2D user pars L Fg. 5. Total transmt power versus number of D2D user pars L wth R = 4. Total transmt power mw 2.5 2 1.5 1.5 DRS Proposed Dstrbuted Algorthm ARS Proposed Centralzed Algorthm 4 5 6 7 8 9 Number of Relays R Fg. 6. Total transmt power versus number of relays R wth L = 13. 26% 37% Total transmt power mw 3 2.5 2 1.5 1.5 18 32% 38% 17 DRS Proposed Dstrbuted Algorthm ARS Proposed Centralzed Algorthm 16 15 14 13 12 11 Loop-nterference channel gan h LI db 1 Fg. 7. Total transmt power versus loop-nterference channel gan h LI. System throughput Gbps 14 12 1 8 6 4 2 Proposed Centralzed Algorthm ARS Proposed Dstrbuted Algorthm DRS 1 2 3 4 5 6 7 8 9 1 11 12 13 Number of D2D user pars L Fg. 8. System throughput versus number of D2D user pars L wth R = 4. 12% 15% ng the transmt power. In Fg. 5 we compare the total transmt power of devces n our proposed algorthms wth that obtaned by ARS and DRS algorthms for dfferent number of D2D user pars. As shown n ths fgure our proposed centralzed algorthm substantally outperforms ARS. The proposed dstrbuted algorthm also outperforms DRS. When the number of D2D user pars s 13 the centralzed algorthm results n 37% less transmt power than ARS and our dstrbuted algorthm acheves 26% less transmt power than DRS. Ths s because the centralzed algorthm acheves the optmal soluton wth mnmum weghted matchng and our proposed dstrbuted algorthm based on stable matchng can obtan a better soluton than heurstc DRS. In Fg. 6 we compare the total transmt power versus dfferent number of relays when the number of D2D user pars s 13. Fg. 6 shows that the total transmt power s decreasng n all algorthms as the number of relays ncreases. Ths s because each D2D user par has more opportunty to select a nearby relay. Results also show that our proposed centralzed algorthm sgnfcantly outperforms ARS whle our proposed dstrbuted algorthm acheves a lower transmt power than DRS. The results ndcates that our algorthms are more effcent especally n the networks wth few relays. In Fg. 7 we compare the total transmt power versus dfferent loop-nterference channel gan. We consder the gan range from 18 db to 1 db [46]. As shown n ths fgure the total transmt power ncreases as the loop-nterference channel gan ncreases. Ths s because a stronger loopnterference wll result n a hgher transmt power. When the loop-nterference channel gan h LI = 18 db our proposed centralzed algorthm acheves 38% lower transmt power compared to ARS. Furthermore our dstrbuted algorthm outperforms DRS by 32%. We now set λ 1 = and λ 2 = 1 to obtan the relay selecton and power allocaton for maxmzng the system throughput. In Fg. 8 we compare the system throughput versus dfferent number of D2D user pars when there are four relays n the network. Results show that our proposed centralzed algorthm acheves a hgher throughput compared to ARS and our proposed dstrbuted algorthm obtans a hgher throughput compared to DRS under dfferent number of D2D user pars. When the number of D2D user pars s 13 the centralzed algorthm acheves 12% hgher throughput than ARS and our dstrbuted algorthm outperforms DRS by 15%. Results also show that the throughput mprovement of our proposed algorthms over ARS and DRS ncreases as the number of D2D user pars ncreases. Ths s because when the the number of D2D user pars ncreases t s more dffcult for ARS and DRS to guarantee that each D2D user par can use ts preferred relay. Ths substantally degrades the performance of the system.
16 7 System throughput Gbps 15 14 13 12 11 1 Proposed Centralzed Algorthm ARS Proposed Dstrbuted Algorthm DRS 9 4 5 6 7 8 9 Number of Relays R Fg. 9. System throughput versus number of relays R wth L = 13. System throughput Gbps 16 15 14 13 12 11 1 9 8 18 Proposed Centralzed Algorthm ARS Proposed Dstrbuted Algorthm DRS 16 14 12 Loop-nterference channel gan h LI db 1 Fg. 1. System throughput versus loop-nterference channel gan h LI. In Fg. 9 we show the system throughput versus dfferent number of relays when the number of D2D user pars s 13. Our proposed centralzed algorthm substantally mproves the system throughput compared to ARS and our dstrbuted algorthm outperforms DRS under dfferent number of relays. Note that the mprovement slghtly decreases as the number of relays ncreases. Ths s because the ARS DRS and our dstrbuted algorthm are more lkely to allocate the optmal relay for each D2D user par when more relays are deployed. In Fg. 1 we compare the system throughput versus dfferent loop-nterference channel gan whch vares from 18 db to 1 db. As shown n ths fgure the system throughput slghtly decreases as the loop-nterference channel gan ncreases. Ths shows that a stronger loop-nterference wll result n a lower throughput. However our proposed centralzed algorthm always acheves a hgher throughput than ARS and our dstrbuted algorthm always outperforms DRS. We now consder the mult-obectve optmzaton problem to study the trade-off between the total transmt power and the system throughput. In Fg. 11 we plot the optmal total transmt power and system throughput obtaned by the centralzed algorthm when λ 1 = 1 and we vary λ 2. From Fg. 11 we can observe that the optmal soluton s senstve to the weght coeffcents. When λ 2 ncreases mprovng the throughput becomes more mportant than reducng the power. In ths case the optmal soluton tends to consume more power Total transmt power mw System throughput Gbps 6 5 4 3 2 1 1 3 1 2 1 1 1 1 1 1 2 1 3 Weght coeffcent λ 2 13.5 13 12.5 12 11.5 11 1.5 1 a Optmal power consumpton 9.5 1 3 1 2 1 1 1 1 1 1 2 1 3 Weght coeffcent λ 2 b Optmal system throughput Fg. 11. Optmal total transmt power and system throughput versus weght coeffcent λ 2 when λ 1 = 1 wth R = 4 and L = 13. to acheve a hgher throughput. Results also show that the ncrement of the throughput s not as fast as the ncrement of the power. In other words the margnal throughput ncrement becomes smaller when the transmt power ncreases. Ths provdes useful nsghts of the system desgn to balance the total transmt power and system throughput. V. CONCLUSION In ths paper we studed the ont relay selecton and power allocaton problem for full-duplex relay-asssted D2D communcaton n mmwave based 5G networks. We frst formulated a mult-obectve optmzaton problem to balance the trade-off between total transmt power and system throughput. The formulated problem characterzes loop-nterference cancellaton n full-duplex relayng systems. It also consders the QoS requrements for dfferent applcatons as well as the physcal constrants of the devces and relays. The problem s a combnatoral optmzaton problem whch s complex to solve usng standard optmzaton technques. To mtgate the complexty of the combnatoral problem we transformed the problem nto a one-to-one matchng problem by constructng a weghted bpartte graph. We then proposed a centralzed algorthm to fnd the soluton n polynomal tme. We proved that
the soluton obtaned by the proposed centralzed algorthm s Pareto optmal. We further proposed a dstrbuted algorthm to reduce the overhead mposed by exchangng control messages. We evaluated the performance of our proposed algorthms through smulatons. Results showed that our proposed algorthms substantally reduce the total transmt power and mprove the system throughput compared to recently proposed algorthms n the lterature. For future work we wll study the resource allocaton problem when the moble devces can also communcate n full-duplex mode. APPENDIX: PROOF OF THEOREM 1 Snce s wthn a closed nterval constrant 31b s a convex constrant. To prove that problem 31 s strctly convex we need to show ts obectve functon s strctly convex. We determne the second-order dervatve of the obectve functon. Snce the frst term of the obectve functon s lnear n the second-order dervatve of the obectve functon s λ 2 C l r. We rewrte C lr as follows: where C lr = B log 2 Ψlr 36 Ψ lr = 1 + h r d P rd + N. 37 Then accordng to 36 C l r can be wrtten as: where and B Ψ 2 l r Ψ 2 l r B Ψ l r Ψ lr 38 Ψ l r N = f rd + N 2h2 r d f rd + N 2h2 r d N h rd + 2h sr h LI + 2h sr h LI N h rd 2h LI hsd + N 2 ln 2 39 Ψ l r = Ω lr 3 3 h LI hsd + N f rd + N 2h2 2 r d ln 2 n whch Ω lr 3 = N h rd h 2 s d f rd + N 2 h 2 2 r d + 8N h 2 s r h 2 LIh 2 r d h 3 s d P 3 s r 4 + N h rd 18h 2 s r h 2 LIN h rd h 2 s d 6h sr h LI N h 2 r d h 3 s d P 2 s r + N h rd 12h 2 s r h 2 LIN 2 h rd 6h sr h LI N 2 h 2 r d h 2 s d N 4 h 4 r d h 2 s d + 2h 2 s r h 2 LIN 4 h 2 r d. 41 Note that Ψ lr > and Ψ l r > snce. If Ψ l r s negatve then 38 s always postve. Note that Ψ l r s negatve f Ω lr >. We now show that Ω lr >. We frst calculate the frst-order dervatve of Ω lr as follows: Ω l r = 6h sr h LI N h 2 r d 2hsd + N f rd + N 2 h2 r d N h rd + 2h sr h LI + 2h sr h LI N. 42 Snce Ω lr = 2h 2 s r h 2 LI N 3 h rd > and Ω l r > we have Ω lr >. Thus Ψ l r <. Gven B > Ψ 2 l r Ψ 2 l r > Ψ l r < and Ψ lr > the second-order dervatve of C lr shown n 38 s always postve whch proves the convexty of C lr and the obectve functon of problem 31. Consderng the constrant s also convex problem 31 s proved to be a convex problem. REFERENCES [1] M. N. Tehran M. Uysal and H. Yankomeroglu Devce-to-devce communcaton n 5G cellular networks: Challenges solutons and future drectons IEEE Commun. Mag. vol. 52 no. 5 pp. 86 92 May 214. [2] X. Chen B. Proulx X. Gong and J. Zhang Explotng socal tes for cooperatve D2D communcatons: A moble socal networkng case IEEE/ACM Trans. Netw. vol. 23 no. 5 pp. 1471 1484 Oct. 215. [3] H. A. Suraweera I. Krkds G. Zheng C. Yuen and P. J. Smth Lowcomplexty end-to-end performance optmzaton n MIMO full-duplex relay systems IEEE Trans. Wreless Commun. vol. 13 no. 2 pp. 913 927 Feb. 214. [4] I. Krkds H. A. Suraweera P. J. Smth and C. Yuen Full-duplex relay selecton for amplfy-and-forward cooperatve networks IEEE Trans. Wreless Commun. vol. 11 no. 12 pp. 4381 4393 Dec. 212. [5] Y. L D. Jn J. Yuan and Z. Han Coaltonal games for resource allocaton n the devce-to-devce uplnk underlayng cellular networks IEEE Trans. Wreless Commun. vol. 13 no. 7 pp. 3965 3977 Jul. 214. [6] F. Wang C. Xu L. Song and Z. Han Energy-effcent resource allocaton for devce-to-devce underlay communcaton IEEE Trans. Wreless Commun. vol. 14 no. 4 pp. 282 292 Apr. 215. [7] J. Qao X. S. Shen J. W. Mark Q. Shen Y. He and L. Le Enablng devce-to-devce communcatons n mllmeter-wave 5G cellular networks IEEE Commun. Mag. vol. 53 no. 1 pp. 29 215 Jan. 215. [8] P. C. Nguyen and B. D. Rao Far schedulng polces explotng multuser dversty n cellular systems wth devce-to-devce communcatons IEEE Trans. Wreless Commun. vol. 14 no. 9 pp. 4757 4771 Sep. 215.
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Boang Ma receved the B.Eng. degree from Southeast Unversty Nanng JangSu Chna n 29 and the M.A.Sc. degree from Unversty of Vctora Vctora BC Canada n 211 and the Ph.D. degree from the Unversty of Brtsh Columba UBC Vancouver BC Canada n 216. Dr. Ma s currently wth Kwantlen Polytechnc Unversty KPU Surrey BC Canada. Hs research nterests nclude nterference management and resource allocaton for small cell networks smart home wreless networks devce-to-devce communcaton and mmwave communcaton systems usng optmzaton theory and game theory.
Hamed Shah-Mansour S 6 M 14 receved the B.Sc. M.Sc. and Ph.D. degrees Hons. from Sharf Unversty of Technology Tehran Iran n 25 27 and 212 respectvely all n electrcal engneerng. From 213 to 218 he was a Post-doctoral Research and Teachng Fellow at the Unversty of Brtsh Columba Vancouver Canada. Dr. Shah- Mansour s currently wth Vancosys Data Securty Inc. Vancouver Canada. Hs research nterests are n the area of stochastc analyss optmzaton and game theory and ther applcatons n moble cloud computng and Internet of Thngs. He has served as the publcaton co char for the IEEE Canadan Conference on Electrcal and Computer Engneerng CCECE 216 and as the techncal program commttee TPC member for several conferences ncludng the IEEE Globecom 215 IEEE VTC Fall 216 218 IEEE PIMRC 217 and IEEE VTC Sprng 218. Vncent W.S. Wong S 94 M SM 7 F 16 receved the B.Sc. degree from the Unversty of Mantoba Wnnpeg MB Canada n 1994 the M.A.Sc. degree from the Unversty of Waterloo Waterloo ON Canada n 1996 and the Ph.D. degree from the Unversty of Brtsh Columba UBC Vancouver BC Canada n 2. From 2 to 21 he worked as a systems engneer at PMC-Serra Inc. now Mcrosem. He oned the Department of Electrcal and Computer Engneerng at UBC n 22 and s currently a Professor. Hs research areas nclude protocol desgn optmzaton and resource management of communcaton networks wth applcatons to wreless networks smart grd moble cloud computng and Internet of Thngs. Dr. Wong s an Edtor of the IEEE Transactons on Communcatons. He has served as a Guest Edtor of IEEE Journal on Selected Areas n Communcatons and IEEE Wreless Communcatons. He has also served on the edtoral boards of IEEE Transactons on Vehcular Technology and Journal of Communcatons and Networks. He was a Techncal Program Co-char of IEEE SmartGrdComm 14 as well as a Symposum Co-char of IEEE ICC 18 IEEE SmartGrdComm 13 17 and IEEE Globecom 13. He s the Char of the IEEE Vancouver Jont Communcatons Chapter and has served as a Char of the IEEE Communcatons Socety Emergng Techncal Sub-Commttee on Smart Grd Communcatons. He receved the 214 UBC Kllam Faculty Research Fellowshp.