1 Optmal Transmsson Schedulng of Cooperatve Communcatons wth A Full-duplex Relay Peng L Member IEEE Song Guo Senor Member IEEE Wehua Zhuang Fellow IEEE Abstract Most exstng research studes n cooperatve communcaton are based on a half-duplex assumpton. Motvated by recent successes n hardware mplementaton of wreless full-duplex transmsson we propose a full-duplex cooperatve communcaton (FDCC approach to maxmze the mnmum transmsson rate among a set of users to a common destnaton wth the help of a dedcated relay. Under the consderaton of hardware cost only the relay node requres full-duplex wreless equpment n our desgn. We derve the achevable transmsson rate for the proposed FDCC scheme under both amplfy-andforward (AF and decode-and-forward (DF modes. Further as the transmsson schedulng of users plays a crtcal role n determnng the achevable transmsson rate n FDCC we formulate the max-mn rate schedulng problem as a nonconvex mxed nteger nonlnear programmng (MINLP problem. By applyng lnearzaton and convex approxmaton technques we propose an optmal algorthm based on a branch-and-bound framework to solve the problem effcently. Extensve smulaton results show that FDCC can sgnfcantly mprove the transmsson rate as compared wth drect transmsson and half-duplex cooperatve communcaton (HDCC. Index Terms cooperatve communcatons full-duplex schedulng. 1 INTRODUCTION Cooperatve communcaton (CC has shown ts great advantages n offerng hgh capacty and relablty by employng several sngle-antenna nodes to form a vrtual antenna array [1] [] [3]. Most exstng research n CC s based on a half-duplex assumpton that any node n a wreless network does not transmt and receve data smultaneously [4] [5]. Although the dea of full-duplex has been proposed to mprove network effcency ts applcatons n a wreless envronment are very lmted due to many mplementaton challenges. Recently Cho et al. [6] have desgned the frst practcal sngle channel wreless full-duplex system by combnng antenna RF and nterference cancellaton technologes. Later Jan et al. [7] have mproved the full-duplex system to support wdeband and hgh power networks. Ther mplementaton successes n the physcal layer stmulate the applcatons of full-duplex technque and nspre the development of effcent algorthms and protocols n upper layers wthout the half-duplex constrant [8] [9]. Recent efforts n explotng the benefts of fullduplex technque n cooperatve networks can be broadly classfed nto two categores. Most work n lterature s n the frst category e.g. [1] [11] [1] wth a smplfed assumpton that there are no drect lnks between source and destnaton nodes. Whle the other category consders drect lnks t requres P. L and S. Guo are wth the School of Computer Scence and Engneerng The Unversty of Azu Japan. W. Zhuang s wth the School of Electrcal and Computer Engneerng Unversty of Waterloo Canada. addtonal technques or mposes extra constrants on the transmsson power. For example Rhonen et al. [13] propose a co-phasng scheme relyng on a dedcated feedback channel for full-duplex cooperatve communcatons. The lmtaton of exstng work motvates us to nvestgate a general and smple scheme that support full-duplex CC n a cooperatve network wth drect lnks for both amplfy-and-forward (AF and decode-and-forward (DF cooperaton modes. The basc dea can be llustrated n Fgure 1 where n users s s 1... s n 1 send data to destnaton d wth the assstance of a relay r. In order to beneft from CC user s frst transmts a sgnal x that reaches both r and d. Then r forwards the heard sgnal to d usng AF or DF. Fnally d explots the spatal dversty of the receved sgnals to decode the orgnal one. In a tradtonal half-duplex model a tme frame s dvded equally nto n tme slots each of whch s dedcated to a user or the relay n order to avod nterference as shown n Fgure (a. When a full-duplex technque s appled at r t can forward sgnal x heard n the last tme slot to d whle recevng sgnal x +1 transmtted by s +1 n the current tme slot as shown n Fgure (b. In such a way a sgnfcantly hgher transmsson rate wll be acheved because each tme frame can be reduced to only n + 1 tme slots or equvalently each source has a longer transmsson tme n our proposed full-duplex CC (FDCC scheme whch results n a hgher transmsson rate. Moreover our approach only requres full-duplex equpment at the relay node wth reasonable hardware cost.
Fg. 1. A wreless network usng CC Fg.. The tme frame structure Our proposed FDCC scheme can be appled n varous scenaros n practce. For example cellular network operators can deploy several relay nodes wth full-duplex capablty n each cell to mprove the performance of uplnks. In a wreless local area network (WLAN full-duplex relay nodes can assst the transmssons from users to the access pont. Even n a mesh network when an ntermedate node needs to forward packets from multple upstream nodes ts packet recevng performance can be mproved by employng a full-duplex relay node to form a FDCC model as shown n Fg. 1. To acheve a practcal and effcent full-duplex CC scheme wth hgh throughput we frst need to deal wth the nterference at destnaton d due to the smultaneous transmssons from s and r. In order to decode the sgnals whle beneftng from FDCC we apply the smlar decodng technque n [14] n whch a receved sgnal at d s treated as a combnaton of two sgnals from s and r. After analyzng the achevable transmsson rate under the proposed decodng scheme we fnd out that the transmsson schedulng of users plays a crtcal role n determnng the transmsson rate. In other words dfferent user transmsson sequences wll result n dfferent performance. The second challenge s to determne an optmal transmsson schedulng to maxmze the mnmum transmsson rate among all users. To conquer the weaknesses of exstng work.e. gnorng drect lnks or requrng addtonal technques we propose a novel FDCC scheme n ths paper. The man contrbutons of ths paper are summarzed as follows. Frst we propose a FDCC scheme that explots the advantages of full-duplex and cooperatve communcaton technques for a set of users sharng common relay and destnaton nodes. Based on the smlar decodng scheme n [14] we derve the achevable transmsson rate of users n close-form expresson under both AF and DF modes and show that transmsson schedulng can affect the transmsson rate n FDCC. Wth the objectve of maxmzng the mnmum transmsson rate among a set of users we formulate the schedulng problems under AF and DF modes as nonconvex mxed nteger nonlnear programmng (MINLP problems. After reformulaton and lnearzaton we propose optmal algorthms based on a branch-and-bound framework to solve these problems. Fnally extensve smulatons are conducted to evaluate the performance of the proposed FDCC scheme. The rest of ths paper s organzed as follows. Secton revews related work. System model s ntroduced n Secton 3. Secton 4 presents the proposed FDCC scheme and ts achevable transmsson rate under both AF and DF modes. Secton 5 presents the optmal algorthms for transmsson schedulng problems. Extensve smulaton results are presented n Secton 6. Fnally Secton 7 concludes the paper. RELATED WORK.1 Full-duplex wreless transmsson The man challenge of full-duplex wreless transmsson s to reduce self-nterference. For ths purpose dgtal cancellaton has been extensvely used n many exstng solutons. Halpern et al. [15] have desgned a practcal nterference-cancellaton algorthm that enables a sngle recever to successfully receve smultaneous overlappng transmssons. ZgZag [16] extends ths approach to decode multple colldng packets from multple collsons. However these technques do not support full-duplex because they cannot subtract enough nterference to decode the orgnal sgnal from the recevng antenna. Recently Cho et al. [6] have mplemented the frst practcal full duplex system by combnng the technques of antenna cancellaton nose cancellaton and dgtal nterference cancellaton. Jan et al. [7] mprove ther mplementaton by usng the novel balun cancellaton such that the bandwdth constrant s elmnated and the number of antennas s reduced from three to two. Inspred by the poneer work much progress has been acheved n the full-duplex technque that can be appled to varous scenaros. Cheng et al. [8] employ full-duplex for secondary users n a cogntve rado network so that they can scan for actve prmary users whle transmttng. Fang et al. [9] study the cross-layer optmzaton for multpath routng n full-duplex wreless networks.
3. Cooperatve communcatons The basc dea of cooperatve communcaton s presented n the poneerng paper [17]. The mutual nformaton and outage probablty between a par of nodes usng CC are studed under both AF and DF n [1]. Bletsas et al. [18] propose a dstrbuted scheme for relay node selecton based on the nstantaneous channel condtons at the relay nodes. Zhao et al.[19] further show that t s suffcent to choose one best relay node nstead of multple ones. Moreover they propose an optmal power allocaton algorthm based on the best relay selecton to mnmze the outage probablty. For multple uncast sessons Sharma et al. [4] consder the relay node assgnment wth the goal of maxmzng the mnmum data rate among all concurrent sessons. Wth the restrcton that any relay node can be assgned to at most one sourcedestnaton par an optmal algorthm call ORA s developed. By relaxng ths constrant to allow multple source-destnaton pars to share one relay node Yang et al. [] prove that the total capacty maxmzaton problem can be also solved wth an optmal soluton wthn polynomal tme. The energy-effcency of CC s studed n [1] []..3 Full-duplex n cooperatve communcatons Recently the advantages of full-duplex technque has been exploted n cooperatve communcatons. A cellular system wth full-duplex amplfy-and-forward relayng s nvestgated n [1]. Ju et al. [11] propose a precoder and decoder for a full duplex relay (FDR system and nvestgate the achevable transmsson rate. Cheng et al. [1] develop a dynamc hybrd resource allocaton polcy for both AF and DF under full-duplex mode to maxmze the network throughput for a gven delay QoS constrant. The most recent paper [3] studes full-duplex operaton from a dversty perspectve and nvestgates several protocols that extract dversty gans over a block fadng channel. All these work assumes that no drect lnks between source and destnaton nodes exst n the networks whch would be not always practcal n realty and cause low sgnal-to-nose rato (SNR at destnaton. On the other hand only few work consders drect lnks n ther proposed full-duplex CC schemes. Rhonen et al. [13] study a wreless full-duplex relay lnk where a destnaton receves superposton of relayed and drect sgnals. A relay protocol that performs cophasng of the two paths s proposed and ts performance n terms of end-to-end sgnal-to-nose rato s analyzed. However the proposed scheme reles on a feedback channel and supports only the AF cooperaton mode. Machado et al. [4] propose a full-duplex CC scheme wthout dedcated relay nodes but mpose constrants on the transmsson power of each node n order to cancel the nterference at the forwardng node. Lu et al. [5] present two dstrbuted lnear convolutonal space-tme codng (DLC-STC schemes for full-duplex CC wth drect lnk. Ther proposal focuses on cancelng the loop nterference at relay node and reples on DLC-STC. Our proposed fullduplex CC operaton works under the support of drect lnks as well but can provde both AF and DF by a general CC scheme that has been wdely adopted e.g. n [18] [19] [6] [14] [7] wthout relyng on any feedback channel transmsson power or sophstcated codng schemes. 3 SYSTEM MODEL We consder that a set of users S = {s s 1... s n 1 } send data to a destnaton d wth the assstance of a dedcated relay node r whch s equpped wth full-duplex hardware. The gven transmsson power of user s and relay r s denoted as P s and P r respectvely. Suppose a channel wth bandwdth W s avalable n the network and s shared by all the nodes n tme dvson. Let h xy denote the effect of path-loss shadowng and fadng between nodes x and y wth a dstance x y. The varances of nose at nodes r and d are denoted as σr and σd respectvely. Typcally there are two cooperatve communcaton modes namely amplfy-and-forward (AF and decode-andforward (DF. The detaled transmsson schemes and correspondng channel capacty between any source s and destnaton d wth the support of a tradtonal half-duplex relay r are presented as follows. Amplfy-and-forward (AF: When source node s transmts data to a destnaton node d wth the help of a relay node r under AF mode each tme frame s dvded nto two tme slots. In the frst tme slot source s transmts a sgnal to destnaton d. Due to the broadcast nature of wreless communcaton ths transmsson s also overheard by relay r. Then relay node r amplfes the receved sgnal wth a multpler α gven n [1]: α = P r P s h s r + σr (1 and forwards t to destnaton d n the second tme slot. Fnally destnaton d decodes out the orgnal sgnal by combnng the two receved ones dssemnated from dfferent paths. Followng the analyss n [1] the channel capacty between s and d wth the assstance of a tradtonal half-duplex relay r can be calculated by: CAF HDCC = W log γ s rγ rd (1 + γ s d + ( γ s r + γ rd + 1 where γ xy denotes the sgnal-to-nose rato (SNR of the transmsson from node x to node y. Decode-and-forward (DF: The transmssons under DF mode follow the smlar process of AF mode except that the relay node r decodes the sgnal receved from source s n the frst tme slot and then
4 transmts t to destnaton d n the second tme slot. The correspondng channel capacty can be calculated by [1]: C HDCC DF = W mn{log (1 + γ s r log (1 + γ s d + γ rd }. (3 Drect transmsson (DT: Under drect transmsson source s transmts data to destnaton d usng a whole tme frame and the channel capacty s calculated by: C DT = W log (1 + γ sd. (4 4 FDCC SCHEMES FOR MULTIPLE USERS Recall the FDCC example llustrated n Fgure 1 n whch any user s ( n 1 and relay r are allowed to transmt smultaneously to explot the full-duplex advantage. Ths makes the nterference at destnaton nevtable. Based on the smlar decodng schemes n [14] for a sngle user we present decodng schemes for multple users under both AF and DF n ths secton. 4.1 Decodng under AF Frst we consder the decodng under the AF mode. Wthout loss of generalty we study the decodng scheme under a gven transmsson order S = {s s 1... s n 1 }. The process s llustrated n Table 1 where yd and y r denote the receved sgnals at destnaton d and relay r respectvely n the -th ( 1 tme slot. Due to the self-cancelng capablty of the full-duplex relay ts receved sgnal yr(x 1 only ncludes the component x 1 whch wll be amplfed by a multpler α 1 and forwarded to the destnaton n the next tme slot. On the other hand destnaton d receves a superposed sgnal yd (x x 1 wth components x and x 1 from r and s 1 respectvely. Our decodng scheme contans two stages: partal decodng and fnal decodng. Partal decodng happens each tme when the destnaton receves a combned sgnal from s and r. After recevng all the sgnals destnaton d conducts the fnal decodng to obtan the orgnal sgnals from all the users. Smlar wth [14] we start by explotng channel dversty to decode a sgnal from last two sgnals wth dfferent attenuaton condtons such that all sgnals wll be decoded n a reverse order. The mutual nformaton between s and d under the gven transmsson order S can be calculated by: ( I AF (s S = log 1 + P s h rd α h sr ( σ AF + P r h rd α h s r h rd α σ r + σ d. (5 In (5 the power ( AF of accumulated nose ncurred by the FDCC scheme for user s can be expressed n a recursve form as: ( σ AF = (γ AF [ ( 1 AF + α 1 h rd σ r + σd where (γ AF s gven by: ] (6 (γ AF = h rdα h sr h s d. (7 Snce a tme frame s dvded evenly nto n + 1 tme slots n FDCC the achevable channel rate of user s can be calculated by: C AF (s S = 4. Decodng under DF W n + 1 I AF (s S. (8 Smlar operatons of FDCC under the DF mode are also llustrated n Table 1. The operatons of partal decodng and fnal decodng can be conducted n a smlar way wth [14]. The correspondng mutual nformaton between s and d s I DF (s S = mn{i 1 DF (s I DF (s S} (9 n whch two terms IDF 1 (s and IDF (s S determne the maxmum rate for successful decodng at relay and destnaton nodes respectvely. The former s obtaned straghtforward from the result of DT (drect transmsson as: ( IDF 1 (s = log 1 + P s h sr (1 and the later can be derved as ( IDF (s S = log 1 + P s h rd ( DF + P r h rd σd σ r (11 by followng the smlar steps n [14]. In (11 the nose power ( DF can be expressed as ( σ DF where (γ DF s gven by: = (γ DF (( 1 DF + σd (1 (γ DF = h rd h s d. (13 The transmsson rate of user s under DF s therefore C DF (s S = W n + 1 I DF (s S. (14 5 OPTIMAL TRANSMISSION SCHEDULING FOR FDCC Be carefully examnng the expresson of achevable transmsson rate derved n last secton under a fxed transmsson schedulng we fnd that dfferent transmsson schedulng wll lead to dfferent performance n FDCC. To guarantee a certan level of qualty-ofservce (QoS for a set of users a natural objectve s to maxmze the mnmum achevable rate among
5 TABLE 1 FDCC scheme under both AF and DF modes r s s 1... s n 1 AF DF d Tx x 1st Rx yr(x 1 yd 1( x Tx x nd 1 α yr(x 1 x Rx yr (x 1 yd (x x 1........................... nth Tx Rx x n 1 α n yr n 1 (x n yr n (x n 1 x n yd n(x n x n 1 (n + 1th Tx α n 1 yr n (x n 1 x n 1 Rx y n+1 (x d n 1 these users whch s defned as max-mn rate n our paper. In ths secton we frst present our motvaton usng a 3-user network nstance. Then we formulate the transmsson schedulng problem n FDCC as a nonconvex mxed nteger nonlnear programmng (MINLP problem. Fnally an effcent branch-andbound algorthm for optmal soluton s proposed. 5.1 Motvaton For a better understandng we present some numercal results on a 3-user network nstance where the locaton of each node s shown n Fg. 3(a. The settngs of transmsson power background nose and channel gan are the same wth ones used n Secton 6.1. To study how the poston of relay node affects the performance we move r along a horzontal lne from (15 to (45. The mnmum transmsson rate that s maxmzed by an optmal transmsson schedulng whch s referred to as max-mn rate under AF and DF are shown n Fg. 3(b. We observe that the transmsson schedulng leadng to an optmal mnmum rate s qute dynamc as relay r moves from left to rght under both AF and DF. When r s placed at center pont (55 of ts movng lne the best performance can be acheved under schedulng (s s 1 s 3. The optmal schedulng goes to (s 3 s s 1 and (s 1 s 3 s when the relay s on the end ponts (15 and (45 respectvely. We attrbute such phenomenon to the fact that dfferent transmsson schedulng can produce vared accumulated noses ( AF and ( DF at each user s whch eventually lead to dstnct transmsson rate performance for AF and DF respectvely. Such mportant observaton motvates us to nvestgate the nfluence of transmsson schedulng to the mnmum rate of a set of users n next secton. 5. Transmsson schedulng under AF 5..1 Problem formulaton To specfy the transmsson order of a set of users we defne a bnary varable u j ( n 1 j 5 45 4 35 3 5 15 1 5 r 1 5 1 15 5 3 35 4 45 5 (a A 3-user network nstance d 3 Max mn rate 11 1 9 831 7 6 5 31 31 31 31 31 13 13 13 13 13 13 13 4 1 15 5 3 35 4 X axs of relay s locaton AF DF 13 (b Max-mn rate versus dfferent locaton of relay r Fg. 3. Numercal results of a 3-user network nstance n 1 as follows { 1 f sj s scheduled mmedately after s u j = otherwse. If we consder a vrtual user s n as both the orgn and termnaton of a crcular schedulng then any user n {s n } S should have exactly one successor and one predecessor. These can be descrbed by the constrants: n u j = 1 n (15 j= n u j = 1 j n. (16 Now we only need to consder the schedulng of users n S by removng s n. In order to guarantee the resultng schedulng acyclc we defne an nteger varable e to denote that user s s scheduled as the e - th one for transmsson. Then we have the followng constrants for e : e n 1 n 1 (17 nu j n + 1 e j e n 1 (n u j j n 1. (18 Note that constrant (18 becomes e j e = 1 f user s s the predecessor of s j.e. u j = 1 and otherwse 1 n e j e n 1(.e. e j e n 1 whch s alway satsfed because of (17.
6 For each user s j S we defne a varable σ j to denote ts nose term ( σ j AF resultng n that (6 can be rewrtten as σ j = (γ AF j [ n 1 u j ( α h rd σ r + + σd ] j n 1. (19 Our objectve s to fnd the optmal schedulng.e. u j ( n 1 j n 1 such that the mnmum transmsson rate s maxmzed. To smplfy the expresson n (5 we let a = P s h rd α h s r ( b = P r h rd α h s r h rd α σr + σd (1 whch are constants for each user s. By defnng a varable c mn the max-mn rate schedulng problem of FDCC under AF mode (SAF can be descrbed as: SAF: s.t. c mn W n + 1 log max c mn ( 1 + a + b n 1 ( (15 (16 (17 (18 and (19. The SAF problem s challengng because of ts NPhardness whch can be seen by a straghtforward reducton from the sequencng problem n [8]. Thus we resort to solvng above nonconvex mxed nteger nonlnear programmng (MINLP problem by removng the nonconvex and nonlnear constrants usng reformulaton and lnearzaton technques. 5.. Optmal soluton We frst consder to lnearze constrant (19. For ths purpose we defne a new varable v j as: v j = u j j n 1 (3 such that constrant (19 can be wrtten n a lnear form as: σ j = (γ AF j ( n 1 n 1 u j α h rd σ r + v j + σ d j n 1. (4 Furthermore (3 can be equvalently replaced by the followng lnear constrants: v j j n 1 (5 M(1 u j v j Mu j j n 1 (6 where M s a suffcently large constant number. Then we consder to remove the logarthm operaton n constrant (. Due to the monotoncty property of the logarthm functon t s equvalent to solve SAF va replacng constrant ( by δ a + b n 1 (7 Fg. 4. Illustraton of the SPCA method. wth objectve δ. To deal wth the nonconvex constrant (7 we explore the SPCA (Sequental Parametrc Convex Approxmaton method [9] whch s partcularly effectve to our formulated problem as shown n later sectons. The basc dea of SPCA s to teratvely solve the resultng lnear programmng (LP problem by replacng the orgnal nonconvex constrants wth lnear ones untl a converged soluton s acheved. At each teraton a new lnear constrant s constructed such that the correspondng lne s tangent to the curve defned by the nonconvex constrant at the pont whch s a soluton obtaned n the prevous teraton. By applyng the SPCA technque the relaxed SAF problem (.e. all nteger varables are relaxed to real ones denoted as SAF R can be quckly solved. Specfcally n the m-th teraton we replace nonconvex constrant (7 by δ a ( σ m 1 ( σ σ m 1 + b. (8 We denote the rght-hand sde of (8 as f m ( n whch σ m 1 means the optmal soluton of varable obtaned n the (m 1-th teraton. As shown n Fg. 4 after solvng the correspondng lnear programmng n the m-th teraton we construct a new lnear constrant δ f m+1 ( to approxmate (7 n the next teraton. The algorthm to solve the SAF R problem s formally descrbed n Algorthm 1 n whch SAF R(m and (m are the problem formulaton and ts optmal soluton n the m-th teraton respectvely. Snce the ntal value of can be set as an arbtrary postve number we set = σ d (γaf by supposng s to transmt frst. Theorem 1: The soluton of the relaxed SAF problem obtaned by Algorthm 1 satsfes the Karush-Kuhn- Tucker (KKT condtons. Proof: For any feasble pont ( σ m 1 a σ m 1 +b we update the lnear constrant for the SAF R formulaton n Algorthm 1. As guaranteed by the analyss n [9] the concluson s acheved when the nonlnear functon a + b and ts approxmated lnear functon f m ( have the same values at = σ m 1 for the orgnal and ther frst-order dfferental functons
7 Algorthm 1 Solvng the SAF R problem 1: m = = ( = = σ d (γaf ( n 1 : whle (m > ϵ do 3: = (m 4: m = m + 1 5: obtan (m as well as m ( n 1 by solvng the followng LP problem wth relaxed varables: SAF R(m: 6: end whle max δ s.t. (15 (18 (4 (6 and (8. respectvely. These can be verfed by: f m ( σ m 1 = a ( f m ( σ m 1 a = σ m 1 + b σ m 1 + b = a ( σ m 1. Note the KKT condtons are satsfed only for the relaxed problem referred to as SAF R here not for the MINLP problem. Although Algorthm 1 returns a soluton satsfyng the KKT condtons we fnd out that t s always the global optmal soluton emprcally through extensve numercal experments. In order to solve the orgnal SAF problem we ntegrate Algorthm 1 nto a branch-and-bound framework. The formal descrpton of the algorthm to solve the SAF problem s shown n Algorthm. In Algorthm we use P to denote a problem set wth an upper bound U and a lower bound L of the optmal soluton that are tghtest found so far. Intally P only ncludes the orgnal problem denoted by p. For any problem p P the optmal soluton of the correspondng relaxed problem can be obtaned by Algorthm 1 and t can serve as an upper bound denoted as u p of the soluton to the orgnal problem. Then the algorthm proceeds teratvely as follows. In each round we fnd a problem p P wth maxmum u p and then set U = u p. Whle any feasble soluton of p can serve as a lower bound the one obtaned usng roundng under the satsfacton of all constrants s used and denoted by l p. The greatest lower bound L s updated from lne 7-14. If the performance gap between L and U s less than a predefned small number ϵ a (1 ϵ-optmal soluton l s returned. Otherwse we replace problem p wth two subproblem p 1 and p by functon SubproblemConstructon whch can accelerate the problem solvng as shown n Algorthm 3. 5..3 Executon acceleraton To accelerate the executon of the branch-and-bound algorthm we explot some problem-specfc characterstcs to reduce the complexty of Algorthm. The Algorthm Solvng the SAF problem 1: P = {p } L = ; : set u p as the optmal soluton of the relaxed problem p usng Algorthm 1; 3: whle P do 4: select a problem p P wth the maxmum u p and let U = u p ; 5: set l p as the soluton of p usng roundng; 6: f l p > L then 7: l = l p L = l p ; 8: f L (1 ϵu then 9: return the (1 ϵ-optmal soluton l ; 1: else 11: remove all problems p P wth L (1 ϵu p ; 1: end f 13: end f 14: select the maxmum unfxed u j from the results of the relaxed problem p and remove p from P; 15: p 1 = SubproblemConstructon(p u j 1; 16: p = SubproblemConstructon(p u j ; 17: solve problems p 1 and p to obtan u p1 and u p respectvely. 18: f L < (1 ϵu p1 then put p 1 nto P; end f 19: f L < (1 ϵu p then put p nto P; end f : end whle 1: return the (1 ϵ-optmal soluton l ; acceleraton process s presented n functon SubproblemConstructon whose formal descrpton s shown n Algorthm 3. Frst we construct a subproblem p accordng to the formulaton of problem p and fx the value of selected varable u j to value. In addton other varables are fxed accordng to constrants (15-(16 such that they wll not be branched n the prospectve teratons. That s because after such assgnment many assocated varables are mmedately fxed as well makng the resultng subproblems to be solved quckly. For example when u j = 1 s fxed many other related varables can be fxed mmedately as well.e. u k = k j and u kj = k. Furthermore f all u kk ( k n 1 have been fxed to for any user s k S.e. user s k s the frst one n the schedulng varable σ k should be also fxed as σd (γaf k. Smlarly for any fxed σ k f the successor of user s k has been determned.e. k ( k n 1 u kk = 1 ts assocated σ k can be fxed as well by the equaton n lne 8. In both cases the correspondng constrant (8 becomes lnear and the convex approxmaton gven by (7 s not requred anymore. It avods teratons ncurred by the SPCA method sgnfcantly. Fnally we return the subproblem p.
8 Algorthm 3 SubproblemConstructon(p u j value 1: construct a subproblem p accordng to problem p wth u j = value; : fx assocated varables accordng to constrants (15-(16; 3: for there exsts any unfxed varable σ k such that all varables u kk ( k n 1 have been fxed to do 4: σ k = σd (γaf k ; 5: replace constrant (8 wth (7 for user s k ; 6: end for 7: for there exsts any unfxed varable σ k such that varables σ k and u kk = 1 have been fxed do ; 9: replace constrant (8 wth (7 for user s k ; 1: end for 11: return subproblem p ; 8: σ k = (γ AF k ( α k h rd σ r + σ k + σ d 5.3 Transmsson schedulng under DF 5.3.1 Problem formulaton The major dfference n problem formulaton under the DF mode s that varable σ j denotes the nose term ( σ DF j such that (1 can be wrtten as σ j = (γ DF j ( n 1 u j + σd j n 1. (9 Therefore the max-mn rate schedulng problem of FDCC under DF (SDF can be smlarly formulated as a nonconvex MINLP problem as follows. c mn SDF: max c mn c mn W n + 1 log (1 + P s h s r W n + 1 log σ r n 1 (3 (1 + P s h rd + P r h rd σ d n 1 (31 (15 (16 (17 (18 and (9. The SDF problem s also NP-hard as shown by followng the smlar reducton for the SAF problem. 5.3. Optmal soluton In the followng we apply smlar approaches to reformulate SDF n an LP problem. Frst we notce that constrant (3 can be removed snce ts rghthand sde s a constant number. Let c be the soluton of the resultng problem wthout (3. The optmal soluton of the orgnal SDF problem can be obtaned as mn s S{c W n+1 log (1+ P s h s r σ } under the same r schedulng order. To lnearze (9 we take exactly the same method as that for (19 by replacng (9 wth (5(6 and σ j = (γ DF j ( n 1 v j + σd j n 1. (3 Mnmum transmsson rate 1 1 8 6 4 DT HDCC FDCC_opt FDCC_rand 6 8 1 1 14 16 18 Number of users (a AF Mnmum transmsson rate 1 1 8 6 4 Drect transmsson HDCC FDCC_opt FDCC_rand 6 8 1 1 14 16 18 Number of users (b DF Fg. 5. The max-mn transmsson rate versus user number Performance rato.8.6.4. 1.8 1.6 1.4 FDCC_opt/DT FDCC_opt/HDCC FDCC_opt/FDCC_rand 6 8 1 1 14 16 18 Number of users (a AF Performance rato.6.4. 1.8 1.6 1.4 FDCC_opt/DT FDCC_opt/HDCC FDCC_opt/FDCC_rand 6 8 1 1 14 16 18 Number of users (b DF Fg. 6. The performance rato versus user number Moreover because the logarthm form n (31 s a monotonc ncreasng functon of Ps SDF can be reformulated as SDF : max δ δ Ps n 1 (33 (15 (16 (17 (18 (5 (6 and (3. Fnally constrant (33 can be lnearzed by lettng δ = 1 δ v j = v j P s = P s and takng SDF n an equvalent mn-max form as follows. SDF : mn δ δ σ n 1 (34 v j σ j n 1 (35 M(1 u j v j Mu j j n 1 (36 σ j = (γj DF ( n 1 v j + σ d P s (37 (15 (16 (17 and (18. Comparng to the reformulaton for the AF case we notce that whle the SPCA technque s not requred for DF Algorthm can be stll appled. The only dfference s that to fnd the optmal soluton of a relaxed problem we just need to solve the relaxed SDF problem nstead of nvokng Algorthm 1. To accelerate the executon of the branch-and-bound algorthm under the DF mode we fx varables u j exactly the same as done by lne n Algorthm 3 whle ts remanng acceleraton for SPCA s not necessary.
9 Mnmum transmsson rate 9 8 7 6 5 4 3 Drect transmsson HDCC FDCC_opt FDCC_rand Mnmum transmsson rate 9 8 7 6 5 4 3 Drect transmsson HDCC FDCC_opt FDCC_rand Performance rato.4. 1.8 1.6 1.4 FDCC_opt/DT FDCC_opt/HDCC FDCC_opt/FDCC_rand Performance rato.4. 1.8 1.6 1.4 FDCC_opt/DT FDCC_opt/HDCC FDCC_opt/FDCC_rand 5 3 35 4 y axs of relay 5 3 35 4 y axs of relay 1. 5 3 35 4 y axs of relay 1. 5 3 35 4 y axs of relay (a AF (b DF (a AF (b DF Fg. 7. The max-mn transmsson rate versus relay s locaton 6 PERFORMANCE EVALUATION 6.1 Smulaton settngs In ths secton we conduct extensve smulatons usng Matlab to evaluate the performance of our proposed algorthms. All users n each network nstance dstrbute randomly wthn a 5 5 square regon and the destnaton s placed at central top (5 5. The transmsson power s set to one unt for each user and the relay. We set the varance of the background nose at the destnaton and the relay to 1 11 W. The channel bandwdth s specfed as W = MHz and the channel gan h j between two nodes wth a dstance j s calculated as h j = j 4 whch has been wdely accepted and used n recent lterature for related topcs such as []. All results n the followng are obtaned by averagng over 5 random network nstances. For comparson we also show smulaton results of the followng four algorthms. (1 Drect transmsson (DT: all users transmt data drectly to the base staton usng tme dvson multplexng. ( Half-duplex cooperatve communcaton (HDCC: the relay node works under half-duplex mode and each user employs t to transmt usng tradtonal CC. (3 FDCC wth optmal schedulng (FDCC opt: the users utlze FDCC wth the optmal transmsson schedulng proposed n ths paper. (4 FDCC wth random schedulng (FDCC rand: the users utlze FDCC wth a random transmsson schedulng. 6. Smulaton results We frst nvestgate the nfluence of user sze to the performance n networks wth a fxed relay at poston (53. As shown n Fgure 5 the performance of all transmsson methods drops as the number of users ncreases from 6 to 18 under both AF and DF. Ths s because the transmsson tme allocated to each user s reduced by sharng wth more users on the same amount of bandwdth. We also show the performance ratos of FDCC opt to other schemes DT HDCC FDCC rand denoted as FDCC opt/dt Fg. 8. The performance rato versus relay s locaton Cumulatve Fracton of teratons 1.9.8.7.6.5.4.3..1 ε=.1 ε=.1 5 1 15 5 3 35 4 Number of teratons Cumulatve Fracton of tme 1.9.8.7.6.5.4.3..1 DF wth ε=.1 DF wth ε=.1 AF wth ε=.1 AF wth ε=.1 3 4 5 6 7 8 9 1 11 Executon tme (sec Fg. 9. The dstrbuton Fg. 1. The dstrbuton of teratons of SPCA of teratons of executon method tme FDCC opt/hdcc and FDCC opt/fdcc rand n Fg. 6 where FDCC opt always outperforms other transmsson schemes. The performance of FDCC opt s 1.3 and 1.51 tmes of HDCC under AF and DF respectvely n 6-node network nstances. When the number of users grows to 18 ts gan ncreases to 1.67 and 1.85 n AF and DF respectvely. The expermental results demonstrate that FDCC opt can use tme slots more effectvely especally n large network nstances. Then we study the nfluence of relay s locaton to performance by changng ts poston from (5 to (54 n network nstances wth 1 users. The larger x-axs value ndcates the relay closer to the destnaton. As shown n Fg. 7 the transmsson rate under drect transmsson shows as a horzontal lne snce t s not affected by relay s locaton. The best performance of HDCC can be acheved when the relay s placed at center (53 of ts movng lne because ts contrbuton becomes smaller as t s ether closer or further to the destnaton. The advantages of FDCC opt over other schemes are obvous by observng the performance rato n Fg. 8. The rato of FDCC opt to DT follows the smlar trend of mnmum transmsson rate n Fg. 7. Under AF the performance gap between FDCC opt and HDCC becomes smaller as the relay s closer to the destnaton because the ncreased nose term reduces the performance of FDCC opt. Under DF on the other hand the relay s locaton has lttle effect to the performance rato snce the the transmsson rate s also constraned by constant IDF 1 n (1. In order to evaluate the effcency of the SPCA method we show the dstrbuton of number of teratons n cumulatve dstrbuton functon (CDF under settngs ϵ =.1 and ϵ =.1 n Fg. 9. We observe
1 that SPCA converges n 8 teratons for all nstances and n 1 teratons for 98% nstances when ϵ =.1 and ϵ =.1 respectvely. Fnally we study the effcency of the proposed algorthms by plottng the CDF of executon tme of 5 15-node network nstances under settngs ϵ =.1 and ϵ =.1 n Fg. 1. When ϵ s set to.1 the upper and lower bounds of all nstances can converge wthn 3.3 seconds. When the performance gap between upper and lower bound s constraned by ϵ =.1 our algorthms n 95% and 86% nstances converge wthn 5 seconds under DF and AF respectvely. 7 CONCLUSION In ths paper we nvestgate the full-duplex cooperatve communcaton n wreless networks where several users send data to a common destnaton under the assstance of a dedcated relay. We derve the achevable transmsson rate for the proposed FDCC scheme under both AF and DF modes. 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11 Peng L receved hs BS degree from Huazhong Unversty of Scence and Technology Chna n 7 the MS and PhD degrees from the Unversty of Azu Japan n 9 and 1 respectvely. He s currently a Postdoctoral Researcher n the Unversty of Azu Japan. Hs research nterests nclude networkng modelng cross-layer optmzaton network codng cooperatve communcatons cloud computng smart grd performance evaluaton of wreless and moble networks for relable energy-effcent and cost-effectve communcatons. Song Guo (M -SM 11 receved the PhD degree n computer scence from the Unversty of Ottawa Canada n 5. He s currently a Senor Assocate Professor at School of Computer Scence and Engneerng the Unversty of Azu Japan. Hs research nterests are manly n the areas of protocol desgn and performance analyss for relable energy-effcent and cost effectve communcatons n wreless networks. Dr. Guo s an assocate edtor of the IEEE Transactons on Parallel and Dstrbuted Systems and an edtor of Wreless Communcatons and Moble Computng. He s a senor member of the IEEE and the ACM. Wehua Zhuang (M 93-SM 1-F 8 has been wth the Department of Electrcal and Computer Engneerng Unversty of Waterloo Canada snce 1993 where she s a Professor and a Ter I Canada Research Char n Wreless Communcaton Networks. Her current research focuses on resource allocaton and QoS provsonng n wreless networks. She s a co-recpent of the Best Paper Awards from the IEEE Internatonal Conference on Communcatons (ICC 7 and 1 IEEE Multmeda Communcatons Techncal Commttee n 11 IEEE Vehcular Technology Conference (VTC Fall 1 IEEE Wreless Communcatons and Networkng Conference (WCNC 7 and 1 and the Internatonal Conference on Heterogeneous Networkng for Qualty Relablty Securty and Robustness (QShne 7 and 8. She receved the Outstandng Performance Award 4 tmes snce 5 from the Unversty of Waterloo and the Premer s Research Excellence Award n 1 from the Ontaro Government. Dr. Zhuang s a Fellow of the IEEE a Fellow of the Canadan Academy of Engneerng (CAE a Fellow of the Engneerng Insttute of Canada (EIC and an elected member n the Board of Governors of the IEEE Vehcular Technology Socety. She was was the Edtorn-Chef of IEEE Transactons onvehcular Technology (7-13 the Techncal Program Symposa Char of the IEEE Globecom 11 andan IEEE Communcatons Socety Dstngushed Lecturer (8-11.