458 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 1, JANUARY 2016

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1 458 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 1, JANUARY 2016 Generlzed Cross-Lyer Desgns for Generc Hlf-Duplex Multcrrer Wreless Networks Wth Frequency-Reuse Rozt Rshtch, Student Member, IEEE, RmyH.Gohry,Senor Member, IEEE, nd Hlm Ynkomeroglu, Senor Member, IEEE Abstrct In ths pper, jont desgns of dt routes nd resource lloctons re developed for generc hlf-duplex multcrrer wreless networks n whch ech subcrrer cn be reused by multple lnks. Two nstnces re consdered. The frst nstnce pertns to the generl cse n whch ech subcrrer cn be tmeshred by multple lnks, wheres the second nstnce pertns to specl cse n whch tme-shrng s not llowed nd subcrrer, once ssgned to set of lnks, s used by those lnks throughout the sgnllng ntervl. Novel frmeworks re developed to optmze the jont desgn of dt routes, subcrrer schedules, nd power lloctons. These desgn problems re nonconvex nd hence dffcult to solve. To crcumvent ths dffculty, effcent technques bsed on geometrc progrmmng re developed to obtn loclly optml solutons. Numercl results show tht the desgns developed n both nstnces yeld performnce tht s superor to tht of ther counterprts n whch frequency-reuse s not llowed. Index Terms Power control, geometrc progrmmng, monoml pproxmton, tme-shrng, self-concordnce. I. INTRODUCTION T HE PROSPECT of hvng ubqutous hgh dt-rte wreless servces s leverged by the verstlty nd portblty of the communcton devces tht wll form the nodes of future wreless networks. These devces wll be ble to perform vrous functons ncludng sendng, recevng nd/or relyng dt to other nodes. As such, t s expected tht future wreless networks wll not possess predetermned topology, but rther n d hoc one tht encompsses mny exstng nd upcomng network structures ncludng current nd rely-ded cellulr networks [1], [2]. Gven the strngent lmttons on the spectrum vlble for wreless communctons, provdng hgh dt-rte servces Mnuscrpt receved July 24, 2014; revsed Mrch 15, 2015 nd June 12, 2015; ccepted August 14, Dte of publcton August 1, 2015; dte of current verson Jnury 7, Ths work ws supported n prt by Dscovery Grnt DG) of the Nturl Scences nd Engneerng Reserch Councl NSERC) of Cnd, n prt by Huwe Cnd Co., Ltd., nd n prt by the Ontro Mnstry of Economc Development nd Innovton s ORF-RE Ontro Reserch Fund Reserch Excellence) progrm. Prelmnry versons of ths work were presented, n prt, t IEEE Globecom, n 201 nd, n prt, t IEEE SPAWC, n The ssocte edtor coordntng the revew of ths pper nd pprovng t for publcton ws Prof. Chn-Byoung Che. The uthors re wth the Deprtment of Systems nd Computer Engneerng, Crleton Unversty, Ottw, ON K1S 5B6, Cnd e-ml: rozt@sce.crleton.c; gohry@sce.crleton.c; hlm@sce.crleton.c). Color versons of one or more of the fgures n ths pper re vlble onlne t Dgtl Object Identfer /TWC bnks on shrng the spectrum by multple users whch results n potentlly sgnfcnt nterference. One opton to mtgte nterference s to use the Orthogonl Frequency Dvson Multple Access OFDMA) technque, wheren set of orthogonl nrrow-bnd subcrrers re exclusvely ssgned to ech user. Ths technque offers severl dvntges ncludng desgn smplcty nd reslence to frequency-selectve fdng. In spte of these dvntges, rte-effectve utlzton of the vlble spectrum my requre the OFDMA subcrrers to be used smultneously, rther thn exclusvely, by multple users. Ths s especlly the cse when the network s composed of essentlly seprted clusters. In contrst, for tghtly coupled networks exclusve usge of subcrrers cn be more benefcl from rte perspectve [], [4]. In order for wreless network to be ble to support the relble communcton of hgh dt rtes, the scrce resources vlble for the network must be crefully exploted. Such resources nclude the spectrum vlble for communcton, tme, nd the typclly low power of the wreless nodes. Proper explotton of these resources nvolves choosng the optml routes of the dt flows, the optml powers to be llocted by the nodes to ech subcrrer, nd the optml schedulng nd possbly durton over whch the subcrrers re ssgned to vrous lnks. Although these tsks hve trdtonlly been performed seprtely, they re nterrelted nd performng them n solton my ncur sgnfcnt loss n performnce. To vod the forementoned drwbck, we devse jont optmzton frmework tht ncorportes dt routng, subcrrer schedulng nd power llocton n the desgn of generc multcrrer network. The network s generc n the sense tht t possesses n d hoc topology nd ts nodes cn ssume multple roles smultneously ncludng beng sources, destntons nd/or relys. Ths frmework s centrlzed, n the sense tht the desgn s performed by centrl entty tht s wre of the network prmeters. Hence, ths frmework cn be seen s benchmrk for dstrbuted nd potentlly less comprehensve desgns. In ths frmework ech subcrrer cn be reused by multple lnks, nd the nodes ctng s relys operte n the hlf-duplex mode,.e., node cnnot send nd receve t the sme tme on the sme subcrrer. The objectve of the desgn s to mxmze weghted-sum of the rtes njected nd relbly communcted over the network. Weghts re ssumed to be known pror, but cn be dpted over tme to ccount for frness ssues nd to mntn desred qulty of servce IEEE. Personl use s permtted, but republcton/redstrbuton requres IEEE permsson. See for more nformton.

2 RASHTCHI et l.: GENERALIZED CROSS-LAYER DESIGNS FOR GENERIC HALF-DUPLEX MULTICARRIER WIRELESS NETWORKS 459 Two nstnces of networks re consdered. In the frst nstnce, ech subcrrer cn be tme-shred by multple lnks, thereby resultng n contnuous subcrrer schedulng vrbles. The role of these vrbles s to determne the frcton of tme durng whch subcrrer s used over prtculr lnk. In contrst wth the frst nstnce, n the second one tme-shrng s not llowed nd subcrrer, once ssgned to set of lnks, s used by those lnks throughout the sgnllng ntervl. Ths nstnce results n bnry subcrrer schedules, whch usng prtculr chnge of vrbles, re ncorported n the power llocton constrnts. Ths results n desgn problem tht s sgnfcntly eser thn ts generl counterprt consdered n the frst nstnce. It s worth notng tht the frst nstnce s generlzton not only of the second nstnce, but lso of nstnces n whch frequency-reuse s not permtted [5]. As such, the frmework consdered n ths nstnce offers sgnfcnt performnce dvntges over currently vlble desgns, but t the expense of ncresng dmensonlty nd desgn complexty. Ths nstnce provdes n nherent trdeoff between performnce nd desgn complexty; the desgn complexty cn be reduced by restrctng the number of lnks tht cn reuse prtculr subcrrer. The optmzton problems rsng from the jont desgn n both nstnces re nonconvex nd hence dffcult to solve. To overcome ths dffculty, logrthmc trnsformton s used to cst the orgnl problem n form tht, for ll but few constrnts, comples wth the geometrc progrmmng GP) stndrd form [6]. The constrnts tht re not comptble wth tht form re pproxmted by monoml expressons tht correspond to ther frst order Tylor expnson round gven ntl pont [7]. Usng n exponentl trnsformton, the resultng pproxmton cn be cst n convex form. A refnement of ths pproxmton cn be obtned by tertve updtng of the ntl pont. In prtculr, we use the so-clled tertve monoml pproxmton technque, wheren the soluton of one convex pproxmton s used s the ntl pont n the followng terton. Under reltvely mld condtons, ths technque s gurnteed to yeld soluton of the Krush-Kuhn- Tucker KKT) system correspondng to the orgnl problem [8]. Numercl results show tht the desgns developed n both nstnces yeld performnce tht s consderbly superor to tht of ther counterprts n whch frequency-reuse s not llowed. In comprson wth currently vlble desgns, the ones presented heren re the frst to ttempt desgnng dt routes, subcrrer schedules nd power llocton jontly when the subcrrers re both tme-shred nd frequency-reused. In prtculr, the contrbutons n ths pper nclude: 1) ntroducng the concept of smultneous tme-shrng nd frequency-reuse of subcrrers; 2) cstng the jont desgn of dt routes, subcrrer schedules nd power llocton n frmework tht s menble to GP-bsed optmzton; ) provdng smplfed pproch tht enbles musterng consderble porton of the gns offered by the full jont desgn, but wth sgnfcntly lower complexty; nd 4) developng n effcent polynoml complexty lgorthm for the specl cse n whch the subcrrers cn be frequency-reused but not tme-shred. Ths work bulds on the results obtned n [9] nd [10]. However, the exposton heren s more comprehensve nd ncludes ddtonl exmples, smplfed pproch nd complexty nlyss. The pper s orgnzed s follows. Secton II provdes n overvew of currently vlble desgn technques. Secton III explns the system model nd desgn objectve. Secton IV consders the jont desgn of dt routes nd power lloctons when tme-shrng of subcrrers s llowed. The complementry nstnce n whch tme-shrng s not llowed s ddressed n Secton V. The complexty of the proposed lgorthms re exmned n Secton VI. Numercl results re provded n Secton VII, nd Secton VIII concludes the pper. For completeness, the GP stndrd form nd the monoml pproxmton technque re provded n Appendx A, nd the dervton of the results pertnng to complexty s provded n Appendx B. II. RELATED WORK In ths secton we provde n overvew of the currently vlble technques for routng nd resource llocton n wreless networks. A plethor of technques s vlble for optmzng ech spect n solton, but sgnfcntly fewer ones consder ther optmzton jontly. Resource llocton n wreless networks consttutes the tsk of determnng the power llocted for ech trnsmsson nd the frcton of tme over whch prtculr subcrrer s ssgned to tht trnsmsson. Instnces n whch resource llocton technques were developed re provded n [6], [11] [15] for vrous network scenros. For nstnce, power llocton technques for sngle-crrer cellulr systems nd d hoc multcrrer systems were developed n [11] nd [6], respectvely. To enble more effectve utlzton of resources, power lloctons were optmzed jontly wth bnry-constrned subcrrer schedules. For nstnce, the desgns developed n [12] nd [1] rely on the premse tht ech subcrrer s exclusvely used by one node nd the solutons obtned theren re potentlly suboptml. When the bnry constrnt on the subcrrer schedules s relxed llowng the subcrrers to be tme-shred by multple nodes, the optml power lloctons cn be shown to be the wter-fllng ones [14]; relted problem ws consdered n [15] for cse n whch the nodes experence self-nose. Further mprovement cn be cheved by jont optmzton of resource lloctons nd routng [5], [16] [19]. For nstnce, method for obtnng jontly optml routes nd power lloctons ws developed n [16] for the cse n whch the nodes were restrcted to use orthogonl chnnels for ther trnsmssons. In complementry fshon, the cse n whch the power lloctons re fxed ws consdered n [17]. Theren, heurstc ws developed for optmzng the dt routes nd subcrrer schedules jontly. Cptlzng on the potentl gns of ncorportng power llocton jontly wth dt routng nd subcrrer schedulng, the uthors consdered generc network n whch the nodes cn ssume multple roles t the sme tme nd ech subcrrer could be ether used exclusvely by one lnk or tmeshred by multple lnks [5]. Although the desgns provded n [5] offer n effectve mens for explotng the resources vlble for the network, these desgns restrct the subcrrers

3 460 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 1, JANUARY 2016 TABLE I RELATED WORK to be used exclusvely by only one lnk t ny gven tme nstnt. Such restrcton my not ncur sgnfcnt performnce loss n tghtly coupled networks [], but n networks wth clustered structures, ths restrcton cn be qute hrmful. For unclustered networks, frequency-reuse my result n substntl ncrese n the nterference levels. However, f properly exploted, frequency-reuse cn yeld vluble performnce gns. The effect of frequency-reuse ws consdered n snglechnnel networks n [18] for the cse n whch the dt rtes re restrcted to ssume dscrete vlues, nd n [19] for the cse n whch the nodes use superposton codng. In the current work, we wll consder the jont optmzton of power lloctons, subcrrer schedules nd dt routes n the desgn of generc multcrrer networks wth frequency-reuse, nd wth nd wthout tme-shrng. As such, these desgns generlze currently vlble ones, nd wll subsequently offer sgnfcnt mprovement over ther performnce. A summry of ths revew nd comprson to our work s presented n Tble I. Fg. 1. An exemplry network wth N = 4, K = 2ndD ={1, 2}. The objectve s to mxmze weghted sum of {s n d) },.e., the rte njected t node n nd ntended for destnton d. III. PROBLEM STATEMENT A. System Model We consder multcrrer wreless network of N nodes, ech wth one trnsmt nd one receve ntenn, nd fxed power budget, P n, n N {1, 2,...,N}. The network opertes over frequency-selectve brodbnd chnnel of bndwdth W 0, whch s prttoned nto K frequency-flt nrrowbnd chnnels, ech of bndwdth W = W 0 K. Node re ssumed to be cpble of smultneously trnsmttng, recevng nd relyng dt. Ths ssumpton s generc, n the sense tht constrnng some nodes to perform subset of tsks cn be redly ncorported n the formultons tht wll be developed herenfter. For trctblty, the nodes re ssumed to lwys hve dt redy for trnsmsson [16], nd for prctcl consdertons, the relyng nodes re ssumed to operte n mult-hop, rther thn coopertve, hlf-duplex mode [17]. The nodes re connected wth L wreless lnks, ech composed of K subcrrers nd the set of ll lnks s denoted by L {1, 2,...,L}. The coeffcent of the k-th subcrrer of lnk l connectng node n to node n s denoted by the complex number h k) nn whch comprses pthloss, shdowng nd fdng. An nstnce of such network wth N = 4 nodes nd K = 2 subcrrers s depcted n Fgure 1. In ddton to ther desred sgnls, the nodes receve superposton of nose nd nterference due to the trnsmssons of other nodes n the network. Denotng the sgnls trnsmtted to nd receved by node n on the k-th subcrrer by u k) n nd y n k), respectvely, we cn wrte y k) n = n N\{n } hk) nn u k) n + v k) n, where \ denotes the set-mnus operton nd v k) n denotes the correspondng zero-men ddtve Gussn nose wth vrnce N 0. Assumng, s before, tht l L s the lnk connectng node n to node n, t cn be seen tht the sgnl-to-nose-plusnterference rto SNIR) observed by node n on subcrrer k of lnk l s gven by SNIRl, k) = p nk h k) nn 2 WN 0 + n N\{n,n } pk) n h k) n n, 1) 2 where p nk s the power llocted by node n to the k-th subcrrer. The second term n the denomntor of 1) represents the ggregte nterference observed by node n on subcrrer k of lnk l. When the nodes trnsmt Gussn dstrbuted sgnls, the mxmum dt rte tht cn be relbly communcted on ths subcrrer s gven by W log SNIRl, k)). For ese of exposton, we dvde both the numertor nd denomntor of 1) by WN 0 nd we use g to denote h k) nn the normlzed chnnel gn, 2 WN 0, between ny two nodes n, n N.

4 RASHTCHI et l.: GENERALIZED CROSS-LAYER DESIGNS FOR GENERIC HALF-DUPLEX MULTICARRIER WIRELESS NETWORKS 461 B. Network Topology The consdered network cn be represented by fullyconnected weghted drected grph wth N vertces nd L = NN 1) lnks. To fcltte enumerton of lnks, the lnk from node n to node n wll be lbelled by l = N 1)n 1) + n 1fn < n nd by l = N 1)n 1) + n f n>n. The sets of ncomng nd outgong lnks of node n N re denoted by L n) nd L + n), respectvely, nd the connectvty of ths grph cn be cptured by n ncdence mtrx, A = [ nl ], where nl = 1fl L + n), nl = 1fl L n) nd nl = 0 otherwse [16]. C. Desgn Objectve Let D {1,...,D} be the set of ll destnton nodes, where D N.Lets n d) be the rte of the dt strem njected nto node n N nd ntended for destnton d D. The objectve of our jont desgn s to mxmze weghted-sum of the rtes njected nto the network,.e., mx d D n N\{d} wd) n s n d), where {w n d) 1 } re non-negtve weghts stsfyng DN 1) d D n N\{d} wd) n =1. Assgnng weghts to the njected rtes provdes convenent mens for controllng the qulty of servce QoS); hgher weght mples hgher prorty. Weghts re typclly ssgned pror, but cn be dpted to meet QoS requrements [15]. Vryng the weghts enbles us to determne the set of ll rtes tht the proposed desgn cn smultneously cheve. Hvng descrbed the system model, n Secton IV we wll chrcterze the constrnts tht must be stsfed by the routes, the subcrrer schedules, the dt rtes nd the power lloctons. IV. GENERAL CASE: ROUTING AND RESOURCE ALLOCATION WITH TIME-SHARING We consder the cse when ech subcrrer cn be both reused nd tme-shred by multple lnks. Ths cse generlzes the cses n whch ether frequency-reuse or tme-shrng of subcrrers s not consdered, e.g., [5]. After chrcterzng the constrnts tht must be stsfed by the network vrbles, we wll formulte the cross-lyer desgn s n optmzton problem. Unfortuntely ths problem s nonconvex nd to obtn soluton of ts KKT system, we wll use n tertve GP-bsed technque tht s gurnteed to converge to such soluton. A. System Constrnts In ths secton, we derve the mthemtcl constrnts tht must be stsfed by ny fesble set of dt routes, tme-shrng schedules nd power lloctons. 1) Routng Constrnts: Let x d) be the dt flow ntended for destnton d D on subcrrer k K of lnk l L. The flows, {x d) }, nd the njected rtes, {sd) n }, re relted by the flow conservton lw, whch must be stsfed t ech node. Ths lw stpultes tht the sum of flows ntended for ny destnton d D t ech node must be equl to zero [16]. Applyng Fg. 2. An exemplry schedulng tble for network wth L =, K = 1. ths lw to the current network nd usng the ncdence mtrx n Secton III-B, t cn be seen tht {x d) } nd {sd) n } must stsfy the followng constrnts: nl x d) = s n d), n N \{d}, d D. 2) l L k K The flow conservton lw mples tht the rte of dt levng the network t d D equls the sum of the dt rtes njected nto the network nd ntended for ths destnton. Hence, we cn wrte s d) d = n N\{d} sd) n. The njected rtes, {s n d) } n =d, re non-negtve, nd snce the network s represented by drected grph, the flows, {x d) },mustbelso non-negtve. Hence, s n d) 0, n N \{d}, d D, ) x d) 0, l L, k K, d D. 4) 2) Schedulng Constrnts: Consderng both tme-shrng nd frequency-reuse requres ntroducng set of vrbles to chrcterze the frcton of tme over whch prtculr subset of lnks utlze the sme subcrrer. To do so, let be the frcton of the sgnllng ntervl durng whch lnks l 1,...,l m L re smultneously ctve on subcrrer k K; the remnng L m lnks n L re slent on ths subcrrer. Wthout loss of generlty, we wll wrte the ndces n n scendng order,.e., l 1 < <l m. For nottonl convenence, let Ɣ be the set of ll the subcrrer tme-shrng schedules. The crdnlty of Ɣ s gven by Ɣ = K L L ) =1 = K 2 L 1). For nstnce, consder network wth L = lnks nd K = 1 subcrrers. In ths cse, Ɣ = γ k) l 1 l m } nd Ɣ =7. To see the role of Ɣ, consder the schedules n Fgure 2. In ths fgure, γ 1) 1 = 0.5, γ 1) 1) 1,2 = 0.2, γ 1,2, = 0., nd ll the other elements n Ɣ re zero. Note tht the fct tht the chnnels re ssumed constnt over the sgnllng ntervl mples tht only the tme-shrng schedules.e., entres of Ɣ) ffect the rte expressons, rrespectve of the prtculr tme ntervl over whch the subcrrers re tmeshred. In other words, horzontl dsplcement of the shded blocks n Fgure 2 does not ffect the rte expressons. The number of vrbles n Ɣ grows exponentlly wth the number of lnks, L. Ths renders the ncorporton of Ɣ n the jont optmzton computtonlly prohbtve. In most cses ths complexty cn be sgnfcntly reduced wthout ncurrng hevy performnce losses. For nstnce, f the network s tghtly coupled, hgh nterference levels render the reuse of subcrrers on multple lnks less benefcl. In such cse, restrctng the reuse of subcrrer to fewer lnks my ncur neglgble deterorton n performnce but reduces the number of vrbles {γ 1) 1,γ 1) 2,γ 1),γ 1) 1,2,γ1) 1,,γ1) 2,,γ1) 1,2,

5 462 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 1, JANUARY 2016 sgnfcntly. To tke dvntge of ths observton, we lmt the number of lnks tht cn reuse prtculr subcrrer to I L. By performng ths restrcton, the number of elements n Ɣ s reduced from K 2 L 1 ) to K I L ) =1, whch, for smll I,s polynoml n L. For nstnce, f t most two lnks re llowed to reuse prtculr subcrrer t ny gven tme,.e., I = 2, the number of elements n Ɣ reduces to LL+1) 2. It s worth notng tht lmtng the number of smultneous trnsmssons, I, nherently offers trde-off between the performnce nd complexty. In prtculr, s I ncreses, the vlble resources re utlzed more effcently. However, our smultons suggest tht most of the gn of tme-shrng nd frequency-reuse s ccrued by only consderng I smultneous trnsmssons. For fesble tme-shrng schedules, the elements n Ɣ must be non-negtve nd, to ensure no overlppng n tme, the totl tme over whch ny subcrrer k K s used must not exceed the length of the sgnllng ntervl. These constrnts mply tht Ɣ 0, elementwse, 5) I γ k) 1, k K. 6) m=1 l 1 l m L Note tht summtons n 6) chrcterze the number of lnks, m, tht reuse prtculr subcrrer k. For nstnce, for the cse n whch t most two lnks reuse ths subcrrer, the left hnd sde LHS) of 6) cn be expressed s l 1 L γ k) l 1 L l 2 L γ k) l 1 l 2. l 1 + Nodes cnnot brodcst dt to multple destntons t the sme tme, tht s, t ny tme nstnt, node n cn hve t most one ctve lnk on subcrrer k. Hence, the tme-shrng schedules correspondng to multple outgong lnks of node n must be zero. Ths cn be represented s nl + 1 nl + 2 γ k) l 1 l 2 + I γ k) m= l l m L = 0, l 1 L,l 2 L \{l 1 }, k K, 7) where nl + = mx{0, nl}, tht s, nl + = 1fl L +n) nd zero, otherwse. To enforce the hlf-duplex constrnt, we must ensure tht no two lnks, l 1 L n) nd l 2 L + n), cn be ctve on the sme subcrrer k K t the sme tme. Ths mples tht ll the tme-shrng schedules tht correspond to l 1 nd l 2,.e., γ k), m = 2,...,I, must be zero. Snce ll the entres n Ɣ re non-negtve, these constrnts cn be wrtten s I nl + 1 nl 2 γ k) l 1 l 2 + γ k) = 0, m= l l m L l 1 L,l 2 L \{l 1 }, k K, 8) where nl = mn{0, nl}, tht s, nl = 1fl L n) nd zero, otherwse. Note tht 7) nd 8) tke effect only when + nl nl = 0 nd + nl + nl = 0, respectvely. ) Power Allocton Constrnts: To fcltte the desgn, we replce the node power vrbles {p nk } wth lnk power vrbles {q }, whch re relted by the followng trnsformton: p nk = mx q, n N, k K. 9) l L + n) To gn better understndng of the trnsformton n 9), we note tht 7) mples tht, of ll the lnks n L + n), only one element n the set {q } l L+ n), n N, k K, cn ssume strctly postve vlue. Now, 9) ndctes tht ths vlue s the power llocted by node n to subcrrer k. Usng 9), we wll formulte our desgn n terms of {q } nsted of {p nk }. These vrbles must stsfy the followng non-negtvty constrnts: q 0, l L, k K. 10) In prctcl network, the nodes re lkely to hve ndvdul power budgets whch bounds the totl power used by ech node on ll subcrrers. To cpture ths constrnt, we note tht only the subcrrers scheduled to outgong lnks contrbute to the power consumpton of ech node. More specfclly, f l 1 L + n), then ll the tme-shrng schedules tht correspond to l 1 contrbute to the power consumpton t node n. Ths constrnt cn be wrtten s q l1 k k K l 1 L + n) γ k) l 1 + I γ k) m=2 l 2 l m L P n, n N. 11) 4) Cpcty Constrnts: To complete the chrcterzton of the network, we pont out tht the dt flows nd the power lloctons re coupled by the mxmum ggregte rte tht cn be supported by the subcrrers of ech lnk. In prtculr, the ggregte rte d D xd) must not exceed the cpcty of the k-th subcrrer of lnk l. To chrcterze the cpcty constrnts, we note tht the trnsmsson on lnk l L nd subcrrer k K s composed of two prts. The frst prt ccounts for the frcton of tme over whch ths trnsmsson s nterference-free, wheres the second prt ccounts for the frcton of tme over whch ths trnsmsson nterferes wth other trnsmssons. To chrcterze the second prt, we dentfy the nterferng lnks nd the frcton of tme over whch these lnks re nterferng. To do so, we note tht, f subcrrer k s tme-shred by lnks l 1,...,l m, then the trnsmssons on lnks l 2,...,l m nterfere wth the trnsmsson on lnk l 1. Hence, the SNIR expresson for the trnsmsson on lnk l 1 s, where l q l1 k g l1 k denotes the ndex of 1+ m =2 q l k g l k the lnk connectng the node t whch lnk l orgntes to the node t whch lnk l 1 ends. Snce lnks l 1,...,l m re smultneously ctve on subcrrer k for frcton of γ k),the expresson for the dt rte tht cn be communcted ) over lnk l 1 s W γ k) q l1 k g l1 k log m. Summng over =2 q l k g l k ll possble combntons of the nterferng lnks, the cpcty

6 RASHTCHI et l.: GENERALIZED CROSS-LAYER DESIGNS FOR GENERIC HALF-DUPLEX MULTICARRIER WIRELESS NETWORKS 46 constrnt on the ggregte flow of lnk l 1 on subcrrer k cn be expressed s d D x d) l 1 k W + γ k) l 1 log q l1 kg l1 k) I m=2 l 2...l m L γ k) log 2 ) q l1 kg l1 k m. =2 q l kg l k 12) B. Problem Formulton To ensure the fesblty of the rtes generted by our desgn, the constrnts n 2) 12) must be stsfed. Combnng these constrnts yelds the followng optmzton problem: mx w n d) sd) n, {s d) n },{x d) },{q },Ɣ d D n N\{d} subject to Routng constrnts n 2) 4), Schedulng constrnts n 5) 8), Power llocton constrnts n 10) nd 11), Cpcty constrnts n 12). 1) The optmzton problem n 1) s nonconvex becuse of the power llocton constrnt n 11) nd the cpcty constrnts n 12). Exmnng 1) revels tht ths problem shres some fetures wth the GP stndrd form, cf. Appendx A-1. To explot ths observton, n the next secton we wll perform chnge of vrbles tht wll enble us to express the objectve nd ll, but one set, of the constrnts n GP-comptble form. The resdul constrnts tht do not comply wth the GP stndrd form re pproxmted usng the monoml pproxmton technque n Appendx A-2. Under reltvely mld condtons [8], tertve pplcton of ths technque s known to yeld soluton of the KKT system correspondng to 1), see e.g., [5], [19]. C. Generlzed GP-Bsed Algorthm To cst 1) n form tht s menble to monoml pproxmton, we defne two sets of vrbles, {t n d) } nd {r d) }, whch re relted to {s n d) } nd {x d) } by the followng mps: s n d) = log 2 t n d), x d) = W log 2 r d), n N \{d}, d D,l L, k K. 14) These mps re bjectve, whch renders recoverng {s n d), x d) } strghtforwrd. The objectve nd the routng constrnts n 1) cn be cst n GP-comptble form. In prtculr, the objectve cn be expressed s ) d D n N\{d} t n d) d) w n nd the routng constrnts cn be expressed s ) Wnl = t d) n, n N \{d}, d D, 15) l L k K r d) r d) 1, l L, k K, d D, 16) t n d) 1, n N \{d}, d D. 17) The non-negtvty constrnts n 5) nd 10) re nherently stsfed n the GP frmework. The constrnts n 6) nd 11) re lredy n GP-comptble form. We now consder the constrnts n 7) nd 8). The rght hnd sde RHS) of these constrnts re zero, whch mkes them ncomptble wth the GP frmework n Appendx A-1. Ths problem cn be llevted by constrnng ther LHS to be less thn n rbtrry smll number ɛ>0,.e., I nl + 1 nl + 2 γ k) l 1 l 2 + γ k) ɛ, nl + 1 nl 2 γ k) l 1 l 2 + m= l l m L I l 1 L,l 2 L \{l 1 }, k K, 18) γ k) ɛ, m= l l m L l 1 L,l 2 L \{l 1 }, k K. 19) The remnng constrnts tht re not GP-comptble re those n 12). Invokng the chnge of vrbles n 14), for l 1 L nd k K, those constrnts cn be expressed s r d) l 1 k 1 + q l 1 kg l1 k) γ k) l 1 d D I m=2 l 2 l m L q l1 kg l1 k m =2 q l kg l k ) γ k) l 1...lm. 20) The RHS of 20) s menble to the monoml pproxmton technque descrbed n Appendx A-2 [7]. One pproch to use ths technque s to pproxmte ll the terms n the RHS of 20) by one monoml. Ths pproch s overly complcted, nd n lterntve s to pproxmte ech term by monoml. The product of these monomls consttutes monoml pproxmton of the RHS of 20). Hence, the constrnt n 20) cn be pproxmted wth ) r d) l 1 k M 1 + q l1 kg l1 k) γ k) l 1 d D ) k) γ I l q l1 kg 1...lm l1 k M m, m=2 l 2 l m L =2 q l kg l k 21) where the functonl M ) s descrbed n Appendx A-2. Note tht, {γ k) l 1 l m } re vrbles nd hence nseprble from the rgument of M ). Now, the problem n 1) cn be pproxmted by the followng GP: ) d) w n mx, {t d) n },{r d) },{q },Ɣ d D n N\{d} t d) n subject to Routng constrnts n 15) 17), Schedulng constrnts n 6), 18) nd 19), Power llocton constrnts n 11), Approxmte cpcty constrnts n 21). 22)

7 464 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 1, JANUARY 2016 TABLE II SUCCESSIVE GP-BASED ALGORITHM FOR SOLVING 22) Note tht the relxtons n 18) nd 19) my result n nfesble subcrrer tme-shres tht do not stsfy the constrnts n 7) nd 8). To construct fesble schedules, the elements of Ɣ tht re less thn or equl to ɛ re set to zero. Usng stndrd exponentl trnsformton, the GP n 22) cn be redly trnsformed nto convex optmzton problem whch cn be solved n polynoml tme usng nteror-pont methods IPMs) [7]. Ths mples tht 22) enbles us to effcently solve 1) pproxmtely n the neghbourhood of ny ntl set {q 0) },Ɣ0)). Fndng the globl soluton for the nonconvex problem n 1) s dffcult, wheres solvng the pproxmted problem n 22) s strghtforwrd. To explot ths fct, we ncorporte the formulton n 22) n n tertve lgorthm, whereby the output of solvng 22) for n ntl pont {q 0) },Ɣ0)) s used s strtng pont for the subsequent terton. Ths technque s usully referred to s the sngle condenston method, e.g., [19], [20], nd under reltvely mld condtons, ts convergence to soluton of the KKT system correspondng to 1) s gurnteed [8]. Snce the orgnl desgn problem s not convex, ths system hs multple locl solutons nd the one to whch the sngle condenston method converges depends on the ntl pont; some of the locl solutons my be globl ones. A summry of ths lgorthm s descrbed n Tble II. In the next secton, we wll dscuss specl cse of ths lgorthm when tme-shrng of subcrrers s not llowed. Before we do tht, we now provde bref dscusson on the mplementton of ths lgorthm. To begn wth, we note tht the lgorthm n Tble II s centrlzed, n the sense tht the desgn s performed by centrl entty tht s wre of the network prmeters. The sgnllng exchnge between the nodes nd the centrl entty, requred to estblsh communcton n the consdered frmework, re descrbed s follows. At the begnnng of ech sgnllng ntervl, the centrl entty prompts the nodes n the network to sequentlly brodcst plot sgnls of prescrbed power levels. Subsequently, ech node computes the subcrrer chnnel gns from ll other nodes n the network. There s totl of LK such gns, where L s the number of lnks nd K s the number of subcrrers. Ech node sends these gns long wth ts destnton nodes, f ny, nd ts prorty weghts to the centrl entty. The centrl entty performs the jont optmzton of the power lloctons, schedulng prmeters nd dt routes s descrbed n Tble II. It then forwrds these decsons to ll the nodes, possbly over dedcted control chnnel. In prtculr, the nformton forwrded by the centrl entty nclude 1) the subcrrer ndex nd the tme llocted to ech trnsmsson. Ths nformton s provded by the set Ɣ. The crdnlty of ths set depends on the number of smultneous trnsmssons llowed n ech subcrrer, I. For nstnce, for I = 2, Ɣ =LKL + 1)/2; 2) The power llocted to ech trnsmsson. Ths nformton s provded by the set {q } nd the crdnlty of ths set s LK; nd ) The dt rtes t ech trnsmttng nd recevng node n the route of the strem ntended for ech destnton. Ths nformton s provded n the set {x d) } nd the crdnlty of ths set s LKD, where D s the number of ntended destntons. V. SPECIAL CASE: ROUTING AND RESOURCE ALLOCATION WITHOUT TIME-SHARING In ths secton, we consder desgn problem smlr to the one descrbed n Secton IV, but for the cse when tmeshrng of subcrrers s not llowed. Ths corresponds to the specl cse n whch the entres of Ɣ n Secton IV-A2 re restrcted to be bnry. Ths restrcton results n mxed nteger progrm whch s generlly dffcult to solve. To overcome ths dffculty, we cpture the effect of the schedulng vrbles n the power llocton constrnts. We wll show tht ths pproch wll enble us to develop desgn lgorthm wth polynoml-complexty. A. System Constrnts 1) Routng Constrnts: These constrnts re dentcl to those descrbed n 2) 4). 2) Power Allocton Constrnts: In chrcterzng these constrnts, we wll use the method descrbed n Secton V-A2 to denote the power llocted for trnsmsson on subcrrer k of lnk l by the vrbles {q }. These vrbles must stsfy the non-negtvty constrnts n 10) nd the power budget constrnt. These constrnts, usng 9), cn be cst s nl + q P n, n N. 2) k K l L Smlr to the cse consdered n Secton IV, the nodes cnnot smultneously brodcst to multple destntons on the sme subcrrer. However, ths requrement n the current cse cn be mplctly cptured by the llocton of the lnk powers. In prtculr, for ny subcrrer k K, ny node n N nd ny two lnks l 1,l 2 L + n), t lest q l1 k = 0orq l2 k = 0,.e., + nl 1 + nl 2 q l1 kq l2 k = 0, l 1,l 2 L, k K, n N. 24) Ths constrnt s sgnfcntly less nvolved thn the one n 7). Smlrly, the hlf-duplex requrement cn be cptured by ensurng tht, for ech node n N, f the power on subcrrer k of L + n) s strctly postve, then the power llocted to ths subcrrer on ll the lnks n L n) s zero, nd vce vers. Hence, the hlf-duplex requrement cn enforced by the followng constrnts: nl 1 + nl 2 q l1 kq l2 k = 0, l 1,l 2 L, k K, n N. 25)

8 RASHTCHI et l.: GENERALIZED CROSS-LAYER DESIGNS FOR GENERIC HALF-DUPLEX MULTICARRIER WIRELESS NETWORKS 465 Note tht these constrnts re smpler thn ther counterprts n 8). Also, note tht 24) nd 25) re trvlly stsfed f ether lnk l 1 or l 2 re not connected to node n. ) Cpcty Constrnts: In ths cse, the constrnts n 12) cn be redly seen to reduce to x d) ) l 1 k W log q l1 kg l1 k l 2 L\{l 1 } q. 26) l 2 kg l 2 k d B. Problem Formulton Usng the chrcterzton descrbed n Secton V-A, the desgn problem cn be cst s: mx w n d) sd) n, {s d) n },{x d) },{q } d D n N\{d} subject to Routng constrnts n 2) 4), Power llocton constrnts n 10), 2) 25), Cpcty constrnts n 26). 27) The optmzton problem n 27) s nonconvex becuse the RHS of 26) s the logrthm of rtonl functon, nd therefore not concve. The equlty constrnts n 24) nd 25) re not ffne nd hence, nonconvex. In the next secton, we wll develop GP-bsed lgorthm, nlogous to the one descrbed n Secton IV-C, to obtn loclly optml soluton. C. Proposed GP-bsed Algorthm The optmzton problem n 27), lthough nonconvex, s menble to the GP-bsed monoml pproxmton n Appendx A-2. To use ths pproxmton, we use 14) to trnsform {s n d) } nd {x d) } to {td) n } nd {r d) }, respectvely. Usng these new vrbles, the routng constrnts re redly expressed n GP-comptble form s descrbed n 15) 17). Substtutng from 14) nto 26) yelds the followng set of equvlent constrnts: 1 + q l2 kg l 2 k r d) l 1 k 1 + q l2 kg l 2 k, l 2 L\{l 1 } d D l 2 L k K, l 1 L, 28) whch re, unfortuntely not GP-comptble. Usng the monoml pproxmton technque n Appendx A-2 yelds the followng pproxmton of 28) n the neghbourhood of {q 0) 1 + l 2 L\{l 1 } q l2 kg l 2 k d D r d) l 1 k c l 1 k l 2 L }: q l2 k/q 0) θl2 k l 2 k), k K,l 1 L, 29) where {q 0) } s the ntl power llocton, c l 1 k = 1 + l 2 L q0) l 2 k g l 2 k, nd θ l2 k = q 0) l 2 k g l 2 k /c l1 k. Anlogous to the cse consdered n Secton IV-C, 24) nd 25) re replced wth the GP-comptble nequlty constrnts. The jont desgn of dt routes nd power lloctons n 27) cn be pproxmted wth the followng GP: {t d) n mx },{r d) },{q } d D n N\{d} t d) n ) d) w n subject to Routng constrnts n 15) 17), Power llocton constrnts n 2) 25)relxed versons), Approxmte cpcty constrnts n 29). 0) A loclly optml soluton of 27) cn be obtned by solvng 0) tertvely usng the sngle condenston method descrbed n Secton IV-C. VI. COMPLEXITY ANALYSIS In ths secton we exmne the computtonl complexty requred for solvng the problems descrbed n Sectons IV-C nd V-C for the cses wth nd wthout tme-shrng, respectvely. The lgorthms n these sectons tertvely solve the fmles of the optmzton problems n 22) nd 0). Beng n GP-comptble form, these problems cn be redly converted nto convex forms nd cn be effcently solved usng IPM-bsed solvers. In IPM, the objectve nd nequlty constrnts re used to construct log-brrer functon whch s mnmzed long centrl pth usng Newton s method. The complexty of ech Newton step grows wth the cube of the number of nequlty constrnts nd the number of Newton steps cn be bounded f the log-brrer functon s self-concordnt [21], cf. Appendx B. In tht cse, the number of Newton steps cn be shown to grow wth the squre root of the number of nequlty constrnts [21]. Unfortuntely, the log-brrer functons relted to the problems n 22) nd 0) re not self-concordnt. To crcumvent ths dffculty, we ntroduce set of uxlry vrbles nd constrnts whch, lthough redundnt, enbles us to construct self-concordnt log-brrer functons. Usng these functons nd the results n [21], we rrve t the followng proposton: Proposton 1: The complexty of solvng 22) wth IPMbsed solvers s of order I O 2LKN + N + DN 1) + 2K L ) ).5, nd the complexty of solvng 0) wth IPM-bsed solvers s of order O LKL + 2) + N + DN 1)).5). Proof: See Appendx B. The frst sttement of Proposton 1 pertns to the generl cse wth tme-shrng nd frequency-reuse. Ths sttement shows tht the complexty of solvng the problem n 22) s polynoml n L for smll vlues of I. The complexty of solvng 22) cn be further reduced by combnng the brodcstng, =1

9 466 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 1, JANUARY 2016 constrnt nd the hlf-duplex constrnt n 7) nd 8), respectvely. In prtculr, exmnng these constrnts revels tht they re relted to the network topology nd do not depend on the chnnel condtons. Hence, these two constrnts cn be enforced by prunng the set Ɣ pror to solvng 1) or ts pproxmted verson n 22). The prunng rule s s follows: For ech l nd l L, f ether nl + 1 nl + 2 = 0or nl + 1 nl 2 = 0, the correspondng tme-shres n 7) nd 8) re removed from the set Ɣ. Unfortuntely, we hve not been ble to obtn closed form of the crdnlty of the resultng Ɣ. However, the reducton n complexty, t lest for smll networks, ppers to be sgnfcnt. For nstnce, for fully connected networks wth N = 4 nodes nd L = NN 1) = 12 lnks, Ɣ s reduced from 4095 to 40. The second sttement of Proposton 1 pertns to the specl cse n whch tme-shrng s not llowed. Ths sttement shows tht the complexty of solvng 0) s polynoml n the number of nodes, N, nd the number of subcrrers, K. In prtculr, t grows s L 7 K.5. Another cse n whch the desgn complexty s polynoml s the one n whch the subcrrers re tme-shred but not frequency-reused [5]. In tht cse the desgn complexty s O LK4 + D) + N + K + DN 1)).5). Hence the specl cses wth ether no frequency-reuse or no tme-shrng hve polynoml complexty. VII. SIMULATION RESULTS In ths secton we provde numercl results to evlute the performnce of jont routng nd resource llocton lgorthms for the cses wth nd wthout tme-shrng. The loctons of the nodes re rndomly generted nd evenly dstrbuted over m 2 squre. The nodes re ssumed to hve dentcl power budgets,.e, P n = P, n N, nd the vlble frequency-selectve chnnel s prttoned nto set of frequency-flt Rylegh fdng chnnels wth the vlues of pthloss PL) nd shdowng components obtned from the non lne-of-sght communcton of ndoor hotspot InH) scenro n the IMT-Advnced document [22]. Accordng to [22], the PL component on lnk l L s gven by PL = 4.log 10 d l ) log 10 f c ), 1) where d l s the length of lnk l n meters nd f c s the crrer frequency n Gghertz whch, n our smultons, s set to f c =.4 GHz. The shdowng component s ssumed to be log-norml dstrbuted wth men of 0 db nd stndrd devton of 4 db. The Rylegh fdng component s generted by the envelope of zero-men unt-vrnce complex Gussndstrbuted rndom vrble. The vlble bndwdth round ech subcrrer s set to W = 200 KHz nd the nose power densty t recevers s set to N 0 = 174 dbm/hz. The results reported heren re obtned usng the CVX pckge [2] wth n underlyng MOSEK solver [24]. The vlue of ɛ n 7) nd 8) s set to Exmple 1: Jont Routng nd Resource Allocton wth Tme-shrng) Consder n exemplry network wth N = 4 nodes. In ths network, nodes nd 4 wsh to communcte TABLE III NORMALIZED CHANNEL GAINS, {g }, IN EXAMPLE 1[DB] wth nodes 2 nd 1, respectvely, over K = 2 subcrrers. In prtculr, for destnton node d = 1, the source s node n = 4 nd nodes {2, } re potentl relys, nd, for destnton node d = 2, the source s node n = nd nodes {1, 4} re potentl relys. The consdered network hs L = 12 drectonl lnks nd therefore the chnnel mtrx hs 12 2 elements. The chnnels re ssumed to be sttc nd ther normlzed gn n db,.e., 10 log 10 g, s gven n Tble III. In ths exmple, the power budget of ech node s set to P = 20 dbm, the number of smultneous trnsmssons s set to I = nd the two rtes, s 2) nd s 1) 4, re ssgned equl weghts,.e., w2) = w 1) 4 = 1. Snce n ths exmple tme-shrng s llowed, the lgorthm n Secton IV-C s used to generte the dt routes, tme-shrng schedules nd power lloctons. The sum-rte yelded by the lgorthm n Secton IV-C s 7.4 b/s/hz. The dt routes generted by ths lgorthm re llustrted n Fgure. For ese of exposton, the network n ths exmple s splt nto the two sub-networks: the one n Fgure ) depcts the routes of the dt ntended for destnton d = 1, nd the one n Fgure b) depcts the routes of the dt ntended for destnton d = 2. The complete network s the superposton of the two sub-networks. For nstnce, the dt trnsmtted over lnk 7, connectng node to node 1, s 4.8 b/s/hz, of whch 2 b/s/hz s ntended for destnton d = 1 nd 2.8 b/s/hz s ntended for destnton d = 2. The tme-shrng schedules of the subcrrers generted by the lgorthm n Secton IV-C re provded n Fgure 4. It cn be seen from ths fgure tht subcrrer k = 1 s both reused nd tme-shred, wheres subcrrer k = 2 s only tme-shred. Fgures nd 4 mply tht lnk 7, connectng node to node 1, nd lnk 11, connectng node 4 to node 2, crry the dt ntended for both destntons on the sme subcrrer, k = 1, durng the sme tme ntervl. The fct tht our desgns enforce hlf-duplex requrement cn be nferred from these fgures. For nstnce, Fgure ) shows tht node uses the sme subcrrer, k = 1, for ts trnsmsson nd recepton on lnks 7 nd 12, respectvely, but Fgure 4 shows tht trnsmsson nd recepton occur durng dfferent tme ntervls. The power lloctons yelded by the lgorthm n Secton IV-C re shown n Tble IV. Ths tble shows tht, becuse of frequency-reuse, the nodes do not necessrly use ther totl power budgets. Ths s due to the fct tht, n ths scenro, when node ncreses ts trnsmsson power, t nflcts hgh nterference on other trnsmssons. Ths s n contrst wth the stuton consdered n [5], wheren frequency-reuse s not llowed nd ncresng the trnsmtted power of node does not ffect the trnsmssons of the other nodes n the network.

10 RASHTCHI et l.: GENERALIZED CROSS-LAYER DESIGNS FOR GENERIC HALF-DUPLEX MULTICARRIER WIRELESS NETWORKS 467 Fg.. Dt routes for ) d = 1, b) d = 2 n Exmple 1. Fg. 5. Dt routes for ) d = 1, b) d = 2 n Exmple 2. TABLE V POWER ALLOCATIONS MW) IN EXAMPLE 2 Fg. 4. Tme-shrng schedules of the subcrrers n Exmple 1. TABLE IV POWER ALLOCATIONS MW) IN EXAMPLE 1 Exmple 2: Jont Routng nd Resource Allocton wthout Tme-shrng) Consder n exemplry network wth N = 6 nodes. In ths network, s before, nodes nd 4 wsh to communcte wth nodes 2 nd 1, respectvely, over K = 4 subcrrers. In prtculr, for destnton node d = 1, the source s node n = 4 nd the other nodes,.e., {2,, 4, 5} re potentl relys, nd, for destnton node d = 2, the source s node n = nd nodes {1, 4, 5, 6} re potentl relys. The consdered network hs L = 0 lnks nd therefore the chnnel mtrx hs 0 4 elements. For spce consdertons, ths mtrx s not provded, but snce the chnnel gn on ech subcrrer s domnted by the PL component, we provde the coordntes of the nodes n the m 2 squre; clcultng the PL components from these coordntes s strghtforwrd, cf. 1). The coordntes of the nodes re {28, 202), 191, 208), 287, 20), 72, 76), 201, 67), 86, 200)}. Settng the node power budgets to P = 20 dbm nd ssumng tht both rtes hve equl weghts, w 2) = w 1) 4 = 1, the jont desgn lgorthm n Secton V-C yelds sum-rte of 9.1 b/s/hz. The dt routes nd power lloctons obtned by ths lgorthm re shown n Fgure 5 nd Tble V, respectvely. For nstnce, n Fgure 5, subcrrer k = 1 s shown to be used twce nd due to the hlf-duplex constrnt, trnsmsson nd recepton tke plce over dstnct subcrrers t ech node. We wll lter show the dvntge of the proposed lgorthm over the lgorthms n whch frequency-reuse s not consdered. Exmple : Averge Weghted-Sum Rte Comprson) In ths exmple, we use Monte Crlo smultons to evlute the verge performnce of the jont desgns wth nd wthout tmeshrng when the chnnels re tme-vryng rther thn sttc s n Exmples 1 nd 2. We consder network wth N = 4 nodes n whch nodes nd 4 wsh to communcte wth nodes 2 nd 1, respectvely, over K = 4 subcrrers. The number of smultneous trnsmssons s set to I = nd the smulton results re verged over 10 ndependent network relztons. The verge weghted-sum rtes yelded by the lgorthms n Sectons V-C nd IV-C for the vlues of P rngng from 0 to 0 dbm re depcted n Fgures 6) nd 6b) for the cses of w 2) = 5w 1) 4 nd w 2) = w 1) 4, respectvely. These fgures lso provde comprson wth the weghted-sum rtes yelded by the desgns n whch frequency-reuse s not consdered [5].

11 468 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 1, JANUARY 2016 Fg. 7. Sum-rte generted by the generlzed lgorthm for dfferent vlues of I. Fg. 6. Averge weghted-sum rte comprson for ) w 2) = 5w 1) 4,nd b) w 2) = w 1) 4. As cn be seen from Fgure 6, the weghted-sum rte yelded by the jont desgn wth both tme-shrng nd frequencyreuse outperforms the desgns n whch ether tme-shrng or frequency-reuse s exclusvely consdered, but t the expense of ncresed complexty. For nstnce, Fgure 6b) suggests tht, t the sum-rte of 12 b/s/hz, the proposed desgn wth both tmeshrng nd frequency-reuse yelds power dvntge of 4 dbm over the desgns n whch ether tme-shrng or frequencyreuse s exclusvely consdered nd power dvntge of 8 dbm over the desgn n whch nether of these technques s consdered. Ths fgure lso suggests tht, for vlues of P less thn 15 dbm, the desgn wth frequency-reuse but wthout tme-shrng yelds better performnce thn the desgn wth tme-shrng but wthout frequency-reuse n [5]. However, for vlues of P hgher thn 15 dbm, the desgn wth tme-shrng but wthout frequency-reuse performs better thn the one wth frequency-reuse but wthout tme-shrng. Ths phenomenon cn be ttrbuted to the effect of nterference. At low powers, the effect of nterference s smll nd frequency-reuse performs generlly better thn tme-shrng. In contrst, t hgh powers, the effect of nterference s more severe nd tme-shrng performs generlly better thn frequency-reuse. As expected, the desgn wth nether tme-shrng nor frequency-reuse hs nferor performnce. Exmple 4: Jont Routng nd Resource Allocton: Generlzed Algorthm) In ths exmple, we evlute the performnce of the lgorthm developed n Secton IV-C. We consder snpshot of network wth N = 5 nodes nd L = 20 lnks lnks wth dstnce more thn 150 m re neglected). In ths network nodes nd 4 wsh to communcte wth nodes 2 nd 1, respectvely, over K = 4 subcrrers. The number of smultneous trnsmssons s set to I = 20, I = nd I = 2, whch results n Ɣ wth , 190 nd 110 vrbles, respectvely. The sum-rte yelded by the generlzed lgorthm wth dfferent vlues of I s depcted n Fgure 7. For comprson, ths fgure lso shows the rtes yelded by the specl cse n Secton V. As cn be seen from Fgure 7, the lgorthm wth I = 2 nd yelds rtes tht re slghtly less thn the rte yelded by the lgorthm wth I = L, however wth sgnfcntly less computtonl complexty. In fct, the complexty of the lgorthm wth I L s polynoml, wheres tht of the lgorthm wth I = L s exponentl n L. Ths feture renders the lgorthm wth I more ttrctve for desgnng lrge networks wth potentlly rpd chnnel vrtons. From Fgure 7 t cn be seen tht the gp between the rtes yelded by dfferent vlues of I decreses s the power budget ncreses. Ths s becuse s power ncreses, nterference becomes more severe, whch cuses the reuse of prtculr subcrrer on multple lnks less benefcl. It cn be lso seen from ths fgure tht, most of the gn of frequency-reuse s mustered by only consderng two or three smultneous trnsmssons,.e., I. Ths mples tht ncresng I trdes complexty for performnce. In prtculr, s I ncreses, the performnce of the lgorthm becomes closer to tht of the one wth I = L, but t the expense of ncresed complexty. In Fgure 8) we nvestgte the convergence behvour of the generlzed lgorthm. We consder n nstnce of network n whch the power budget of ll nodes s set to P = 10 dbm. It cn be seen from ths fgure tht, n ddton to beng sgnfcntly less computtonlly demndng, the lgorthm wth lower vlue of I exhbts consderbly fster convergence thn tht of the one wth hgher vlue of I. Ths convergence cn be further melorted by choosng the ntl pont more crefully, for nstnce, by choosng ths pont to be the soluton yelded by lgorthm n [5] for the cse wth no frequency-reuse. To llustrte the effect of rndom ntlzton of the generlzed lgorthm, n Fgure 8b) the vlue of the objectve to whch the generlzed lgorthm wth I = 2 converged s shown for 80 rndom nstnces of fesble ntl ponts, q 0),Ɣ0) ) [0, P] LK [0, 1] Ɣ. It cn be seen from ths fgure tht lthough the lgorthm s reltvely senstve to the choce of the ntl pont, fndng ntl ponts tht result n good locl mxm s generlly esy.

12 RASHTCHI et l.: GENERALIZED CROSS-LAYER DESIGNS FOR GENERIC HALF-DUPLEX MULTICARRIER WIRELESS NETWORKS 469 VIII. CONCLUSION In ths pper we focused on the jont optmzton of dt routes, subcrrer schedules nd power llocton n hlfduplex multcrrer network when ech subcrrer cn be reused by multple lnks. The gol s to mxmze weghted-sum of the rtes communcted over the network. The consdered network s generc n the sense tht t subsumes mny structures ncludng cellulr nd devce-to-devce communctons s specl cses. We consdered two nstnces of ths problem: 1) when ech subcrrer cn be tme-shred by multple lnks; nd 2) when tme-shrng s not llowed nd subcrrer, once ssgned to set of lnks, wll be used by those lnks throughout the sgnllng ntervl. The jont desgn n the frst nstnce results n superor performnce but wth hgh complexty. The second nstnce s specl cse of the frst one nd cn be prmeterzed usng sgnfcntly smller number of vrbles. The jont desgn problem n both nstnces s nonconvex nd loclly optml solutons re obtned usng GP-bsed monoml pproxmton technque. Numercl results show tht the desgns developed n both nstnces yeld performnce tht s sgnfcntly better thn tht of ther counterprts n whch frequency-reuse s not llowed. Fg. 8. ) Convergence behvour nd b) performnce of the generlzed lgorthm wth dfferent ntl ponts. Fg. 9. Rte-regon comprson. Exmple 5: Averge Rte-Regon Comprson) In ths exmple we provde the rte regons tht cn be cheved by the lgorthms n Sectons V-C nd IV-C, when P = 10 dbm. These regons re obtned by vryng the weghts s 1) 4, s2) ) over the unt smplex,.e., {w1) 4,w2) ) w1) 4 0, w 2) 0, w 1) 4 + w 2) = 1}, nd re depcted n Fgure 9. A comprson between these rte regons nd the ones correspondng to the cse when frequency-reuse s not consdered [5] s lso provded n ths fgure. As cn be seen from Fgure 9, the rte regon correspondng to the desgn wth both tmeshrng nd frequency-reuse properly contns the rte regons correspondng to the desgns n whch ether tme-shrng or frequency-reuse s exclusvely used. It cn be lso seen tht restrctng the number of smultneous trnsmssons to be less thn three suffces to cheve most of the frequency-reuse gn nd wth less computtonl complexty. APPENDIX A THE GP STANDARD FORM AND MONOMIAL APPROXIMATION 1) The GP Stndrd Form: For self-contnment, n ths ppendx we wll revew the stndrd GP form. A GP optmzton problem cn be redly trnsformed to n effcently solvble convex one. To provde the stndrd form of GP, let z R n be vector of postve entres. A monoml n z s defned to be functon of the form c 0 zα nd posynoml n z s defned to be functon of the form J j=1 c n=1 j z α j, where c j > 0, {α } nd {α j }, re rbtrry constnts, j = 0, 1,...,J, nd = 1,...,n. A stndrd GP [6], [7], [21] s n optmzton of the form: mn f 0 z), z subject to f z) 1, = 1,...,m, 2) g z) = 1, = 1,...,p, where { f } re posynomls nd {g } re monomls. 2) Monoml Approxmton: A monoml pproxmton of dfferentble functon hz) 0 ner z 0) s gven by ts frst order Tylor expnson n the logrthmc domn [6], [7]. Defnng β = z0) h hz 0) ) z 0) z=z,wehvemhz)) = hz 0) ) n =1 z ) β,where M ) s the monoml pproxmton. Ths pproxmton wll be used to provde locl GP z 0) pproxmtons n the neghbourhood of gven ntl pont. APPENDIX B PROOF OF PROPOSITION 1 For smplcty, we wll begn by provng the second sttement of Proposton 1. In ths proof, we wll show tht, by ncludng

13 470 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 1, JANUARY 2016 redundnt constrnts, the log-brrer functon of the problems n 0) nd 22) cn be cst n self-concordnt form, whch hs the followng defnton [21]: Defnton 1: A functon f : R n R s sd to be selfconcordnt f, for ll x, v R n, s R such tht x + sv s n the domn of f nd f x + sv) 2 2 f x + sv) /2. s s 2 A. Proof of the Second Sttement of Proposton 1 To determne the complexty of solvng the problem n 0), we begn by convertng ths problem nto convex one. Usng stndrd exponentl trnsformtons, we wrte ) t n d) = exp ln2)s n d), n N \{d}, d D, ) r d) = exp ln2) xd), l L, k K, d D, W y = expq ), l L, k K. ) Substtutng the vrbles n 0) wth the ones n ) nd tkng the logrthm of the obtned objectve nd constrnts result n convex optmzton whch cn be solved effcently usng the IPM technque. To use ths technque, log-brrer functon s syntheszed from the objectve nd nequlty constrnts. The complexty nlyss of the IPM technque s smplfed when the log-brrer functon s self-concordnt [21], cf., Defnton 1. The log-brrer functon correspondng to the convex form of 0) cn be wrtten s φ = t n d w n d) sd) n + ψ, 4) where ψ represents the component of the log-brrer functon ssocted wth the nequlty constrnts n the convex form of 0). To exmne whether φ s self-concordnt, we note tht the converted objectve nd the nequlty constrnts correspondng to 16), 17) nd the relxed versons of 24) nd 25) re lner nd therefore ther correspondng components n the logbrrer functon re self-concordnt [21]. Hence t remns to consder the self-concordnce for the constrnts n 29) nd 2). For smplcty, we wrte the posynoml constrnt n 29) n the stndrd form n 2). After chngng the vrbles nd tkng the logrthm of both sdes, ths constrnt cn be wrtten n generl form s ) log exp α + b β + c ) 0, 5) where {α }, {β } re the optmzton vrbles nd { }, {b }, {c } re constnts. The component correspondng to the constrnt n 5) n the log-brrer functon cn now be expressed s log log ) exp α + b β + c ). 6) To ensure tht 6) s self-concordnt, we ntroduce uxlry vrbles, λ, to bound the exponentlly trnsformed vrbles n 5). Usng these new vrbles, the constrnt n 5) cn be replced wth the followng set of constrnts [21]: λ 1, λ 0, α + b β + c log λ 0. 7) Now the ssocted log-brrer functon of the constrnts n 7) cn be shown to be self-concordnt, cf., [21, Exmple 9.8]. For the constrnts n 2), we follow the steps nlogous to the ones used wth the constrnts n 29). In prtculr, by ntroducng new uxlry vrbles, we construct self-concordnt log-brrer functon. Usng ths functon, the complexty cn be shown to be proportonl to m.5, where m s the number of nequlty constrnts. Hence, the complexty of solvng 0) cn be bounded by O LKL + 2) + N + DN 1)).5), whch completes the proof of the second sttement of Proposton 1. B. Proof of the Frst Sttement of Proposton 1 The proof of the frst sttement of Proposton 1 follows from rguments smlr to the one used n the proof of the second sttement nd s omtted for brevty. For the frst sttement, the number of nequlty constrnts cn be redly verfed to be 2LKN + N + DN 1) + 2K I =1 L ), whch yelds the frst sttement of Proposton 1. REFERENCES [1] H. Ynkomeroglu, Fxed nd moble relyng technologes for cellulr networks, n Proc. 2nd Workshop Appl. Serv. Wreless Netw., Jul. 2002, pp [2] X. Bngnn, S. Hschke, nd B. We, The role of d hoc networkng n future wreless communctons, n Proc.Int. Conf. Commun. Tech., Apr. 200, pp [] S. Hysh nd Z.-Q. Luo, Spectrum mngement for nterferencelmted multuser communcton systems, IEEE Trns. Inf. Theory, vol. 55, no., pp , Mr [4] H. L nd H. Lu, An nlyss of uplnk OFDMA optmlty, IEEE Trns. Wreless Commun., vol. 6, no. 8, pp , Aug [5] R. Rshtch, R. H. Gohry, nd H. Ynkomeroglu, Routng, schedulng nd power llocton n generc OFDMA wreless networks: Optml desgn nd effcently computble bounds, IEEE Trns. Wreless Commun., vol. 1, no. 4, pp , Apr [6] M. Chng, C. W. Tn, D. P. Plomr, D. O Nel, nd D. Juln, Power control by geometrc progrmmng, IEEE Trns. Wreless Commun., vol. 6, no. 7, pp , Jul [7] S. Boyd, S.-J. Km, L. Vndenberghe, nd A. Hssb, A tutorl on geometrc progrmmng, Optm. Eng., vol. 8, pp , Mr [8] B. R. Mrks nd G. P. Wrght, A generl nner pproxmton lgorthm for nonconvex mthemtcl progrms, Oper. Res., vol.26,pp , Aug [9] R. Rshtch, R. Gohry, nd H. Ynkomeroglu, A cross-lyer desgn for generc hlf-duplex nterference-lmted multcrrer networks, n Proc. IEEE Int. Workshop Sgnl Process. Adv. Wreless Commun., Jun. 2014, pp [10] R. Rshtch, R. H. Gohry, nd H. Ynkomeroglu, An effcent cross lyer desgn n OFDMA-bsed wreless networks wth chnnel reuse, n Proc. IEEE Globl Commun. Conf., Dec. 201, pp [11] H. Inltekn nd S. V. Hnly, Optmlty of bnry power control for the sngle cell uplnk, IEEE Trns. Inf. Theory, vol. 58, no. 10, pp , Oct [12] K. Km, Y. Hn, nd S.-L. Km, Jont subcrrer nd power llocton n uplnk OFDMA systems, IEEE Commun. Lett., vol. 9, no. 6, pp , Jun

14 RASHTCHI et l.: GENERALIZED CROSS-LAYER DESIGNS FOR GENERIC HALF-DUPLEX MULTICARRIER WIRELESS NETWORKS 471 [1] C. Y. Ng nd C. W. Sung, Low complexty subcrrer nd power llocton for utlty mxmzton n uplnk OFDMA systems, IEEE Trns. Wreless Commun., vol. 7, no. 5, pp , My [14] J. Jng nd K. B. Lee, Trnsmt power dptton for multuser OFDM systems, IEEE J. Sel. Ares Commun., vol. 21, no. 2, pp , Feb [15] J. Hung, V. G. Subrmnn, R. Agrwl, nd R. A. Berry, Downlnk schedulng nd resource llocton for OFDM systems, IEEE Trns. Wreless Commun., vol. 8, no. 1, pp , Jn [16] L. Xo, M. Johnsson, nd S. P. Boyd, Smultneous routng nd resource llocton v dul decomposton, IEEE Trns. Commun., vol. 52, no. 7, pp , Jul [17] K. Krkyl, J. Kng, M. Kodlm, nd K. Blchndrn, Cross-lyer optmzton for OFDMA-bsed wreless mesh bckhul networks, n Proc. IEEE Wreless Commun. Netw. Conf., Mr. 2007, pp [18] M. Johnsson nd L. Xo, Cross-lyer optmzton of wreless networks usng nonlner column generton, IEEE Trns. Wreless Commun., vol. 5, no. 2, pp , Feb [19] R. H. Gohry nd T. J. Wllnk, Jont routng nd resource llocton v superposton codng for wreless dt networks, IEEE Trns. Sgnl Process., vol. 58, no. 12, pp , Dec [20] M. Chrfeddne nd A. Pulrj, Sequentl geometrc progrmmng for 2 2 nterference chnnel power control, n Proc. IEEE Conf. Inf. Sc. Syst., Mr. 2007, pp [21] S. Boyd nd L. Vndenberghe, Convex Optmzton. Cmbrdge, U.K.: Cmbrdge Unv. Press, [22] Int. Telecommun. Unon ITU), Gudelnes for evluton of rdo nterfce technologes for IMT-dvnced, Genev, Swtzerlnd, ITU-R: TR M.2151, Dec [Onlne]. Avlble: M [2] M. Grnt nd S. Boyd. 2011, Jn.). CVX: Mtlb Softwre for Dscplned Convex Progrmmng, Verson 1.21 [Onlne]. Avlble: [24] MOSEK Apps. 2012). The MOSEK Optmzton Toolbox for Mtlb Mnul, Verson 6.0 [Onlne]. Avlble: Rozt Rshtch S 12 M 14) receved the B.S. nd M.Sc. degrees n electrcl engneerng from Isfhn Unversty of Technology, Isfhn, Irn, n 2005 nd 2008, respectvely. Snce 2010, she hs been Ph.D. cnddte nd Reserch Assstnt wth the Deprtment of Systems nd Computer Engneerng, Crleton Unversty, Ottw, ON, Cnd. Her reserch nterests nclude resource llocton, power control nd multhop routng n wreless networks, pplcton of optmzton n wreless communcton, nd cross-lyer desgn n rely-sssted networks. She ws member of Crleton-BlckBerry nd Crleton-Huwe projects durng nd , respectvely. Durng her Ph.D. studes, she ws the recpent of the Crleton Unversty Presdent 2010 Doctorl Fellowshp Awrd nd Grdute Scholrshp Awrd for the cdemc yers She lso receved Ontro Grdute Scholrshp for the cdemc yer Hlm Ynkomeroglu S 96 M 98 SM 12) ws born n Gresun, Turkey, n He receved the B.Sc. degree n electrcl nd electroncs engneerng from the Mddle Est Techncl Unversty, Ankr, Turkey, the M.A.Sc. degree n electrcl engneerng now ECE) nd the Ph.D. degree n electrcl nd computer engneerng from the Unversty of Toronto, Toronto, ON, Cnd, n 1990, 1992, nd 1998, respectvely. From 199 to 1994, he ws wth the R&D Group, Mrcon Komnksyon A.S. Snce 1998, he hs been wth the Deprtment of Systems nd Computer Engneerng, Crleton Unversty, Ottw, ON, Cnd, where he s now Full Professor. From 2011 to 2012, he ws Vstng Professor t the TOBB Unversty of Economcs nd Technology, Ankr. Hs reserch nterests nclude wreless technologes wth specl emphss on cellulr networks. He hs gven hgh number of tutorls nd nvted ts on wreless technologes n the ledng nterntonl conferences. In recent yers, hs reserch hs been funded by Huwe, Telus, Blckberry, Smsung, Communctons Reserch Centre of Cnd CRC), nd Nortel. Ths collbortve reserch resulted n over 20 ptents grnted nd ppled). He hs been nvolved n the orgnzton of the IEEE Wreless Communctons nd Networkng Conference WCNC) from ts ncepton, ncludng servng s Steerng Commttee Member s well s the Techncl Progrm Chr or Co-Chr of WCNC 2004 Atlnt, GA, USA), WCNC 2008 Ls Vegs, NV, USA), nd WCNC 2014 Istnbul, Turkey). He ws the Generl Co- Chr of the IEEE Vehculr Technology Conference Fll 2010 held n Ottw, ON, Cnd. He hs served on the edtorl bords of the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, nd the IEEE COMMUNICATIONS SURVEYS AND TUTORIALS. He ws the Chr of the IEEE Techncl Commttee on Personl Communctons now clled Wreless Techncl Commttee). He s Dstngushed Lecturer for the IEEE Communctons Socety ) nd the IEEE Vehculr Technology Socety ). He ws the recpent of the IEEE Ottw Secton Outstndng Eductor Awrd n 2014, Crleton Unversty Fculty Grdute Mentorng Awrd n 2010, the Crleton Unversty Grdute Students Assocton Excellence Awrd n Grdute Techng n 2010, nd the Crleton Unversty Reserch Achevement Awrd n He s Regstered Professonl Engneer n the provnce of Ontro, Cnd. Rmy H. Gohry S 02 M 06 SM 1) receved the B.Sc. degree Hons.) from Assut Unversty, Asyut, Egypt, the M.Sc. degree from Cro Unversty, Gz, Egypt, nd the Ph.D. degree from the McMster Unversty, Hmlton, ON, Cnd, ll n electroncs nd communctons engneerng, n 1996, 2000, nd 2006, respectvely. He s n Adjunct Reserch Professor nd Senor Reserch Assocte wth the Deprtment of Systems nd Computer Engneerng, Crleton Unversty, Ottw, ON, Cnd. He s the Co-Inventor of four U.S. ptents, nd the uthor of more thn 0 well-cted IEEE journl ppers. He s lso the referee for more thn ten scentfc IEEE journls, nd member of the techncl progrm commttees of seven nterntonl IEEE conferences. Hs reserch nterests nclude nlyss nd desgn of MIMO nd coopertve wreless communcton systems, pplctons of optmzton nd geometry n sgnl processng nd communctons, nformton theoretc spects of multuser communcton systems, nd pplctons of tertve detecton nd decodng technques n multple ntenn nd multuser systems. He s Regstered Lmted Engneerng Lcensee LEL) n the provnce of Ontro, Cnd.