Resource allocation based on integer programming and game theory in uplink multi-cell cooperative OFDMA systems

Transcription

1 RESEARCH Oe Access Resource allocatio based o iteger rogrammig ad game theory i ulik multi-cell cooerative OFDMA systems Zhao Hou *, Yuemig Cai ad Da Wu Abstract I this article, we roose a semi-distributed resource allocatio framework for the resource otimizatio i multicell ulik cooerative orthogoal frequecy divisio multilexig systems. Secifically, we model the resource allocatio framework as a otimal roblem. This otimizatio roblem is divided ito two stes. First, usig iteger rogrammig, we achieve the joit relay selectio ad subcarrier allocatio based o maximizig system sum rate i a cetralized way. Secod, the distributed ower allocatio is achieved based o game theory, for cooerative ad o-cooerative users, resectively. For cooerative mobile statios, a imroved utility is roosed to regulate ower allocatio i the two time slots. esides the existece of Nash equilibrium NE), a ew aroach for the strict mathematical roof of the uiqueess of NE is roosed. Simulatio results demostrate that the roosed algorithm successfully combies the merits of cetralized ad distributed framework. It ca effectively make use of relays to ehace the sum rate of users as well as achieve the fairess amog users. Keywords: cooerative OFDMA systems, iter-cell iterferece, semi-distributed resource allocatio framework, game theory, rogressive otimizatio, Nash equilibrium. Itroductio Owig to its otetial to realize high-rate ad reliable commuicatios over wireless chaels, cooerative orthogoal frequecy divisio multilexig OFDM) techology has draw extesive attetio i recet years as a critical 4G techology. The erformace of a OFDMA system deeds o careful resource maagemet, icludig subcarrier allocatio ad ower cotrol. However, the itroductio of cooerative techology, which ehaces the system erformace by achievig the beefits of satial diversity, imoses more comlexity o resource maagemet. Curret resource allocatio algorithms i cooerative OFDMA systems ca be divided ito cetralized ad distributed categories [-9]. The former caot esure fairess amog users ad may lead to the ear-far effect, the disroortio of resource allocatio caused by the chael state associated with the differet locatios of the users [4,5]. The latter is costraied with worse efficiecy ad more comlexity, ad asks for more chael overheads. * Corresodece: koghoumigfeg@hotmail.com Istitute of Commuicatio Egieerig, PLA Uiversity of Sciece ad Techology, Najig, Chia Keuyoug s [] scheme is ursig for maximizig the OFDMA system sum rate uder the ower costrait. It has bee acceted as the otimal solutio that each subcarrier was allocated to the user with the margial rate. Zhag et al. [2] study the relay selectio, subcarrier ad ower allocatio i cooerative OFDMA system uder a QoS requiremet. ut, as cetralized allocatio schemes, these two schemes result i ear-far effect because they aim at otimizig the system erformace rather tha the erformace of each user. Therefore, subcarrier ad ower are allocated to users whose chael states are better, ad users who have worse chael state achieve oor wireless trasmissio sice they obtai oor wireless resources [4]. Game theory, a mathematical methodology traditioally alied i micro-ecoomic studies, has foud its alicatios i various asects of commuicatio egieerig as a effective tool for studyig coflictive ad cooerative issues amog several actors [0], icludig distributed resource allocatio i OFDMA systems [3-9]. Ha et al. [3] itroduce a game-based ower miimizatio model i a multi-cell OFDMA system. Wu et al. [4] 20 Hou et al; licesee Sriger. This is a Oe Access article distributed uder the terms of the Creative Commos Attributio Licese htt://creativecommos.org/liceses/by/2.0), which ermits urestricted use, distributio, ad reroductio i ay medium, rovided the origial work is roerly cited.

2 Page 2 of 0 ad Yu et al. [5] focus o a subcarrier ad ower allocatio scheme i ulik OFDMA systems, where each user ot oly has the differet ower costrait, but also is limited by differet rate requiremets. I [6], uder a cooerative OFDMA sceario, the authors build u the iterferece chael model based o amlify-ad-forward AF) ad decode-ad-forward DF) modes, ad study ower cotrol scheme based o a two-stage game. ut, the studies i [3,6] do ot take subcarrier allocatio ito accout, ad i [3] the authors do ot itroduce the relay odes. Authors of [4-6] cosider a sigle-cell sceario, i.e. absece of cosideratio of iter-cell iterferece. Lee ad Yum [7] itroduce a ovel framework to fid the ecessary ad sufficiet coditio for Paretoefficiecy. The resource allocatio algorithm develoed by Lag et al. [8] aims at miimizig the users trasmittig ower i a multi-cell OFDMA system based o game theory uder the costrait of the eak value of ower. However, this algorithm caot esure covergece. Yu et al. [9] roose a game aroach for distributed ower allocatio i a multi-cell dowlik cooerative OFDMA system, but the relay selectio ad subcarrier allocatio are ot cosidered. I additio, existig aroaches [] to rove the uiqueess of a game-based resource allocatio scheme are of high comlexity. I this article, we roose a semi-distributed resource allocatio framework for the resource otimizatio i multi-cell ulik cooerative OFDM systems i DF mode. Secifically, we model the resource allocatio framework as a otimal roblem. This otimizatio roblem is divided ito two stes. First, usig iteger rogrammig, we achieve the joit relay selectio ad subcarrier allocatio based o maximizig system sum rate i a cetralized way. Secod, the distributed ower allocatio is achieved based o game theory, for cooerative ad o-cooerative users, resectively. For cooerative mobile statios, a imroved utility is roosed to regulate ower allocatio i the two time slots. esides the existece of Nash equilibrium NE), a ew aroach for the strict mathematical roof of the uiqueess of NE is roosed. Simulatio results demostrate that the roosed algorithm successfully combies the merits of cetralized ad distributed framework. It ca effectively make use of relays to ehace the sum rate of users as well as achieve the fairess amog users. The rest of the article is orgaized as follows: Sectio 2 describes the chael ad system model. Sectio 3 states the framework of the rogressive otimizatio, which is comosed of the joit relay selectio ad subcarrier allocatio, the game aroach for distributed ower allocatio scheme, the existece ad uiqueess of NE as well as the algorithm framework. Simulatio results are give ad aalyzed i Sectio 4. Fially, cocludig remarks i Sectio 5 ed the article. 2. System model We cosider a multi-cell cooerative OFDMA sceario. I each cell, there is oe base statio, multile users ad multile relays. Each user ca commuicate with the base statio directly or through a certai relay. All of the orthogoal subcarriers are available i each cell, ad each of them ca exclusively be allocated to a certai user [9]. Therefore, iter-cell iterferece origiates from co-chael users i adjacet cells. Assume that there is a cooerative OFDMA hexago cell, i which the base statio is i the cetre ad K relays are equally located o a circle which has the same cetre with the cell. esides, the radius of this circle r 2 is smaller tha the hexago cell radius r. A total umber of M users are deloyed uiformly i the cell, M d of which coect directly to the base statio while the rest M r oes commuicate through a certai relay, M d, M r = 0,,..., M. Iother words, users ca commuicate either i a o-cooerative or i a cooerative mode. The total badwidth of the system is, which is equally divided ito N orthogoal subcarrier. The chael is frequecy-selective Rayleigh fadig. Aroud this cell there are L co-chael cells. The distributio of relays ad users of all cells is similar. Figure shows the cell layout. We assume all chael state iformatio CSI) ca be erfectly be kow by the base statio, relays ad users. Thus, k Î {,2,..., K}, m Î {,2,..., M}, Î {,2,..., N}, for the th subcarrier, the chael coefficiets betwee base statio ad the mth user, the kth relay ad the mth user, thebasestatioadthekth relay are deoted by h d,m, h a,km ad h b,k, resectively. The otatio a meas the first time slot, while the otatio b deotes the secod oe. Meawhile, the otatio d corresods with the users which are trasmittig directly with the base statio. The oise variace i the chael is s 2. I this article, the relays work uder DF mode. I the odd time slot, a cooerative user trasmits iformatio to its corresodig relay. Ad i the ext eve slot, the relay decodes the sigal received i the odd slot ad forwards it to the base statio. For the th subcarrier, the trasmittig ower betwee the mth user ad the kth relay i the odd slot ca be deoted as km ), ad the trasmittig ower betwee the kth relay ad the base statio i the ext eve slot is km 2). Therefore, for the cooerative users, i DF mode, the istataeous rate of relay-user k, m) otheth subcarrier ca be deoted as c km = 2N mi h km ) 2 a,km h km 2) 2 b,k log 2 + σ 2,log + I 2 + a σ 2 + I b )

3 Page 3 of 0 Figure Demostratio of the system structure ad the workig rocess of the multi-cell ulik OFDMA system. where I a = ow hh km, l) a,km l) 2 ad i L I b = ow hh k 2, l) b,k l) 2 deote the co-chael iter- i L ferece from the lth adjacet cell cooerative users i the odd ad eve time slots, resectively, k Î {,2,..., K}. For the th subcarrier, hh a,km l) is the chael coefficiet betwee the kth relay ad the mth user i the lth adjacet cell, ad hh b,k l) is the chael coefficiet betwee the base statio ad the kth relay, i additio of ow km i, l) which deotes the trasmit ower from the lth adjacet cell i the ith slot, i =,2. O the other had, by deotig 0m ) ad 0m 2) as the trasmittig ower of o-cooerative users i the odd ad eve time slots, resectively, we ca exted the direct users istataeous rate. c 0m = h 0m ) 2 d,m h 0m 2) 2 d,m log2 + 2N σ 2 +log + I σ 2 + I 0 2) where I 0 = ow hh 0m i, l) d,m l) 2 deotes the co- i L chael iterferece from the lth adjacet cell ocooerative users. Similarly, the trasmittig ower of o-cooerative user i the ith slot ca be deoted as ow 0m i, l), ad hh d,m l) is the chael coefficiet betwee the base statio ad the mth user, i =,2. 3. Solutio o relay selectio, subcarrier ad ower allocatio As is kow to us, subcarrier ad ower allocatio imose great ifluece o system erformace. I ulik cooerative OFDMA system, subcarrier is ublic resource which is i cometitio amog differet users. However, ower of each user ca be cotrolled ideedetly accordig to their ow demad ad ower caacity. So, we develo a semi-distributed allocatio framework, i which we are ursuig for the combiatio of merits of cetralized ad distributed schemes, ad ca effectively achieve the subcarrier ad ower

4 Page 4 of 0 allocatio. I the roosed scheme, relay selectio ca be achieved with subcarrier allocatio uder the cotrol of a cetre, i.e. the base statio, while the ext ste is achieved by each user. O the other had, resource allocatio, which cotais relay selectio, subcarrier ad ower allocatio, is a NP-hard roblem which has great comlexity. For simlicity, the divisio of the origial roblem ito two rogressive sub-otimal oes, which is at the cost of a loss of otimality, ad is coveiet for us to solve the roblem. 3.. Cetralized joit relay selectio ad subcarrier allocatio The relay selectio ad ower allocatio framework ca be rereseted by biary assigmet variables x km,which ca alteratively equals to 0 or. While x km =, theth subcarrier is matched for the commuicatios betwee the mth user ad the kth relay, vice versa. Ifk =0,the value of x km demostrates whether the th subcarrier is matched directly for the commuicatios betwee the mth user ad the base statio. The K +) M N dimesio 0- matrix X =x km ) ca be defied as the relay-user-subcarrier matchig matrix, i.e. the relay selectio ad subcarrier allocatio matrix. Therefore, we ca otimize the framework aimig at maximizig the system caacity. However, water-fillig algorithm caot esure the QoS of farther users whose CSI is ot as good as the earer oes, i.e. the ear-far effect is more obvious [4]. Hece, we ca itroduce QoS requiremet to esure that each user ca reach a miimum rate. The otimizatio roblem for joit relay selectio ad subcarrier allocatio i cooerative OFDMA etworks ca be formulated as followig [2]. arg max x km s.t. x km K K k=0 m= k=0 m= = {0, } C sum = K M x km M c km c mi M k=0 m= = N c km x km 3) where the m vector c mi deotes the miimum rate of each user. The otimizig framework is therefore formulated as a iteger rogrammig roblem. y itroducig a m dual vector l m ad itroducig sub-gradiet method, we ca make use of the iterative algorithm roosed i [2] with regardless of ower allocatio as followig., k, m) =k, m ) = arg max [+λ m iter)] c x km iter) = km k,m 0, o.w. 4) λ m iter +) = [ λ m iter) + α iter) c mi c km x km iter))]+ 5) where iter reresets the iterative times, ad aiter) meas a roer iterative ste legth which is related to the iterative times. Theorem [2] If α iter),adaiter) 0 as iter iter, the the otimizig goal ca coverge uo the otimal value. Accordig to Theorem, we ca etitle a o-coverget ifiite series to aiter), whose items iclie towards zero while the iterative time goes to ifiite. Therefore, we ca evaluate α iter) = iter.yiteratioof Equatios 4 ad 5, the otimal solutio ca be obtaied accordigly. It is achieved by the base statio i the cetre Distributed ower allocatio scheme Whe subcarriers allocatio has bee fiished, we ca focus o ower allocatio with the goal of maximizig each user s rate. We ca ay more attetio to the eve time slot, i which we ca distiguish cooerative ad o-cooerative modes. y itroducig game theory, the ower allocatio sceario ca be corresoded with the three basic comoets of game theory as followig. No-cooerative layers: the M d o-cooerative users ad the K relays which coflict with each other show i Equatios ad 2. The actio rofile: the trasmittig strategy of each layer. The set of utility fuctios: a icome which is related to the rate of each layer. Therefore, the game ca be exressed as arg max x km u s s 2) ) 6) s.t. s 2) max, s = {, 2,..., K, K +,..., K + M d } where s 2) is the trasmittig ower of ode s i the eve time slot, ad the elemets of max are the eak values of the K relays ad the M d o-cooerative users. The desig of the utility fuctio exerts great ifluece o the system erformace. For o-cooerative users, with the urose of adjustig the trasmittig ower i the eve time slot, we ca desig the utility fuctio of the mth o-cooerative user as

5 Page 5 of 0 u m 0m 2) ) = N ψd,m ) log 2 +SINR d,m 2) α d s.t. 0m 2) m,max ψ d,m h d,m 2 0m 2) 7) ψd,m where ψ d, m is the subcarrier set of the mth o-cooerative user obtaied i Sectio 3., SINR d,m 2) = 0m 2) h d,m 2 is the sigal-to-oise-aditerferece-ratio SINR) of the mth o-cooerative σ 2 + I 0 user uo the th subcarrier i the eve time slot, a d is a o-egative cost factor whose uit is bs/w, i which the otatio W deotes the ower uit Watt, ad m, max deotes the ower costrait of the mth ocooerative user. I the utility fuctio above, the first term reresets the goal to maximize the mth ocooerative users rate, ad the secod term is the cost fuctio which aims at reducig the trasmittig ower. Particularly, ath atteuatio is a imortat factor to determie the cost origiatig from ower cosumtio. I light of this, the 2-orm of chael coefficiets is itroduced ito the cost fuctio to regulate the ower cosumtio i differet chael coditios. Similarly, for relays which are alied to forward cooerative users iformatio i the odd slot, the utility fuctio ca be desiged as followig. uk km 2) ) = log N 2 +SINR k,m 2) ) αr km 2) h 2 b,k β ψr,k ψr,k ψr,k km 2) km ) ) 2 8) 3.3. The existece of NE Whe it comes to a game model, NE is always our exectatio. NE is a imortat cocet i a o-cooerative game, i which each actor is cotet to kee the reset strategy rather tha chage it. I other words, NE is the rofile of each actor s otimal strategy. Theorem 2 [0] The fuctio must be quasi-cocave or quasi-covex) if it is cocave or covex). Theorem 3 [0] There is a NE at least i game G = [N, {P i }, u i )], if ) P i are the sub-sets of R ad are o-emty, comact ad covex; 2) u i are cotiuous i P i ; 3) u i are quasi-cocave i P i. where N is the set of layers, P i ad u i are the actio rofile ad the utility fuctio of layer i, i =,2,..., N. Accordig to these theorems, we ca give the roof of theexisteceofneithegamescearioithisarticle as follow. Proof: It is obviously that the first ad the secod coditios are satisfied accordig to the exressios of P i ad u i ). We ca focus o the rest oe. I our game model, for the mth o-cooerative user, we ca get the first-order ad secod-order derivative fuctios of its utility of Equatio 7 as u m 0m 2) ) 0m 2) = N l 2 SINR d,m 2) +SINR d,m 2) 0m 2) α d h d,m 2 9) s.t. ψ r,k km 2) k,max where ψ r, k is the subcarrier set of the kth relay for the commuicatio betwee the kth relay ad its corresodig cooerative users, h SINR km 2) 2 b,k d,m 2) = σ 2 + I b is the SINR of the mth direct user uo the th subcarrier i the eve time slot ad k,max deotes the ower costrait of the kth relay. a r is a o-egative cost factor whose uit is bs/w, while b is a o-egative weight factor for the ricig item whose uit is bs. Emloyig a utility fuctio i the form of 7) will lead to a uilateral SINR icrease i eve time slot, ad the caacityofarelayuseriscofiedbythemiimum SINR of the two kids of time slots. To get the balace i the two slots, the itroductio of the third item is ecessary. The squared term i the exressio grows fast whe the SINR i the eve slot icreases ad exceeds that i the odd slot. Thus, the rice rises to uish the relay from allocatig too much ower o that subcarrier [9]. u 2 m 0m 2)) 0m 2)) = 2 N l 2 +SINR d,m 2) +SINR d,m 2) ) 2 0m 2) ) 2 < 0 0) Similarly, for the kth relay, we ca get the first-order ad secod-order derivative fuctios of its utility of Equatio 9 as uk km 2) ) km 2) = SINR k,m N l 2 2) +SINR 2) k,m h 2 km 2) αr b,k 2β km 2) ) 2 km 2) km )) ) u 2 k km 2)) km 2)) = 2 N l 2 SINR ) +SINR k,m 2) ) 2 k,m 2) 2 km 2) 2β km 2) ) 2 < 0 2) Therefore, for both relays ad direct users, their utility fuctios are quasi-cocave i km 2) ad 0m 2), resectively. Cosequetly, the NE exists i the roosed framework The uiqueess of NE The uiqueess of NE demostrates that each ode ca reach a covergig strategy rofile. The authors of [] roosed a aroach to rove it. As a matter of fact, the uiqueess ca also be obtaied from its origial meaig.

6 Page 6 of 0 Cosiderig the ower strategies rofile P ca be obtaied by iteratio, whether the ositive term series Pt +) Pt) ca reach its covergece determies the uiqueess of NE. Proof: For the mth o-cooerative user, let Equatio 9 be zero, we ca get 0m 2) = SINR d,m 2) α d N h 2 +SINR d,m l 2 d,m 2) 3) Therefore, we ca get the value of 0m 2) by iteratio of Equatio 3 ad h 0m 2) 2 SINR d,m.wheit d,m 2) = σ 2 + I 0 comes to the iterative exressio, as the iterative time t, we ca get the ositive term series 0m t +) 0m t) SINR d,m = 2)t) h 2 +SINR α d d,m N l 2 d,m 2)t) SINR d,m 2)t ) h 2 +SINR α d d,m N l 2 d,m 2)t ) SINR d,m 2)t) SINR d,m 2)t ) = h 2 α d d,m N l 2 Let δt) = +SINR d,m 2)t) ) +SINR d,m 2)t )) +SINR d,m 2)t) ) +SINR d,m 2)t )), the equatio above ca be exressed as 0m t +) 0m t) = δt) SINR h 2 d,m 2)t) SINR d,m 2)t ) α d d,m N l 2 h d,m 2 0m 2)t) h d,m 2 0m 2)t ) = δt) h 2 σ α d d,m N l I 0 σ 2 + I 0 = α d N l 2 δt) σ 2 + I 0m 2)t ) 0m 2)t 2) 0 ) 2 δt)δt 2) = ) α d N l 2 σ 2 2 0m + I 2)t 3) 0m 2)t 4) 0 = ) t α d N l 2 2 ) σ 2 + I 0 t δi) 0m 2)2) 0m 2)) i=3 4) SINR d,m 2)t) 0, δt)<. So, the secod term of the equatio above iclies to zero as t. Ifweadjust the factor a d to a adequate value to esure α d N l 2 σ 2 + I 0 ) <, i.e. α d > N l 2 σ 2 + I 0 ), ) t lim t α d N l 2 2 ) = 0. Ad σ 2 + I 0 0m 2)2) 0m 2)) is a defiite value. Thus, the right-had side of Equatio 4 iclies towards zero. So, does it if α d N l 2 ) σ 2 2 =. I a word, while + I 0 α d N l 2 ), σ 2 the right-had side of Equatio 4 + I 0 is icliig to zero as t, i.e. the series 0m t +) 0m t) is absolutely coverget. It meas that by iteratio, the solutio of NE 0m 2) ca reach a certai value. NE is therefore uique. Similarly, whe it comes to the kth relay, let Equatio be zero, we ca get the iterative exressio of km 2) as followig. b + c + d SINR k,m 2)t) km 2)t +)= +SINR k,m 2)t) 5) 2a where a = 4β 2 4βα r h c = km ) ) 2 b,k km ) d = 2β km ) ) 2 N. Therefore, 2β km ) ) 2, b = 2 km 2)t +) km 2)t) = c + d SINR k,m 2)t) 2a +SINR k,m 2)t) c + d SINR k,m 2)t ) +SINR k,m 2)t ) + α r h b,k = 2a c + d SINR k,m 2)t) +SINR k,m 2)t) + c + d SINR k,m 2)t ) +SINR k,m 2)t ) = d h b,k 2 2a c σ 2 + I0 c + d SINR k,m 2)t) +SINR k,m 2)t) + c + d SINR k,m 2)t ) +SINR k,m 2)t ) t = d h b,k 2 2 2a c σ 2 + km 2)2) km 2)) I0 4 2β km ) α r SINR k,m 2)t) SINR k,m 2)t ) +SINR k,m 2)t) ) +SINR k,m 2)t )) km 2)t ) km 2)t 2) h b,k 2, ad +SINR k,m 2)t) ) +SINR k,m 2)t )) t t i=2 c + d SINR k,m 2)i) ) +SINR k,m 2)i) + c + d SINR k,m 2)i ) i=2 +SINR +SINR k,m )+SINR 2)i ) k,m 2)i ) k,m 2)i ) Imitatig the exlaatio of the couterart of direct users, the secod term is a fiite value, ad the last two terms are smaller tha. y adjustig the relatioshi amog a, c ad d to make sure d h b,k 2 2a c σ 2,the + I 0 series km t +) km t) is absolutely coverget. NE is therefore uique. Coclusively, NE of our game aroach is uique. I other words, all of the relays ad o-cooerative users ca automatically choose a strategy of trasmittig ower, ad o oe will uilaterally chage the ower value. ut, as far as we are cocered, the Pareto efficiecy caot be esured because this is a sub-otimal scheme Relay selectio, subcarrier ad ower allocatio algorithm Accordig to the discussio above, we ca get the rogressive otimizatio of relay selectio, subcarrier ad ower allocatio mechaism as Figure 2.

7 Page 7 of 0 Figure 2 Demostratio of the rogressive otimizatio o relay selectio, subcarrier ad ower allocatio. Phase is the joit relay selectio ad subcarrier allocatio based o iteger rogrammig, while hase 2 deicts the iterative stes of the game-based ower allocatio scheme. Figure 3 Demostratio of the simulatio results of users sum rate versus umber of relays.

8 Page 8 of 0 4. Simulatio results We cosider a cooerative OFDMA cell with N =64 orthogoal subcarrier available ad a radius of r = km. DF relays are deloyed o a circle with a radius of r 2 = 0.5 km at equal agular distace ad users are radomly distributed i the cell. The frequecy-selective chael cosists of six ideedet aths. Rayleigh multiath adots Clarke s model. A modified COST23-Hata roagatio model is utilized with ath loss log R) where R is the distace whose uit is kilometre. The total badwidth is.25 MHz ad the ower sectral desity of oise is -55 dm/hz [2]. The total ower available at all users is 0.05 W, while that at all relays is 0. W. a d ad a r are evaluate to 40 ad 00 kbs/w, resectively. The ower is uiformly allocated o each subcarrier. Six similar cells are allocated aroud this cell, which have the same subcarriers. Figure 3 demostrates the relatio of average sum rate versus umber of relays uder the algorithm i [2,4] ad the roosed oe, resectively. The total umber of the users is 20. We ca see that as the umber of relays icreases, the sum rate of users icreases more obviously uder the roosed algorithm. Accordig to our aalysis i Sectio 3., o oe had, uilateral ursuit of maximizig system rate uder cetralized cotrol caot always meet the requiremet of QoS for all distributed users. O the other had, distributed resource allocatio may lead to disorder ad low efficiecy. For resource which is i cometitio amog users such as subcarrier, it is ufair for the later choosig users to choose subcarriers by turs. Simulatio results are cosistet with the aalysis above. I additio, it demostrates that the roosed algorithm ca efficietly make use of relays to romote the sum rate of users. Figure 4 discusses the relatio of each user s rate versus the distace betwee base statio ad users uder the algorithm i [2,4] ad the roosed oe, resectively. The total umber of the users is eight with three relays assistig their trasmissio. As Figure 4 deicts, each user s rate uder the roosed algorithm decreases more smoothly tha that i the other two algorithms as the distace icreases. Therefore, the fairess is well achieved. It is also cosistet with the aalysis i the aragrah above. Figure 4 Demostratio of the simulatio results of each users rate versus distace betwee the base statio ad the users.

9 Page 9 of 0 Figure 5 exresses the relatio of average sum rate versus umber of relays uder the algorithm i [2,4] ad the roosed oe, resectively. The total umber of the relays is 6. We ca see the sum rate of users icreases the most sharly uder cetralized algorithm. It is because that the algorithm i [8] aims at maximizig the system rate, regardless of ower cosumtio of a certai user. The roosed algorithm is a rogressive otimal framework, i the secod ste of which we cosider the comromise of user rate maximizatio ad ower cosumtio. Therefore, fairess is achieved at the cost of reductio o the sum rate of users. It also demostrates that regardless of fairess, cetralized framework erforms better if the umber of user icreases. Whe it comes to the coverget seed of the roosed scheme, we ca discuss this questio i the two stes. I the first ste, usig sub-gradiet algorithm, its coverget seed is ot so fast [2], totally about iterative times. The legth of iterative ste is related with the coverget seed as aalysis i Sectio 3.2. I the secod ste, the roosed scheme whose utility fuctio structure similar with [5] ca covergece i 0 times. I a word, the roosed algorithm, which combies the merits of cetralized ad distributed framework, ca efficietly make use of relays to romote the sum rate of users. Meawhile, fairess imrovemet is well achieved. Strict mathematic roof of Pareto efficiecy is ot give, but simulatio results demostrate that the roosed scheme erformaces romotio o the base of the cetralized ad distributed schemes. I additio, whe it comes to the feasibility of the roosed algorithm, the CSI should be obtaied by the base statio i the joit relay selectio ad subcarrier allocatio, ad the be coscious of each relay ad user. I other words, some overheads are ecessary for trasmittig the CSI from base statio ad these distributed odes. Therefore, the roosed algorithm is suitable for the resource allocatio i cooerative OFDMA systems if the cost of chael overhead ca be afforded. Otherwise, i eergysavig systems, CSI overheads reduce the feasibility of the roosed algorithm. 5. Coclusio I this article, we roose a semi-distributed resource allocatio framework for the resource otimizatio i multi- Figure 5 Demostratio of the simulatio results of users sum rate versus umber of users.

10 Page 0 of 0 cell ulik cooerative OFDM systems i DF mode. Secifically, we model the resource allocatio framework as a otimal roblem. This otimizatio roblem is divided ito two stes. First, usig iteger rogrammig, we achieve the joit relay selectio ad subcarrier allocatio based o maximizig system sum rate i a cetralized way. Secod, the distributed ower allocatio is achieved based o game theory, for cooerative ad o-cooerative users, resectively. For cooerative mobile statios, a imroved utility is roosed to regulate ower allocatio i the two time slots. esides the existece of NE, a ew aroach for the strict mathematical roof of the uiqueess of NE is roosed. Simulatio results demostrate that the roosed algorithm successfully combies the merits of cetralized ad distributed framework. It ca effectively make use of relays to ehace the sum rate of users as well as achieve the fairess amog users. As a rogressive otimizatio, the semi-distributed scheme combies the merits of cetralized ad distributed framework i a low comlexity. I the future, we are lauchig o further study about a ovel utility fuctio desig for system rate elevatio, ad develo our study towards a trasmittig scheme with lower iterferece, ower cosumtio ad CSI overheads. The roof of the Pareto efficiecy of trasmittig ower strategy, as well as the certificatio of the idetity of NE ad the otimal solutio, eeds to be develoed as well. 8. W Lag, X Yi-sheg, S Ego, Resource allocatio i multi-cell OFDM systems based o o-cooerative game, i IEEE Pers Idoor Mobile Radio Commu, Helsiki, Filad -5 Setember 2006) 9. X Yu, T Wu, J Huag, Y Wag, A o-cooerative game aroach for distributed ower allocatio i multi-cell OFDMA-relay etworks. IEEE Tras Commu. 430), ) 0. A Mackezie, LA DaSilva, Game Theory for Wireless Egieers Morga & Clayool Publishers, Washigto, 2006). RD Yates, A framework for ulik ower cotrol i cellular radio systems. IEEE Sel Areas Commu. 37), ). doi:0.09/ L Wolsey, Iteger Programmig Wiley-Itersciece Publicatio, Sa Fracisco, 998) doi:0.86/ Cite this article as: Hou et al.: Resource allocatio based o iteger rogrammig ad game theory i ulik multi-cell cooerative OFDMA systems. EURASIP Joural o Wireless Commuicatios ad Networkig 20 20:69. Ackowledgemets This study was suorted by the Natioal Natural Sciece Foudatio of Jiagsu Provice No. K2000), the Major Natioal Sciece & Techology Secific Project No. 200ZX ) ad the Natioal Natural Sciece Foudatio of Chia No ad No ). Cometig iterests The authors declare that they have o cometig iterests. Received: 7 July 20 Acceted: 5 November 20 Published: 5 November 20 Refereces. K Keuyoug, H Yougam, K Seog-Lyu, Joit subcarrier ad ower allocatio i ulik OFDMA systems. IEEE Commu Lett. 69), ) 2. D Zhag, Y Wag, J Lu, QoS aware resource allocatio i cooerative OFDMA systems with service differetiatio, i IEEE It Cof Commu., Cae Tow South Africa May 200) 3. Z Ha, Z Ji, KJR Liu, Power miimizatio for multi-cell OFDM etworks usig distributed o-cooerative game aroach, i IEEE Globecom, Texas America. 5, November 2004) 4. D Wu, D Yu, Y Cai, Subcarrier ad ower allocatio i ulik OFDMA systems based o game theory, i IEEE It Cof Neural Netw Sigal Process, Wuha Chia Jue 2008) 5. D Yu, D Wu, Y Cai, Subcarrier ad ower allocatio based o game theory i ulik OFDMA systems. J Electro If Techol. 4, ) 6. D Wu, Y Cai, Power allocatio i iterferece relay chaels based o ocooerative game theory, i IEEE It Cof Wireless Commu Sigal Process, Najig, Chia November 2009) 7. K-D Lee, T-SP Yum, O Pareto-efficiecy betwee rofit ad utility i OFDM resource allocatio. IEEE Tras Commu. 58), ) Submit your mauscrit to a joural ad beefit from: 7 Coveiet olie submissio 7 Rigorous eer review 7 Immediate ublicatio o accetace 7 Oe access: articles freely available olie 7 High visibility withi the field 7 Retaiig the coyright to your article Submit your ext mauscrit at 7 srigeroe.com