Evolution of Spatial and Multicarrier Scheduling: Towards Multi-cell Scenario

Ev olution of Spatial and Multicarrier Sc heduling: T o w ards Multi-cell Scenario P ol Henarejos 1 ? , Ana P erez-Neira 1 , 2 , V elio T ralli 3 , Marco Moretti 4 , Nikos Dimitriou 5 , and Giulio Dainelli 5 1 Cen tre T ecn` ologic de T elecom unicacions de Catalun ya (CTTC), Barcelona, Spain {pol.henarejos, ana.perez}@cttc.es 2 Univ ersitat Polit ` ecnica de Cataluny a (UPC), Barcelona, Spain, 3 CNIT, Universit y of F errara, F errara, Italy , velio.tralli@unife.it 4 Dipartimen to di Ingegneria dell’Informazione, Universita di Pisa, Italy , marco.moretti@iet.unipi.it 5 Institute of Acelerating Systems & Applications, National Kap odistrian Universit y of Athens, Athens, Greece, nikodim@phys.uoa.gr Abstract. OFDMA systems are considered as the promising multiple access sc heme of next generation m ulti-cellular wireless systems. In order to ensure the optimum usage of radio resources, OFDMA radio resource managemen t algorithms hav e to maximize the allo cated p o wer and rate of the differen t subchannels to the users taking also in to accoun t the gen- erated co-c hannel interference betw een neigh b oring cells, whic h affects the received Quality of Service. This paper discusses v arious schemes for p o w er distribution schemes in multiple co-channel cells. These sc hemes include centralized and distributed solutions, which may inv olve v arious degrees of complexity and related ov erhead and may emplo y pro cedures suc h as linear programming. Finally , the pap er introduces a new solu- tion that uses a netw ork flow mo del to solve the maximization of the m ulti-cell system sum rate. The application of spatial beamforming at eac h cell is suggested in order to b etter cop e with interference. 1 In tro duction Most of the existing literature on resource allo cation fo cuses on the single cell scenario, where all users are assigned to a different p ortion of the av ailable sp ec- trum. How ever, mobile comm unication systems are b etter describ ed as multi- cellular systems where either co ordinated or uncoordinated cells transmit on the same bandwidth and are therefore the capacity is increased, but not unaffected b y Multiple Access Interference (MAI). MAI in particular deteriorates the p er- formance of users near cell b oundaries. Th us, any resource allo cation problem in ? This w ork was supp orted by the Europ ean Commission under the Netw ork of Ex- cellence NEWCOM++ (216715). The authors thank the partners of WPR8 in New- com++ NoE for their v aluable technical help. 2 P . Henarejos, A. Perez-Neira, V. T ralli, M. Moretti, N. Dimitriou, G. Dainelli a m ulti-cell environmen t has to tak e into account the impact of the MAI on the system. A frequency reuse factor larger than one guarantees a large reduction of the interference at the cost of a reduction of the efficiency in the usage of sp ec- tral resources. F or this reason in recen t literature several works hav e fo cused on Orthogonal F requency Division Multiple Access (OFDMA) allo cation in multi- carrier cellular systems with a frequency reuse factor equal to one. Due to the strong impact of MAI in this scenario, it is imp ortan t to take full adv antage of frequency and multi-user diversit y of the system. The authors from [1] set the initial p oin t to start the current research, in which is addressed this manuscript. In the past years, several approaches relying on the concept of in ter-cell co- ordination ha ve emerged, which can b e distinguished in tw o categories: pac ket- based co ordination and resource-allocation based co ordination. In the first one, data pac kets destined to the users are replicated at several base stations, b e- fore jointly preco ding/beamforming, and transmitting from all the Base Station an tennas (BS) [2], [3], [4]. The dra wback of this approach is a large ov erhead in inter-cell signaling, pack et routing, and feedbac k for exc hanging the channel state information required to compute the preco ders. In the second approac h, the in terference is tac kled b y means of coordinated resource con trol (pow er, sc heduling, etc.) b et ween the cells [5], [6] which make low er complexity and dis- tributed co ordination techniques are possible. P o wer control and smart soft reuse partitioning are p ossible strategies that can b e applied [7], [8], [9]. Dynamic m ulti-cell p o wer control targeted at maximizing the sum of user rates in the net w ork is a very difficult task and does not lend itself easily to a distributed (across the cells) implementation, except for some particular cases with a large num ber of users [10]. The reason is as follo ws: dynamic pow er control affects the Signal to Interference plus Noise Ratio (SINR) of all users in all cells in a fully coupled manner making in terference unpredictable. OFDMA resource allo cation is a viable solution to exploit channel and multi- user div ersity in wireless communication systems. In a multi-user scenario, with an OFDMA multiple access scheme, each user is assigned a subset of orthogonal sub carriers. If the transmitter has full knowledge of the Channel State Infor- mation (CSI), sub carriers can b e assigned with the goal of maximizing some optimalit y criterion. Since the radio propagation channels are statistically inde- p enden t among the users, what is a bad c hannel for one user ma y b e a go o d one for another and th us, thanks to the effect of multi-user diversit y , dynamic resource allo cation largely increases the system sp ectral efficiency . Resource Alloca tion schemes in a multi-cell scenario can b e divided into cen tralized and distributed algorithms. Cen tralized schemes p erform allo cation through a central unit like the Radio Netw ork Controller (RNC) that collects CSI and in terference level for each user in the system. Ideally , the RNC de- cides whic h sub c hannels (or sub carriers) to assign to each single user with the suitable format and pow er level. On the other hand, in a distributed algorithm resource allo cation is p erformed autonomously b y each single BS in its cell. The main problem for cen tralized sc hemes is the large amoun t of signaling needed for exc hanging CSI and allocation feedback. Moreov er, the allocation complexity Ev olution of Spatial and Multicarrier Scheduling 3 gro ws exp onen tially with the num ber of users in the netw ork, since resource as- signmen t is realized by a single unit, which has to process large amounts of data. Th us, centralized algorithms are often studied to pro vide an ideal b ound for the p erformance of others schemes. Distributed algorithms require low er complexit y and signaling, since they p erform allo cation locally at each BS and therefore require only the information ab out the users in the cell. Ho wev er, such solutions v ery often lead to iterative algorithms, whic h may hav e con v ergence problems. Sometimes the differences b et ween distributed and centralized sc hemes blur aw ay since distributed schemes may require a limited amount of centralized informa- tion to impro ve their p erformance. Recen tly , the spatial diversit y has b een in tro duced to reach an acceptable p erformance. Man y solutions of the current State of Art (SoA) prop ose Multiple Input Multiple Output (MIMO) techniques, widely extended for the single cell scenarios. Nevertheless, these sc hemes treat inter-cell in terference as noise, where the p erformance is limited, sp ecially for edge-cell users. The authors in [11], [12] and [13] deal with this kind of problem. Unfortunately , computational p o wer and complexity raise up with the num b er of cells. Thus, distribution forms of co operation among the user terminals and BS app ear with great interest. One alternativ e is co op erativ e MIMO to minimize total p o w er with QoS constraints [14]. This scheme implies that BS are going to change their p eak pow er and may b e not suitable in several scenarios. This man uscript presen t a second alternativ e of co op erativ e MIMO to maximize a cost function of rate. If the cost function is the sum rate function, the problem becomes NP-hard [15]. How ev er, other cost functions are p ossible to decrease its complexity [16]. This do cumen t aims to organize the SoA according to the requirements of the system, if the interference managemen t is relev an t or not. Moreov er, tw o algorithms to distribute users in the several cells and p o wer allo cation are also presen ted. Thereby , section 2 introduces the different techniques used in the OFDMA p o wer distribution in multi-cell scenarios. Section 3 presents a sc heme to distribute users in the several cells, with centralized or distributed complex- it y , based on the Linear Programming (LP). F urthermore, section 4 describ es a framew ork to p erform the pow er allo cation and user selection from the approach of a single-cell m ulti-antenna scenario. Finally , this do cumen t is ended by the conclusions. 2 P o wer distribution for OFDMA multi-cell systems P ow er has an important role in m ulti-cell systems not only for the rate optimiza- tion, but for the interference management. Netw orks where interference is not a big problem (such as those where there is frequency planning and adjacent cells use differen t OFDMA sub carriers) may striv e to optimize the throughput. On the con trary , in net works where in terference plays an imp ortan t aspect, p o w er strategy ma y b e orien ted to limit this interference. The following algorithms co ver b oth categories. 4 P . Henarejos, A. Perez-Neira, V. T ralli, M. Moretti, N. Dimitriou, G. Dainelli 2.1 La yered and distributed dynamic resource allo cation algorithm A downlink communication system in a cellular netw ork, where all cells adopt a frequency reuse factor equal to one, is considered. Eac h user CSI and the lev el of interference on each sub carrier are assumed to b e p erfectly known by eac h BS. Since in eac h cell a sub c hannel is allo cated to at most one user, the effects of MAI depend on the users (and specifically on their lo cation and pow er/rate allo cation) that are allo cated the same channel in adjacen t cells. The MAI on c hannel m affecting user k in cell q is: I k,m,q = Q X j =1 ,j 6 = q p m,q G k,m,j (1) where p m,q indicates the pow er transmitted b y cell q on subcarrier m and G k,m,j = | h k,m,q | 2 is the channel gain b et w een user k and cell q on sub c han- nel m . Th us, the p o wer required for ac hieving a certain target SINR is: p k,m,q = S I N R B N 0 + I k,m,q G k,m,q (2) where B stands for the bandwidth of the signal and N 0 is the noise pow er in W /H z . A la yered architecture that in tegrates in eac h cell a Pac ket Scheduler (PS) with an adaptiv e resource allo cator (RA) is considered. First, the RA allo cates the resources with the goal of minimizing the transmitted p o wer in each cell sub ject to user’s rate constraints to k eep low the MAI. T o exploit multi-user div ersity , the RA tends to assign most of the resources to the users that ha v e go od channel condition. Second, the PS enforces long-term fairness in order to comp ensate the short term displacement of resources due to the RA. Moreo ver, a load control mech- anism is in tro duced to force the con vergence of the allo cation. T o reduce the complexit y of the allo cation phase, all users adopt only one transmission format with sp ectral efficiency η 0 for all the sub carriers so that rate constraints are translated into a num b er of resources, i.e., R k,q = B η 0 m k,q , where m k,q stands for the num b er of channels allo cated to user k at cell q . The cost of a resource for a giv en user is the p o wer required for ac hieving sp ectral efficiency η 0 . Whenev er a cell modifies its allo cation, it changes also the interference ex- p erienced b y users in neigh b oring cells, which in turn change their allo cation. Th us, the allo cation phase is iterated until a stable allo cation is reac hed in all cells. T o help conv ergence, if the system is not able to reach a stable allo cation, the load is progressively reduced in all cells. After a stable allocation is reached, the PS up dates the maxim um rate re- quiremen ts m k,q for each user in eac h cell with the goal of ac hieving long-term fairness. The conv ergence is not alwa ys guaranteed but the combined actions of load control and pac k et scheduling push to wards con vergence and fairness at the same time. An additional action to ensure conv ergence is pro vided b y a mecha- nism where the most pow er consuming users can b e progressiv ely switched off. Ev olution of Spatial and Multicarrier Scheduling 5 This approac h tends to be unfair because users near the cell b oundary risk to b e to o p enalized in terms of the reduction of the av ailable bandwidth. If not carefully designed, the main drawbac k of this scheme is the num b er of iterations required for ac hieving a stable allo cation. Minim um feedback la yered and distributed dynamic resource allo ca- tion algorithm A minimum feedback scheduling tec hnique extends the con- cepts of distributed allo cation that was outlined in the previous Section. It is assumed that eac h user measures the in terference of eac h sub carrier and sends to the BS only the interference v alues that corresp ond to the “b est” subcarriers. The num b er of those “best” subcarriers, i.e. those which exp erience the low est MAI, is a parameter to b e determined b y the system op erator. F or the other sub carriers, the RA algorithm assumes the worst case scenario and assigns to them a fixed high v alue of interference. With this approac h the required feed- bac k is reduced while the RA algorithm has still the necessary inputs in order to b e able to provide results. Of course, in this case the allo cations will not b e the optimal since the algorithm is forced to work without the actual interference v alues for all the sub carriers. How ev er, it can b e argued that it may pro vide similar throughput results as the previous algorithm with muc h less ov erhead and complexity [17]. Random sub carrier allo cation algorithm Another p ossible wa y to allo cate resources to the users consists of a random resource allo cation to the users in eac h cell, without any kind of optimization criteria. In this case the PS sets again the maximum num b er of sub carriers which can b e assigned to the users, and then the allo cator assigns randomly the sub carriers with resp ect to the constrain ts set by the PS. In this case, after random allo cation, only p o w er con trol tak es place and after that the sub carriers which hav e not ac hieved their SINR target are switched off. Additionally , users in the outer region of a cell will use a p ortion of the bandwidth whic h is differen t from that one utilized by the users in the outer region of the adjacent cells. 2.2 P ow er planning The ob jectiv e is to achiev e a fully distributed implementation of resource allo- cation ov er a multi-cell OFDMA netw ork, whose aim is minimizing the net work outage capacity . T o reach this goal, the selection of the user to be sc heduled and of the resources (here defined as the couple sub carrier/transmit p ow er level) to b e assigned to him, should b e p erformed taking into account the c hannel gain and the receiv ed interference p o w er. If a fully distributed approac h is pursued, eac h BS can only rely on lo cal information provided via a feedback c hannel b y its own set of users. So, in this work some structuring inside the system is in tro duced, in order to mak e in terference lev el inside the net work predictable. Though in principle pow er lev els can contin uously v ary inside a predefined range, only a certain set of p ossible pow er lev els are assumed, and these are 6 P . Henarejos, A. Perez-Neira, V. T ralli, M. Moretti, N. Dimitriou, G. Dainelli distributed among cells and sub carriers according to a predefined pattern. This concept will b e denoted from now on as “p o wer planning”. In particular, the netw ork is organized in groups of Q adjacent cells according to a regular pattern as done for frequency planning in 2G systems and, for analogy , this group of cells is denoted as “cluster”. Then, the M equally spaced OFDMA sub carriers assigned to each cell are arranged in Q groups of M /Q adjacen t sub carriers, from now on denoted also as “sub-bands”. It is clear that the larger the v alue of Q , the smaller the frequency div ersity if correlation b et w een subcarriers is taken into accoun t. A p o wer vector P = [ p 1 . . . p Q ] is introduced, which is comp osed of the Q p o w er lev els, also denoted as elements of the “p o wer profile”. Hence, in the allo- cation pro cess only these Q p o wer v alues are usable. F rom no w on, this vector will b e denoted as “m ulti-cell transmit p o w er v ector”. Thus, the terms “p o w er profile” and “multi-cell transmit p o wer v ector” are considered to represent iden- tical things. In each c ell, every sub-band is assigned with one of the v alues belonging to p o w er vector P , and o ver all sub-bands inside a cell all v alues of P are exploited. Nev ertheless, looking at a sp ecific sub-band, the set of cells b elonging to the same cluster use all p o w er lev els a v ailable in P . So, each cell in the net work is assigned with a tag j ranging from 1 to Q denoting the cell type. Then, since each tag is assigned with a sp ecific p o wer v ector (i.e., with a sp ecific order of the p ossible Q p o wer levels in v ector P ), cells with the same tag will be assigned with the same p o w er v ector, whereas cells belonging to the same cluster are assigned with p erm utations of the original p o w er v ector. 3 Multi-cell user assignmen t Previous section was addressed to manage the pow er budget and choose whic h strategy can result more effective. How ev er, this asp ect also separates the p o wer v ariable from the others. One of the adv an tages of decoupling the pow er v ariable dep ending on scenario requirements is the fact that users can b e sc heduled a p os- teriori following the same requirements. Even though the p o wer can b e used to sc hedule users, i.e. p o wer equal to zero implies no user is scheduled, LP metho ds has b een considered to b e an efficient tool to solve this kind of problems. On the con trary , since pow er and user scheduling is p erformed in separated steps, solution b ecomes sub optimal. Authors in [18] presen t a resource allocator for the uplink of m ulti-cell OFDMA systems. That concept is also applicable for the do wnlink c hannel. Pre- vious section has defined the p o wer strategy and p o w er is solved in this p oin t. Hence, user scheduling can b e p erformed through LP . Even though authors in [18] minimize the p ow er, the maximization of sum rate can also b e pursued. Moreo ver, LP offers the c hance to in tro duce more v ariables to the problem to mak e it as so general as it is desired Ev olution of Spatial and Multicarrier Scheduling 7 Consider a do wnlink OFDMA system with Q cells with one BS eac h, K users distributed ov er all cells and M carriers a v ailable in each BS. Since p o wer is defined in the previous sections, all rates of users are pre-defined in the follo wing manner. The rate of k th user at m th carrier and q th cell is: r k,m,q = log 1 + G k,m,q p m,q P Q q 0 6 = q G k,m,q 0 p m,q 0 + B N 0 ! (3) where p m,q is the p o w er serv ed b y q th cell at carrier m , defined previously . This scheme can b e easily combined with b eamforming to decrease the effect of the interference. Thus, the BS may use a b eam to increase the SINR and the o verall sum rate. 3.1 Cen tralized algorithm A centralized approach could be done b y modeling the m ulti-cell system as a single netw ork with one central control unit which computes the access parame- ters for all users in all cells and the wa y with which users are scheduled ov er the net work. In order to assign users to cells and carriers, LP algorithms are used and they pro ve to b e a go od wa y to exploit this kind of scenarios. The problem can b e stated as: b = arg max b Q X q =1 K X k =1 M X m =1 b k,m,q r k,m,q s.t. K X k =1 b k,m,q ≤ 1 , q = 1 , . . . , Q, m = 1 , . . . , M Q X q =1 b k,m,q ≤ 1 , k = 1 , . . . , K , m = 1 , . . . , M (4) where b k,m,q = 1 if user k is scheduled at m th carrier and in the q th cell and 0 otherwise. The first constraint implies that only one user can b e sc heduled in eac h carrier and cell. The second constraint, implies that only one cell can b e assigned to one user at same carrier. This problem can b e solved by LP easily . Ho wev er, it requires centralized sc hemes and feedbac k corresponding to all h k,m,q that must be known at the transmitters. Although complexity is prop ortional to eac h v ariable that is introduced, results are near optimal. 3.2 Distributed algorithm The centralized algorithm is optimal compared to the distributed algorithm since it has more information ab out the channels of all users in all cells. On the other hand, having a centralized algorithm requires a h uge amount of feedbac k infor- mation pro cessing complexit y and in tro duces large amoun ts of o v erhead in the 8 P . Henarejos, A. Perez-Neira, V. T ralli, M. Moretti, N. Dimitriou, G. Dainelli calculations. The idea is to distribute the complexit y in each cell, i.e. removing q index. Hence, each BS should execute the following algorithm separately . A distributed algorithm could b e derived from the ab o ve as: b = arg max b K X k =1 M X m =1 b k,m,q r k,m,q s.t. K X k =1 b k,m,q ≤ 1 , m = 1 , . . . , M . (5) Note that the second constraint is remov ed since it requires a centralized w ay of con trolling all users scheduled in all cells. Thereby , one user can b e scheduled in differen t cells at the same carrier. This simplification distributes complexit y ov er the netw ork and do es not require any centralized pro cessing. Additionally , the amoun t of feedbac k can be reduced if in terference is assumed to be equal to all users in all cells. That is equiv alen t to approximate the rate of k th user at m th carrier and q th cell as: r k,m,q = log  1 + G k,m,q p m,q I m,q + B N 0  . (6) It is easy to show that the q th cell only requires c hannel gains G k,m,q of its K users at eac h carrier. 4 F rom spatial to multi-cell sc heduling In [19] the authors present a spatial sc heduler for m ulticarrier systems in a single- cell scenario. The basis is a net work flo w formulation for maximization of the system sum rate. T o sumarize it, the spatial diversit y is solved using Multiuser Opp ortunistic Beamforming and choosing the user p erm utation, and its corre- sp onding b eam set, that achiev e the b est sum rate; then, the pow er allocation is performed from this spatial allocation. This section prop oses a mo dification of the algorithm in [19] and distributes the spatial dimension, separating the an tennas one from each others and distributing one per BS. F or this reason, instead of b eam-user selection, cell-user selection has to be carried out. The w ork in [19] considers ergodic sum rate maximization for contin uous rates. The ergo dic framework also allo ws the optimization in the time domain. In fact, if [1 , . . . , N ] is the time interv al of the optimization, for any generic system or user metric, R [ n ], under ergo dic assumption for random pro cesses in the system, the appro ximation (1 / N ) P n R [ n ] ≈ E { R [ n ] } = E { R } = R holds, where R does not dep end on time n . Hence, optimizing R means optimizing R [ n ] ov er time in terv al [1 , . . . , N ]. The discrete v ariable or index u m,q ∈ K 0 = { 0 , 1 , . . . , K } indicates the user (i.e. 0 means no user) that is scheduled to use cell q on sub carrier m . Note that only one user or none can b e sc heduled for each carrier and each cell. The whole Ev olution of Spatial and Multicarrier Scheduling 9 set of these v ariables is the matrix U ∈ K M × Q 0 , whereas the whole set of p o w ers is the matrix P ∈ R + ,M × Q ∪ { 0 } . It is implicitly assumed that if u m,q = 0 then p m,q = 0 2 . The aim of resource allo cation is to dynamically assign radio in terface re- sources to the different users, i.e. to determine optimal v alues of U and P . The problem can b e formulated as max U , P K X k =1 R k ( U , P ) s.t. P q ( U , P ) ≤ ¯ P , ∀ q R k ( U , P ) ≥ φ k K X s =1 R s ( U , P ) , ∀ k (7) where R k ( U , P ) = E { R k ( U , P ) } = P M m =1 P Q q =1 E  δ u m,q k r k,m,q  is the rate pro vided to user k from (3), P q ( U , P ) = P M m =1 E { p m,q } is the total av erage p o w er sp en t b y cell q to serve the allo cated users and δ u k is the Kronec ker’s delta 3 . The first constraint refers to the total p o wer used which must b e less than a maximum amount ¯ P . The second constraint implies that users ought to obtain the prop ortional φ k part of the sum rate, which determines the share of throughput finally ac hiev ed b y eac h user. Therefore, φ must satisfy the condition P K k =1 φ k = 1. It is imp ortan t to underline that in this problem rate and p o w er constraints are referred to as av erage v alues. In this wa y , the instantaneous constraints are relaxed leading to a reduction in the complexit y of the resulting optimization algorithm. 4.1 Dual optimization framework and adaptive algorithms The optimization problem is non conv ex and Lagrangian dualit y [20] is used to solv e the problem. It enables each user to adapt their resources lo cally with the aid of limited information exchange. An interesting p oint of the Lagrangian is the dual decomp osition into individual user and cell terms. This fact motiv ates decen tralized algorithms as in [20] and allows to distribute the complexity ov er the netw ork. How ever, to obtain a distributed algorithm as seen later, it is nec- essary to decouple user assignment from p o w er assignment. The dual ob jective of problem (7) is defined as min λ > 0 , µ ≥ 0 g ( λ , µ ) = min λ > 0 , µ ≥ 0  max U , P L ( U , P , λ , µ )  = min λ > 0 , µ ≥ 0 L ( U ∗ , P ∗ , λ , µ ) (8) 2 This also means that P has an implicit dep endence on U and vice versa as shown afterw ards. 3 δ u k = 1 if u = k and 0 otherwise. 10 P . Henarejos, A. Perez-Neira, V. T ralli, M. Moretti, N. Dimitriou, G. Dainelli where L ( U , P , λ , µ ) is the Lagrangian function of the problem (7) and λ , µ are the Lagrangian m ultipliers. It is imp ortan t to remark that while the primal problem is a non-concav e maximization, the dual problem becomes a con vex optimization. How ev er, the dual problem is not differen tiable and an iterativ e subgradien t metho d is used to up date the K + Q solutions of the dual problem at each discrete time in- stan t. Starting from initial solutions λ 0 and µ 0 , the up date equations at the i th iteration derive from subgradient expressions and are: λ i +1 =  λ i − δ λ  ¯ P q − P q ( U ∗ i , P ∗ i )  +  µ i +1 = " µ i − δ µ R k ( U ∗ i , P ∗ i ) − φ k K X s =1 R s ( U ∗ i , P ∗ i ) !# + (9) where [ x ] +  = max( , x ) , 0 <   1 and δ λ , δ µ are p ositiv e step-size parameters. U ∗ i , P ∗ i indicate the optimal solutions of the Lagrangian at the i th iteration, i.e. those whic h maximize L ( U , P , λ i , µ i ). 4.2 Solutions for the allo cation problem The optimal p o wer and user solutions are difficult in that case due to the cross dep endence of user and p o w er allocation. Therefore, the dual ob jective can be rewritten as follo ws: g ( λ , µ ) = max U , P L ( U , P , λ , µ ) = Q X q =1 λ q ¯ P + M E  max u m  max p m ≥ 0 M ( u m , p m )  (10) with M ( u m , p m ) = Q X q =1 ,u m,q 6 =0  ( µ u m,q − µ T φ ) log 2 (1 + r u m,q ,m,q ( p m )) − λ q p m,q  . (11) The optimal solution, given λ , µ , b ecomes, for eac h frequency m , u ∗ m = arg max u m M ∗ ( u m ) (12) with M ∗ ( u m ) = max p m ≥ 0 M ( u m , p m ) . (13) This sho ws that user selection (12) and pow er allocation (13) are decoupled from the dual optimization and b oth them can b e computed separately . In fact, user selection is computed b efore the pow er solution is found. Concerning spatial allo cation, the main issue is to reduce the searc h space. This issue can b e faced by using suboptimal greedy selection pro cedures. The Ev olution of Spatial and Multicarrier Scheduling 11 simplest among them is the opp ortunistic selection. Thereb y , each user selects the b est cell b y assuming that all base stations are transmitting with a preassigned p o w er and feeds back the selected cell with its SINR, while each cell allo cates its resources to the best user selected among those comp eting for that cell. This can b e done helped b y spatial b eamforming at each base station. Next sub-section commen ts further on that. 4.3 Discussion on Centralized and Distributed solutions The optimal solution requires a centralized controller that runs all or parts of the algorithms. Even though the p o wer allo cation algorithm based on the up date of λ i can b e distributed on eac h base station, user alloc ation algorithm requires a cen tralized solution, i.e. a con troller which knows all channel gains determines, for all sub carriers, the vector u m and send it to base stations through signaling. A decentralized implementation can b e set up by using the opp ortunistic sub optimal solution of p o w er allo cation. In this case user allo cation algorithm has tw o steps (for each sub carrier): – Eac h user selects the best cell b y assuming that all base stations are trans- mitting with a preassigned pow er and feeds back the selected cell with its SINR. – Eac h base station allo cates its resources to the b est user selected among those comp eting for that cell. When user allo cation is decen tralized, t wo p oin ts need to b e remarked. The first one is related to the up date of µ i .This can b e performed at the base stations, if users are served b y only one base station, or it can b e performed b y the users after that the information on the resource allo cation is sent to them. The second one is related to the ev aluation of the user rates. This can b e done based on the SINR ev aluated b y the user, whic h do es not take into account the p o w ers actually allo cated to interfering users, because they are not kno wn. Therefore, the allocated rate is not the actual rate supported b y the transmission leading to p ossible outages. This can be av oided only by ev aluating the SINR b y using the worst-case v alues of interfering p o wer in the vector V m , whic h can b e further constrained to be less than a maximum v alue P max on each resource unit. T o counteract the losses that opp ortunistic schemes presen t when the num b er of users is mo derate or lo w 4 , while still preserving a decentralized implemen ta- tion, the “p o wer planning” concept (as [21] for time-division multiple access systems) can b e introduced to preassign suitable pow er v alues to vector V m with the additional constraint p m,q ≤ v m,q . 4 When the num b er of users is not very large, sum rate is maximized by allocating a n umber of users on each sub carrier and slot usually smaller than Q . 12 P . Henarejos, A. Perez-Neira, V. T ralli, M. Moretti, N. Dimitriou, G. Dainelli 5 Conclusions Multi-cell scenarios are present in a v ery large n umber of standards and systems. The trends of technology and the increased requirements in bandwidth usage ef- ficiency dictate the tight re-use of the frequency bands in neighboring cells. T o do that, effective interference management schemes are required to regulate opti- mally the transmitted p o wer in eac h subcarrier in all cells. In this con text, pow er planning w as presen ted as a suitable to ol to extract effectiv e net work parameters and requirements. Additionally , la y ered and distributed dynamic resource allo- cation algorithms w ere in tro duced in those scenarios that hav e predefined rate requiremen ts and p o w er may b e adjusted to guaran tee the pro vided QoS. Fi- nally , cross-laying for m ulti-cell user scheduling is focused by Lagrangian dualit y to solve the same problem and ensure the QoS constraints. References 1. G. 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