Assumptions � We consider the downlink of a wireless multi- carrier communication system with K users and N subcarriers � Perfect CSI at the base station � Ideal feedback channel to signal the assignment decision. � We introduce a binary allocation variable � 1 if channel is assigned to user n k = � � k n , � 0 otherwise -26-
Resource allocation: constraints � Univocal assignment: � Subcarriers are allocated univocally to the users: only one user at the time can occupy a given subcarrier K � 1 n 1,..., N � � = k , n k 1 = � The presence of an integer assignment variable greatly complicates the allocation problem since it makes the problem NOT convex -27-
Resource allocation schemes Rate adaptive � Objective : maximize the overall rate r tot subject to a global power constraint � n P n = P 0 � � = �� P G K N n k n , r log � 1 � � + t ot k , n 2 2 � � � k 1 n 1 = = Margin adaptive � Objective : minimize the overall power P tot subject to the different users’ rate constraints. = �� K N P P � n k n , tot k 1 n 1 = = -28-
Rate adaptive schemes -29-
Sum-rate maximization � The sum-rate maximization problem is solved by � � P G 1. assigning each sub-carrier to �� n k n , max log � 1 � + � the user that maximizes its 2 k n , 2 � � � P , � k n gain s . . t 2. performing waterfilling over � all the sub-carriers allocated. 1 n � � � k n , � Such a solution maximizes the k � cell throughput but is P P � n 0 extremely unfair since it n privileges the users closest to {0,1} k n , � � � k n , the BS and starves all the others. -30-
Max-min rate allocation � Fairness is introduced by � � PG � allocating resources with n k n , � � maxmin log 1 + � 2 k n , 2 � � � k P , � the goal of maximizing the n minimum capacity offered st . . to each user, thus � 1 n � � � introducing fairness among k n , k the users. � P P � � In general, fairness comes n 0 n at the cost of a reduction {0,1 } k n , � � � of the overall throughput k n , of the cell. -31-
Max-min rate allocation � Problem is NOT convex � A heuristic solution is implemented: 1. Uniform power allocation on all sub- channels P = P 0 / N 2. A greedy assignment strategy that iteratively allocates the sub- carriers to the user with the smallest rate -32-
Sum-rate maximization with proportional rate constraints � Different users may � � PG � � n k n , � � max log 1 + � require different data 2 k n , 2 � � � P , � k n rates. In this case, a fair st . . solution is to allocate � 1 n � � � k n , radio resources k � proportionally to the P P � n 0 n users’ different rate r : r :...: r : :...: = � � � constraints. 1 2 K 1 2 K {0,1 } k n , � � � k n , -33-
Sum rate maximization with proportional rate constraints � The optimization problem is a mixed binary integer programming problem and as such is not convex and in general very hard to solve. We follow an heuristic approach: � Subcarrier allocation phase : assuming an uniform power distribution, the subcarriers are allocated complying as much as possible with the proportional rate constraints. � Power allocation phase : once the subcarrier are allocated to the users, the power is distributed so that the proportional rate constraints are exactly met. -34-
Sum rate maximization with proportional rate constraints: subcarrier allocation � At each iteration, the user with the lowest proportional capacity has the option to pick the best sub-channel. � The sub-channel allocation algorithm is suboptimal, because it is greedy and assumes a uniform distribution of power. -35-
Sum rate maximization with proportional rate constraints: power allocation � The proportional rate constraints are enforced by allocating the power to the users in two steps 1. Find for each user the expression of P k ( tot ) the total power allocated to each user k . 2. Distribute the power among users in such a way that the proportional rate constraints are met and the overall power does not exceed P 0 -36-
Sum rate maximization with proportional rate constraints: power allocation � The Lagrangian is ( Z k , n = G k , n / � � ) : � K � K � � � � P k,n � P 0 L ( P, � ) = log 2 (1 + P k,n Z k,n ) + � 1 k =1 n � � k k =1 n � � k � � � K � � log 2 (1 + P 1 ,n Z 1 ,n ) � � 1 + � k log 2 (1 + P k,n Z k,n ) � k k =2 n � � 1 n � � k � Leading to the power per user P k(tot) � N k Z k,n � Z k, 1 P tot = N k P k, 1 + k Z k,n Z k, 1 n =1 � The optimal distribution of the P k(tot) is found iteratively with the Newton-Raphson method. -37-
Rate adaptive results � Simulation parameters � Number of cells: 1 � Maximum BS transmission power: 1 W � Cell radius: 500 m � MT speed: static � Carrier frequency: 2 GHz � Number of sub-carriers: 192 � Sub-carrier bandwidth 15 kHz � Path loss exponent: 4 � Log-normal shadowing standard dev. 8 dB � Small-scale fading Typical Urban (TU) -38-
Rate adaptive results � Algorithms simulated � Sum rate maximization � Max-min � Sum rate maximization with proportional rate constraints � Equal rate constraints � = � = ‧= � 1 2 K � Rate constraints proportional to the pathloss -39-
Rate adaptive results 4e7 Sum rate Prop rate 2 Prop rate 1 3.5e7 Max � min 3e7 Throughput 2.5e7 2.0e7 1.5e7 4 6 8 10 12 14 16 Number of users -40-
Rate adaptive results Sum rate 1 Max � min Prop rate 2 Prop rate 1 0.8 Fairness 0.6 0.4 0.2 0 4 6 8 10 12 14 16 Number of users -41-
Margin adaptive schemes -42-
Resource allocation: constraints � Each user has to meet a certain target rate, which constraints the task of resource allocation N � r k ( ) r k 1,..., K = = k n , n 1 = � The power necessary for user k to transmit with rate r k , n on subcarrier n is ( ) r P 2 1 / Z = � k n , k n , k n , -43-
WCLM: formulation � The allocator solves the optimization problem by � �� ( ) r k n , min 2 1 k n , � assigning the subcarriers Z r , � k n k n , and the rate on each s t . . subcarrier. � 1 n � � � � Address the fairness issue k n , k � but there are no explicit r r k ( ) � � k n , k , n limits to the transmitted n {0,1} k n , � � � power. k , n � Problem is NOT convex -44-
WCLM: convexification � � � � sk,n N K 1. Relax the integer allocation � k,n � 1 1 min � k,n 2 Z k,n s, � n =1 variable � k , n . k =1 s.t. � Another way to interpret � N n =1 s k,n = r ( k ) the optimization is to � K consider � k , n as the time- � k,n = 1 k =1 sharing factor for the k � k,n � [0 , 1] user of subcarrier n. 3. Once the problem is 2. Introduce a new rate convex, it can be solved in variable s k , n = r k , n � k , n so that the dual domain the objective function becomes convex (positive semidefinite Hessian ) -45-
WCLM dual function � Lagrangian of the problem is � � s � � � � k n , N K K N N K 1 �� � � � � � � � � L s ( , , , ) 2 1 s r k ( ) 1 � � µ = � k n , � � � � � � µ � � � � � � k n , k k n , n k n , � � � � Z � � n = 1 k = 1 k = 1 n = 1 n = 1 k = 1 k , n � Given the Lagrangian multipliers � yields � 0 log 2 / Z � � � k k n , = � s k n , ( ) log Z / log 2 log 2 / Z � � � � > � k n , 2 k k n , k k n , � While for � k , n holds � � � � � Z Z � � 1 k k n , k k n , � 1 if k = arg min � � 1 � � � log � � = � k 2 Z � log 2 � � log 2 � � k n , k n , � � 0 else -46-
WCLM: algorithm flow chart � The algorithm is iterative � Start from some arbitrary values of � and computes � k , n and s k , n . � If the users’ rate constrains are not satisfied iteratively increase the values of � until all the rate constraints are met. This procedure requires the inversion of non –linear functions to converge. -47-
WCLM algorithm � Complexity is a major issue: due to the nature of the allocation problem, the solution proposed is iterative : depending on system parameters, convergence may be extremely slow. � Solution admits non-integer values of the allocation variable. � It is necessary to implement a heuristic (multi-user adaptive OFDM, MAO) to set the vector of allocation variables to integer values -48-
Linear programming � Resource allocation can be formulated as a linear programming problem: � Linear objective function � Linear constraints -49-
Linear programming: multiple tx formats � All rate requests are expressed as a multiple integer of a certain fixed rate corresponding to a spectral efficiency � ( ) log 1 P Z � = + 2 k n , k n , � The power necessary for user k to transmit the rate b � ( b =0, � , B ) on the subcarrier n is a fixed cost ( b ) b � P (2 1) / Z = � k n , k n , -50-
Linear programming: multiple tx formats � After linearization of the ��� ( ) b ( ) b min P � k n , k n , objective function and of � k n b the constraints, resource st . . �� allocation can be ( ) b 1 n 1,..., N � � = k n , formulated as a linear k b �� integer programming ( ) b b r k ( ) k 1 ,. .., K �� = = k n , (LIP) problem k b ( ) b {0,1 } b k n , , � � � � Combinatorial problem k n , with exponential complexity in N, K, and B -51-
Linear programming: single tx format � Provided that there is enough �� min P � multi-user diversity, it is k n , k n , � k n possible adopt only one s t . . transmission format ( B =1) with � very limited performance loss. 1 n 1,..., N � � = k n , � The rate requirements r ( k ) are k � translated into a minimum n k ( ) k 1,. .. , K � = = k n , number n ( k ) of subcarriers to k be allocated per user {0, } 1 � � k n , � By relaxing the integer constraint, RRA turns into a standard LP problem -52-
Linear programming: single tx format � The relaxed LP RRA problem has the characteristic that it can be modeled as a a network flow problem. � The network simplex method (NSM) is the most efficient solver for min-cost-max-flow network problems and outperforms other existing techniques � Because of its topology, the solution of the relaxed LP RRA is integral and thus, regardless of the relaxation, always a combination of 0 and 1 � The single format choice allows a great simplification of the solution of the RRA problem at the cost of only a modest worsening of system performance. Dynamical assignment of subcarriers already provides a great deal of diversity! -53-
Linear programming: power allocation � After having solved the LP single-format resource allocation, power can be further reduced by solving a single-user waterfilling problem for each user on the assigned subcarriers. � For user k, who is allocated the set � k of subcarriers , the problem is formulated as 1 � ( ) r min 2 1 k n , � Z r n �� k , n k s t . . � r r k ( ) = k , n n �� k -54-
Optimal allocation � The solution of an optimization problem can be bounded by resorting to the Lagrange dual � Duality gap � is the difference between the solution of the primal problem and the solution of the dual problem � Qualification conditions . It has been showed that in multi-carrier applications, even if the original RRA problem is non-convex , the duality gap tends to zero as the number of tones goes to infinity. -55-
Optimal rate allocation: primal � The primal problem is N K � �� ( ) r k n , min 2 1 k n , � formulated as a Z � r , n 1 k 1 = = k n , minimization problem with s t . . standard rate and exclusive N � r r k ( ) k � � � allocation constraints k n , k n , n 1 = � � � The problem is K � � � � = � | � = 1 ; � � {0,1} k n , k n , k , n � � combinatorial (i.e. all k 1 = possible allocations should be evaluated!) and its complexity grows exponentially with K and N -56-
Optimal allocation: Lagrange dual � The Lagrangian is � � N K � N N �� � � ( ) r k n , L r ( , , ) 2 1 r r k ( ) � � = k n , � � � � � � � k k n , k n , � � Z n 1 k 1 n 1 n 1 = = = = k , n defined over the set R and all the positive rates r k , n � The Lagrangian dual function is � � N � K N � � � ( ) r � k n , g ( ) min 2 1 r r k ( ) = k n , � � � � � � � k k n , k n , � � Z r � , n = 1 k = 1 n = 1 k n , s t . . K � 1 n � = � k , n k = 1 -57-
Optimal allocation: Lagrange dual � The Lagrange dual of the RRA can be written as the sum of N reduced-complexity minimization problems � � K � � ( ) r � k n , � g ( ) min 2 1 r � = k n , � � � � � n k k n , Z r , � � � k 1 = k n , s t . . K � 1 n � = � k n , k 1 = � Solving the per-carrier problem still involves an exhaustive search over the whole set of users. -58-
Optimal allocation: dual update method � The solution to the Lagrange dual problem is found following an iterative process: � Given the multiplier vector � , we find g ( � ) and the rates allocated for each user � The rate results for the different users contribute to the subgradient N � � � � d ( ) k r ( ) k r ( ) k 1, , K = = k n , n 1 = � The subgradient is employed to update the multiplier vector � ( Ellipsoid method ) -59-
Optimal allocation: ellipsoid method � It is the multi-dimenional � An ellipsoid with a center z and a extension of the bisection shape defined by positive method semidefinite matrix A is defined as � The idea is to localize the set E ( A, z ) = � x | ( x � z ) T A ( x � z ) � 1 � of candidate � s within some closed and bounded set. � Then, by evaluating the � The update rule is the following subgradient of g ( � ) at an appropriately chosen center of such a region, roughly half of the region may be eliminated from the candidate set. � The iterations continue as the size of the candidate set diminishes until it converges to an optimal -60-
Optimal power allocation � The same approach can be K N � � used to solve the ( ) max w log 1 P Z + � k 2 n k n , k n , � , P maximization of the sum k = 1 n = 1 s t . . of weighted rates N � P P � n 0 n 1 = � � K � � � R | 1 ; { 0,1} = � � = � � k n , k n , k , n � � k 1 = -61-
Simulation setup � Number of cells = 1 � Radius of each cell R = 500 m � Total available bandwidth W = 5MHz � Center frequency = 2 GHz � Number of subcarriers N = 64 � Number of users K = 8 -62-
Power vs. spectral efficiency 14 Optimal WCLM 12 LP 10 Power [W] 8 6 4 2 0 0.5 1 1.5 2 2.5 3 3.5 4 Spectral efficiency (bit/s/Hz) -63-
Power vs. number of users 8 Optimal WCLM LP 7 Average tx power [W] 6 5 4 3 2 2 4 8 16 Number of users -64-
MIMO resource allocation -65-
MIMO system � We are considering a N T × N R MIMO system with N T > N R so that at least Q = � N T / N R � users can transmit on the same frequency channel. � Users signals are separated by the implementation of linear precoding and receving filters � Signal model � � � H H H � � z W y W H B s W H x = = + + � k n , k n , k n , k n , k n , k n k n , , k n , k n , j n , k n , � � U j k j , � � n Desired signal Multiple access interference and noise -66-
MMO optimal allocation � Margin adaptive optimization problem: � Optimize linear precoder, transmit power distribution and channel allocation to minimize the overall transmit power � Problem is NOT convex and prohibitively complex N � � ( ) min tr R x x � k n , , U n 1 k = � n s t . . N � ( ) H � 1 log det I H R H R r k ( ) 1 k K � + � � � k n , 2 N k n , x k n , i R k n , k n , n 1 = U Q 1 n N � � � n -67-
Block diagonalization (BD) approach � To simplify the problem, we first decide the precoding strategy and then allocate remaining resources (channels and power). � By projecting each user’s MIMO channel on the interference null space, users’ channels are decoupled so that the users transmit on the same channel do not interfere with each other � Allocation task is greatly simplified in absence of interference -68-
BD-based RA � Problem is still not convex N � � min P but as the number of k n , p � , U n = 1 k � n subcarriers increase the s t . . N duality gap tends to zero � P � ( ) r , r k ( ) 1 k K � � � � k n k n , , and can be solved in the n 1 = U Q 1 n N � � � dual domain. n � � k n , ( ) l P P = k n , k n , l = 1 � � � ( ) l ( ) l P g k n , � P � k n , k n , ( ) � � r , = log 1 + � � k n , 2 2 � � � l 1 = 0 -69-
BD-based RA – Dual domain � � On each subcarrier the µ P � ( ) ( ) g min P r , = � µ k , k k , � � � P � , U k � � dual function must be s t . . evaluated over all the U Q � � possible combinations of 1. Exhaustively compute all user Q users combination � For each combination 2. For each user combination the precoding and evaluate + � � 2 µ � receive linear filters ( ) l � � k 0 P = � � � k n , ( ) l � ( ) log 2 g � � k n , must be evaluated!! 3. Choose the combination that minimizes the metric -70-
Successive channel assignment � To reduce allocation complexity, we first group the users on the base of their channel quality and then sequentially solve the RA problem. � The implementation of a sequential allocation strategy forces a change in the design of the linear precoder. � The users of a group do not interfere with the users already allocated but do generate interference versus the sets of user allocated successively. � MAI is treated as spatially colored noise -71-
Successive channel assignment � Taking into account the colored interference the allocation problem can be formulated as the solution of Q successive problems N � � ( ) min tr R x x � k n , , K k n 1 � = q s t . . N � ( ) ( ) q ( ) q H � 1 K log det I H R H R r k ( ) k � + � � k n , k n , k n , 2 N x i q R k n , k n , n = 1 � 1 1 n N � � � � k n , K k � q -72-
Successive channel assignment � Problem becomes more N � � min P tractable by whithening k n , p a , K k � n = 1 q the colored noise at the s t . . receiver multiplying the N � 1/2 P � K � ( ) � r , r k ( ) k R � � � received signal by k n k n , , q i n 1 = k n , � 1 1 n N � � � � k n , K k � q -73-
LP-based channel assignment � Supposing that each user N transmits with a fixed spectral � � min P � k n , k n , efficiency on all his channels: � K n = 1 k � q � power needed becomes a cost N � K n k ( ) k � = � � rate requirements r ( k ) translates k n , q n 1 = into requesting a certain number � 1 n 1, , N � � = ‧ of channels n ( k ) k n , K k � q � Each successive problem can be formulated as a LP problem -74-
Simulation setup � Number of cells = 1 � Radius of each cell R = 500 m � Total available bandwidth W = 5MHz � Center frequency = 2 GHz � Number of subcarriers N = 64 � Number of users K = 8 � Two scenarios: 4x2 and 2x1 -75-
Computational complexity 9 10 BDRAA SCAA LPSCA 8 10 7 10 Complexity 6 10 5 10 4 10 2 4 8 16 32 Number of users -76-
Power vs number of users 4 4 BDRA BDRA SCAA SCAA LPSCA LPSCA 3.5 3.5 LPOA LPOA 3 3 Power [W] Power [W] 2.5 2.5 2 2 1.5 1.5 1 1 2 4 8 16 32 2 4 8 16 32 Number of users Number of users -77-
Power vs. spectral efficiency 10 10 BDRA BDRA 9 SCAA 9 SCAA LPSCA LPSCA 8 8 LPOA LPOA 7 7 6 6 Power [W] Power [W] 5 5 4 4 3 3 2 2 1 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4 Average spectral efficiency [bit/s/Hz] Average spectral efficiency [bit/s/Hz] -78-
Multi-cell algorithms -79-
Multi-cellular RRA schemes � We consider a downlink communication in an OFDMA-based multi-cell network with full reuse of the frequency spectrum among cells � The most limiting factor for this systems is represented by multiple access interference (MAI), caused by users in adjacent cells that share the same spectrum -80-
Multi-cell RRA: interference � With respect to the conventional single-cell scenario, the main problem is the feasibility of the allocation. � Given a certain traffic configuration, there might be no solution that satisfies the rate requirements of all users in the system. � It is equivalent to the problem of power control for single-carrier cellular networks. � RRA schemes need to enforce strategies designed to control the users’ requirements in order to meet a feasible solution. -81-
Multi-cell RRA: interference � MAI depends on the users allocated on the same channel in other cells. � Interference power is computed as N cells � I ( i ) P ( j ) n G ( j ) k,n = k,n,i j =1 ,j � = i � The required transmitted power to achieve a certain target � k , n ( i ) is: � 2 + I ( i ) P ( i ) k,n = � ( i ) k,n k,n G ( i ) k,n,i -82-
Multi-cell power control � Let us focus on subcarrier n � Suppose that in cell i it is allocated to user k ( i ). � Let ( i ) ( i ) ( ) j ( ) i ( ) i , and . G G P P = = � = � n i , j k i ( ), , n i k n , � Power control consists in solving a set of linear equations in P � � � N 1 � c � ( ) � (1) (1) 2 ( ) j � � P G P = � � + I G P u � � = 1, j G � � � j = 2 1,1 � � � 2 G � � � � � � l j , i u = ; G = � � � � N � 1 1 � i � c G G l j , ( N ) ( N ) 2 ( ) j � � P G P = � � + c c i l � N , j G � c � � ( ) � j = 1 diag , , N , N = � ‧ � c c 1 N c -83-
Multi-cell power control � � The matrix has non-negative elements and it is � G by its nature irreducible. Invoking the Perron- Frobenius theorem, these three statements are equivalent * : 1 � � � It exist a power vector ( ) P P I G u = � � � � � ( ) ( ) G 1, G � � � < � � � G max eigenvalue of M M � � k � � ˝ lim G = 0 k � + � -84-
Multi-cellular RRA schemes � In a multi-cell system, RRA algorithms can be classified as distributed and centralized � In a distributed scheme, resource allocation is performed locally by each base station, exploiting the knowledge of channel conditions of only the users in the cell � In a centralized approach, a radio network controller ( god ), that ideally knows the channel state information of all users in the system, assigns the radio resources aiming at a global optimum -85-
Distributed schemes � Attractive because of the limited (!!) amount of feedback and computational complexity. � Each cell has its own controller: RRA is performed on the base of the information available in the cell. � Hybrid schemes allow a certain amount of information exchanged on the network backbone. � The lack of centralized information is partially compensated by the implementation of iterative algorithms. -86-
Centralized schemes � Centralized solutions aim at optimizing the system performance globally: � Controller possesses full information about all users in the system � Unfortunately, they are practically unfeasible due to � the large amount of signaling they require � their complexity, which grows exponentially with the number of users in the system (scale with the number of cells) -87-
Multi-cell algorithms: distributed schemes -88-
Distributed schemes: the PB algorithm � This distributed algorithm addresses the problem by dividing it in three steps. 1. Set max SINR bounds per subcarrier per user per cell 2. Solve the allocation problem 3. Implement an admission control strategy -89-
PB algorithm: SINR bounds � � The spectral radius of matrix is lower ( ) � G � M � � G than any sub-multiplicative matrix norm of � � � ( ) i ( ) j G � � � k n , k n i , , � � ( ) j j i , � G � G � max � � = � � M ( ) i G � 1 � � i N � � c k n i , , � ( ) � By imposing , we can set a bound for the � G 1 � < M max target SINR per user per subcarrier, i.e. � ( ) i ( ) j G � ( ) i G k n , k n i , , � j j i , � k n i , , ( ) i ( ) i 1 E < � < = � k n , k n , ( ) i ( ) j G G k n i , , k n i , , j j i , � -90-
PB algorithm: the allocation problem � PB propose an iterative scheme that 1. implements a heuristic that allocates the subcarriers to users ( ) i 2 I + � k n , ( ) i min � 2. solves a convex problem in the k n , ( ) i G � SINR variable designed to k n i , , minimize the transmit power, s t . . having assumed that the N interference power is fixed � ( ) i log (1 ) r k ( ) k + � � � 2 k n , 3. performs power control so that n = 1 each user meets its target SINR ( ) i ( ) i E k , n � � � � Algorithm is iterated until k n , k n , convergence -91-
PB algorithm: the allocation problem � The admission control strategy consists in switching off those users that: � due the max SINR bounds, do not reach their target rates � exceed a certain predetermined power limit � There is a fairness problem!! -92-
Distributed schemes: a LP approach � As in the single-cell scenario, we have formulated a linear programming approach with just a single transmission format for all users. � The distributed approach leads to an iterative procedure: at the beginning of each new iteration the resources’ costs in each cell are updated taking into account the interference levels of the previous iteration. � Due to interference, allocation in one cell perturbs the allocation in all neighboring cells -93-
LP algorithm: load control � Allocation convergence is not guaranteed and the algorithm needs to be modified to reach a stable allocation. � We implement a load control mechanism that progressively reduce the total amount of resources allocated in each cell until a stable allocation is achieved � LP formulation still maintains the network flow topology -94-
LP algorithm: formulation � In each cell the LP K N �� problem is formulated so ( ) i ( ) i min P � k n , k n , � that a certain number N ( i ) k 1 n 1 = = s . . t of subcarriers has to be K � allocated. ( ) i 1 n � � � k n , k 1 = � Each user k can get at N � ( ) i ( ) i n ( ) k k � � � most n ( i ) ( k ) resources. k n , n = 1 K N �� ( ) i ( ) i N � = k n , k = 1 n = 1 -95-
LP algorithm: packet scheduler � By assigning a different number of subcarriers to users, the LP RRA sets the actual rate offered by the system to each user. � Exploiting multi-user diversity, it tends to assign most of the resources to users with the best channels. � In order to compensate the displacement of resources due to the RRA, in each cell we implement a Packet Scheduler (PS) that aims at maximizing fairness among users by setting the max number of resources for each user -96-
LP algorithm: architecture � At the beginning of each frame, in each cell the PS sets the maximum rate per user � Given the requirements dictated by the PS, the RRA LP iterates until it finds a stable allocation in each cell � If, after a certain number of iterations, a stable allocation has not been found, the load of the cells is progressively reduced until allocation converges. � The allocation results are fed-back to the PS so that it updates the requirements to enforce fairness -97-
LP algorithm: packet scheduler Feedback on last Packet resource allocation Scheduler Feedback on max. rate constraints ( ) i � k n , ( ) i n k Resource Allocator -98-
Multi-cell algorithms: centralized schemes -99-
Centralized approach � The maximum achievable performance of multi-cell resource allocation is currently unknown. � In analogy with the bound developed for the single- cell scenario, we develop a bound on the performance of centralized resource allocation in the dual domain. � Pros: Analytically sound, useful bound to compare other algorithms’ performance � Cons: Exponential complexity, requires the knowledge of all the users channel gains at the central controller -100-
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