Centralized Multi-Cell Resource And Power Allocation For Multiuser .

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Centralized Multi-Cell Resource and PowerAllocation for Multiuser OFDMA NetworksMohamad Yassin‡ , Samer Lahoud‡ , Marc Ibrahim , Kinda Khawam§ , Dany Mezher , Bernard Cousin‡‡ University Saintof Rennes 1, IRISA, Campus de Beaulieu, 35042 Rennes, FranceJoseph University of Beirut, ESIB, Campus des Sciences et Technologies, Mar Roukoz, Lebanon§ University of Versailles, PRISM, 45 Avenue des Etats-Unis, 78035 Versailles, FranceAbstract—Multiuser Orthogonal Frequency Division MultipleAccess (OFDMA) networks, such as Long Term Evolution networks, use the frequency reuse-1 model to face the tremendousincrease of mobile traffic demands, and to increase networkcapacity. However, inter-cell interference problems are generated,and they have a negative impact on cell-edge users performance.Resource and power allocation should be managed in a mannerthat alleviates the negative impact of inter-cell interferenceon system performance. In this paper, we formulate a novelcentralized multi-cell resource and power allocation problem formultiuser OFDMA networks. The objective is to maximize systemthroughput while guaranteeing a proportional fair rate for allthe users. We decompose the joint problem into two independentproblems: a resource allocation problem and a power allocationproblem. We prove that each of these problems is a convexoptimization problem, and that their optimal solution is also anoptimal solution to the original joint problem. Lagrange dualitytheory and subgradient projection method are used to solve thecentralized power allocation problem. We study the convergenceof our centralized approach, and we find out that it reduces intercell interference, and increases system throughput and spectralefficiency in comparison with the frequency reuse-1 model,reuse-3 model, fractional frequency reuse, and soft frequencyreuse techniques.Index Terms—Convex optimization, resource and power allocation, inter-cell interference, ICIC, OFDMA.I. I NTRODUCTIONMultiuser Orthogonal Frequency Division Multiple Access(OFDMA) networks, such as the Third Generation PartnershipProject (3GPP) Long Term Evolution (LTE) [1] and LTEAdvanced (LTE-A) [2] networks, are able to avoid the negativeimpact of multipath fading and intra-cell interference, byvirtue of the orthogonality between subcarrier frequencies.Nevertheless, Inter-Cell Interference (ICI) problems arise indense frequency reuse networks due to simultaneous transmissions on the same frequency resources. System performanceis interference-limited, since the achievable throughput isreduced due to ICI.Fractional Frequency Reuse (FFR) [3] and Soft FrequencyReuse (SFR) [4] were introduced to avoid the harmful impactof ICI on system performance, by applying static rules onResource Block (RB) usage and power allocation between cellzones. Heuristic Inter-Cell Interference Coordination (ICIC)techniques are proposed to achieve ICI mitigation withoutsevere degradation of the overall system throughput. In [5],a heuristic power allocation algorithm is introduced to reduceenergy consumption and to improve cell-edge UEs throughput. It has been proven that the proposed algorithm reducespower consumption without reducing the achievable throughput. Moreover, it mitigates ICI and increases the achievablethroughput for cell-edge UEs.Beside heuristic resource and power allocation algorithms[6], convex optimization is used to improve the performanceof multiuser OFDMA networks, and to alleviate the negativeimpact of ICI on UE throughput. Resource and power allocation problems are usually formulated as nonlinear optimizationproblems, where the objective consists in maximizing systemthroughput, spectral efficiency, or energy efficiency, with constraints on the minimum throughput per UE or other Qualityof Service (QoS) parameters [7].The majority of state-of-the-art contributions formulate theresource and power allocation problem for a single cellnetwork [8–10]. Moreover, low-complexity suboptimal algorithms are proposed to perform resource and power allocation[10]. Therefore, the optimal solution is not always guaranteed.In this paper, we formulate the joint resource and powerallocation problem for multiuser OFDMA networks, as acentralized optimization problem. We demonstrate that theoriginal problem is separable into two independent optimization problems: a resource allocation problem and a powerallocation problem. Our objective is to maximize systemthroughput while guaranteeing proportional fair rate amongthe UEs, under constraints related to resource usage, Signalto-Interference and Noise Ratio (SINR), and power allocation.Our major contributions are summarized as follows: Propose an original formulation of the centralized jointresource and power allocation problem: instead of considering a single cell OFDMA network, we formulate ourproblem for a multi-cell OFDMA network. Moreover, ICIproblems are taken into account. Maximize the mean rate per UE, and ensure a proportional fair rate for all the active UEs. Prove the convexity of our centralized problem by applying an adequate variable change. Decompose the joint resource and power allocation problem into two independent problems. Solve the centralized power allocation problem using Lagrange duality theory and subgradient projection method. Validate the convergence of our proposed approach andevaluate its performance in comparison with the fre-

quency reuse-1 model, reuse-3 model, FFR, and SFRtechniques.The remainder of this paper is organized as follows. Insection II, we describe the limitations of the existing stateof-the-art approaches. In section III, system model is presented followed by our joint resource and power allocationproblem formulation. The joint problem is decomposed intotwo independent problems in section IV: a resource allocationproblem and a power allocation problem. We also demonstratethe convexity of the formulated problems. In section V, wesolve both resource and power allocation problems. Then weinvestigate the convergence of the centralized approach insection VI, where we also provide comparisons with stateof-the-art ICIC approaches. Section VII concludes this paperand summarizes our main contributions.II. R ELATED W ORKFor a given multiuser OFDMA network, resource and powerallocation problem is formulated as a centralized optimization problem. Centralized inter-cell coordination is thereforerequired to find the optimal solution, where the necessaryinformation about SINR, power allocation, and resource usageare sent to a centralized coordination entity.In [11], the multi-cell optimization problem is decomposedinto two distributed optimization problems. The objectiveof the first problem is to minimize the transmission powerallocated for cell-edge UEs, while guaranteeing a minimumthroughput for each UE. RB and power are allocated to celledge UEs so that they satisfy their minimum required throughput. The remaining RBs and the remaining transmission powerare uniformly allocated to cell-center UEs. At this stage, thesecond problem finds the resource allocation strategy thatmaximizes cell-center zone throughput. An improved versionof this adaptive ICIC technique is proposed in [12], whereresource allocation for cell-edge UEs is performed dependingon their individual channel conditions. However, the maindisadvantage of this adaptive ICIC technique and the proposedimprovement is that they do not consider the impact of ICIbetween adjacent cells when power allocation is performed.Resource and power allocation for a cluster of coordinatedOFDMA cells are studied in [13]. Energy efficiency is maximized under constraints related to the downlink transmissionpower. However, noise-limited regime is considered, and ICIis neglected. Moreover, energy-efficient resource allocation forOFDMA systems is investigated in [14], where generalizedand individual energy efficiencies are defined for the downlinkand the uplink of the OFDMA system, respectively. Propertiesof the energy efficiency objective function are studied, then alow-complexity suboptimal algorithm is introduced to reducethe computational burden of the optimal solution. Subcarrierassignment is made easier using heuristic algorithms. Authorsof [15] consider the joint resource allocation, power allocation,and Modulation and Coding Scheme (MCS) selection problem.The joint optimization problem is separated into resourceallocation and power allocation problems, and suboptimalalgorithms are proposed. Another low complexity suboptimalresource allocation algorithm is proposed in [16]. The objective consists in maximizing the achievable throughput, underconstraints related to resource usage in the different cells.Cooperation between adjacent cells is needed.The majority of state-of-the-art contributions that formulatespectral efficiency or energy efficiency problems as centralizedoptimization problems, neglect the impact of ICI on system performance [8–10], or introduce suboptimal approachesto solve resource and power allocation problems [17–19].Moreover, performance comparisons are not made with otherdistributed heuristic ICIC algorithms with a lower complexity.In the next section, we formulate our multi-cell resource andpower allocation problem that takes inter-cell interference intoaccount.III. S YSTEM M ODEL AND P ROBLEM F ORMULATIONA. System ModelWe consider the downlink of a multiuser OFDMA systemthat consists of I adjacent cells and K active UEs. Let I {1, 2, ., I} denote the the set of cells, and K {1, 2, ., K}the total set of active UEs. We also define K(i)PI as the numberof UEs served by cell i. Thus, we havei 1 K(i) K.The set of available RBs in each cell is denoted by N {1, 2, ., N }.In OFDMA networks, system spectrum is divided intoseveral channels, where each channel consists of a number ofconsecutive orthogonal OFDM subcarriers [20]. An RB is thesmallest scheduling unit. It consists of 12 consecutive subcarriers in the frequency domain, and seven OFDM symbols withnormal cyclic prefix in the time domain [21] (or six OFDMsymbols with extended cyclic prefix). Resources are allocatedto UEs each Transmit Time Interval (TTI), which is equal to1 ms. When the frequency reuse-1 model is applied along withhomogeneous power allocation, each RB is allocated the samedownlink transmission power PmaxN , where Pmax denotes themaximum downlink transmission power per cell. The signalto interference and noise ratio for a UE k attached to cell iand allocated RB n is given by:σk,i,n N0 π GPi,n k,i,n,00i0 6 i πi ,n Gk,i ,n(1)where πi,n is the downlink transmission power allocated bycell i to RB n, Gk,i,n denotes channel gain for UE k attachedto cell i and allocated RB n, and N0 is the thermal noisepower. Indexes i and i0 refer to useful and interfering signalsrespectively. Notations, symbols, parameters, and variablesused within this paper are reported in Table I.B. Problem Formulation1) Centralized Multi-Cell Optimization Problem: We define θk,n as the percentage of time during which UE k isassociated with RB n. θk,n and πi,n are the optimizationvariables of the joint resource and power allocation problem.Our objective is to manage resource and power allocation ina manner that maximizes system throughput and guarantees

TABLE I: Sets, parameters and variables in the xπminI 0 (i)minimum power constraint of the transmit power allocated toeach RB. θk,n , k K, n N , and πi,n , i I, n Nare the optimization variables of the joint resource and powerallocation problem.Index of cellIndex of UEIndex of RBSet of cellsTotal set of UEsSet of UEs associated to cell iSet of RBsRate of UE k associated with RB n on cell iTransmit power of cell i on RB nChannel gain for UE k over RB n on cell iThermal noise densityPercentage of time RB n is allocated to UE kSINR for UE k over RB n on cell iMaximum DL transmission power per cellMinimum DL transmission power per RBSet of neighboring cells for cell i2) Upper Bound of the Objective Functions Difference: Inorder to reduce the complexity of the joint resource and powerallocation problem (4), we prove that this problem is separableinto two independent problems: a resource allocation problemand a power allocation problem. Given Jensen’s inequality andthe concavity of the log function, we have:P Plog (θk,n .ρk,i,n )n N θk,n .ρk,i,n n Nlog N N (5a)! logXθk,n .ρk,i,nn NP n Nlog (θk,n .ρk,i,n ) N log ( N ) ,(5b)1 N throughput fairness between the different UEs. The peak rateof UE k when associated with RB n on cell i is given by:!πi,n Gk,i,nPρk,i,n log 1 .(2)N0 i0 6 i πi0 ,n Gk,i0 ,nThen, the mean rate of UE k is given by:X(θk,n .ρk,i,n ) .Sinceand K . log ( N ) are constant terms, maximizingthe objective function of problem (4) is achieved by maximizing the following term:X X X(log (θk,n ) log (ρk,i,n )) .(6)i I k K(i) n NIV. P ROBLEM D ECOMPOSITION(3)We tackle ICIC as an optimization problem, where weintendto maximize the mean rate of UEs in a multiuserOur centralized resource and power allocation problem seeksOFDMAsystem. We consider a system of I cells, having K(i)rate maximization. We make use of the logarithmic functionUEspercelli. According to (6), and due to the absence ofthat is intimately associated with the concept of proportionalbindingconstraints,the optimization problem (4) is linearlyfairness [22]. Our problem is formulated as follows:separable into two independent problems: a power allocationmaximize η problem and a resource allocation problem.θ,π!!X XXπ GPi,n k,i,nlogθk,n . log 1 N0 i0 6 i πi0 ,n Gk,i0 ,ni I k K(i)n NA. Centralized Multi-Cell Power Allocation Problem(4a)Xsubject toθk,n 1, n N ,(4b)In the first problem, the optimization variable π is considk K(i)ered, and the problem is formulated as follows:Xθk,n 1, k K(i),(4c)maximize η1 πn N!!XX X Xπi,n Gk,i,nπi,n Pmax , i I,(4d)Plog log 1 N0 i0 6 i πi0 ,n Gk,i0 ,nn Ni I k K(i) n Nπi,n πmin , i I, n N ,(4e)(7a)X0 θk,n 1, k K(i), n N .(4f)subject toπi,n Pmax , i I,(7b)n NThe objective function η ensures a proportional fair rate for allπi,n πmin , i I, n N .(7c)UEs in the network. Constraints (4b) ensure that an RB is usedat most 100% of the time, and constraints (4c) ensure that a UEProblem (7) consists in finding the optimal power allocation.shares its time on the available RBs. Constraints (4d) guaranteethat the total downlink transmission power allocated to the In the following, we introduce a variable change that allowsavailable RBs does not exceed the maximum transmission to formulate problem (7) as a convex optimization problem.The power allocation problem (7) can be written as follows:power Pmax for each cell i, and constraints (4e) represent then N

η1 maximizeρX X Xlog (ρk,i,n )(8a)i I k K(i) n Nsubject to ρk,i,nπ GPi,n k,i,n log 1 N0 i0 6 i πi0 ,n Gk,i0 ,n! i I, k K(i), n N ,Xπi,n Pmax , i I,,B. Centralized Resource Allocation Problem(8b)(8c)n Nπi,n πmin , i I, n N .b,The objective function of problem (10) is concave in ρand constraints (10b), (10c), and (10d) are convex functions.Thus, the power allocation problem is a convex optimizationproblem.(8d)Let us consider the following variable change:The optimization variable θ is considered in the secondoptimization problem that is given in the following:X X Xmaximizeη2 log (θk,n )(11a)θsubject toi I k K(i) n NXθk,n 1, n N ,(11b)θk,n 1, k K(i),(11c)ρbk,i,n log (exp (ρk,i,n ) 1) , i I, k K(i), n N ,(9a)k K(i)πbi,n log(πi,n ), i I, n N .n NX(9b)To show that the optimization problem (8) is a convex optimization problem, we need to show that the objective functionis concave and the inequality constraint functions define aconvex set. After applying the variable change on UE peak rateconstraints (8b), these constraints can be written as follows:N0 Gk,i,nXGk,i0 ,n) 0,exp (bρk.i.n πbi0 ,n πbi,n )Gk,i,n0log(exp (bρk.i.n πbi,n )i 6 iwhich are the logarithmic of the sum of exponential functions.Therefore, they are convex functions [23]. When we applythe variable change on power constraints (8c), we get thefollowing:Xπi,n Pmax , i I0 θk,n 1, k K(i), n N .As demonstrated for the power allocation problem (7),we prove that problem (11) is indeed a convex optimizationproblem in θ. The objective function (11a) of the resourceallocation problem (11) is concave in θ, since the log functionis concave for θ ]0; 1]. Moreover, constraints (11b), (11c),and (11d) are linear and separable constraints. Hence, theresource allocation problem (11) is a convex optimizationproblem, and it is separable into I subproblems. For eachcell i, the ith optimization problem is written as follows:X Xmaximize(η2 )i log (θk,n )(12a)θsubject tok K(i) n NXexp (bπi,n ) log (Pmax ) 0, i I.n NPSince log( exp) is convex [23], the constraints at handare therefore convex. Using the variable change, the powerallocation problem (8) can be written as follows:X X Xmaximize η1 log (log (exp (bρk,i,n ) 1))bρsubject toi I k K(i) n N(10a)N0log(exp (bρk.i.n πbi,n ) Gk,i,nXGk,i0 ,nexp (bρk.i.n πbi0 ,n πbi,n )) 0,Gk,i,n0i 6 i(12b)Xθk,n 1, k K(i),(12c)n N!Xθk,n 1, n N ,k K(i)n N log(11d)0 θk,n 1, k K(i), n N .(12d)V. C ENTRALIZED M ULTI -C ELL R ESOURCE AND P OWERA LLOCATIONAs proven in the previous section, the joint resource andpower allocation problem (4) is separable into two independentconvex optimization problems: a power allocation problem,and a resource allocation problem. In this section, we solvethe resource and power allocation problems using Lagrangeduality theory and subgradient projection method.A. Solving the Centralized Power Allocation Problem1) Lagrange-Based Method: Since the power allocation i I, k K(i), n N ,(10b) problem (10) is a convex optimization problem, we can!Xmake use of Lagrange duality properties, which also leadlogexp (bπi,n ) log (Pmax ) 0, i I, to decomposability structures [24]. Lagrange duality theoryn Nlinks the original problem, or primal problem, with a dual(10c)maximization problem. The primal problem (10) is relaxedπbi,n log (πmin ) , i I, n N .(10d) by transferring the constraints to the objective in the formof weighted sum. The Lagrangian is formed by relaxing the

coupling constraints (10b) and (10c) in problem (10):X X Xb , λ, ν) L (bρ, πlog (log (exp (bρk,i,n ) 1))i I k K(i) n N X X Xλk,i,n (log(exp(bρk.i.n πbi,n )i I k K(i) n N Xexp(bρk.i.n πbi0 ,n πbi,n )i0 Ni0 6 iGk,i0 ,n))Gk,i,n! Xνilogi IXexp (bπi,n )N0Gk,i,n! log (Pmax ) .n N(13)b and πb are called the primalThe optimization variables ρvariables. λk,i,n and νi are the Lagrange multipliers or pricesassociated with the (k, i, n)th inequality constraint (10b) andwith the ith inequality constraint (10c), respectively. λ and νare also termed the dual variables.After relaxing the coupling constraints, the optimizationproblem separates into two levels of optimization: lower levelb , λ, ν) is theand higher level. At the lower level, L(bρ, πobjective function to be maximized. ρbk,i,n and πbi,n are theoptimization variables to be found, and the primal problem isgiven by:maximizeb , λ, ν)L (bρ, π(14a)subject toπbi,n log(πmin ), i I, n N .(14b)b,bρπIn order to solve the primal optimization problem (14), weuse the subgradient projection method. It starts with someinitial feasible values of ρbk,i,n and πbi,n that satisfy the constraints (14b). Then, the next iteration is generated by takinga step along the subgradient direction of ρbk,i,n and πbi,n . Forthe primal optimization variables, iterations of the subgradientprojection are given by: L, ρbk,i,n k K(i), i I, n N ,(15a) Lπbi,n (t 1) πbi,n (t) δ(t) , i I, n N . bπi,n(15b)ρbk,i,n (t 1) ρbk,i,n (t) δ(t) The scalar δ(t) is a step size that guarantees the convergence ofthe optimization problem (14). At the higher level, we have themaster dual problem in charge of updating the dual variables λand ν by solving the dual problem:minimizeb , λ, ν))max (L (bρ, π(16a)subject toλ 0,(16b)ν 0.(16c)λ,νb,bρπb , λ, ν)) is differenThe dual function g (λ, ν) max (L (bρ, πb,bρπtiable. Thus, the master dual problem (16) can be solved usingthe following gradient method: L, λk,i,n k K(i), i I, n N ,(17a) Lνi (t 1) νi (t) δ(t) , i I, n N , (17b) νiλk,i,n (t 1) λk,i,n (t) δ(t) where t is the iteration index, and δ(t) is the step size atiteration t. Appropriate choice of the step size [25] leads to?convergence of the dual algorithm. πbi,nand ρb?k,i,n denotethe solution to the primal optimization problem (14). Whent the dual variables λ(t) and ν(t) converge to thedual optimal λ and ν , respectively. The difference betweenthe optimal primal objective and the optimal dual objective,called duality gap, reduces to zero at optimality, since theproblem (10) is convex and the KKT conditions are satisfied.We define bρ, bπ , λ, and ν as the differences betweenthe optimization variables obtained at the current iteration andtheir values at the previous iteration. They are given by:b(t)k, bρ(t 1) kbρ(t 1) ρb (t)k, bπ (t 1) kbπ (t 1) π(18a)(18b) λ(t 1) kλ(t 1) λ(t)k,(18c) ν(t 1) kν(t 1) ν(t)k.(18d)2) Iterative Power Allocation Algorithm: The procedurefor solving the centralized power allocation problem is described in Algorithm 1. Initially, the primal optimization variables ρbk,i,n and πbi,n as well as the dual variables λk,i,n and νistart with some initial feasible values. t, tprimal , and tdualdenote the number of rounds required for the centralized powerallocation problem to converge, the number of iterations forthe primal problem, and the number of iterations for the dualproblem, respectively. At each round t, we start by updatingthe primal optimization variables, using the P RIMAL P ROBLEMfunction given in Algorithm 2. The solution to the primaloptimization problem at the current round t is denoted by?πbi,n(t 1) and ρb?k,i,n (t 1). The P RIMAL P ROBLEM functionupdates πbi,n (tprimal 1) and ρbk,i,n (tprimal 1), and increments tprimal until bπ (tprimal 1) and bρ(tprimal 1)become less than .Then, the solution to the dual optimization problem at thecurrent round t, denoted by νi? (t 1) and λ?k,i,n (t 1)is calculated using the D UAL P ROBLEM function given inAlgorithm 3. νi and λk,i,n are updated using the primal?solution πbi,n(t 1) and ρb?k,i,n (t 1), until ν(tdual 1)and λ(tdual 1) become less than . An additional roundof calculations is performed, and t is incremented as long as bπ ? (t 1) or bπ ? (t 1) or ν ? (t 1) or λ? (t 1) isgreater than . Otherwise, the current solution is the optimalsolution to the centralized power allocation problem.B. Solving the Resource Allocation ProblemIn this subsection, we search for the optimal solution tothe resource allocation problem (12). For each cell i, theproblem (12) is a convex optimization problem.

Algorithm 1 Dual algorithm for centralized power allocation1:2:b , λ, ν), Pmax , and πmin .Parameters: L(bρ, πInitialization: set t tprimal Ptdual 0, and πi,n πmin , i I, n N , such asπi,n Pmax , i I.n NCalculate πbi,n (0) and ρbk,i,n (0) accordingly.3: Set λk,i,n (0) and νi (0) equal to some non negative value.b? (t 1)) P RIMAL P ROBLEM(ν ? (t), λ? (t))4: (bπ ? (t 1), ρ5:6:7:8:9:(ν ? (t 1), λ? (t 1)) D UAL P ROBLEM(bπ ? (t ?b (t 1))1), ρif ( bπ ? (t 1) ) or ( bρ? (t 1) ) or ( ν ? (t 1) ? ) or ( λ (t 1) ) thent t 1go to 4end ifAlgorithm 2 Primal problem on P RIMAL P ROBLEM(ν ? (t), λ? (t))for i 1 to I dofor n 1 to N doπbi,n (tprimal 1) max log (πmin ) ; πbi,n (tprimal ) δ (t) b Lπi,nfor k 1 to K(i) doρbk,i,n (tprimal 1) ρbk,i,n (tprimal ) δ(t) ρb Lk,i,nend forend forend forif ( bπ (tprimal 1) ) or ( bρ(tprimal 1) )thentprimal tprimal 1go to 2end ifb (tprimal 1), ρb(tprimal 1)return πend functionAlgorithm 3 Dual problem function1:2:3:4:5:6:7:8:9:10:11:12:13:14:15:b? (t 1))function D UAL P ROBLEM(bπ ? (t 1), ρfor i 1 to I do Lνi (tdual 1) max(0; νi (tdual ) δ(t) ν)ifor n 1 to N dofor k 1 to K(i) doλk,i,n (tdual 1) max(0; λk,i,n (tdual ) δ(t) λ L)k,i,nend forend forend forif ( ν(tdual 1) ) or ( λ(tdual 1) ) thentdual tdual 1go to 2end ifreturn ν(tdual 1), λ(tdual 1)end functionnetworks, that consists of seven adjacent hexagonal cells, withone UE served by each cell. UE positions and radio conditionsare randomly generated, and the initial power allocation foreach RB equals πmin . System bandwidth equals 5 MHz. Thus,25 RBs are available in each cell. The maximum transmissionpower per cell Pmax is set to 43 dBm or 20 W. At the firstiteration, the dual variables λk,i,n (0), k K(i), i I, n N , and νi (0), i I, are assigned initial positive values. Theevolution of πbi,1 along with the number of iterations is shownin Fig. 1, where πbi,1 is the logarithm of the transmission powerallocated by the cell i to the RB 1. In addition, the number ofprimal iterations and the number of dual iterations per roundare shown in Fig. 2. 2.1 2.15VI. P ERFORMANCE E VALUATIONIn this section, we evaluate the convergence and the performance of our proposed centralized joint resource and powerallocation problem.A. Centralized Resource and Power AllocationTo verify the convergence of the centralized solution, weconsider a multi-user OFDMA network, such as LTE/LTE-A 2.2πbi,nTheorem 5.1: For each cell i, the optimal solution tothe resource allocation problem (12) is given by: θk,n 1max( K(i) , N ) , k K(i), n N .The proof of Theorem 5.1 is given in Appendix I. Whenthe number of active UEs is less than the number of availableresources, θk,n N1 , k K(i), n N . Thus, theavailable resources are not fully used over time, and each UEis permanently served. Otherwise, when K(i) N , the1optimal solution is: θk,n K(i) , k K(i), n N . Inthis case, each RB is fully used over time, while UEs are notpermanently served over time.πb1,1πb2,1πb3,1πb4,1πb5,1πb6,1πb7,1 2.254 2.25 2.256 2.258 2.3 2.26 2.350 2.26211201000114011601180200030004000Number of Iterations1200500012206000Fig. 1: Primal variables πbi,nWe notice that for the centralized power allocation approach,the primal problem requires approximately 6000 iterations toconverge. As shown in Fig. 2, 1100 rounds are required toreach the optimal values of the primal and the dual variables.The zoomed box within Fig. 1 shows πbi,n versus the numberof primal iterations for a given round t. The values of πbi,n arecalculated using the dual variables obtained at the round (t

12020Primal problemDual 019.73919.860νiNumber of 83028402850286028704019.62000200400600800Round Number1000120019.50100020003000 4000 5000 6000Number of Iterations70008000Fig. 2: Primal and dual iterations per roundFig. 4: Lagrange prices νi1). We also notice that the number of primal iterations and thenumber of dual iterations decreases with the number of rounds.When t increases, the impact of Lagrange prices λk,i,n (t) andνi (t) on the primal variables calculation is reduced, and thenumber of primal iterations required for convergence becomeslower. The same behavior is noticed for the number of dualiterations when the number of rounds increases.difference between the updated primal variables at round t andtheir values at round (t 1) is less than ,5,1λ1,6,1λ1,7,10.60.550.50100020003000 4000 5000 6000Number of Iterations70008000Fig. 3: Lagrange prices λk,i,nFor the same simulated scenario, we also show the dualvariables λk,i,n and νi versus the number of dual iterations inFig. 3 and Fig. 4, respectively. We notice that approximately8000 iterations are required for the dual problem to converge.At a given round t, the Lagrange prices λk,i,n and νi areupdated using the most recent values of the primal variables.The zoomed boxes within Fig. 3 and Fig. 4 show the evolutionof λk,i,n and νi versus the number of iterations, respectively.These values are updated until λk,i,n and νi becomeless than . Convergence of the centralized power allocationproblem occurs when two conditions are satisfied: first, thedifference between the updated primal variables at round tand their values at round (t 1) is less than . Second, theB. Comparison with State-of-the-Art ApproachesWe also compare the performance of our proposed centralized resource and power allocation approach with that of stateof-the-art resource and power allocation approaches such asthe frequency reuse-1 model, the frequency reuse-3 model,FFR, and SFR techniques [26]. The frequency reuse-1 modelallows the usage of the same frequency spectrum simultaneously in all the network cells. Moreover, homogeneous powerallocation is performed.In the frequency reuse-3 model, one third of the availablespectrum is used in each cell in a cluster of three adjacentcells. Interference problems are eliminated, but the spectralefficiency is reduced. FFR and SFR techniques divide each cellinto a cell-center and a cell-edge zones, and set restrictionson resource usage and power allocation in each zone. Forall the compared techniques, resource allocation is performedaccording to Theorem 5.1.1) System Throughput: For several simulation runs, weshow the total system throughput for our proposed centralizedresource and power allocation approach, for the frequencyreuse-1 model, reuse-3 model, FFR, and SFR techniques underthe same simulation scenarios. Simulation results, includingthe 95% confidence interval, are illustrated in Fig. 5.It is shown that the centralized resource allocation approachoff

resource and power allocation problem for a single cell network [8-10]. Moreover, low-complexity suboptimal algo-rithms are proposed to perform resource and power allocation [10]. Therefore, the optimal solution is not always guaranteed. In this paper, we formulate the joint resource and power allocation problem for multiuser OFDMA networks, as a

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