Centralized Versus Decentralized Multi-Cell Resource And Power .
Centralized versus Decentralized Multi-Cell Resourceand Power Allocation for Multiuser OFDMA NetworksMohamad Yassina,b, , Samer Lahouda , Kinda Khawamc , Marc Ibrahimb ,Dany Mezherb , Bernard Cousinaa Universityof Rennes 1, IRISA, Campus de Beaulieu, 35042 Rennes, FranceJoseph University of Beirut, ESIB, CST, Mar Roukoz, Lebanonc University of Versailles, 45 Av. des Etats-Unis, 78035 Versailles, Franceb SaintAbstractThe exponential growth in the usage of mobile networks along with the increasing number of User Equipments (UEs) are exacerbating the scarcity offrequency resources. Dense frequency reuse on the downlink of multiuser Orthogonal Frequency Division Multiple Access networks leads to severe Inter-CellInterference (ICI) problems. Resource and power allocation techniques are required to alleviate the harmful impact of ICI. Contrarily to the existing techniques that consider single-cell resource and power allocation problem withouttaking ICI into account, we formulate a centralized downlink multi-cell jointresource and power allocation problem. The objective is to maximize systemthroughput while guaranteeing throughput fairness between UEs. We demonstrate that the joint problem is separable into two independent problems: aresource allocation problem and a power allocation problem. Lagrange duality theory is used to solve the centralized power allocation problem. We alsotackle the resource and power allocation problem differently by addressing it ina decentralized manner. We propose a non-cooperative downlink power allocation approach based on game theory. The players are the base stations, andeach base station seeks to maximize its own utility function. We investigate theconvergence of our proposed centralized and decentralized approaches, and we CorrespondingauthorEmail address: mohamad.yassin@usj.edu.lb (Mohamad Yassin)Preprint submitted to Elsevier Computer CommunicationsFebruary 27, 2017
compare their performance with that of state-of-the-art approaches.Keywords: Convex optimization, resource and power allocation, inter-cellinterference, ICIC, OFDMA1. IntroductionMultiuser Orthogonal Frequency Division Multiple Access (OFDMA) networks, such as the Third Generation Partnership Project (3GPP) Long TermEvolution (LTE) [1] and LTE-Advanced (LTE-A) [2] networks, are able to avoid5the negative impact of multipath fading and intra-cell interference, by virtue ofthe orthogonality between subcarrier frequencies. Nevertheless, Inter-Cell Interference (ICI) problems arise on the downlink of dense frequency reuse networksdue to simultaneous transmissions on the same frequency resources. Systemperformance is interference-limited, since the achievable throughput is reduced10due to ICI.Fractional Frequency Reuse (FFR) [3] and Soft Frequency Reuse (SFR) [4]were introduced to avoid the harmful impact of ICI on system performance,by applying static rules on Resource Block (RB) usage and power allocationbetween cell-center and cell-edge users. Heuristic Inter-Cell Interference Co-15ordination (ICIC) techniques are proposed to achieve ICI mitigation withoutsevere degradation of the overall system throughput. For instance, authorsof [5, 6] propose suboptimal solution for the resource allocation problem. Theobjective is to minimize ICI by exploiting User Equipment (UE) diversity tomaximize system throughput. They propose a two-level algorithm that oper-20ates at the evolved-NodeBs (eNodeBs) and at a central controller connected toseveral eNodeBs. In [7], a heuristic power allocation algorithm is introduced toreduce energy consumption and to improve cell-edge UEs throughput. It hasbeen proven that the proposed algorithm reduces power consumption withoutreducing the achievable throughput. Moreover, it mitigates ICI and increases25the achievable throughput for cell-edge UEs.Beside heuristic resource and power allocation algorithms [8], convex opti-2
mization is used to improve the performance of multiuser OFDMA networks,and to alleviate the negative impact of ICI on UE throughput. Resource andpower allocation problem is usually formulated as nonlinear optimization prob30lem, where the objective consists in maximizing system throughput, spectralefficiency, or energy efficiency, with constraints on the minimum throughput perUE or other Quality of Service (QoS) parameters [9]. The exponential growthin the usage of mobile networks along with the increasing number of UEs areexacerbating the scarcity of frequency resources.35The majority of state-of-the-art contributions formulate the resource andpower allocation problem for a single cell network [10, 11, 12], or do not consider the impact of ICI on system performance. For instance, the tradeoffbetween spectral efficiency and energy efficiency is addressed in [12], and alow-complexity suboptimal algorithm is proposed to allocate RBs for practical40applications of the tradeoff. However, the system model consists of a single cellOFDMA network, where one subcarrier is assigned to at most one UE. Therefore, ICI problems are not considered. In this article, we formulate the jointresource and power allocation problem for the downlink of multiuser OFDMAnetworks, as a centralized multi-cell optimization problem. Inter-cell interfer-45ence is taken into account, and throughput fairness between the different users isguaranteed. We prove that our joint problem is separable into two independentoptimization problems: a resource allocation problem and a power allocationproblem. Our objective is to maximize system throughput, while satisfyingconstraints related to resource usage, Signal-to-Interference and Noise Ratio50(SINR), and power allocation. We also propose a decentralized power allocation approach that does not rely on centralized controllers. Each base stationmaximizes its own utility function in a distributed manner. We evaluate theperformance of the proposed approaches, and we compare their performancewith state-of-the-art resource and power allocation approaches.55The remainder of this article is organized as follows. In section II, we describe the limitations of the existing state-of-the-art approaches. In section III,system model is presented followed by our joint resource and power allocation3
problem formulation. The joint problem is decomposed into two independentproblems in section IV: a resource allocation problem and a power allocation60problem. We also demonstrate the convexity of the formulated problems. Insection V, we solve both resource and power allocation problems using the Lagrange duality theory. Our decentralized power allocation approach is introduced in section VI. Then, we investigate the convergence of the centralizedand the decentralized approaches in section VII, where we also provide compar-65isons with other approaches. Section VIII concludes this article and summarizesour main contributions.2. Related Work2.1. State-of-the-Art ContributionsFor a given multiuser OFDMA network, resource and power allocation prob70lem is formulated as a centralized optimization problem [10, 11, 12]. Centralizedinter-cell coordination is therefore required to find the optimal solution, wherethe necessary information about SINR, power allocation, and resource usage aresent to a centralized coordination entity.In [13], the multi-cell optimization problem is decomposed into two dis-75tributed optimization problems. The objective of the first problem is to minimize the transmission power allocated for cell-edge UEs, while guaranteeing aminimum throughput for each UE. RB and power are allocated to cell-edge UEsso that they satisfy their minimum required throughput. The remaining RBsand the remaining transmission power are uniformly allocated to cell-center80UEs. At this stage, the second problem aims at finding the resource allocation strategy that maximizes the achievable throughput for cell-center UEs. Animproved version of this adaptive ICIC technique is proposed in [14], where resource allocation for cell-edge UEs is performed depending on their individualchannel conditions. However, the main disadvantage of this adaptive ICIC tech-85nique and the proposed improvement is that they do not consider the impact ofICI between adjacent cells when power allocation is performed. Each cell solves4
its own optimization problem without requesting additional information fromits neighboring cells.Resource and power allocation for a cluster of coordinated OFDMA cells90are studied in [15]. Energy efficiency is maximized under constraints related tothe downlink transmission power. However, noise-limited regime is considered,and ICI is neglected. Moreover, energy-efficient resource allocation for OFDMAsystems is investigated in [16], where generalized and individual energy efficiencies are defined for the downlink and the uplink of the OFDMA system,95respectively. Properties of the energy efficiency objective function are studied,then a low-complexity suboptimal algorithm is introduced to reduce the computational burden of the optimal solution. Subcarrier assignment is made easierusing heuristic algorithms. Authors of [17] consider the joint resource allocation, power allocation, and Modulation and Coding Scheme (MCS) selection100problem. The joint optimization problem is separated into resource allocationand power allocation problems, and suboptimal algorithms are proposed. Another low complexity suboptimal resource allocation algorithm is proposed in[18]. The objective consists in maximizing the achievable throughput, underconstraints related to resource usage in the different cells. Cooperation between105adjacent cells is needed. A multi-cell resource allocation approach for OFDMAsystems with decode-and-forward relaying is proposed in [19], where an interference constraint is introduced along with time sharing variables. Althoughthis approach guarantees throughput fairness between the different users, thespectral efficiency is reduced since the cells are not allowed to use the available110spectrum during 100% of the time due to time sharing between base stationsand relays.Minimizing energy consumption and maximizing spectral efficiency in multiuser OFDMA networks cannot be achieved simultaneously. Energy-bandwidthtradeoff is studied in [20], where authors consider the total energy consumption115versus the end-to-end rate in wireless multihop networks. For an arbitrary placement of wireless nodes, resource and power allocation that minimizes the energylevel required to achieve a given data rate is found. However, interference-free5
resource allocation is considered, and the impact of ICI on system performanceis not taken into account.1202.2. Our ContributionsThe majority of state-of-the-art contributions that formulate spectral efficiency or energy efficiency problems as centralized optimization problems, neglect the impact of ICI on system performance [10, 11, 12], or introduce suboptimal approaches to solve resource and power allocation problems [21, 22, 23].125Moreover, performance comparisons are not made with other distributed heuristic ICIC algorithms, that are usually characterized by a lower computationalcomplexity. In our work, we consider the multi-cell downlink resource and powerallocation problem, where the objective is to maximize system throughput whileguaranteeing throughput fairness between the different UEs. Moreover, ICI is130taken into account when solving the centralized resource and power allocationproblem. We also formulate a decentralized non-cooperative power allocationapproach based on game theory. The players are the cells, and each cell seeksmaximizing its own utility function independently of the other cells in the network. We investigate the convergence of both centralized and decentralized ap-135proaches, and we compare their performance with that of the frequency reuse-1model, the frequency reuse-3 model, FFR, and SFR techniques. Our majorcontributions are summarized as follows: Propose an original formulation of the centralized joint resource and powerallocation problem: instead of considering a single cell OFDMA network,140we formulate our problem for a multi-cell OFDMA network, taking ICIproblems into account. The objective is to maximize the mean rate perUE, and ensure a proportional fair rate for all the active UEs. Decompose the joint downlink resource and power allocation probleminto two independent problems, and solve the centralized power alloca-145tion problem using Lagrange duality theory and subgradient projectionmethod.6
Formulate a novel decentralized super-modular game for resource andpower allocation, and propose a best response algorithm to attain theNash Equilibrium. Then, solve the decentralized power allocation prob150lem using subgradient projection method. Validate the convergence of the proposed centralized and decentralizedapproaches and evaluate their performance in comparison with broadlyadopted state-of-the-art approaches.3. System Model and Problem Formulation1553.1. System ModelWe consider the downlink of a multiuser OFDMA system that consists of Iadjacent 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) as thePInumber of UEs served by cell i. Thus, we have i 1 K(i) K. The set of160available RBs in each cell is denoted by N {1, 2, ., N }.In OFDMA networks, system spectrum is divided into several channels,where each channel consists of a number of consecutive orthogonal OFDM subcarriers [24]. An RB is the smallest scheduling unit. It consists of 12 consecutive subcarriers in the frequency domain, and seven OFDM symbols withnormal cyclic prefix in the time domain [25] (or six OFDM symbols with extended cyclic prefix). Frequency resources are allocated to UEs each TransmitTime Interval (TTI), which is equal to 1 ms. When the frequency reuse-1 modelis applied along with homogeneous power allocation, each RB is allocated thesame downlink transmission powerPmaxN ,where Pmax denotes the maximumdownlink transmission power per cell. The signal to interference and noise ratiofor a UE k attached to cell i and allocated RB n is given by:σk,i,n N0 π GPi,n k,i,n,i0 6 i πi0 ,n Gk,i0 ,n7(1)
where πi,n is the downlink transmission power allocated by cell i to RB n, Gk,i,ndenotes channel gain for UE k attached to cell i and allocated RB n, and N0 isthe thermal noise power. Indexes i and i0 refer to useful and interfering signalsrespectively. In our work, we rely on perfect channel state information to infer165the SINR. Authors of [26] provide models to account for imperfect channel stateand study the impact on energy efficiency. Notations, symbols, parameters, andvariables used within this document are reported in Table 1.Table 1: Sets, parameters and variables in the nPmaxπminI 0 (i)Index of cellIndex of UEIndex of RBSet of cellsTotal set of UEsSet of UEs associated to cell iSet of RBsPeak rate 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 UE k is associated with RB nTotal system achievable mean rateSINR for UE k over RB n on cell iMaximum DL transmission power per cellMinimum DL transmission power per RBSet of neighboring cells for cell i3.2. Problem Formulation3.2.1. Centralized Multi-Cell Optimization Problem170We define θk,n as the percentage of time during which UE k is associatedwith RB n. θk,n , k K, n N , and πi,n , i I, n N , are the optimizationvariables of the joint resource and power allocation problem. Our objective isto manage resource and power allocation in a manner that maximizes systemthroughput and guarantees throughput fairness between the different UEs. The175standard approach is to have integer scheduling variables, while in our problemformulation, θk,n and πi,n are continuous variables. In fact, using continuousvariables will decrease the computation time and the complexity of the problem8
without losing generality. A simple way of implementing the solution is toextend the Round-Robin scheduler in a way to allocate equal time shares to the180users in the cell on each RB.The peak rate of UE k when associated with RB n on cell i is given by:ρk,i,nπ GPi,n k,i,n log 1 N0 i0 6 i πi0 ,n Gk,i0 ,n!.(2)!!(3)Then, the mean rate of UE k is given by:X(θk,n .ρk,i,n ) n NXn Nπ GPi,n k,i,nθk,n . log 1 N0 i0 6 i πi0 ,n Gk,i0 ,n.Our centralized multi-cell joint resource and power allocation problem seeks ratemaximization in a proportional fair manner. We make use of the logarithmicfunction that is intimately associated with the concept of proportional fairness[27]. Our problem is formulated in the following:maximizeθ,πX Xi I k K(i)subject toη Xlogn NXπ GPi,n k,i,nθk,n . log 1 N0 i0 6 i πi0 ,n Gk,i0 ,n!!(4a)θk,n 1, n N ,(4b)θk,n 1, k K(i),(4c)πi,n Pmax , i I,(4d)k K(i)Xn NXn Nπi,n πmin , i I, n N ,(4e)0 θk,n 1, k K(i), n N .(4f)The objective function η ensures a proportional fair rate for all UEs in thenetwork. Constraints (4b) ensure that an RB is used at most 100% of the time,9
and constraints (4c) ensure that a UE shares its time on the available RBs.Constraints (4d) guarantee that the total downlink transmission power allocated185to the available RBs does not exceed the maximum transmission power Pmax foreach cell i, and constraints (4e) represent the minimum power constraint of thetransmit power allocated to each RB. In fact, the power allocated to each RB islarger than a predefined value denoted πmin , and the transmit power of cell i islower than Pmax . In practice, these bounds are related to hardware limitations.190θk,n , k K, n N , and πi,n , i I, n N are the optimization variables ofthe joint resource and power allocation problem.In order to reduce the complexity of the joint resource and power allocationproblem (4), we prove that this problem is separable into two independent problems: a resource allocation problem and a power allocation problem. In fact,maximizing the objective function of problem (4) is achieved by maximizing thefollowing term:X X X(log (θk,n ) log (ρk,i,n )) .(5)i I k K(i) n NThe proof of this hypothesis is given in Appendix I.4. Problem DecompositionWe tackle ICIC as an optimization problem, where we intend to maximize195the mean rate of UEs in a multiuser OFDMA system. We consider a system ofI cells, having K(i) UEs per cell i. According to (5), and due to the absence ofbinding constraints, the optimization problem (4) is linearly separable into twoindependent problems: a power allocation problem and a resource allocationproblem.2004.1. Centralized Multi-Cell Power Allocation ProblemIn the first problem, the optimization variable π is considered, and theproblem is formulated as follows:10
maximizeπη1 π GPi,n k,i,nlog log 1 N0 i0 6 i πi0 ,n Gk,i0 ,ni I k K(i) n NXsubject toπi,n Pmax , i I,!!X X X(6a)(6b)n Nπi,n πmin , i I, n N .(6c)Problem (6) consists in finding the optimal power allocation. However, it isnot a convex optimization problem as formulated in (6). In the following, we205introduce a variable change that allows to formulate problem (6) as a convexoptimization problem as follows:X X Xmaximizeη1 log (ρk,i,n )ρ,πsubject to(7a)i I k K(i) n Nρk,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,,(7b)(7c)n Nπi,n πmin , i I, n N .(7d)Let us consider the following variable change:ρbk,i,n log (exp (ρk,i,n ) 1) , i I, k K(i), n N ,(8a)πbi,n log(πi,n ), i I, n N .(8b)Hence, the original variables are given by:ρk,i,n log (exp (bρk,i,n ) 1) , i I, k K(i), n N ,(9a)πi,n exp (bπi,n ) , i I, n N .(9b)To show that the optimization problem (7) is a convex optimization problem, we need to show that the objective function is concave and the inequality210constraint functions define a convex set. After applying the variable change on11
peak rate constraints (7b), they can be written as: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 log(exp(bρk,i,n ) 1) log(1 exp(bρk.i.n ) 1 1 exp(bρk.i.n ).(N0 N0 exp(bπi,n )Gk,i,nP)πi0 ,n )Gk,i0 ,ni0 6 i exp(bexp(bπi,n )Gk,i,nPN0 i0 6 i exp(bπi0 ,n )Gk,i0 ,nPi0 6 iexp(bπi0 ,n )Gk,i0 ,n )exp(bπi,n )Gk,i,n 1 XN0Gk,i0 ,n 0, log exp(bρk.i.n bπi,n ) exp(bρk.i.n bπi0 ,n bπi,n )Gk,i,n 0Gk,i,n i 6 i i I, k K(i), n N .These constraints are the logarithmic of the sum of exponential functions.Thus, they are convex functions [28]. When we apply the variable change onpower constraints (7c), we get:Xπi,n Pmax , i In N! logXexp (bπi,n ) log (Pmax ) 0, i I.n N215PSince log( exp) is convex [28], the constraints at hand are therefore convex.Using the variable change, the power allocation problem (7) can be written asfollows:maximizeb,bρπη1 X X Xlog (log (exp (bρk,i,n ) 1))(10a)i I k K(i) n Nsubject to XN0Gk,i0 ,n log exp(bρk.i.n bπi,n) exp(bρk.i.n bπi0 ,n bπi,n) 0,Gk,i,n 0Gk,i,ni 6 i i I, k K(i), n N ,!Xlogexp (bπi,n ) log (Pmax ) 0, i I,(10b)(10c)n Nπbi,n log (πmin ) , i I, n N .12(10d)
b and πb , and conThe objective function of problem (10) is concave in ρstraints (10b), (10c), and (10d) are convex functions. Thus, the power allocation220problem is a convex optimization problem.4.2. Centralized Resource Allocation ProblemThe optimization variable θ is considered in the second optimization problemthat is given in the following:maximizeθsubject toX X Xη2 log (θk,n )(11a)i I k K(i) n NXθk,n 1, n N ,(11b)θk,n 1, k K(i),(11c)k K(i)Xn N0 θk,n 1, k K(i), n N .(11d)As demonstrated for the power allocation problem (6), we prove that prob225lem (11) is indeed a convex optimization problem in θ. The objective function (11a) of the resource allocation problem (11) is concave in θ, since the logfunction is concave for θ ]0; 1]. Moreover, constraints (11b), (11c), and (11d)are linear and separable constraints. Hence, the resource allocation problem (11)is a convex optimization problem, and it is separable into I subproblems. For230each cell i, the ith optimization problem is written as follows:maximizeθsubject to(η2 )i X Xlog (θk,n )(12a)k K(i) n NXθk,n 1, n N ,(12b)θk,n 1, k K(i),(12c)k K(i)Xn N0 θk,n 1, k K(i), n N .(12d)5. Centralized Multi-Cell Resource and Power AllocationAs stated in the previous section and proven in Appendix I, the joint resourceand power allocation problem (4) is separable into two independent convex13
optimization problems: a power allocation problem, and a resource allocation235problem. In this section, we solve the resource and power allocation problemsusing Lagrange duality theory and subgradient projection method.5.1. Solving the Centralized Power Allocation Problem5.1.1. Lagrange-Based MethodSince the power allocation problem (10) is a convex optimization problem, we240can make use of Lagrange duality properties, which also lead to decomposabilitystructures [29]. Lagrange duality theory links the original problem, or primalproblem, with a dual maximization problem. The Lagrangian of problem (10)is given as follows:b , λ, ν) L (bρ, πX X Xlog (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 i! Xi IνilogXexp (bπi,n )N0Gk,i,nGk,i0 ,n))Gk,i,n(13)! log (Pmax ) .n Nb and πb are called the primal variables. λk,i,nThe optimization variables ρ245and νi are the dual variables associated with the (k, i, n)th inequality constraint (10b) and with the ith inequality constraint (10c), respectively.After relaxing the coupling constraints (10b) and (10c), the optimizationproblem separates into two levels of optimization: lower level and higher level.b , λ, ν) is the objective function to be maximized. ρbk,i,nAt the lower level, L(bρ, πand πbi,n are the optimization variables to be found. At the higher level, we havethe master dual problem in charge of updating the dual variables λ and ν by14
solving the dual problem:minimizeb , λ, ν))max (L (bρ, π(14a)subject toλ 0,(14b)ν 0.(14c)λ,νb,bρπIn order to solve the primal optimization problem (lower level of optimization), we use the subgradient projection method. It starts with some initial feasible values of ρbk,i,n and πbi,n that satisfy the constraints (10d). Then, the nextiteration is generated by taking a step along the subgradient direction of ρbk,i,nand πbi,n . For the primal optimization variables, iterations of the subgradientprojection are given by:ρbk,i,n (t 1) ρbk,i,n (t) δ(t) L, ρbk,i,n k K(i), i I, n N ,πbi,n (t 1) πbi,n (t) δ(t) L, i I, n N . bπi,n(15a)(15b)The scalar δ(t) is a step size that guarantees the convergence of the primal optib , λ, ν)mization problem [29]. The partial derivatives of the objective function L(bρ, πwith respect to ρbk,i,n and πbi,n , are given in the following: Lexp (bρk,i,n ) λk,i,n , ρbk,i,n(exp (bρk,i,n ) 1) log (exp (bρk,i,n ) 1) k K(i), i I, n N ,X Lexp (bπi,n ), i I, n N . λk,i,n νi P bπi,nexp (bπi,n )k K(i)(16a)(16b)n Nb , λ, ν)) is differentiable. Thus, atThe dual function g (λ, ν) max (L (bρ, πb,bρπthe higher optimization level, the master dual problem (14) can be solved using15
the following gradient method:?λk,i,n (t 1) λk,i,n (t) δ(t)(log(exp(bρ?k.i.n πbi,n) X?exp(bρ?k.i.n πbi?0 ,n πbi,n)i0 Ni0 6 iN0Gk,i,nGk,i0 ,n)),Gk,i,n k K(i), i I, n N ,νi (t 1) νi (t) δ(t) log(17a)! !X?exp πbi,n log(Pmax ) ,n N i I, n N ,(17b)where t is the iteration index, and δ(t) is the step size at iteration t. Appropriate?choice of the step size [30] leads to convergence of the dual algorithm. πbi,nandρb?k,i,n denote the solution to the primal optimization problem. When t thedual variables λ(t) and ν(t) converge to the dual optimal λ and ν , respectively. The difference between the optimal primal objective and the optimal dualobjective, called duality gap, reduces to zero at optimality, since the problem (10)is convex and the KKT conditions are satisfied. We define bρ, bπ , λ, and νas the differences between the optimization variables obtained at the current iteration and their values at the previous iteration. They are given by:b(t)k, bρ(t 1) kbρ(t 1) ρ(18a)b (t)k, bπ (t 1) kbπ (t 1) π(18b) λ(t 1) kλ(t 1) λ(t)k,(18c) ν(t 1) kν(t 1) ν(t)k.(18d)5.1.2. Iterative Power Allocation AlgorithmThe procedure for solving the centralized power allocation problem is described in Algorithm 1. Initially, the primal optimization variables ρbk,i,n and πbi,n250as well as the dual variables λk,i,n and νi start with some initial feasible values.t, tprimal , and tdual denote the number of rounds required for the centralized16
Algorithm 1 Dual algorithm for centralized power allocation1:2:b , λ, ν), Pmax , and πmin .Parameters: the utility function L(bρ, πInitialization:sett t t 0, and πi,n πmin , i I, n N ,primaldualPsuch asπi,n Pmax , i I. Calculate πbi,n (0) and ρbk,i,n (0) accordingly,n N3:4:5:6:7:8:9: k K(i), i I, n N .Set λk,i,n (0) and νi (0) equal to some non negative value, k K(i), i I, n N .b? (t 1)) PrimalProblem(ν ? (t), λ? (t))(bπ ? (t 1), ρ?b? (t 1))(ν (t 1), λ? (t 1)) DualProblem(bπ ? (t 1), ρ?if ( bπ (t 1) ) or ( bρ (t 1) ) or ( ν (t 1) ) or ( λ? (t 1) ) thent t 1go to 4end ifpower allocation problem to converge, the number of iterations for the primal problem, and the number of iterations for the dual problem, respectively.At each round t, we start by updating the primal optimization variables, us255ing the PrimalProblem function given in Algorithm 2. The solution to the?primal optimization problem at the current round t is denoted by πbi,n(t 1)and ρb?k,i,n (t 1). The PrimalProblem function updates πbi,n (tprimal 1) andρbk,i,n (tprimal 1), and increments tprimal until bπ (tprimal 1) and bρ(tprimal 1) become less than .260Then, the solution to the dual optimization problem at the current round t,denoted by νi? (t 1) and λ?k,i,n (t 1) is calculated using the DualProblemfunction given in Algorithm 3. νi and λk,i,n are updated using the obtained?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 round of calculations is performed, and t is265incremented as long as bπ ? (t 1) or bπ ? (t 1) or ν ? (t 1) or λ? (t 1)is greater than . Otherwise, the obtained solution at the current round is theoptimal solution to the centralized power allocation problem.5.2. Solving the Resource Allocation ProblemIn this subsection, we search for the optimal solution to the resource allo-270cation problem (12). For each cell i, the problem (12) is a convex optimizationproblem, as proven previously.17
Algorithm 2 Primal problem on PrimalProblem(ν ? (t), λ? (t))for i 1 to I dofor n 1 to N do πbi,n (tprimal 1) max log (πmin ) ; πbi,n (tprimal ) δ (t) L bπi,n for 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 functionTheorem 5.1. For each cell i, the optimal solution to the resource allocationproblem (12) is given by:θk,n 1, k K(i), n N .max ( K(i) , N )(19)The proof of Theorem 5.1 is given in Appendix II. When the number ofactive UEs is less than the number of available resources, θk,n 2751 N , k K(i), n N . Thus, the available resources are not fully used over time, andeach UE is permanently served. Otherwise, when K(i) N , the optimalsolution is:
resource allocation problem and a power allocation problem. Lagrange dual-ity theory is used to solve the centralized power allocation problem. We also tackle the resource and power allocation problem di erently by addressing it in a decentralized manner. We propose a non-cooperative downlink power alloca-tion approach based on game theory.
to communicate effectively is that a multi-layered centralized bank needs to put in more effort to maintain the quality of the soft information that has been gathered. This adds an extra cost to the client targeting activity in a centralized organizational structure. A consequence of the arguments presented above is that decentralized banks are .
of the cell and eventually divides into two daughter cells is termed cell cycle. Cell cycle includes three processes cell division, DNA replication and cell growth in coordinated way. Duration of cell cycle can vary from organism to organism and also from cell type to cell type. (e.g., in Yeast cell cycle is of 90 minutes, in human 24 hrs.)
and environmental effects. As a result, the true environmental and social costs of wastewater treatment are often not included in decision making. Many communities face decisions regarding centralized versus decentralized wastewater treatment as well as numerous strategies and technologies available within the centralized and decentralized sectors.
Section 8. 2. RELATED WORK Several groups have designed centralized and decentral-ized shape formation algorithms for modular robots [2, 10]. Centralized algorithms suffer from scalability problems, but decentralized solutions have been shown to be more scalable as system size increases. Decentralized algorithms often uti-
UNIT-V:CELL STRUCTURE AND FUNCTION: 9. Cell- The Unit of Life: Cell- Cell theory and cell as the basic unit of life- overview of the cell. Prokaryotic and Eukoryotic cells, Ultra Structure of Plant cell (structure in detail and functions in brief), Cell membrane, Cell wall, Cell organelles: Endoplasmic reticulum, Mitochondria, Plastids,
[The building block of thunderstorms is the thunderstorm cell. A thunderstorm can be made of one cell or multiple cells. A single-cell thunderstorm can be an ordinary cell or a supercell thunderstorm. Thunderstorms with more than one cell can be multi-cell clusters or multi-cell lines, which are also called squall lines.] Ordinary Cell As the .
The Cell Cycle The cell cycle is the series of events in the growth and division of a cell. In the prokaryotic cell cycle, the cell grows, duplicates its DNA, and divides by pinching in the cell membrane. The eukaryotic cell cycle has four stages (the first three of which are referred to as interphase): In the G 1 phase, the cell grows.
Massachusetts Curriculum Framework for English Language Arts and Literacy 3 Grade 5 Language Standards [L]. 71 Resources for Implementing the Pre-K–5 Standards. 74 Range, Quality, and Complexity of Student Reading Pre-K–5 . 79 Qualitative Analysis of Literary Texts for Pre-K–5: A Continuum of Complexity. 80 Qualitative Analysis of Informational Texts for Pre-K–5: A .
