Optimal Resource Allocation for Energy-Efficient OFDMA Networks

This paper focuses on radio resource allocation in OFDMA networks for maximizing the energy efficiency subject to the data rate requirements of users. We propose the energy-efficient water-filling structure to obtain the closed-form optimal energy-efficient power allocation for a given subcarrier assignment. Moreover, we establish a new sufficient condition for the optimal energyefficient subcarrier assignment. Based on the theoretical analysis, we develop a joint energy-efficient resource allocation (JERA) algorithm to maximize the energy efficiency. Simulation results show that the JERA algorithm can yield optimal solution with significantly low computational complexity.


Introduction
As the high data rate applications are going to dominate the mobile services, energy efficiency (EE) is becoming more crucial in wireless communication networks.Energy-efficient radio resource allocation is one of the effective ways to improve the EE of the OFDMA (orthogonal frequency division multiple access) networks [1].Although radio resource allocation in OFDMA networks has been extensively studied, the major focus is on improving spectral efficiency (SE) which may not always coincide with EE [2].
Different from SE-based resource allocation schemes in which the total transmitting power is fixed, the EEbased schemes adjust the power level adaptively based on the channel conditions [3].Accordingly, the classic waterfilling power allocation method cannot be applied directly due to the unknown total transmitting power.In order to determine the proper transmitting power, the perturbation functions of EE have been studied in [4,5].It has been shown that EE is strictly quasi-concave in SE [4] and in the total transmitting power [5].However, the perturbation functions of EE cannot be expressed in an analytic expression; only the approximation algorithms have been proposed to find the near-optimal solutions [4][5][6].
In this paper, we focus on joint subcarrier assignment and power allocation in OFDMA networks for maximizing the energy efficiency problem subject to the data rate requirements of users.The main contributions of our work are summarized as follows.
(i) We prove that EE is strictly pseudo-concave with respect to power vector for a given subcarrier allocation, which guarantees that the solution satisfying the KKT conditions is also the global optimal.Using this property, we show that the optimal solution has a special EE water-filling structure that is determined by only one variable.Based on this observation, we further provide the first closed-form expression for the optimal energy-efficient power allocation.
(ii) According to the analysis, we propose an optimal energy-efficient power allocation algorithm by sequentially searching within a finite number of water-level intervals.The computational complexity of the proposed algorithm is much lower than that of

System Model
Consider a downlink OFDMA network with one base station and  users.Let K = {1, . . ., } and N = {1, . . ., } be the set of users and subcarriers, respectively.Define the subcarrier assignment matrix  = ( , ) × where  , = 1 means that subcarrier  is allocated to user , and otherwise  , = 0.The transmitting power allocation matrix is defined as p = ( , ) × , where  , ≥ 0 represents the transmitting power allocated to user  on subcarrier .Then, the maximum achievable data rate of user  on subcarrier  is given by where  is the bandwidth of subcarrier and  , denotes the normalized channel power gain of user  on subcarrier .
Accordingly, the overall system data rate is and the total transmitting power is In addition to transmitting power, the energy consumption also includes circuit power which is consumed by device electronics.The circuit power is modeled as a constant   , which is independent of data transmission rate [7].Accordingly, we define EE as the amount of bits transmitted per Joule of energy; that is,  EE ≜ /(  +   ), where  is the reciprocal of drain efficiency of power amplifier.
In our work, we consider maximizing EE under the minimum data rate requirements,   , and the total transmitting power constraint,  max .Accordingly, this optimization problem can be formulated as where (4b) indicates the minimum data rate requirement of each user, (4c) is the total transmitting power constraint, and (4d) is the constraint on subcarrier assignment to ensure that each subcarrier is only assigned to one user.Similar to the traditional spectral-efficient resource model, P1 is a mixed integer nonlinear programming problem, and it is not trivial to obtain the global optimal solution to this problem.To solve the problem, we first decompose P1 into two subproblems, which include (1) the energy-efficient power allocation for a fixed subcarrier assignment  and (2) the energy-efficient subcarrier assignment for a given total transmitting power   .Then, based on the properties of the subproblems, we develop an algorithm to find the solution of joint energy-efficient power allocation and subcarrier assignment to maximize EE.

Optimal Energy-Efficient Power Allocation
In this section, we analyze the optimal energy-efficient power allocation (EPA) based on the EE water-filling structure.All the major results are given by some theorems.In particular, Theorems 2 and 5 demonstrate that the global optimal solution to the energy-efficient power optimization problem is with the EE water-filling structure.Theorem 3 provides the corresponding closed-form water-level, whose optimality is proved by Theorem 4.

EE Water-Filling
Structure.Given the subcarrier assignment matrix , the set of subcarriers assigned to user  can be denoted by N  () = { |  , = 1} and the power vector p = ( 1 ⋅ ⋅ ⋅   )  .Then P1 is reduced to the following EPA problem:  [8].Besides,   (p) is differentiable and concave for all , and   (p) is positive and affine.Therefore, according to the KKT sufficient optimality theorem [9], any feasible solution p satisfying the KKT conditions is also globally optimal for P2.
When the feasible solution set of P2 is nonempty, the minimum power vector p to guarantee the minimum datarate requirement of each user must be a feasible solution to P2, which can be obtained by solving the following margin adaptive (MA) problem [10]: Resorting to the Lagrange dual theory, the optimal solution to P3 is given by p = (( x () − 1/  ) + ) ×1 , where x  is the root of the equation We call x  the lowest power water-level of user .More importantly, a series of feasible solutions to P2 can be constructed based on p by raising the water-levels of some users and maintaining that of the others.To be specific, by sorting all the users in ascending order of their lowest power water-levels such that x 1 ≤ ⋅ ⋅ ⋅ ≤ x +1 with x +1 = +∞, the region of the promotable water-level can be divided into  intervals, that is, ( x where The corresponding data rate of each user satisfies It can be found that the data rate of each user with the same water-level is greater than the minimum requirement, while that of the other users is equal to the minimum requirement. It is noteworthy that since x  in ( 8) can be obtained by solving (7) and hence p () can be expressed solely as a function of the water-level , accordingly, P2 can be transformed into a single variable problem.Furthermore, for any given total transmitting power   , P2 is equivalent to P4 shown in the following.If the optimal solution to P4 has the EE water-filling structure, we can deduce that P2 must be maximized at a power vector with the EE water-filling structure.It can be proved by Theorem 1: where  () (p) ≜ ∑  =1 log 2 (1 +     ).
Theorem 1.Given   ≥  min , the optimal solution p * to P4 has the EE water-filling structure (8).
Proof.Since P4 is a convex programming, the optimal solution p * must satisfy the KKT conditions; that is, there exist scalars   ≥ 0 ( = 1 ⋅ ⋅ ⋅ ) and  such that According to (11), we have Besides, based on the complementary slackness conditions (12), we can get that   = 0 if   (p * ) >   .Let  ≜ /(ln 2 ⋅ ) and   ≜ (1 +  () ).Therefore, p * can be further expressed as follows: where To maximize the overall data rate, the power allocated to the users in K  2 should be minimized.And then   = x  , ∀ ∈ K  2 .Since   ≥ 0, we get x  =   ≥ .On the other hand,  > x  (∀ ∈ K  1 ) in order to satisfy   (p * ) >   .Consequently, We can now conclude that p * has the EE water-filling structure as (8).
According to the water-levels of p (), the subcarrier set can be further divided into three subsets: EE (p) have the following properties, which will be used in the proof of the following theorems.[ x , x].The minimum power vector p = p ( x ) is also with the EE water-filling structure whose water-level is x 1 and the total transmitting power is  min .On the other hand, suppose p is the optimal solution to P4 when   =  max ; according to Theorem 1, we have p = p ( x), where x is the EE water-level.Since the feasible region of P1 is nonempty, the total transmitting power   must satisfy  min ≤   ≤  max .Hence, the corresponding power vector with the EE water-filling structure in the feasible region of P1 should be subject to p () ∈ P = { p (),  ∈ [ x , x]}.Based on the strict pseudo-concavity of  () EE (p), we have the following theorem.

EE Water-Level Interval
Theorem 2. Assume ď =   ( p ) and d =   (p).p is the optimal solution to P2 if and only if ď1 ≤ 0, and p is the optimal solution to P2 if and only if d1 ≥ 0.
However, when ď1 > 0 and d1 < 0, whether there exists a EE water-level to make p () optimal is still not answered.We should study the relation between  () EE (p) and the EE waterlevel .
Similar to  () EE () is also strictly pseudo-concave, and the first-order derivative is where According to the first-order optimality condition, a stationary point  0 of

EE (𝑥), its closed-form expression is given by
where (⋅) represents the Lambert- function.
The proof of the theorem can be found in [11].

EE-Optimal Water-Level.
According to the strict pseudoconcavity,  () EE () is maximized at the stationary point  0 .However, whether the corresponding p ( 0 ) is the global optimal solution to P2 still needs to be verified.EE ( p ( 0 ))/  ≤ 0 ( ∈ N 0 or N 2 ).Then it can be verified that p ( 0 ) satisfies the KKT conditions.Hence p ( 0 ) is the optimal solution to P2.
In addition, the existence of  0 is proved by Theorem 5.
Theorem 5.If neither p nor p is the optimal solution to P2, there must exist  0 ∈ ( x , x) such that p ( 0 ) is the optimal solution to P2.
Proof.According to the intermediate value theorem, to prove Theorem 5, we should show that there must exist a continuous power interval (, ] such that () > 0, () ≤ 0.
In fact, if neither p nor p is the optimal solution to P2, we can verify that ( x ) > 0 and ( x) < 0 according to Property 1. Assume ( x , x) is divided into  water-level rise intervals.It can be proved that there must exist a water-level rise interval (, ] such that () > 0, () ≤ 0. If there does not exist such an interval, it can be deduced that ( x )⋅( x) > 0, which yields a contradiction.

Optimal Energy-Efficient Subcarrier Assignment
In this section, we will provide a sufficient condition for the optimal energy-efficient subcarrier assignment (ESA) based on the relation between EE and the total transmitting power   .By utilizing this sufficient condition, a quick search method can be devised to obtain the optimal ESA, which will be described in the next section.
According to (4a)-(4e), a feasible ESA can be obtained by solving a rate adaptive (RA) problem for a given total transmitting power   .Moreover, the maximum EE can only be achieved at one of three different total transmitting powers, including two boundary points ( min and  max ) and a stationary point   of the perturbation function of P1 [5].To obtain the optimal ESA, it should first determine   , which is an unknown value.Unfortunately,   is difficult to be determined and only an approximation can be found by the iterative algorithms [5].Therefore, only the suboptimal ESA can be obtained according to the approximate   .
On the other hand, based on the EE water-filling structure discussed in the previous section, the optimal ESA can be obtained by calculating the exact optimal EE for every feasible subcarrier assignment  and then selecting the one with the maximum value.This exhaustive search is prohibitive for large  and  in a practical system.However, combining the EE water-filling framework and the property of the perturbation function of P1, a sufficient condition for the optimal ESA can be established to greatly simplify the search.
EE , where   is the subcarrier assignment obtained by solving the RA problem with   =   ( 0 ).

Joint Energy-Efficient Resource Allocation Algorithm
Based on the analysis in the previous sections, we develop an optimal energy-efficient resource allocation algorithm with low complexity to solve P1, named as joint energy-efficient resource allocation (JERA) algorithm.Different from the existing algorithms proposed in [4][5][6], the JERA algorithm consists of two layers to iteratively perform subcarrier assignment and power allocation so as to achieve the optimal solution.The aim of the outer layer is to find a feasible subcarrier assignment for a given total transmitting power, and the inner layer is in charge of energy-efficient power allocation based on the obtained subcarrier assignment.
Based on the EE water-filling framework, the optimal EPA

Performance Evaluation
In this section, simulation results are given to verify the theoretical analysis and the performance of the proposed algorithms.In our simulation, the number of data subcarriers is set to be 72 and the bandwidth of each subcarrier is 15 kHz [5].The block Rayleigh fading channel model is considered and the Okumura-Hata path loss model is followed; that is, () = 137.74+ 5.22 log() in decibels, where  is the distance between transmitter and receiver in kilometers.The standard deviation of shadowing is 7 dB, and the thermal noise spectral density is −174 dBm/Hz [4].The circuit power is 20 W and the maximum transmitting power is 40 W for the base station [12].The drain efficiency of power amplifier is assumed to be 38% [4].Each user in the simulation has the same minimum rate requirement of 100 kbps.First, we compare the performance of the OEPA algorithm with the other two algorithms: the BPA algorithm [5] and the MWF (multilevel water filling) algorithm [10] for a fixed subcarrier allocation.Although the MWF algorithm is a classical SE-based scheme rather than a EE-based scheme, it is used as a benchmark to measure the difference in the energy efficiency between the two classes of scheme.In this simulation example, the number of users is set to 30.The users are uniformly distributed in a circle centered at the BS with a variable radius.The results in Figures 1(a) and 1(b) show that the average EE and the system throughput of all the three algorithms decrease with the channel power gain.This is due to the fact that the average channel-gain-to-noise ratio (CNR) of each user decreases with the increase of the distance between the user and the BS, such that more power to that of the BPA algorithm.In addition, the complexity of the BPA algorithm increases evidently when the error tolerance becomes tighter.This is due to the fact that the OEPA algorithm can obtain the exact optimal solution by checking at most  +  continuous power intervals in the worst case based on the closed-form expression of EPA.On the contrary, the BPA algorithm is to search bidirectionally for the optimal transmitting power, which results in a higher computational complexity.On average, the CPU time for convergence of the OEPA algorithm is about 15.64% and 13.03% of that of the BPA algorithm with  = 0.1 and  = 0.001, respectively.
Moreover, in order to verify the optimality of the proposed JERA algorithm, we compare the EE obtained by JERA with the global optimum obtained by exhaustive search.In this case, we consider a system with 9 subcarriers and 3 users to reduce the complexity of exhaustive search.As shown in Figure 3, the achieved optimal EE in both algorithms decrease with the distance between the BS and users.Moreover, the two curves match with each other very well.It demonstrates the proposed ESA search method based on Theorem 6 is effective and the JERA algorithm can obtain the global optimal ESA and EPA simultaneously.
In addition, we compare the performance of the JERA algorithm with that of the JIOO algorithm [5] with different number of users and subcarriers.The result is shown in Table 1; it can be observed that the EE of the JERA algorithm is superior to that of the JIOO algorithm due to the optimality of the solution obtained by the JERA algorithm.More importantly, the convergence rate of the JERA algorithm is significantly faster than that of the JIOO algorithm.Specifically, the number of iterations for convergence of the JERA algorithm is less than 5 in average, while the JIOO algorithm requires at least 29 iterations in average to approximate the optimum.It is worth noting that each iteration in both algorithms needs to solve a RA problem with rate requirements and total transmitting power constraint.Despite of the nonconvexity of this type RA problem, it has been proven that it can be solved   efficiently by the Lagrange dual decomposition method with zero duality gap [13].

Conclusion
In this paper, we investigated the EE maximization problem under both the user rate requirements and the transmitting power constraint.Utilizing the EE water-filling structure, we obtain the closed-form of the optimal EPA.The sufficient condition for optimal ESA is also derived based on the relation between EE and power.Furthermore, we propose a low-complexity algorithm with joint ESA and EPA to address the energy-efficient resource allocation in downlink OFDMA-based networks.Simulation results show that the proposed algorithm achieves the optimal energy-efficient resource allocation with significantly reduced computational complexity compared with the iterative methods.

Theorem 3 .
EE () is the root of the equation () = 0.The closed-form expression of  0 is given by Theorem 3. If there exists a stationary point  0 in the domain of  ()

Figure 1 :
Figure 1: Performance comparison of different algorithms.

Figure 2 :
Figure 2: Comparison of convergence performance of different algorithms.

Figure 3 :
Figure 3: Evaluation and comparison of average EE.

Table 1 :
Performance comparison of the two algorithms.