1 Introduction
Next generation cellular networks Wu et al. (2017)Agiwal et al. (2016) are envisioned to have some unique beneficial characteristics, among which ultradense deployment Ge et al. (2016) of network nodes is one of the most crucial paradigms. Ideally, ultradense deployment of nodes enhances the coverage and capacity of the system Zhou et al. (2014). However, there are still some problems with this concept if the deployment and system parameters are not properly optimized. Since the number of transmitters in such networks is very large, this causes huge interference. Consequently, energyefficient transmission is one of the important research endeavors in this arena. In order to obtain fruitful outcome in this context, designing an effective and efficient radio access technology Edler and Lundberg (2014) is one of the possible solutions. Nonorthogonal multiple access (NOMA) is considered as the most promising radio access technique for next generation wireless systems Ding et al. (2014)HIGUCHI and BENJEBBOUR (2015)Wong et al. (2017). Through experimentation and theoretical analysis Higuchi and Kishiyama (2013)Endo et al. (2012)Umehara et al. (2012)Otao et al. (2012), it is proved that the NOMA technique is able to provide enhanced performance in different sectors of wireless communication systems comparing with other orthogonal multiple access (OMA) techniques. Conceptually, powerdomain NOMA Islam et al. (2017) allows multiple users to occupy the same resource. This leads to additional interference for NOMAequipped networks and their neighboring networks. Consequently, existing resource management techniques Feng et al. (2013) in conventional networks, especially the energyefficient ones, need to be revisited due to the incorporation of additional interference this new technology brings.
Although the NOMA technique allows multiple users to be superimposed on the same frequency channel, due to the usage of practical, finitelength and possibly nonGaussian codes, it is not an optimal design to assign large number users to the same channel. Consequently, dedicated spectrum of a system needs to be subdivided into multiple subchannels in order to support increased number of users Dai et al. (2015)
. At the same time, how to allocate these subchannels among the users in a multiplexed manner given the maximum allowable number of users that can utilize a subchannel simultaneously, is an important problem. While targeting on the maximization of the overall throughput, some research has been conducted on the downlink and uplink subchannel and power allocation in such NOMA systems. Based on some assumption of having constant power on the subchannels, existing works typically provide some heuristic solutions for the subchanneluser mapping problem. Once the subchanneluser mapping is done, in order to enhance the performance of the system further, different existing works have provided different schemes for the power allocation. For example, in
Saito et al. (2013); Benjebbour et al. (2013), the authors use the fractional transmit power allocation technique among users and apply the equal power allocation technique across subchannels. Hojeij et al. (2015) uses the water fillingbased approach for the power allocation, and in Parida and Das (2014), the authors use difference of convex (DC) programmingbased approach Vucic et al. (2010) for the power allocation at both the user and subchannel levels. For uplink systems, there are some works as well, such as Kumaran and Qian (2003); AlImari et al. (2015); Mollanoori and Ghaderi (2014). In AlImari et al. (2015), the authors used the iterative water filling idea Yu et al. (2004) for both the subchanneluser mapping problem and their granular power allocation. On the other hand, in Mollanoori and Ghaderi (2014), the authors assumed that the resources are time slots instead of frequency channels. Although the work in AlImari et al. (2015) can ensure that users can be assigned to multiple subchannels, each user is assigned to at most one resource block via the scheme in Mollanoori and Ghaderi (2014). While the base station (BS) of all these works is assumed to be equipped with the halfduplex functionality, the authors in Sun et al. (2017) have proposed a joint subchannel and power allocation scheme for such a system based on the assumption that the BS is equipped with the fullduplex functionality to serve uplink and downlink users simultaneously.With the increasing desire to have green communications in the recent years, reducing energy consumption has become the prime concern for researchers, and the fifth generation (5G) systems have also targeted energy efficiency as one of the major milestones to achieve Abrol and Jha (2016). Although the minimization of used power reported in Lei et al. (2016) could be one of the objectives of energyefficient transmissions, the resultant solution of such a type of formulation is not spectrumefficient. In order to achieve spectrum efficiency with the minimal power, the ideal objective of an energyefficient transmission is to maximize the achievable bits of a channel under the unit Joule of power consumption Jumira and Zeadally (2012). For a NOMAequipped single cell, which has two users, an energyefficient power allocation scheme is studied in Han et al. (2013). In this work, the authors found the relationship between energy efficiency and spectrum efficiency under the total system power constraint. At the same time, energy efficiency is also studied for NOMAequipped MIMO systems in Sun et al. (2015). The closest to ours and the most recent energyefficient resource allocation scheme appeared in Fang et al. (2016). The system model that this paper considered is similar to ours, and our target is to enhance the performance of the proposed scheme in this paper further. This work has proposed an energyefficient downlink subchannel and power allocation scheme for a cellular network under total power constraint at the BS with a number of drawbacks. For example, through their scheme, each user cannot be assigned to multiple subchannels, the concept of which fails to exploit the multiuser diversity of wireless systems Tse and Viswanath (2005). Moreover, each subchannel can be assigned to at most two users, which also conflicts with the NOMA concept. Furthermore, they have used the DC programmingbased approach in order to allocate power across all the subchannels and users of the system. Actually, through this power allocation, the entire power at the BS is used up, and hence the system fails to reach the optimal energyefficient state. On the other hand, they also have utilized the DC programmingbased approach in order to calculate the optimal instantaneous rate of a subchannel, which is an essential intermediate step of their subchanneluser mapping algorithm. However, the DC programmingbased approach is appropriate only in this case because of provisioning at most two users to a subchannel. If the allowable number of users per subchannel is increased, the DC programmingbased approach is not appropriate to solve this problem any more as the problem can no longer be expressed as a difference of convex functions under this realistic consideration.
The contribution of this paper is the design of an energyefficient downlink subchannel and power allocation scheme in a cellular network with enhanced performance compared to the work in Fang et al. (2016). Since this problem considers subchannels assignment which are associated with discrete variables in the formulated problem, the problem is NPhard in general. Moreover, even if the subchannel assignment information is known, the power allocation problem is nonconvex Bertsekas (1999)
. As a result, joint subchannel assignment and power allocation of this problem can be considered as an mixed integer nonlinear programming (MINLP) problem. Considering the importance of the overall energy efficiency for such a system from the perspective of green communications, since the existing solution of
Fang et al. (2016) is approximate, the investigation of the better ones is required. Consequently, we find that decomposing the problem into a subchannel allocation problem followed by a power loading problem and then solving each problem individually while considering interdependency, is an elegant approach to solve this problem. In the first step, under the assumption that total power of the BS is subdivided equally among all the subchannels, we solve the subchanneluser mapping problem via a manytomany matching model Hamidouche et al. (2014); Roth (1984). However, unlike the scheme in Fang et al. (2016), our proposed subchanneluser mapping scheme can utilize the multiuser diversity efficiently. On the other hand, one of the assumptions of the subchannelusers mapping algorithm in Fang et al. (2016) is, at most users can be assigned to a subchannel. Because of this assumption, the power allocation problem of a subchannel is amenable to the DC programmingbased approach. If a subchannel is assigned to more than users, the power allocation problem is no longer amenable to the DC programmingbased approach. In contrast, our proposed scheme is general, and the number of users assigned to a subchannel is arbitrary. Our discovery to solve this problem via the geometric programming (GP)based Boyd et al. (2007); Chiang (2005) approach is one of the specific contributions of this paper. Then, in the second step, once the subchanneluser mapping information are known, we adopt the GPbased approach in order to allocate power across all subchanneluser slots upon approximating the energy efficiency of system by a ratio of two posynomials. The DC programmingbased approach in Fang et al. (2016) allocates power across the subchannels while ignoring the detailed granularity of the user power level. Our proposed power loading scheme using the GPbased approach allocates power to each user’s data stream on each of its allocated subchannels. Compared to the most relevant existing work Fang et al. (2016), the major contributions of the paper are summarized as follows.
For the subchanneluser mapping task, unlike the one in Fang et al. (2016), we have adopted a manytomany matching model that exploits the multiuser diversity of wireless systems efficiently. Through this scheme, not only each subchannel can be used to serve multiple users, but also each user can utilize multiple subchannels. However, assigning more and more users to a subchannel cannot necessarily enhance the energy efficiency of the system further. Our proposed algorithm can smartly adapt user assignment to each subchannel given the information of some maximal allowable number of users to each subchannel.

While solving the subchanneluser mapping problem, it is required to determine the optimal instantaneous sumrate of a subchannel in the intermediate decision making steps. When a subchannel can serve at most users, the approach in Fang et al. (2016) is appropriate to determine the instantaneous sumrate of a subchannel. However, in practice, having more than users on a subchannel can enhance the system performance. On the other hand, when a subchannel is assigned to more than users, the DC programmingbased approach is no longer a suitable method to determine the optimal instantaneous sumrate of that subchannel. This is because the objective function cannot be expressed as a difference of convex functions in this case. However, this problem is amenable to GP via the single condensation heuristic method after approximating the objective function of the problem by a ratio of two posynomials. Consequently, upon the approximation, we have adopted the single condensation method to determine the optimal instantaneous sumrate of each subchannel while making the decision about each subchanneluser assignment.

Once the subchanneluser mapping information are known from the first step, the DC programmingbased approach in Fang et al. (2016) can allocate power optimally across the subchannels. However, this power allocation scheme ignores the detailed granularity of each subchanneluser slot, and the entire power of the BS is used up. While targeting on the optimality and considering the detailed granularity of each subchanneluser slot, we approximate the original problem via replacing the log function by the first term of its approximated series Nave to facilitate the problem solving via the GPbased approach. Upon approximating the objective function of the problem via a ratio of two posynomials, we adopt the GPbased single condensation heuristic to solve the approximated problem. The proposed scheme allocates power to each user’s data stream on each of its allocated subchannels. As a result, the total required power, to reach the optimal overall energyefficient state, is much less than the total power of the BS. From the perspective of green communications, this phenomenon of our power loading scheme is a big advantage over the DC programmingbased approach. Although the proposed GPbased approach provides finegrained power allocation across each subchannel and user with higher energy efficiency, this approach is computationally intensive with the growing number of users and subchannels due to the implementation limitation of the offtheshelf GP solvers Grant and Boyd (2014). Therefore, based on the insights of the optimal solution, we also have proposed a computationallyefficient suboptimal solution for the finegrained energyefficient power loading problem, which has a polynomial time complexity.

Extensive simulation has been conducted in order to verify the effectiveness of our proposed resource allocation scheme. The results demonstrate that our scheme always outperforms the scheme in Fang et al. (2016) under various realistic scenarios.
The rest of the paper is organized as follows. Along with the background information and the description of the system, in Section 2, we formulate our energyefficient resource allocation problem. The detailed solution approach is provided in Section 3. Followed by the simulation methodology, we evaluate the performance of our proposed resource allocation schemes in Section 4. Finally, Section 5 concludes the paper with some implication.
2 System Model and Problem Formulation
In this paper, we consider a downlink scenario of a cellular network, which has one BS. Time is divided into frames, and the entire preassigned spectrum for the system is divided into subchannels in each frame. The resultant subchannels are the elements of a set, denoted by N. There are users in the system, and the corresponding set holding these users is denoted by M. Using the subchannels in set N as the transmission media, the BS transmits data to the users in set M. Both the BS and the users in the system are equipped with NOMA technologies. The BS transmits its data to a set of users using the superposition coding (SC) technique over a set of subchannels. Whereas, the receivers (i.e., the users) apply the SIC technique on each subchannel to decode the superimposed signals for extracting their own individual signal. However, before the downlink transmission operation, it is required to schedule subchannels and power across the users optimally so that the overall energy efficiency of the system is maximized. We assume that the scheduling scheme in the system is centralized, and the BS is appointed to conduct the entire scheduling operation. To develop this scheduling scheme, since the entire channel state information (CSI) of the system is required, the BS is aware of all these information. At the beginning of each time frame, all users send their CSI to the BS via some reliable control channels. The CSI of the subchannels in set N follows the block fading model Tse and Viswanath (2005), where the CSI of each subchannel is constant for a time slot and varies in an i.i.d. manner from time slot to time slot. The BS has the maximal power constraint, denoted by .
We assume that the BS assigns users to the th subchannel, and the corresponding set holding these users is denoted by . Given this, let denote the symbol transmitted by the BS on subchannel as , and it is given by . Here, is the modulated symbol of the th user on subchannel and is the power level assigned to user on subchannel . Consequently, the received signal of user on subchannel can be represented as
where is the channel gain of user on the th subchannel. is the noise power over subchannel , which follows Additive White Gaussian Noise (AWGN) McClaning (2000)
distribution with mean zero and variance
, i.e., . The noise power of subchannel is statistically same for all users. In NOMA systems, each subchannel is shared by multiple users. Consequently, each user on subchannel receives its signal as well as the the interference signals from the other users that share the same subchannel. Let us denote the set holding the users that cause interference to user on subchannel as . Given the information of set , the received signaltointerferenceratio (SINR) of the th user on subchannel is given by(1) 
where is the noise power on subchannel , and represents the power gain of the th user on subchannel .
Since the NOMA users are equipped with the SIC technique, the way a NOMA user retrieves information depends on the order of the allocated power level among all the NOMA users in the corresponding subchannel. In other words, a particular NOMA user can decode the information of other users that are assigned to larger power level compared to itself. On the other hand, this user can decode its own information considering the power level of other remaining users (that are assigned to lower power level compared to itself) as noise. Therefore, if the order of the power level assigned to the NOMA users are known, the decoding order is straightforward for a particular NOMA user and vice versa. It is clear that the optimal decoding order of the NOMA users and their power assignment are coupled to each other and is not a straightforward problem. However, in order to reduced the complexity of the proposed solution scheme in this paper, we intend to fix the order of the power level among the NOMA users while solving the entire energy efficiency problem. Ideally, in order to maximize the energy efficiency of the system, the optimal power allocation among the NOMA users of a subchannel should follow the water fillingbased approach. On the other hand, when the fairness of the system is considered, the order of the ideal power allocation should be inbetween the water fillingbased approach and the opposite of the water filling one. If we would fix the order of the power level among the NOMA users according to the water filling norm, it is very likely that an ideal power loading scheme^{2}^{2}2The proposed power loading scheme in this paper is particularly optimal for the applications that work in the low SNR regime. assigns very low power to the users with worse channel condition. In this case, those users (with worse channel condition) may not be able to decode their information or this is not a fair attitude to those users. Therefore, while considering the fairness of the system, we have adopted the opposite norm of the water filling technique so that the users with worse channel have fair provision. In other words, in order to reduce the overall computational complexity of the solution schemes and considering the fairness of the system, similar to Fang et al. (2016), the users with larger gain are assigned to lower power level compared to the users with lower gain. As a result, following the norm of the SIC technique, the set of users that cause interference to user on subchannel is given by . Consequently, the sumrate of subchannel , denoted by , can be obtained by the Shannon’s capacity formula Hazewinkel (2001).
If the circuit power consumption on subchannel is denoted by , the energy efficiency of subchannel is given by
In this work, given the power constraint of the BS, our objective is to allocate the subchannels in set N across all the users in set M so that the aggregate energy efficiency of all subchannels,
, is maximized. Clearly, this is an optimization problem. To formulate this problem, we define binary variables
. implies that subchannel is allocated to user , and means the other case. Ideally, more and more users are assigned to a subchannel, the better the system performance. However, due to the increasing level of interference with more and more assigned users and the decoding complication of the SIC technique, not necessarily more users assigned to a subchannel will enhance the system energy efficiency. While giving weight to this observation and insight, we assume that maximum users can be assigned to a subchannel^{3}^{3}3Setting to , the multiuser diversity of a subchannel can be fully exploited. Due to the limited power level and the superpositioncoded interference level caused by other users, an ideal resource allocation scheme should smartly select the users that are beneficial for that subchannel in terms of energy efficiency. Therefore, if the achieved instantaneous rate of user is and its minimum required rate is (to satisfy its qualityofservice requirement), the energyefficient downlink subchannel and power allocation problem in this context can be formulated as (2)(6). Note that the aggregate power level of subchannel is denoted by , and .(2)  
subject to  
(3)  
(4)  
(5)  
(6) 
In the above formulation, there are two types of variables, i.e., and . are the set of discrete variables, and the problem is NPhard because of these variables. The NPhardness property of this problem can be proved by mapping it to the classical relaxed binpacking problem Martello and Toth (1990). Let us assume that the subchannels in set N are the possible bins, the users in set M are items and the capacity of each bin is . Now, if we assume that an item can reside into multiple bins and if we transform the objective of the problem from minimizing the number of used bins to maximizing the system energy efficiency, reduction of our problem to the relaxed bin packing problem is completed. Thus, the NPhardness property of our problem is proved. On the other hand, even if the information of set are known, it is straightforward to prove that the problem is nonconvex with respect to (w.r.t.) . Moreover, in order to reduce the complexity of the solution method proposed in the following section, the ordering of the power level and gain of the users in set are considered to be fixed and given as follows. For any users and in set , if they follow , they must hold .
3 Solution Approach
In this section, we explore the solution approach of the energyefficient downlink resource allocation problem of a NOMA system described in (2)(6). Apparently, due to the discrete nature of subchannel assignment (i.e., variables ) and the continuous nature of power assignment (i.e., variables ), this is an MINLP problem. Furthermore, the assignment issue of users on the same subchannel brings further complication in the solution strategy. Therefore, we realize that decomposing the problem into a subchannel allocation subproblem followed by a power loading subproblem and then solving each subproblem based on the insights of the optimal solution, is a good strategy to solve the joint problem. Consequently, based on the assumption that the maximal power of the BS () is equally subdivided among all the subchannels, we solve the subchanneluser mapping subproblem using a manytomany matching model. Compared to the onetomany matching model adopted in Fang et al. (2016), we believe that a manytomany one can capture the structure of the problem well. This is because one user can be assigned to multiple subchannels in order to exploit the multiuser diversity of wireless systems, and one subchannel can be assigned to multiple users to take the advantages of the NOMA technology. While solving the subchanneluser mapping subproblem, given the power constraint of each subchannel, it is required to assign power optimally among the allocated users so that the overall rate of that subchannel is maximized. For this purpose, we have noticed that if the value of is larger than , the resultant optimization problem cannot be written as the difference of convex functions, and hence the DC programmingbased approach is not appropriate to solve this problem. However, we found that GP is a wellfit technique to solve this problem after approximating the objective function of the original problem by a ratio of two posynomials. For the solution of the second part of the problem, we assume that the subchanneluser mapping information are available. Given this information, Fang et al. (2016) has utilized the DC programmingbased approach to allocate power across the subchannels while ignoring the detailed granularity of the user power level. On the attempt of finding an elegant solution for the power loading subproblem, we find that the GPbased approach is a good one as well upon approximating the problem by a ratio of two posynomials.
3.1 Subchannel and User Mapping Scheme
Intuitively, assignment of many subchannels to a user and allocating multiple users to a subchannel (to follow the guidelines of the NOMA technique) is expected to enhance the overall energy efficiency of the system. However, due to the usage of practical, finitelength and possibly nonGaussian codes, we assume that maximum users can be assigned to a subchannel. Given the total power constraint of the BS, this problem is NPhard. The nature of the problem implies that a manytomany matching model Hamidouche et al. (2014); Roth (1984) is appropriate to capture the aforementioned behavior. Given that maximum users can be multiplexed to a subchannel, users in set M and subchannels in set N are two sets of players of this manytomany matching model. Note that each user can have infinite ( in practice) subchannels if possible. In this case, since the BS has the maximal power constraint , this should be subdivided equally among all the subchannels.
Definition 1: A manytomany matching model is a mapping from set M to set N such that every and satisfy the following properties:

and



if and only if
where is the set of partners for user and is the set of partners for subchannel under the matching model . The definition states that each user in set M is matched to a subset of subchannels in set N, and vice versa. However, before accomplishing these assignment operations, each user needs to have a preference list based on some criteria. The criterion of constructing the preference list for the users is based on their gain from the subchannels, which is similar to that in Fang et al. (2016). The preference of each subchannel is based on the overall benefit (i.e., energy efficiency) of the system. For example, if user chooses subchannel , this subchannel only accepts this user if and only the energy efficiency of the system is enhanced by this allocation.
To solve our subchanneluser mapping problem, we are interested to look at a stable solution, in which there are no players that are not matched to one another but they all prefer to be partners. Since the subchannel players give preference to the overall energy efficiency of the system while choosing partners from set M, a stable solution is considered to be an elegant solution of this problem^{4}^{4}4At this point, none of the players can enhance their performance further by choosing alternative partners, and hence this point is considered as a stable point.. In the manytomany matching models Roth (1984), many stability concepts can be considered based on the number of players that can improve their utility by forming new partners among one another. However, due to the large number of players () in our problem, identifying optimal subset of partners for a player is computationally intractable. Consequently, we choose to solve the matching problem by identifying the partners one by one from the opposite set. This way of choosing the partners in the matching relation brings pairwise stability. In Definition 1 and Definition 2, we highlight some properties of a pairwise stable matching relation. For the sake of these definitions, we define some notations. For example, denotes the choice set of user , which basically follows . Moreover, implies that subchannel prefers the users in set over that in set .
Definition 1: Consider that pair is not the element of matching model , that is and . Now consider another pair , that satisfies and . For the matching relation to be pairwise stable, it is not possible that both and are satisfied.
Definition 2: Let () be the set of matched users of subchannel under the matching relation , and . Consider is one of the matched users in set . Now, another user has come to be matched with subchannel . Note that . However, if user is replaced by user , the performance of subchannel is enhanced. Therefore, in order to have stable matching, should hold. This phenomenon of stable matching is called substitutability.
While satisfying the properties of a stable manytomany matching relation (e.g, substitutability Roth (1984)), we have proposed an algorithm in Algorithm 1. In order to bring the stability in this matching relation or enhance the overall energy efficiency of the system, we have adopted a few heuristics or strategies. The description of the algorithm is as follows. In order to realize the outcome of each step of the algorithm, we introduce one more type of set variable: that holds the allocated subchannels for user . First, and are initialized with . These sets are gradually filled up as we go through the iterations in between step and step . At the initialization phase, each user also constructs its subchannel preference list based on the descending order of their gain. Then, inside the outermost loop (between step and step ), if no assignment is possible, the algorithm terminates^{5}^{5}5At this point, it is assumed that the system has reached a stable situation or the improvement of energy efficiency is no longer possible.. Inside the inner loop (between step and step ), each user chooses its most preferred unallocated (and not rejected already) subchannel . At this point, two conditions are possible. The first condition is that the number of allocated users to subchannel can be less than (maximum allowable number of users per subchannel), and the second condition is the other case. If the first condition is true, we can apply the addition strategy for this subchanneluser assignment: user can be added to this subchannel and this strategy is added to set S (which was initialized before sorting out the possible strategies for use ). Whereas, for the second condition, only substitution strategy (already allocated user can be replaced by new user ) is possible. After filling the strategy set S, no matter the number of allocated users to subchannel is less than or equal to , the elements of S are filtered based on the previous sumrate of subchannel before this new possible assignment. The filtered strategy set is CS in Algorithm 1. Finally, strategy is chosen based on the sumrate ( in Algorithm 1) each strategy achieves. If is empty, the innermost loop continues, and the next user is chosen from set M for building its possible strategy set. For the other case, the corresponding strategy is executed. As a result, set , set (only for the substitution strategy) and set are updated. By analyzing the algorithm, we conclude Proposition 1, Proposition 2 and Theorem 1.
Proposition 1: Given a certain value of , allocating more and more users to a subchannel cannot enhance the energy efficiency further.
Proof: See Appendix A.
Proposition 2: Once a user is rejected while to be assigned with a subchannel through the addition or the substitution strategy, that rejection is final. More precisely, even if that user wants to be assigned to that particular subchannel later, this assignment no longer enhances the system energy efficiency.
Proof: See Appendix B.
Theorem 1: The subchanneluser mapping algorithm (i.e., Algorithm 1) is guaranteed to converge to a pairwise stable matching relation.
Proof: See Appendix C.
Computational Complexity of Algorithm 1: As discussed and proved in Proposition 2, if user is rejected by subchannel once, that user does not propose subchannel to be matched with anymore. This is because the energy efficiency of this subchannel is not enhanced in any way if previously rejected user is given preference in the following iterations. The enhancement of the energy efficiency for each subchannel implies the enhancement of the overall energy efficiency in the system. The termination of the outermost loop (between step and step ) depends on the improvement of the overall energy efficiency. To be precise, if user in the inner loop proposes a subchannel and the corresponding results in empty value, the next user in set M builds its possible strategy set. If none of the users in set M can be matched with any of the subchannel in set N, the outermost loop terminates. Therefore, intuitively, each user approaches at most subchannels to be matched with. Consequently, the outermost loop runs at most times. On the other hand, the complexity of finding the best strategy for each user inside the inner loop is , and the corresponding reason is as follows. If the number of users assigned to subchannel is less than , only one strategy (the addition strategy) is possible. If the number of users assigned to subchannel is exactly , then strategies are possible due to the possible replacement policies. Since the value of is usually not large in practice, we consider the complexity of finding the best strategy as constant. The remaining operations inside the innermost loop happen in constant time, and so we can ignore the complexity of these operations as well. Consequently, the overall complexity of the algorithm is .
3.2 Power Allocation Scheme
From Algorithm 1, we know the subchanneluser mapping information, i.e., and . This information is derived based on the assumption that the maximal power level of the BS is equally subdivided among all subchannels, i.e., . However, in (2), we see that the instantaneous rate of user assigned to subchannel is the increasing function of and the decreasing function of the interference power level caused by other users, i.e., . Consequently, even if the information about and are known, the power allocation across all subchanneluser slots, i.e., , is a nonconvex optimization problem. In the literature, the transformation of such a type of problem to convex form is not reported yet, and hence the dualbased method Wang et al. (2016) or the bisection search Sun et al. (2015) cannot be adopted to solve this problem. Therefore, our next objective is to design a method that allocates power across all the subchanneluser slots in such a manner that nearoptimality is achieved. We have adopted the GPbased optimization technique to solve this power allocation problem, the description of which is provided in the following discussions.
A GP Boyd et al. (2007); Chiang (2005)
is a type of mathematical optimization problem with the characteristics that the objective and constraint functions must be monomial(s) or posynomial(s). Two problems that we have discussed in the following have a certain form, such as
. Here, both functions and are posynomials; are the variables in the problem; and is the number of variables. This type of function is still not amenable to GP. To make this problem amenable to GP, there are some heuristics, such as the single condensation method and the double condensation method Boyd et al. (2007); Chiang (2005). We plan to apply the single condensation method in this context, which requires the denominator (i.e., ) to be approximated via some monomial . Given the values of , the unknown variables of this monomial can be approximated by and . As a result, the approximated objective function becomes the ratio of a posynomial and a monomial, which is a posynomial. Consequently, we require an iterative process in order to solve this problem step by step. The steps of this iterative process are provided in Section IIIA of Ruby et al. (2015).3.2.1 Joint Power Allocation among all Subchannels
Once the subchanneluser assignment information, i.e., , in order to obtain the energyefficient power level of each subchanneluser tuple given the power constraint at the BS, we need to solve an optimization problem, which is as follows.
(7) 
(8) 
The relations in (6) basically are associated with equations. The number of variables (i.e., ) in all these equations is . However, using (18) Fang et al. (2016), the number of variables in all these equations can be reduced to . Therefore, for , intuitively, by solving all these equations jointly, the relations in the last constraint of (3.2.1) are achievable. For the other case (i.e., ), the last constraint of (3.2.1) can be approximated as well based on some simple assumption. Moreover, on subchannel , the set of users, , that cause interference to user is determined based on the decoding order described in Section 2. Clearly, the objective function here is not amenable to GP (not even any GPbased heuristic method) because of the energy efficiency factor . Therefore, we have decided to expand the log series. Since the SINR of user on subchannel is larger than zero, we can expand the log function according to Nave . The accuracy of the log expansion is verified in Fig. 1. For the sake of simplicity, using the first term of the expanded log function, we transform the original objective function to a form, which becomes the ratio of two posynomials. As shown in Fig. 1, both the original log function and the first term of the expanded series are increasing with the increasing value of . Moreover, the difference between the original log function and the first term of the log series increases with the increasing value of . However, this difference is negligible when the value of is . If we look at the objective function in (3.2.1), is mapped to the SINR term of each user on a particular subchannel. The SINR of a user on a subchannel is a function of its and other users’ gain and assigned power level. If the values of gain and maximal power level of the system are such that the SINR of a user on a subchannel does not exceed dB (the absolute value of which is around 3), the resultant approximated problem using the first term of the log series does not deviate much from the original problem. On the other hand, the search range of the SINR for a user over a subchannel may exceed dB at higher gains and a larger power level of the system. However, the first term of the log series dominates the entire log function. Moreover, since the nature of the log function as well as the first term of the log series are increasing, the sum of the log function as well as the sum of the first term (of the log series) are increasing. Therefore, when the search range of a SINR exceeds dB, although the approximated function is deviated from the original function, the resultant solution of the approximated problem should not be very bad. The accuracy of the resultant solution can be improved by approximating the problem using the first two terms of the log series instead of only the first one. Since the problem is a maximization problem, we apply the inverse function^{6}^{6}6The inverse function of is . to transform it to the minimization one. Consequently, the problem becomes solvable via GP, as described in Section 3.2. The outcome of the proposed finegrained power allocation scheme strengthens the wellestablished truth Han et al. (2013) that the two objectives of maximizing the energy efficiency and maximizing the spectrum efficiency cannot be achieved simultaneously at the same operating point.
3.2.2 Power Allocation of a Subchannel
For subchannel , once and are known, we would like to determine . If , Fang et al. (2016) has proved that this problem can be solved using the DC programmingbased approach. However, if , this problem can no longer be solvable via this approach. To prove this statement, let us assume , and holds for subchannel . Under this circumstances, the instantaneous rate of the rd user on subchannel can be given by . The interference term in the instantaneous rate of the rd user is a barrier to express the problem in terms of a difference of convex functions. If the value of is even larger, the nonconforming nature of the problem to the DC programmingbased approach becomes even more severe. In general, the energyefficient power allocation problem of subchannel can be written as follows.
(9)  
(10) 
3.2.3 Joint ComputationallyEfficient Suboptimal Power Allocation among all Subchannels
The solution of the finegrained power allocation problem in Section 3.2.1 is computationally intensive due to the cumbersome iterative process of the single condensation method as well as the implementation complication of the offtheshelf GP solvers Grant and Boyd (2014)
. The complexity of this iterative process is proportional to the number of degrees of freedom in the system. If the number of users and subchannels in the system is moderately large, the solution of the problem cannot be obtained in a reasonable amount of time. Consequently, in this subsection, we have proposed another suboptimal solution for the finegrained power allocation across the subchannels and users with much less computational overhead even with the offtheshelf GP solvers
Grant and Boyd (2014). The mechanism of this suboptimal solution is developed based on the insights of the optimal solution, the description of which is given as follows. The idea of this suboptimal solution is to obtain the energyefficient solution of the subchannels one by one. For this, using (18) Fang et al. (2016), we plan to approximate the problem in (9)(10) to a onevariable optimization problem no matter the value of is. If the energyefficient power level of subchannel is , using (18) Fang et al. (2016), we have , where . Then, we can transform the problem in (9)(10) to either the problem in (11)(12) or that in (13)(14). Since we do not know the allocated power level of subchannel in advance, we assume that the maximal power level at the BS, , is the upper limit to be allocated at the beginning. It is proved in Fang et al. (2016) that is a concave function, and hence the DCprogrammingbased approach can be used to solve the problem in (11)(12). On the other hand, in (13)(14), both the and are the polynomial functions of variable . Using the first term of the log series, this problem can be transformed to a form which is the ratio of two posynomials as well. Consequently, we can adopt the GPbased single condensation method to obtain an elegant energy efficiency of subchannel . From our experiments, the latter solution approach provides better solution for this problem, and so we adopt it as a part of our proposed suboptimal energyefficient power allocation scheme. The number of iterations required for the single condensation method is constant as the number of optimization variable is one in this case no matter the value of for subchannel is.(11)  
(12) 
(13)  
(14) 
After solving the problem in (13)(14) for all the subchannels, the resultant total power can be or . For the latter case, the resultant total power, , can be considered as an elegant solution. On the other hand, for the former case, we need to revisit the solution of the optimization problem further. In order to develop a good solution for this case, we need to know the energy efficiency rate per unit of power level for all subchannels. Once these information are known, we can develop a suboptimal scheme, that assigns unit power to the subchannels one by one until the total allocated power to all the subchannels results in . In order to know which subchannel has higher rate of energy efficiency, we construct an energy efficiency rate matrix EEM, the dimension of which is . The st row of this EEM matrix contains the energy efficiency of the subchannels given that each subchannel is assigned with level of power. Each element of the nd row contains the subtracted energy efficiency when the corresponding subchannel is allocated with and level of power. Finally, each element of the last row contains the subtracted energy efficiency of the corresponding subchannel when it is assigned to and level of power. The energy efficiency of each subchannel is computed using the formulation in (13)(14) given the maximal power constraint designated for each element in matrix EEM. Based on the information of this matrix, power is assigned to each subchannel one by one in an iterative manner. For example, in one iteration, tuple in EEM matrix has been selected since it has the highest value. In this case, subchannel is assigned to level of power. This process continues until the allocated power level to all subchannels, , is equal to . The detailed steps of the entire iterative process are provided in Algorithm 2.
Computational complexity of Algorithm 2: Ideally, the GPbased single condensation method is supposed to adopt a conventional optimization solver, such as the interiorpoint method to solve the transformed convex problem in each iteration. Since the number of iterations is constant for the single condensation method to solve the problem in (13)(14), its computational complexity is equivalent to the complexity of the adopted convex optimization solver. In Nesterov and Nemirovskii (1994), it is shown that the interiorpoint method can obtain the optimal solution in polynomial time, and hence we can say that the complexity of the single condensation method to solve the problem in (13)(14) has a polynomial time complexity. Moreover, although the offtheshelf GP solvers do not solve a GP problem in a native manner, since the number of variable(s) of the problem in (13)(14) is one, the computational complexity of the resultant solution is constant and quick unlike that in Section 3.2.1. The steps in between and , the problem in (13)(14) needs to be solved times. If the resultant total power does not satisfy the power constraint, we need to construct matrix EEM. Except the st row, for each tuple of matrix EEM, the problem in (13)(14) needs to be solved twice while replacing by different values. Therefore, the process of constructing matrix EEM takes a polynomial time multiplied with . Since each tuple is marked once it is selected for the assignment of power, the steps in between and run at most times. As a result, the complexity of constructing matrix EEM is dominated by the rest of the other steps in Algorithm 2.
4 Performance Evaluation
In this section, via simulation, we evaluate the performance of our proposed energyefficient downlink resource allocation schemes. Followed by the methodology, we exhibit the detailed outcome of the simulation in order to verify the effectiveness of the proposed schemes.
4.1 Simulation Setup
The cellular network, that we consider, is isolated from the neighboring networks. It has a circularlike shape and suitable for an office environment. We place the BS at the center of the network, and the users are uniformly spread surrounding the BS within m distance. Typically, the minimum relative distance between two emploees in an office environment could be m, and hence we set the minimum distance between two users as m. Although we set a minimum relative distance between two users, the performance of each NOMA user is independent of this distance Shahab and Shin (2018). We also set the minimum distance from each user to the BS as m. As mentioned previously, continuous time is divided into frames. Each time frame is equivalent to s. During each time frame, the designated spectrum is equally subdivided among subchannels, and these subchannels are available to be allocated among users in the system. Each subchannel is assumed to have KHz bandwidth. According to Qiu and Chawla (1999), the theoretical limit of the channel capacity is given by , where denotes the Bit Error Rate (BER). The BER for each subchannel is configured as
. We consider that the shadow and Rayleigh are two main fading components of a wireless channel between the BS and a user in the system. The shadow fading follows the lognormal distribution with variance
.In order to calculate the quantity of the shadow fading over a subchannel, we assume the reference distance as km and the SNR for this reference distance is dB. The Rayleigh fading for all users over all subchannels follows the Rayleigh distribution with mean and variance. Using all these parameters, the gain of each subchannel for a user towards the BS is computed following () in Ruby et al. (2015). We employ the SIC technique in Vanka et al. (2012); Miridakis and Vergados (2013) for a user in order to decode the SCcoded signal of each subchannel transmitted by the BS.
In addition to implementing our proposed resource allocation schemes, we have implemented the scheme in Fang et al. (2016). In the following, this scheme is referred as “Scheme in Fang et al. (2016)”. In order to show that the manytomany matching model applied in the subchanneluser mapping algorithm outperforms the onetomany matching model, we have one version (i.e., Proposed Scheme1) which uses Algorithm 1 to solve the subchanneluser mapping problem and the DC programmingbased approach for the final power allocation across the subchannels and users. Note that in this version, for step of Algorithm 1, () Fang et al. (2016) is used. On the other hand, in order to show the superiority of our finegrained power allocation scheme via the GP technique over the DC programmingbased power allocation approach, we have another version (i.e., Proposed Scheme). This version adopts the subchanneluser mapping algorithm in Fang et al. (2016) for the first step, and adopts the scheme in Section 3.2.1 in order to have power allocation across each allocated subchanneluser slot. We have three more versions of our proposed schemes, namely Scheme, Scheme and Scheme, in order to exhibit a tradeoff between optimality and computational complexity. Both Scheme and Scheme adopt the GP technique in Section 3.2.1 for the final power allocation. However, Scheme uses () Fang et al. (2016) for step in Algorithm 1, and Scheme uses the GP technique in Section 3.2.2. Consequently, step in Algorithm 1 happens in constant time for Scheme version. On the other hand, this operation of Scheme has polynomial time complexity w.r.t. . Since the maximum value of is (e.g., in our simulation), this complexity of Scheme can be considered negligible, but provides better solution compared to Scheme. Moreover, our Scheme version uses the technique in Section 3.2.2 for step of the subchanneluser mapping algorithm, and uses Algorithm 2 for the final finegrained power allocation across the subchannels and users. The complexity of the suboptimal power allocation algorithm (i.e., Algorithm 2) is elaborately explained in Section 3.2.3. Apparently, both Algorithm 2 and the finegrained power allocation scheme in Section 3.2.1 have polynomial time complexity. However, since the complexity of the GP technique is computationally intensive with the large number of optimization variables (e.g., ), the proposed Scheme or Scheme may run very slowly with the growing number of users and subchannels in the system. On the other hand, although step of Algorithm 2 is associated with solving the problem in (13)(14) via the GP technique, the corresponding problem has only one optimization variable which results in constant time complexity for this step. Other steps of Algorithm 2 have constant time complexity, and hence the complexity of Scheme can be considered negligible compared to Scheme or Scheme. However, because of the intense search of the finegrained power allocation scheme in Section 3.2.1, Scheme and Scheme (especially Scheme) achieve much better performance compared to Scheme, which has been verified in the subsequent discussions. Finally, in order to demonstrate the global optimal performance, we apply the bruteforce search on both the subchanneluser mapping and power allocation problems for a system with and users. In the following subsection, for each data point, we conduct the simulation over times.
4.2 Simulation Results
For , and , Fig. 2 presents the increasing energy efficiency with the increasing number of users. Given the constant power level at the BS, the more the users, the higher the sumrate, and hence the larger the efficiency is. This trend is similar for all the schemes no matter it is ours or the scheme in Fang et al. (2016). Via the manytomany matching model, if a user has better gain over many subchannels compared to other users, that user can be assigned to as many subchannels as possible if such assignments increase the system energy efficiency. On the other hand, via the onetomany matching model, it is possible that a user with worse gain is forced to be assigned to a subchannel, despite such assignment is not necessarily beneficial for the system energy efficiency. This is because each user can be assigned to at most one subchannel via this matching model. As a result, both intuitively and empirically, we see the evidence of enhanced performance for our Scheme compared to that in Fang et al. (2016), even when we use (18) Fang et al. (2016) for step in Algorithm 1 and the DC programmingbased approach for the final power allocation. To provide further evidence, for , we plot Fig. (a)a and Fig. (b)b, which are the outcome of the subchanneluser mapping algorithms. In Fig. (b)b, we compare the number of allocated subchannels to each individual user (the users are sorted in the ascending order of their distance from the BS) between the onetomany and our model. From our observation, we see that via the manytomany matching model, a user can be assigned to multiple subchannels, and this number depends on the subchannels over which that particular user can achieve enhanced energy efficiency compared to other users. Our channel model is such that the quantity of the shadow fading is dominated by that of the Rayleigh fading. Therefore, a user that is closer to the BS is likely to have better gain over more subchannels compared to a farther user. Therefore, in Fig. (b)b, we see that a user that is the closest to the BS is assigned to the largest number of subchannels. On the other hand, based on the dynamics of users’ gain, in order to enhance the energy efficiency further, some subchannels accommodate more users compared to other subchannels.
In Fig. (a)a, we compare the number of assigned users to each individual subchannel between two subchaneluser mapping algorithms. In this figure, although , not necessarily all subchannels have allocated users. In general, it is seen that if the difference of gain between any two assigned users over any subchannel is larger compared to other subchannels, that subchannel achieves better sumrate as well as better energy efficiency. The numerical degree of users’ gain over a subchannel also plays a crucial role in the decision whether additional user will be allocated to that subchannel or not. Because of these user dynamics, some subchannels cannot be assigned to many users as the allocation of more and more users may decrease the sumrate as well as its energy efficiency, which is proved in Proposition 1. Due to the structure of the formulation, unlike the DC programmingbased approach, the GP technique can provide finegrained energyefficient power allocation across all subchanneluser tuples. Hence, the energy efficiency is much better for this case (i.e., Scheme) even if we use (18) Fang et al. (2016) for step of Algorithm 1. If we use the GP technique instead of (18) Fang et al. (2016) for step in Algorithm 1 (i.e., Scheme), we obtain better organization in the subchanneluser map due to the better power allocation among the users of each subchannel. Therefore, this scheme has the best performance compared to all the others. The elegance of the finegrained GP technique is further evident from the enhanced performance of Scheme over the scheme in Fang et al. (2016). In Scheme, we take the same subchanneluser mapping algorithm (via the onetomany matching model) as that in Fang et al. (2016). On the other hand, we argued previously that the finegrained power allocation using the GP technique is computationally quite intensive. Consequently, we developed Algorithm 2 in order to have finegrained energyefficient power allocation across all the allocated subchanneluser tuples in a low complexity manner. Because of adopting the insights of the optimal solution, the resource allocation scheme via this algorithm (i.e., Scheme) has very close performance to that of the Scheme and Scheme versions, and obviously outperforms the scheme with the DC programmingbased approach.
Given and , Fig. 6 presents total power consumption with the increasing number of users. This is natural that the more the users in the system, the diversity of the users that spread among all the allocated subchannels (i.e., multiuser diversity) increases. Therefore, with the increasing number of users, it is more likely that each subchannel is assigned to at least one user with good channel. On the other hand, from our observation, it is seen that if a subchannel has at least one user with good channel, it requires less power for that subchannel to reach the optimal energyefficient state. Consequently, a lower power level should be required to reach the optimal energyefficient state for a system with more users compared to that with less users. This observation and insights hold for both the GP technique and Algorithm 2. However, since the mechanism of Algorithm 2 is somewhat suboptimal (although developed based on the insights of the optimal solution) compared to that in Section 3.2.1, total consumed power in this case is slightly larger compared to the other one. The findings of this figure are quite interesting in a sense that the schemes with the finegrained power allocation via the GP technique and Algorithm 2 use much less power compared to that with the DC programmingbased approach. If we compare Fig. 2 with this figure, it becomes even more interesting as our schemes achieve much better energy efficiency using much less power compared to that with the DC programmingbased approach. On the other hand, the schemes with the DC programmingbased approach use full power of the BS, but incur much less energy efficiency. From the perspective of green communications, the results presented in this figure verify our original motivation towards pursuing this work.
Similar to our work, the conventional definition of energy efficiency is achievable blocks of bits from a channel under the usage of unit power level. The Shannon’s information capacity theorem has already established that two objectives, i.e., minimizing the consumed energy and maximizing the spectral efficiency are not achievable simultaneously at the same operating point. Consequently, under the consideration of fixed circuit power, there always exist two separate optimal points in the energy efficiency versus spectrum efficiency curve. In Fig. 7, we compare the energyefficient total throughput acheievd by all aforementioned schemes. The more the users, the better the utilization of limited resources because of the enhanced multiuser diversity. Hence, we see the increasing trend in the energyefficient total throughput, achieved by all the schemes, with the increasing number of users. However, due to the fact in the Shannon’s information capacity theorem, since the scheme in Fang et al. (2016) does not achieve the optimal energyefficient state, it is possible that the total energyefficient throughput achieved by this scheme is larger than our schemes. This is what observed in this figure.
In Fig. (a)a, given and , we show the energy efficiency with the increasing power at the BS. As we saw in the previous results that the scheme in Fang et al. (2016) fails to exploit the multiuser diversity of wireless systems as well as uses the DC programmingbased approach to obtain the coarsegrained power allocation, this scheme has the lowest energy efficiency no matter the power constraint of the BS is. For this case, we see that with the increasing power level, the trend of energy efficiency is decreasing. This is because there is a tradeoff between the transmission capacity and the energyefficient power consumption. Whereas, for our case, since the GP technique provides a finegrained elegant power allocation, the overall energy efficiency is much better compared to the benchmark scheme. Moreover, since the GP technique provides the unique solution while consuming much less power, no matter we increase the power level of the BS, the energy efficiency remains same at the unique point. Similar trend is observed in the case of our suboptimal Algorithm 2 although the energy efficiency achieved by this scheme is slightly lower compared to the finegrained GPbased power allocation scheme. In order to show the power consumption for this case, we plot Fig. (b)b. Since the optimal energyefficient state of the system is unique and our proposed GP technique is able to search this state to some extent, we see the constant level of used power no matter how much power the BS has. In the similar manner, given the subchanneluser mapping matrix, via our suboptimal Algorithm 2, we obtain an elegant energyefficient point for each individual subchannel. Consequently, although this suboptimal scheme does not achieve as good solution as by that in Section 3.2.1, the suboptimal unique energy efficiency is still achieved at the unique power level. Whereas, the schemes, which use the DC programmingbased approach, consume full power of the BS. Therefore, in this case, we see the increasing power consumption with the increasing total power at the BS.
In Fig. (a)a, given and , we show the decreasing overall energy efficiency with the increasing circuit power consumption. This is a natural trend achieved by all the schemes. Since circuit power is not used in enhancing system throughput but is added as consumed power, the overall energy efficiency is decreasing with the increasing circuit power. However, our subchanneluser mapping scheme can better exploit the multiuser diversity of wireless systems and we use the GP technique for the final finegrained energyefficient power allocation, our schemes, even via our suboptimal Algorithm 2, always outperform the scheme in Fang et al. (2016). In order to show the total power consumption in this case, we plot Fig. (b)b with the increasing . Increasing means, a subchannel requires higher power to reach its energyefficient state. Consequently, the optimal energy efficiency is achieved at larger power level with the increasing . For the similar reason, via our suboptimal Algorithm 2, the same trend is observed with the increasing . On the other hand, since the DC programmingbased approach uses full power of the BS no matter the value of is, the total power consumption is constant (i.e., ) via the scheme in Fang et al. (2016) and our Scheme version.
5 Conclusion
In this paper, we proposed an energyefficient downlink subchannel and power allocation scheme for NOMA systems with enhanced performance compared to the most relevant existing work in the literature. Due to the discrete nature of subchannel assignment and the characteristics of the NOMA technique, this is an MINLP problem. Therefore, similar to an existing work, we solved the problem via decomposing it into a subchannel allocation subproblem followed by a power loading subproblem. However, unlike the existing work, via a manytomany matching model, we better exploited the multiuser diversity of wireless systems in the solution of the subchanneluser mapping subproblem. In the second step, unlike the DC programmingbased approach, via the GP technique, we were able to allocate the power level across the allocated subchanneluser slots in a finegrained manner such that better energy efficiency is achieved compared to the benchmark scheme. Since the finegrained power allocation via the GP technique is computationally intensive using the offtheshelf GP solvers, we also proposed a suboptimal finegrained power allocation algorithm with much lower computational complexity. Under various realistic scenarios, extensive simulation had been conducted to verify that our scheme (even via our computationallyefficient suboptimal power allocation algorithm) can outperform the existing scheme while consuming much less power in the system.
Besides achieving an elegant energyefficient state via the better resource allocation schemes, the implication of this work is extended to a certain extent. Via our schemes (even the suboptimal one), since better energy efficiency is achieved at a lower power level, the interference effect to the neighboring networks is expected to be mitigated. At the same time, unused power in the system can be used for other purposes.
Appendix A Proof of Proposition 1
Consider a subchannel , and it has two allocated users, indexed by and . Moreover, we set , and there is one assumption, i.e., . It implies that the sumrate of user and user is larger than that of only user . Now, user has come to be assigned with subchannel . There are two possible conditions for this assignment, which are and . According to step of Algorithm 1, we only consider this user to construct an addition strategy if and only if the former case is true. In practice, both the conditions for any subchannel can happen. This statement can be proved from the result of Fig. 14. In this figure, we plot w.r.t. . It is obvious that can be both positive and negative. The energy efficiency of subchannel is enhanced if and only if the aforementioned value is positive, and user is not added to subchannel for the negative case. This completes the proof. Note that and in Fig. 14.
Appendix B Proof of Proposition 2
Without loss of generality, let us assume . Consider that subchannel has already users, indexed by and , and holds. At this point, user has come to be assigned with subchannel with . Since the nature of function is increasing w.r.t. (as shown in Fig. 15), we can conclude . Moreover, using (18) Fang et al. (2016) and then simplifying, and are given by
(15)  
(16)  
Since
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