Energy efficient OFDMA networks maintaining statistical QoS guarantees for delay-sensitive traffic

An energy-efficient design is proposed under specific statistical quality-of-service (QoS) guarantees for delay-sensitive traffic in the downlink orthogonal frequency-division multiple-access (OFDMA) networks. This design is based on Wu's $\textit{effective capacity}$ (EC) concept [1], which characterizes the maximum throughput of a system subject to statistical delay-QoS requirements at the data-link layer. In the particular context considered, our main contributions consist of quantifying the $\textit{effective energy-efficiency}$ (EEE)-versus-EC tradeoff and characterizing the delay-sensitive traffic as a function of the QoS-exponent $\theta$, which expresses the exponential decay rate of the delay-QoS violation probabilities. Upon exploiting the properties of fractional programming, the originally quasi-concave EEE optimization problem having a fractional form is transformed into a subtractive optimization problem by applying Dinkelbach's method. As a result, an iterative inner-outer loop based resource allocation algorithm is conceived for efficiently solving the transformed EEE optimization problem. Our simulation results demonstrate that the proposed scheme converges within a few Dinkelbach algorithm's iterations to the desired solution accuracy. Furthermore, the impact of the circuitry power, of the QoS-exponent and of the power amplifier inefficiency is characterized numerically. These results reveal that the optimally allocated power maximizing the EEE decays exponentially with respect to both the circuitry power and the QoS-exponent, whilst decaying linearly with respect to the power amplifier inefficiency.

ing fifth generation (5G) mobile communication systems [2], [3], which are expected to support bandwidth-thirsty delaysensitive multimedia services, such as ultra high-definition (UHD) video streaming [4].Meanwhile, the economical, environmental and societal pressures require a significant reduction of the carbon-footprint of the ubiquitous information and communication technologies (ICT), which will be responsible for 4 -6% of the annual global greenhouse gas emissions by 2020, unless the energy-consumption-per-bit is sharply reduced [5].Conventional designs of wireless communication networks have been dominated by improving the attainable spectral efficiency (SE), which was achieved by degrading the 5G design objectives concerning the energy-efficiency (EE) and delay.Therefore, an important research challenge for sustainable future wireless communication systems has been how to achieve significantly higher throughput (bits/second), while simultaneously improving the energy-efficiency (EE) and the delay.
According to the Shannon-Hartley theorem [6], in a pointto-point signal link having a given bandwidth W and additive white Gaussian noise (AWGN) power spectral density (PSD) N 0 , the maximum achievable transmission rate R [bits/second] of this link is logarithmically proportional to the transmit power P : Therefore, the relationship between the SE η SE = R W [bits/second/Hz] and EE η EE = R P [bits/second/Watt or bits/Joule] can be expressed as1 It is plausible that when η SE approaches zero, η EE converges to a constant 1 N0 ln 2 ; while if η SE tends to infinity, η EE approaches zero [7].As a result, in general the SE and EE of a communication system conflict with each other.
In order to achieve a desirable EE-SE tradeoff (EST), radio resources such as the available transmit power and bandwidth (e.g. the subcarriers in orthogonal frequency-division multipleaccess (OFDMA), which has been used in LTE-family of wireless standards), have to be appropriately allocated to different users.

B. Related Works
The SE-maximization problem has been studied in various contexts during the last few decades.By contrast, the EEmaximization became a hot topic in the resource allocation (RA) of wireless communication systems only recently.For instance, in [8], a general EST framework was proposed for the downlink OFDMA networks, where the overall EE, SE and per-user rate constraints were jointly considered, while a tight upper bound and lower bound on the optimal EST relationship were obtained based on Lagrangian dual decomposition.Additionally, it was demonstrated under this framework that the EE is a strictly quasi-concave function of the SE [8].Furthermore, energy-efficient RA in both the downlink and uplink of cellular OFDMA networks has been studied in [9].Explicitly, for the downlink transmission the weighted EE was maximized, while for the uplink it was the minimum individual EE that was maximized, both under certain prescribed per-user rate requirements.As a further advance, a series of optimization problems concerning both the SE and the spectral-normalized EE [bits/Joule/Hz] maximization in the context of multirelay aided OFDMA networks subject to a maximum total network transmit power budget were studied in [10]- [13].To elaborate a little further, [10], [11] considered the scenario where each network entity has only a single antenna, and the classic Dinkelbach's method was invoked for solving the resultant fractional programming problem.By contrast, [12] considered the more complex and generalized context where each network entity is equipped with multiple antennas, and the low-complexity Charnes-Cooper transformation method was employed for solving the resultant fractional programming problem.Furthermore, the EE optimization problem for the most complicated multi-cell multi-antenna multi-relay OFDMA networks was studied in [13].To achieve the optimum SE and/or EE, the emerging interference alignment (IA) technique was adopted for managing the multi-cell co-channel interference, which represents the first work having studied the EE of IA techniques.Another interesting contribution was provided in [14], where a multi-cell OFDMA network was considered, and a novel EST metric capable of simultaneously capturing both the EST relationship and the individual cells' preferences for the EE or SE performance, was introduced as the utility function for each base station (BS).However, the system's delay, which is a vitally important quality-of-service (QoS) metric for delay-sensitive multimedia applications in 5G communications, was not considered in [8]- [14].Since the achievable data rate varies as a function of the fading channel's quality, satisfying deterministic delay-QoS constraints is quite challenging, even impossible in some cases.As a result, satisfying statistical delay-QoS specifications for transmission over wireless channels becomes relevant, when the delay of certain services must be lower than a specific threshold for at least a certain percentage of time [15], [16].
Most of existing delay-QoS related contributions did not consider the system's EE [17]- [22].For example, in [17] the data-link layer's delay-QoS performance was characterized using a cross-layer model relying on the effective capacity (EC) concept [1], which has been recognized as a critically important metric for the statistical delay-QoS guarantees in wireless mobile networks.Based on this cross-layer model, a pair of adaptive RA schemes aiming for achieving the maximum EC over single-hop fading wireless links were proposed in [18], [19].Additionally, the authors of [20] investigated the EC of a cognitive radio relay network, when the secondary user transmission is subject to satisfying spectrumsharing restrictions imposed by a primary user.The authors of [21] proposed a delay-QoS-driven power allocation scheme for two-hop wireless relay links, while a delay-QoS-driven BS selection algorithm was proposed in [22] for satisfying multiple downlink users' delay-bound violation probabilities.
Nonetheless, there are a few seminal contributions related to the EE of delay-constrained systems.For example, in [23] the overall transmit power of vehicle-to-roadside infrastructure communication networks was minimized by jointly assigning power and subcarriers under delay-aware QoS requirements.More specifically, the authors of [23] developed a cross-layer framework where orthogonal frequency-division multiplexing (OFDM), which may be regarded as a special case of OFDMA, was employed at the physical layer, while the power-and the subcarrier-assignment policy operates at the data-link layer.Additionally, in [24] an energy-efficient RA scheme was proposed for multiuser cooperation aided OFDMA networks under a specific rate-QoS provision.To elaborate a little further, in [24] a joint power allocation, subcarrier allocation as well as mobile-relay selection algorithm was developed, aiming for maximizing the system's overall EE by taking into account different rate-QoS requirements.The authors of [25] indeed investigated the effective energy efficiency (EEE) maximization under the EC-based statistical delay-QoS constraint.However, they considered a simple point-topoint communication system, where only power allocation is involved [25].

C. Contributions of This Paper
Against the above background, in this paper we propose an energy-efficient RA strategy under a specific statistical delay-QoS provision for delay-sensitive applications in the downlink of OFDMA cellular networks.Furthermore, the impact of the circuitry power, of the QoS-exponent and of the power amplifier inefficiency is characterized numerically.These results reveal that the optimally allocated power maximizing the EEE decays exponentially both with the circuitry power and with the QoS-exponent, whilst decaying linearly with respect to the power amplifier inefficiency.The main contributions of this paper are significantly different from those of [24], although it is probably the most closely related work to ours.
• We consider a non-cooperative OFDMA network, while the RA in [24] was carried out by considering a usercooperation aided OFDMA network relying on timedivision duplex (TDD).• In the cross-layer optimization problem considered, only channel statistics are needed for obtaining both powerand subcarrier-allocation solutions, while the instantaneous channel state information (CSI) was required by the RA scheme of [24].As a result, our approach significantly simplifies the RA strategy to be used in the OFDMA networks that are capable of supporting delay-sensitive traffic.
• Our work invokes the EC concept instead of Shannon's channel capacity.As a result, we investigate the tradeoff between the EEE and the EC.By contrast, in most existing literature, such as [24], the tradeoff between the traditional EE and SE was studied.• In the particular optimization problem solved in this paper, the maximum delay bound and the probability of delay-QoS violation are characterized jointly with the aid of the statistical QoS-exponent θ.Furthermore, the minimum EC constraint is also investigated and incorporated in our optimization problem (not as a delay constraint though).By contrast, statistical delay-QoS concept was not considered in [24], where the delay tolerance was in fact implicitly mapped to a traditional minimum-rate requirement.The remainder of this paper is organized as follows.The preliminaries and an OFDMA power consumption model are introduced in Section II.In Section III, the EEE optimization problem is formulated.The solution approach combining Dinkelbach's method and Lagrangian dual decomposition is presented in Section IV.Our numerical simulation results are provided in Section V, which demonstrated the efficacy of the proposed algorithm.Finally, Section VI concludes the paper.

II. PRELIMINARIES
In this section, the data-link layer queueing model, the major concepts regarding the statistical delay-QoS guarantee, and the power consumption model invoked are briefly revisited for making the paper self-contained.

A. Queueing, Effective Bandwidth (EB) and Effective Capacity (EC)
There are two important concepts associated with the datalink layer's delay-bound violation probability, namely the EB [26] and the EC [1].Both of them rely on the queueing (first-in first-out buffering) model, which is employed for matching the source traffic arrival process and the network service process.As a benefit of the buffer, the queue prevents the loss of packets that could take place when the source rate is higher than the service rate, which is achieved at the expense of an increased delay.
1) Queuing-induced Delay: Assuming stationary arrival and service processes, at a given time instant t, the parameter θ, which is the so-called "QoS-exponent" representing the decay rate of the tail distribution of the queue length Q(t), satisfies [15], [16]: In other words, the probability of the queue length exceeding a certain threshold q decays exponentially as the threshold q increases.As a consequence, given a sufficiently large maximum tolerable stationary queue length q max , the following approximation is valid for the buffer-overflow probability [15]: By contrast, for a small q max , the following approximation was shown to be more accurate [1]: where α = Pr[Q(t) ≥ 0] denotes the probability that the buffer is not empty, which is approximated by the ratio of the average arrival rate over the average service rate [16].
Similarly, when the QoS metric of interest is delay, with D(t) denoting the delay experienced by a source packet arriving at time instant t with respect to the buffer, and upon assuming a maximum tolerable delay of d max [second], the following approximation holds: where δ is the fixed rate [bits/second] jointly determined by the arrival and service processes relying on a relationship between EB and EC, as detailed later.Explicitly, (6) indicates that the delay-bound violation probability must not be higher than ε.
To elaborate a little further, a smaller θ implies a slower rate of decay, which indicates that the system can only provide a looser delay-QoS guarantee.By contrast, a larger θ results in a faster rate of decay, which implies that a more stringent delay-QoS requirement can be supported.In particular, when θ → ∞, the system can tolerate an arbitrarily long delay.On the other hand, when θ → 0, the system cannot tolerate "any" delay, which corresponds to an extremely stringent delaybound.The statistical delay-QoS constraint of (6) may also be interpreted as the packet loss rate (PLR) requirement [27], because once the buffer is full and the delay is in excess of its maximum, the packets have to be dropped.Based on this relationship, from ( 6), the QoS-exponent for a certain user can be bounded as: When the delay bound d max is the main QoS metric of interest, we can further define the delay-QoS-exponent as θ D = θδ = − ln ε dmax .

2) Concepts of EB and EC:
The QoS-exponent θ > 0 or the delay-QoS-exponent θ D is of paramount importance in terms of characterizing the statistical delay-QoS guarantees, since they both characterize the exponential decay rate of the delay-QoS violation probabilities.
The stochastic behavior of a source traffic arrival process can be modeled asymptotically by its EB function B e (θ).More specifically, let us consider an arrival process {A(t), t ≥ 0}, where A(t) represents the amount of source data [bits] arriving over the time interval [0, t).Let us assume that the Gärtner-Ellis limit of the arrival process A(t), which is defined as the asymptotic log-moment generating function of A(t): does exist for all θ ≥ 0 and that Λ B (θ) is differentiable.Then, the EB function of A(t) is defined as [1], [26]: Analogously to the arrival process A(t), let the sequence does exist for all θ ≥ 0 and that it is differentiable for all θ ∈ R [15], where E(•) is the expectation operator with respect to R[t].Additionally, we assume that Λ C (θ) is a convex function.Then, the EC function of the service process R[t] under a given statistical delay-QoS requirement specified by the exponent θ > 0 is defined as [1]: It should be noted that when the sequence {R[i], i = 1, 2, . ..} associated with the service process R[t] is a statistically uncorrelated process 3 , the EC expression of ( 11) may be simplified as: It is important to note that the EC in ( 12) is a monotonically decreasing function of θ [17], [23].
Remark: The QoS of a user may be uniquely and unambiguously specified by the statistical QoS-triplet (δ, d max , α), and the EB may be interpreted as the minimum constant service rate required by a given arrival process for which the QoSexponent θ is fulfilled [18].Hence, the EC may be regarded as the dual concept of the EB.Since its inception, the EC has become an important data-link layer metric that provides unique insights into the entire network's performance in the presence of statistical delay-QoS limitations.
The classic large deviations theory was employed for the formulation of the EC, which incorporates the statistical delay-QoS constraints by capturing the decay rate of the buffer occupancy probability for large queue lengths.Since the average arrival rate is equal to the average departure/service rate when the queue is in its steady-state 4 , the EC can be physically interpreted as the maximum throughput of a system whose queue is in its steady-state [28], subject to the constraints imposed on the queue length/buffer-overflow probability of (4) or similarly on the delay-bound violation probability of ( 6), where α is almost surely equal to one.Viewed from a different perspective, the EC may also be interpreted as the maximum attainable service-rate as a function of the QoSexponent θ ≥ 0, or as the maximum constant arrival rate that a given service process is capable of coping with, whilst guaranteeing a statistical delay-QoS requirement specified by θ ≥ 0.
It is worth noting that the EC characterizes the attainable performance in the large-queue-length regime.By contrast, if the maximum tolerable queue length is finite and short, the maximum supported arrival rates δ will be smaller than that predicted by the EC.In such cases, packet loss events occur when the queue is full.As a result, packet retransmission may be required.Hence, systems having a limited queue length in general require more energy.On the other hand, the largequeue-length regime may be regarded as a fundamental limit that can be used as an important benchmark of buffer-aided wireless transmission systems [28].
Finally, in general the derivation of an analytical expression for the EC of an arbitrary stochastic service process remains an open challenge.However, when the service process can be characterized by an independent identically distributed (i.i.d.) process, the EC expression will be substantially simplified [27].

B. EC of OFDMA Systems
Using the result concerning ΛC (−θ) θ in [15] and [16,Sec. 7.2], the EC of a given statistical delay-QoS constraint θ was analyzed for a simple ON-OFF communication channel in [28].Herein, the analysis is extended to realistic OFDMA communication channels.
Let the sequence {R[i], 1, 2, . ..} be a statistically uncorrelated process.Then, R k can be invoked for representing the total amount of data bits delivered on the subcarriers occupied by user k within each frame-duration T f [second], i.e. we have , where φ k,n ∈ {1, 0} indicates whether the nth subcarrier is assigned to user k or not, and r k,n , as defined formally in (15), is the number of bits per frameduration T f .Furthermore, a feasible subcarrier assignment indicator matrix (K × N dimension) should satisfy: where K is the number of OFDMA users and N is the number of orthogonal subcarriers.The condition (13) indicates that at most only a single user is allowed to activate the nth subcarrier.
Hence, for the kth user, the EC corresponding to an OFDMA frame-duration can be formulated as: , where is the kth row of the power allocation matrix P defined in (18), while is the kth row of the subcarrier assignment indicator matrix φ φ φ.Furthermore, p k,n and g k,n respectively represent the transmit power and the channelpower-gain on the nth subcarrier, which is used for transmission to the kth user, with N 0 being the single-sided noisepower spectral density and B the bandwidth of a single OFDM subcarrier.The maximum instantaneous transmission rate for the kth user on the nth subcarrier in a single frame with duration T f is: Hereafter we assume that the statistical distribution of the channel-power-gain g k,n is known at the transmitter side.Therefore, the probability density distribution (pdf) of g k,n , namely f (g k,n ) is also known at the transmitter side.Furthermore, herein f (g k,n ) is assumed to be continuously differentiable with respect to g k,n .Hence, the expected value in ( 14) may be computed as:

C. OFDMA Power Consumption Model
In order to deal with the RA strategy of energy-efficient communication systems, every single term of the OFDMA system's power consumption must be taken into account, when formulating the optimization objective function.Herein, the total power consumption, which includes a static term and two dynamic terms, is expressed as where R represents the data rate, while P CS is the static circuit power consumption of electronic devices such as mixers, filters and digital-to-analog converters.The second term is associated with the power consumption of the radio frequency (RF) power amplifier (PA), where ̺ is the PA inefficiency.The third term in (17) represents a linear sum-rate dependent power dissipation, where the value of β ≥ 0 reflects the relative importance of this term.Depending on the specific values of β, the third term may represent the baseband back-end signal-processing power dissipation of the transmitter only, of the receivers only, or of both the transmitter and receivers [30].Note that herein a linear relationship between the data rate and the signalprocessing power consumption has been assumed.These three terms associated with the total power consumption are detailed below.
The total transmit power of a base station (BS) must be bounded and be nonnegative for any feasible power allocation policy.The corresponding power allocation matrix is described by: where P max represents the maximum total transmit power available at the BS's transmitter, while the instantaneous power p k,n transmitted on the nth subcarrier for the kth user can be mapped into the maximum instantaneous transmission rate r k,n .More specifically, from (15) we obtain: Furthermore, the static power consumption of the circuitry, namely P CS in (17), is determined by the active circuit blocks, such as the analog-to-digital converter (ADC), digital-to-analog converter (DAC), synthesizer (syn), mixer (mix), low power amplifier (LPA), intermediate frequency amplifier (IFA) as well as the transmitter and receiver filters (filt, filr) [31].Hence, the static power consumption of the circuitry can be decomposed into several terms as follows: P CS = 2P syn + P mix + P LPA + P filt + P filr + P IFA + P ADC .
As a result, the overall power consumption at the BS, namely (17), may be reformulated as: where the PA inefficiency ̺ is expressed as ̺ = PAPR ξ − 1 [31], with the numerator being the peak-to-average power ratio (PAPR) and ξ the drain efficiency of the PA.The parameter PAPR depends on the specific modulation scheme.Explicitly, the circuit power P C (φ φ φ, R) is modeled as a function of the data rate and the subcarrier allocation policy, yielding Linear sum-rate dependent power , (21) which contains a static term and a dynamic term, corroborating the power consumption model of (17).
Observe that the last term in ( 21) represents a second-order effect, which leads to slowly increasing values as the information rate increases.As a result, P CS of (21) becomes dominant.Hence, for the sake of simplicity, in this paper a constant circuitry power consumption model has been assumed, i.e.
It is worth noting that in this paper we mainly aim for maximizing the EEE subject to a given delay-QoS constraint of a realistic OFDMA network.Note that the EC C e (θ) can be considered as the maximal throughput per frame-duration under the QoS-exponent θ.Therefore, by interpreting θ as the delay-QoS constraint, it is possible to formulate an equivalent problem, which aims for maximizing the EC for a given statistical delay-QoS constraint.As a result, we can further maximize the EEE, which can be simply formulated as the ratio of the EC to the total network's energy consumption, in bits Joule .In Sec.III and Sec.IV, we will focus our attention on the problem formulation, as well as on designing the corresponding iterative RA algorithms, respectively.

III. FORMULATION OF THE DOWNLINK OFDMA EEE MAXIMIZATION PROBLEM
In this paper, the downlink of an OFDMA network having N subcarriers and a total bandwidth of N B is considered.As shown in ( 4) and ( 5), since the approach adopted is based on asymptotic analysis, the buffers at the BS are assumed to be large enough and always full, so that no empty scheduling slot is caused by having insufficient source packets in the buffers.

A. The Original EEE-Maximization Problem
Before presenting our EEE-optimal design, let us formally define the EEE for the downlink OFDMA network as the ratio of the overall EC to the total consumed energy in [bits/Joule]: where P T (φ φ φ, R, P) is given by ( 17) and (20).Note that the EEE definition of ( 22) considers the delay-QoS requirements specified by θ.In this definition, the EEE is described as a delay-QoS-guaranteed metric.Hence, our EEE-optimal design conceived for the downlink of OFDMA systems can be formulated as the EEE maximization under statistical delay-QoS guarantees according to: s.t.C1: Constraint C1 holds for the minimum EC that the kth user should achieve.C2 ensures that the total power allocated to the N subcarriers of K users does not exceed the maximum transmit power P max available at the BS.Constraints C3 and C4 are imposed in order to guarantee that each subcarrier is used at most by one user, hence avoiding inter-user interference.The feasible region for the optimization variables φ φ φ and P is described by the constraints C1 − C5.
Additionally, at a given time instant, the channel-powergains of the different OFDMA subcarriers belonging to a specific user, for example g k,n , n = 1 . . .N for the kth user, may be modelled by independent identically distributed (i.i.d.) random variables.As a result, we can simply use f (g k,n ) or f k,n , n = 1 . . .N to represent the pdf of the channel-powergain on each subcarrier.
The EEE optimization problem (23) can be classified as a nonlinear fractional program [32], [33], whose objective function is the ratio of two functions and it is generally a nonconvex (non-concave) function.In the following, we will show that the EC of Rayleigh fading channels (RFC) is a concave function, while the EEE function is quasi-concave, which is consistent with the above statement.More specifically, the numerator of the objective function of ( 23) is concave with respect to (w.r.t.) the variables φ k,n and p k,n , since it is the non-negative sum of multiple concave functions.Furthermore, the denominator is affine, i.e. convex as well as concave.It is well known that for this kind of objective function, the problem is quasi-concave [34].The proof of these properties is offered in Lemma 2 and Appendix A.

B. Relaxations of the EEE-Optimal Design
In order to conceive an EEE-optimal design we have to solve Problem (23) to find the optimal subcarrier and power allocation.In fact, the subcarrier allocation itself is a combinatorial integer programming problem, which is in general NP-hard.Hence, introducing a relaxation into the subcarrier constraints makes Problem (23) more tractable.The approach adopted herein for the mixed-integer programming problem5 of (23) relies on approximating the integer part of Problem (23) by its continuous relaxation, since in general continuousvariable based optimization problems are easier to solve than discrete-variable based combinatorial optimization problems.The idea of continuous relaxation is to enlarge the feasible set, while making sure that it includes, but is not limited to, all feasible solutions that satisfy the original constraints [35], [36].Therefore, instead of forcing the optimization variable (subcarrier occupancy indicator) to be either 0 or 1, the constraint (C4) in Problem ( 23) can be relaxed to The relaxation of the subcarrier assignment variables, allowing them to take continuous values over the [0, 1] interval, is equivalent to the multi-user time-sharing of each subcarrier over a large number of OFDM symbols [11], [36], [37] and generally does not solve exactly the original problem.Wireless communication channels are typically time varying and the channels may not stay unchanged long enough for timesharing to be feasible [36].Fortunately, it has been shown that the solution of the relaxed problem under the time-sharing condition is arbitrarily close to the solution of the original problem, when the number of subcarriers tends to infinity [37].
In fact, the gap between the two solutions can be small even for a small number of subcarriers [11], [37], [38].
Hence, this relaxation is applied to the subcarrier assignment indicator set of (13), to the power allocation set of (18), to the achievable rate of (15) and to the overall OFDMA EC of ( 14), respectively as follows: where the new subcarrier assignment index φ k,n is a continuous variable in the interval [0, 1], and it can be interpreted as the portion of subcarrier n assigned to user k, i.e. we have [39], [40], or interpreted as the timesharing factor of subcarrier assignment [11].Hence, instead of restricting the boundaries of the partitions between the two users to align with the bin boundaries as the integer programming does, in [40] the boundary is allowed to be anywhere in the bin, hence relaxing the integer programming problem into a continuous-variable optimization problem.As a beneficial result of the φ φ φ-relaxation, the following variable transformations can be introduced: Then, a modified version of the original EEE-maximization problem of (23) may be obtained as: s.t.C1:

C. Calculation of the EC for NLOS Rayleigh Fading Channels
When a non-line-of-sight (NLOS) Rayleigh fading propagation channel is considered, the channel-power-gain g k,n is an exponentially distributed random variable.As a result, the expectation in ( 14) is readily obtained by: ℓe −ℓg k,n dg k,n .
Employing the following substitutions: and assuming ℓ = 1, while A k , D > 0, the integral may be calculated as: where E n (x) is the exponential integral function.From ( 14), ( 29) and ( 30) the EC of the kth user can be calculated for a RFC as: and the system's total EC is written as: while the relaxed form of (32) may be directly defined as: where The concavity of the system's EC is discussed in the Proof of Lemma 1.
Lemma 1: For NLOS Rayleigh fading channels, the relaxed EC function (33) of the system is concave in both p k,n and φ k,n .
Proof: See Appendix A.

IV. AN ALGORITHM FOR SOLVING THE OFDMA EEE-MAXIMIZATION PROBLEM
The energy efficiency of wireless networks may be defined as the number of transmitted bits per unit of energy [Joule].Hence, given the EC defined for Rayleigh fading channels in (32), we may define the system's EEE in [bits/Joule] as: , where again, ̺ is the PA inefficiency and P C the circuitry power dissipation at the BS.Therefore, the EEE optimization problem of OFDMA systems operating in NLOS Rayleigh fading channels under a specific statistical delay-QoS provision is formulated as: Observe in (35) that the EEE is the ratio of a nonnegative weighted sum of concave functions over a nonnegative affine function.Therefore, the following Lemma holds: Lemma 2: The EEE function η RFC E of ( 35) is quasi-concave.Proof: From Lemma 1 we infer that η RFC E is the ratio of a concave function to an affine positive function.According to [41, Table 5.5 on P. 165] this ratio results in a semi-strictly quasi-concave function.
Therefore, the EEE optimization problem (36) and its relaxed form relying on ( 33)-( 34) are concave fractional programming problems, whose objective functions are cast in a fractional form.In order to solve the above fractional programming problem, Dinkelbach's classic method [32], [33] may be invoked.

A. Dinkelbach's Method
Since concave-convex fractional programs share important properties with concave optimization problems, it is possible to solve concave-convex fractional programs with the aid of standard methods developed for concave optimization problems.Here, we use Dinkelbach's method [32], [33], which operates in an inner-outer iteration manner.
Upon using Dinkelbach's iterative method [32], [33], the quasi-concave problem posed in (23) can be solved in a parameterized concave form.To elaborate a little further, the original concave-convex fractional program has a form similar to maximize where F is a compact, connected set and z(x) > 0 is assumed.
For the sake of notational simplicity, we define F ⊃ {Φ, ℘} as the set of feasible solutions of the original optimization problem described by (23).The original problem can be associated with the following parametric concave problem [32], [34]: where q ∈ R is treated as a parameter.The objective function, which is denoted hereafter by F (q) for this parametric problem, is convex, continuous-valued and strictly decreasing.Additionally, without loss of generality, we define the maximum EEE q * of the system considered as: It is plausible that we have    F (q) > 0 ⇔ q < q * F (q) = 0 ⇔ q = q * F (q) < 0 ⇔ q > q * .Hence, Dinkelbach's method presented in Algorithm 1 solves the following problem: C e (φ φ φ, P) − q U P (φ φ φ, P), (38) which is equivalent to finding the root of the nonlinear equation F (q) = 0.

Algorithm 1 Dinkelbach's Method
Input: Solve Problem (38) with q = qn to obtain φ φ φ * and P * ; qn+1 ← Ce(φ φ φ * ,P * ) UP(φ φ φ * ,P * ) ; Dinkelbach's method in fact constitutes the application of Newton's method to a nonlinear fractional program [42].As a result, the sequence converges to the optimal point at a superlinear convergence rate [33].In summary, Dinkelbach's method [32] is an iterative technique of finding the increasing values of feasible q by solving the parameterized problem of max φ φ φ,P F (q n ) = max φ φ φ,P {C e (φ φ φ, P) − q n U P (φ φ φ, P)} at the nth iteration.This iterative process continues until the absolute difference value |F (q n )| becomes less than or equal to a prespecified tolerance threshold ǫ.
The parametric version of the relaxed EEE-maximization problem of ( 28) is described as: maximize C e (φ φ φ, P) − q U P (φ φ φ, P), s.t.C1: Since this is a concave problem and the conditions (C1), (C2) and (C3) satisfy Slater's conditions [43], one can solve the dual problem to obtain the primal solution with zero duality gap.Therefore, the Lagrangian over P and φ φ φ for the optimization problem of ( 39) is presented in (40).Additionally, the following relationship holds: where the Lagrange dual function is given by sup L, i.e. by the supremum of the Lagrangian.The relationship in ( 41) is further developed in (42), which leads to the conclusion that the dual problem of arg min ν,λ sup P,φ φ φ L(P, φ φ φ, ν, λ) may be solved by solving KN subproblems of the form presented in (43), while the dual variables ν and λ can be updated by applying the subgradient method of [44].Since Problem ( 43) is in a standard concave form, the Karush-Kuhn-Tucker (KKT) first order optimality conditions of [44] may be used for finding the problem's optimal solution.The next two subsections deal with the updating process of the primal and dual variables.
1) Updating the Power and Subcarrier Allocation: For a fixed λ, ν k and q i−1 , we may solve max P,φ φ φ L(P, φ φ φ, ν, λ) in order to obtain the optimal power and subcarrier allocation.Therefore, the following condition is both necessary and sufficient for the power allocation's optimality: which is equivalent to finding the specific point given by ( 45).This point can be computed using Newton's method.
Once the optimal power allocation (P * ) has been calculated, the optimal subcarrier allocation may be obtained through: where we have (46) the derivative is independent of φ φ φ.Therefore, its value means that either the optimal value occurs at the boundaries of the feasible region, and thus L(P, φ φ φ, ν, λ) must be a decreasing function within the feasible region, or the derivative is null and hence the optimal subcarrier allocation is obtained inside the feasible region.Since only a single user is allowed to transmit on each subcarrier, the following condition may be applied in a Gauss-Seidel fashion [45] when designing the iterative algorithm, where Φ n is the nth column of Φ.Indeed, in the Gauss-Seidel-type iterative algorithms only a single dimension is considered at each iteration.Hence, this type of iterative algorithms are said to be sequential.For example, the iterative algorithm designed herein applies the condition (47) to each subcarrier sequentially, rather than in parallel.This process is illustrated by the loop starting from Line 7 in Algorithm 2, where the power allocation procedure is executed for each subcarrier of every user in the system and then the condition ( 47) is applied to this particular subcarrier.
It is worth noting that since φ k,n only assumes binary values and the condition (47) implies that only a single user is assigned to each subcarrier.As a consequence, the constraints C3 and C5 are implicitly satisfied.Furthermore, the condition C4 is satisfied by the assumption that U p (φ φ φ, P) is a positive affine function, as shown in Lemma 2. Thus, the conditions C3-C5 may be omitted in the Lagrangian function (40).
2) Updating the Dual Variables: In order to update the dual variables λ and ν, one may use the subgradient algorithm, whose equations are presented in (48) and (49).The parameters α λ and α ν are the appropriate step sizes of the subgradient algorithm.

B. Dinkelbach-Lagrange Dual Decomposition Algorithm
The algorithm developed for our EEE-optimal design of OFDMA networks under statistical delay-QoS provision is summarized in this section.The main loop of the proposed algorithm is composed by Dinkelbach's algorithm illustrated in Algorithm 1.The Lagrange dual decomposition procedure is used for solving the inner loop, which is equivalent to solving Problem (38).The pseudo-code in Algorithm 2 implements the entire power and subcarrier allocation policies for our EEE maximization problem.The variables used throughout each algorithm are presented at the end of Algorithm 2, while the identifiers in round brackets indicate to which procedure the variable belongs: (L) for the Lagrange dual decomposition method and (D) for Dinkelbach's method.
The underlined while in Line 3 represents the main Dinkelbach loop, while the non-underlined while in Line 6 corresponds to the Lagrange dual decomposition method's main loop or, alternatively to Dinkelbach's method's inner loop.

V. SIMULATIONS AND NUMERICAL RESULTS
In order to illustrate the algorithm's performance in solving our EEE-maximization problem, numerical simulations were conducted.The adopted simulation parameter values for the downlink of the OFDMA system considered are presented in Table I.
Aiming for comparing different scenarios associated with different solutions and for evaluating the impact of the parameter values on the solution of the EEE-maximization problem, in Table II we summarize four different scenarios: the first while j ≤ I dd or |λ(j + 1) − λ(i)| > ǫ λ and | min(ν(j + 1) − ν(j))| > ǫν 7.
for n from 1 to N 8.
for k from 1 to K 9.
end for 11.
end for 13.
i ← i + 1; 17. end while --------------P = initial power allocation matrix; P * = optimal power allocation; φ φ φ = initial subcarrier allocation matrix; φ φ φ * = optimal subcarrier allocation; I dd = maximum number of iterations (L); I Dink = maximum number of iterations (D); ǫ Dink = Dinkelbach algorithm precision (D); ǫ λ = power allocation precision (L); ǫν = subcarrier allocation precision (L).two scenarios are simple and were used for investigating the impact of each system parameter on the result of the EEEoptimization problem.The third and fourth scenarios are more realistic, with a larger number of subcarriers, wider subcarrier bandwidth and more users.Therefore, they are more complex to deal with.
In order to observe the relationship between the EEE and the EC, we present Fig. 1 which illustrates the contour plot of the EEE surface with respect to the total transmission power of User 1 and 2 in Scenario 1 of Table II.Since we have θ 1 < θ 2 , two subcarriers are allocated to User 1, while User 2 only has a single subcarrier to transmit information.The figure also presents the maximum EC line (black dashed line).For a given P max the black dashed line shows the optimal power allocation policy that achieves the maximum EC.It is noteworthy that the   sup P,φ φ φ L(P, φ φ φ, ν, λ) = arg max

Parameters
maximum EC line is not far from the maximum EEE point, as seen in the zoomed-in part of Fig. 1.Fig. 2 shows the convergence of Algorithm 2 in terms of the total transmission power [Fig.2(a)] and EEE [Fig.2(b)] for Scenario 1 of Table II.Note that the maximum EEE curve in Fig. 2 was found through an exhaustive search method considering both the subcarrier allocation and power allocation domains.We observe from Fig. 2 that the algorithm requires only 6 iterations to converge 6 .Moreover, Fig. 3 depicts the typical convergence profile for the proposed Algorithm 2 in terms of the total transmission power and EEE for a more realistic system configuration, namely for Scenario 4 of Table II, with a product of KN = 6400.The maximum achievable EEE is not shown in Fig. 3, since there are 2 N possible subcarrier allocation matrices and hence an exhaustive search becomes computationally prohibitive.We can see that full convergence to the EEE-optimal design is achieved by the proposed algorithm after 5 Dinkelbach iterations within a precision of ǫ = 10 −6 .Additionally, it is noteworthy that the EEE values achieved for this realistic scenario are significantly lower than those of the less realistic Scenario 1.For example, by comparing the achievable EEE shown in Fig. 2 (under Scenario 1, with product KN = 6) and Fig. 3 (under the realistic Scenario 4), we can see that the EEE of Scenario 1 is almost 100 times higher than that of Scenario 4. This difference is mainly due to the different circuitry power consumption values P C , which has been increased from 20dBm = 100mW in Scenario 1 to 50dBm = 100W in Scenario 4. This result corroborates our previous discussions concerning (21) on the importance of the fixed circuitry power consumption P C at BSs.Furthermore, by jointly considering multiple representative simulation scenarios, it is possible to evaluate the average number of iterations required by the proposed algorithm for achieving convergence.To elaborate a little further, Table III presents the average number of iterations (over 100 realizations) at which the proposed algorithm converges (i.e., ǫ ≤ 10 −6 ) under different values of (K, N ).It is noteworthy that the increase in problem dimensions, represented by the product KN of Scenario 3, only imposes a modest impact on the number of Dinkelbach iterations required.As seen from Algorithm 1 and ( 22), the computational complexity per Dinkelbach iteration is roughly the same in terms of complexity order, with K and N only slightly affecting the number of simple summations.Hence, the computational complexity required by Algorithm 2 to achieve convergence also increases modestly with KN .
In order to gain further insights into the EEE-maximization problem considered, the impact of three parameters of paramount importance are evaluated by considering the associated optimal transmit power of User 2, i.e., p * 2 .The first parameter examined was the QoS-exponent θ, which has a direct relationship both to the maximum delay bound d max and to the probability of not exceeding this bound, ε.Hence, in Fig. 4 we show the optimal power allocation for User 2 considering different values of its QoS-exponent θ 2 in the interval of [0.1; 2], while keeping the QoS-exponent of User 1 at θ 1 = 0.1.The values of the other parameters are the same as those of both Scenario 1 and Scenario 2. Note that for Scenario 1, which has three subcarriers, two of them are allocated to User 1, while User 2 receives information through only one of the OFDMA subcarriers.The results of Fig. 4 demonstrate the impact of different values of θ on the optimal power allocation that achieves the maximum EEE.Since θ is related to both the maximum delay bound d max and its violation probability ε, physically it can characterize both stringent delay-QoS requirement (larger θ) as well as loose delay-QoS requirement (smaller θ).For instance, θ = 1 can represent ε = 10% probability of violating a delay-limit of d max = 2.3 seconds,   II.The black dashed line in (a) or the black solid line in (b) represent the optimal power allocation policy that achieves the maximum effective capacity.The red circle shows the optimal power allocation that achieves the maximum EEE.Note that p 1 represents the total power that is used for transmission from the BS to User 1 over all the subcarriers allocated to this user.
or ε = 50% probability of violating d max = 0.69 seconds; θ = 0.23 can indicate ε = 10% probability of violating d max = 10 seconds, as predicted by (7).The following conclusion can be drawn directly from Fig. 4: the optimal power allocation policy has an exponential decay dependence with respect to the QoS-exponent θ, implying that a lower delay tolerance, i.e, a smaller d max or a larger θ in (7), requires a lower transmit power to achieve the optimal EEE and viceversa.
The second parameter studied is the circuitry power P C .Fig. 5 depicts the optimal power allocation value for User 2 in both Scenario 1 and Scenario 2, where the EEE is maximized.In contrast to the impact of the QoS-exponent, as the circuitry power consumption increases, the optimal power allocation value increases linearly with it.
The third parameter investigated is the PA inefficiency ̺.From its definition given in Section II-C, we know that  II.
̺ is directly proportional to the PAPR value and inversely proportional to the drain efficiency of the PA.Fig. 6 shows the optimal power allocation policy for User 2 in both Scenario 1 and Scenario 2. As we may observe in this figure, the optimal power allocation value decays exponentially with the PA inefficiency, but smoother than the trend is for the QoSexponent.In fact, if we consider the scenario of a single user and a single subcarrier, the EEE function will result in a well-known bell-shaped curve.Hence, increasing either the PA inefficiency or the QoS-exponent basically shifts the optimum point to the left of the original optimum point, while increasing the circuitry power shifts the optimum point to the right of the original one.

VI. CONCLUSIONS
In this paper we have demonstrated the concavity of the EC function and the quasi-concavity of the EEE function.The relaxed EEE-maximization problem was reformulated for using Dinkelbach's method, which is capable of solving a more tractable parameterized version of the original fractional programming problem.The Lagrange dual decomposition method was invoked to solve the sub-optimization-problem  II.
that emerges in the inner loop of Dinkelbach's method.Our numerical simulation results have demonstrated that the proposed algorithm is capable of converging to the optimal solution in a small number of iterations, even under realistic scenarios associated with large system dimensions quantified in terms of the product of the number of users and subcarriers, i.e.KN .We also offered an investigation concerning the system parameters in order to quantify how each of the three key parameters impacts the EEE function maximization, which facilitates a deeper understanding of the importance of these parameters in circuitry and infrastructure design.

APPENDIX A PROOF OF LEMMA 1
Proof: We commence the proof by showing that: is concave.Since C(•) is twice differentiable, the second-order test may be applied to verify its concavity.Thus, the Hessian  matrix H of C(•) is: where C pp , C pφ , C φp and C φφ are defined in (50).From (50) we may conclude that H is a (2×2)-element symmetric matrix and the following statements are equivalent [47, Theorem 1.10, p. 11]: 1) H is semidefinite negative; 2) All principal minors of H are nonpositive.
In fact it may be easily verified that both principal minors M i of H are nonpositive for any p k,n ≥ 0 and φ k,n ∈ [0, 1]: Naturally, M 2 is nonpositive since it is the negative counterpart of a quadratic term.However, it is not easy to observe that C pp ≤ 0 by simply checking the expression in (50).In order to show that the inequality holds, let us consider, without loss of generality, that φ k,n = 1.Fig. 7 illustrates the regions, specified by N 0 B, p k,n and A k , where M 1 is satisfied.Other alternative methods for demonstrating the validity of this inequality include demonstrating that the second derivatives regarding p k,n , N 0 B and A k are negative.However, this is omitted here due to space limitations.As shown in Fig. 7, M 1 holds for any p k,n ≥ 0. Since M 1 grows linearly with φ k,n and φ k,n is nonnegative, the only condition for M 1 to hold is p k,n ≥ 0. Therefore, the Hessian is semidefinite negative, which implies that C(p k,n , φ k,n ) is concave.
Finally, according to [44, p.79] (operations that preserve convexity), the following statement is true: if C(p k,n , φ k,n ) is concave, then C e (P, φ, θ) is concave, because it is the nonnegative weighted sum of concave functions.

Fig. 1 .
Fig. 1.EEE contour (a) and surface (b) for Scenario 1 of TableII.The black dashed line in (a) or the black solid line in (b) represent the optimal power allocation policy that achieves the maximum effective capacity.The red circle shows the optimal power allocation that achieves the maximum EEE.Note that p 1 represents the total power that is used for transmission from the BS to User 1 over all the subcarriers allocated to this user.

Fig. 2 .
Fig. 2. Typical convergence profile of Algorithm 2 in terms of the total transmission power (a) and EEE (b) for Scenario 1 of TableII.

Fig. 3 .
Fig. 3. Typical convergence profile of Algorithm 2 in terms of total transmission power (a) and achievable EEE (b) for a realistic system configuration of Scenario 4 in TableII.

Fig. 4 .
Fig. 4. Optimal power allocation for User 2 considering different values of θ 2 within the interval [0.1; 2], while θ 1 is fixed to 0.1, and the values of the other parameters are the same as those of Scenario 1 and Scenario 2.

Fig. 5 .
Fig. 5. Optimal power allocation for User 2 considering different values of PC in both Scenario 1 and Scenario 2.

N0B p k,n 2 θ k p 3 k,n E A k N0B p k,n 2 ,C 1 N0BFig. 6 .
Fig. 6.Optimal power allocation for User 2 considering different values of ̺ in both Scenario 1 and Scenario 2.

Fig. 7 .
Fig. 7. Illustration of the 3D region plot, where the inequality Cpp ≤ 0 holds, i.e. we have a polyhedric convex set.The polyhedron base is formed by N 0 B and p k,n axes, while its height is A k .
2, . ..} represent a discrete-time stationary and ergodic stochastic service process and R[t]