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Statistical Delay QoS Driven Energy Efficiency andEffective Capacity Tradeoff for Uplink Multi-User

Multi-Carrier SystemsWenjuan Yu, Leila Musavian and Qiang Ni

Abstract—In this paper, the total system effective capacity(EC) maximization problem for the uplink transmission, in amulti-user multi-carrier OFDMA system, is formulated as acombinatorial integer programming problem, subject to eachuser’s link-layer energy efficiency (EE) requirement as wellas the individual’s average transmission power limit. To solvethis challenging problem, we first decouple it into a frequencyprovisioning problem and an independent multi-carrier link-layer EE-EC tradeoff problem for each user. In order to obtainthe subcarrier assignment solution, a low-complexity heuristicalgorithm is proposed, which not only offers close-to-optimalsolutions, while serving as many users as possible, but alsohasa complexity linearly relating to the size of the problem. Afterobtaining the subcarrier assignment matrix, the multi-carrierlink-layer EE-EC tradeoff problem for each user is formulatedand solved by using Karush-Kuhn-Tucker (KKT) conditions.The per-user optimal power allocation strategy, which is acrossboth frequency and time domains, is then derived. Further, wetheoretically investigate the impact of the circuit power and theEE requirement factor on each user’s EE level and optimalaverage power value. The low-complexity heuristic algorithmis then simulated to compare with the traditional exhaustivealgorithm and a fair-exhaustive algorithm. Simulation resultsconfirm our proofs and design intentions, and further showthe effects of delay quality-of-service (QoS) exponent, the totalnumber of users and the number of subcarriers on the systemtradeoff performance.

Index Terms—Link-layer energy-rate tradeoff, delay-outageprobability, effective capacity, energy efficiency.

I. I NTRODUCTION

Green communication networks, which not only emphasizeon spectrum efficiency (SE), but also promise high energyefficiency (EE), have become imperative needs of futurecommunication systems. However, by nature, EE and SE couldrequire conflicting design approaches. From the information-theoretic point of view, the EE-SE tradeoff problem in a down-link orthogonal frequency division multiple access (OFDMA)network was analyzed in [2], in which the impact of thechannel power gain and the circuit power on the EE-SErelation was discussed. Considering the cognitive radio net-works, a multi-objective optimization was formulated in [3],in which the ergodic capacity was maximized and the total

This work was supported in part by the China Scholarship Council, UKEPSRC under grant number EP/K011693/1, and grant number EP/N032268/1,and the EU FP7 under grant number PIRSES-GA-2013-610524. Part of thiswork was presented in [1].(Corresponding author: Qiang Ni)

W. Yu and Q. Ni are with the School of Computing and Communica-tions, InfoLab21, Lancaster University, LA1 4WA, UK (Emails: {w.yu1,q.ni}@lancaster.ac.uk). L. Musavian is with the School of Computer Scienceand Electronic Engineering, University of Essex, CO4 3SQ, UK (Email:leila.musavian@essex.ac.uk).

transmission power of femtocell base stations was minimized.A general power consumption model in multi-user OFDMAsystems, including the transmission power, signal processingpower, and circuit power from both the transmitter and thereceiver sides, was first established in [4]. Then the authorsin [4] proposed a joint optimization method to iterativelyfind the optimal solution for the EE-maximization problem,subject to a peak transmit power constraint and a minimumsystem data rate requirement. The EE and SE tradeoff problemhas also been extensively studied for other kinds of wirelesscommunication networks, such as energy-constrained wire-less multi-hop networks with a single source-destination pair[5], general narrowband interference-limited systems [6]andOFDMA-based cooperative cognitive radio networks [7]. Inthe aforementioned studies, however, the system throughputwas given by Shannon limit, without taking into accountdelay constraints. For systems with delay-sensitive applica-tions, such as video conferencing and online gaming, thephysical-layer based power and rate adaptation techniquesmaynot be efficient. In fact, 5G, the next generation of mobilecommunication technology, has been anticipated to offer >1Gbps downlink data rate, sub-1ms end-to-end latency and90% reduction in network energy usage [8]. This infers thatthe future wireless communication networks are targeted atsatisfying the end-user applications’ delay quality-of-service(QoS) requirements, while at the same time increasing EEand SE for green communications.

In order to fulfill these requirements, extensive studiesin the context of power control, scheduling, and admissioncontrol have been widely provided in [9]–[21]. A cross-layeroptimization framework for delay-sensitive applicationsovera single wireless link was formulated in [9], in which somecharacteristics, e.g., delay deadlines, dependencies, distortionimpacts, are considered and discussed. The authors in [10]provided energy-efficient transmission techniques for a groupof M packets subject to individual packet transmission delayconstraints. The above works all characterize the delay QoSrequirement for a dynamic queuing system in a deterministicway, where the delay is bounded within a certain threshold[11]. Although this sounds reasonable for real-time services,satisfying fixed QoS guarantees is especially challenging infading communication scenarios, due to the random varia-tions experienced in channel conditions, user mobility andchanging environment [12], which could lead to settling fornon-necessarily low data rates. In contrast to the above de-terministic delay QoS bounds, in this paper we concentrateon the delay QoS requirement in a statistical way, which

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considers and confines the delay bound violation probabilityto a required value range. In this direction, the authors in [13]introduced a link-layer capacity notion supporting statisticaldelay QoS requirements, which is the concept of effective ca-pacity (EC). Formulated as the dual of the effective bandwidth,EC specifies the maximum arrival rate that can be supported bya wireless channel given that a target delay-outage probabilityrequirement is guaranteed [13] [19]. Therefore, EC can beregarded as the link-layer SE. The link-layer EE, henceforth,can be formulated as the ratio of the EC to the total powerexpenditure [14].

Due to the inconsistent property of the link-layer EE andEC, many researchers have elaborately studied how to balancethe two metrics. Considering frequency flat-fading channels,an optimal power allocation strategy to maximize EC sub-ject to a link-layer EE constraint, for delay-limited mobilemultimedia applications was obtained in [15]. For a Rayleighflat-fading channel under delay-outage probability constraints,a multi-objective optimization problem to jointly maximizeEE and EC was formulated and solved in [16]. The abovementioned papers, however, focus on a point-to-point single-channel communication system.

We note that based on the theory of Shannon limit, thetotal average rate of a multi-carrier system is a linear sum-mation of each subcarrier’s achievable average rate. This,however, does not apply to systems with limited statisticaldelay requirements. Specifically, in delay-constrained systems,the concavity and monotonicity of the EC do not remainhomogeneous for single-carrier and multi-carrier systems[17].In addition, for systems with statistical delay QoS constraints,it has been proven that the optimal power allocation strategyfor single-carrier communications cannot be simply extendedto the multi-carrier communications [17]. Hence, considering asingle-user multi-carrier link over a frequency-selective fadingchannel, the delay-constrained EC maximization and EE max-imization problem were separately addressed in [17] and [18],respectively. However, the link-layer EE-EC tradeoff problemfor the multi-carrier communications is not investigated andanalyzed in the literature. Especially, when we consider amulti-user multi-carrier network, the link-layer EE-EC tradeoffproblem becomes more challenging. The formulated problemwill be a complex combinatorial integer programming prob-lem, rather than a convex optimization problem in [17] whichwas solved using Lagrangian method. In [20] and [21], anEE optimization problem with statistical delay provisioningand per-user’s EC requirement constraint was analyzed for adownlink multi-user OFDMA network. In these papers, thepower allocation for each subcarrier is assumed to be onlyrelated to its subcarrier’s channel power gain, and not relatedto the same user’s other subcarriers’ channel power gains.Therefore, based on this assumption and the independent andidentically distributed (i.i.d.) property of all subcarriers, the ECvalue for a single-user multi-carrier system can be formulatedas a linear summation of the EC values of all subcarriers.While this independent optimization approach is optimal inmaximizing the Shannon capacity (e.g., water-filling powercontrol for multi-carrier transmissions), it is not the optimalpolicy to maximize the EC-based problems for an arbitrarystatistical delay provisioning [17]. In this paper, we willnot

make this assumption, and aim to derive the optimal powerallocation strategy for each user, which is not only across thetime domain, but also across the frequency domain.

In this paper, we target to maximize the system totalEC for the uplink transmission in a multi-user multi-carrierOFDMA network, subject to each user’s required link-layerEE performance level and its individual resource limits. Wedecouple the problem into two parts and provide the subcarrierassignment solution and optimal power allocation strategyfor each user. In more detail, we propose a low-complexityheuristic algorithm, which first allocates each served usertheexact number of its required subcarriers, and then implementsthe optimal per-user power allocation strategy to calculate eachuser’s current EC value. Finally, the remaining subcarriers willbe allocated by adopting the strategy that the user with currentminimum EC value has the allocation priority.

To sum up, this paper has the following contributions:

• A novel total EC maximization problem for the uplinktransmission, in a multi-user multi-carrier OFDMA sys-tem, is formulated as a complex combinatorial integerprogramming problem, subject to each user’s link-layerEE requirement and the individual’s average input powerlimit. A new adjustable EE requirement factor is definedto further tune each user’s EE constraint value, whichtransforms the formulated problem into a tradeoff prob-lem between the system total EC and the users’ individualEE achievements.

• The formulated challenging problem is first decoupledinto a frequency provisioning problem and an indepen-dent link-layer multi-carrier EE-EC tradeoff problem foreach user. The traditional exhaustive algorithm and a fair-exhaustive algorithm are introduced first, followed by alow-complexity heuristic algorithm, which cares aboutuser fairness, offers a close-to-optimal performance, andalso has a complexity linearly relating to the size of theproblem.

• The independent multi-carrier power-constrained link-layer EE-EC tradeoff problem is then solved and analyzedfor each user, given a subcarrier assignment matrix.The optimal power allocation strategy, which is acrossfrequency and time domains, and the Pseudocode of thepower allocation process are derived and proposed.

• We prove that each user’s average optimal power levelmonotonically decreases with its EE requirement factor.Furthermore, we prove that each user’s link-layer EEvalue monotonically decreases with its circuit powervalue, but increases with its EE requirement factor.

• Simulation results reveal that when there is a link-layerEE constraint, each user’s operational tradeoff EC value1 will not show a monotonic trend with its delay QoSexponent. Further, the tradeoff EC value achieved witha smaller number of available subcarriers may be higherthan the one obtained with larger number of subcarriers.

1Here each user’s operational tradeoff EC value is the calculated final ECvalue achieved at its EE requirement equality.

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Fig. 1: Uplink transmission in a multi-user multi-carrier network.

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. Multi-user Multi-carrier System Model

We consider the uplink transmission, where theK activeusers send their own information to the base station, in amulti-user multi-carrier OFDMA system depicted in Fig. 1.A total bandwidth ofB is divided intoN subcarriers, each

with a bandwidth ofB

N. Assume that each subcarrier is

exclusively assigned to at most one user at each time toavoid interference among different users. The total numberof allocated subcarriers for all users does not exceed theavailable frequency resources. Therefore, a feasible subcarrierassignment indicator matrix can be denoted asφ, whichsatisfies

φ ∈ Φ ,

{[φk,n]K×N | φk,n ∈ {0, 1},

K∑

k=1

φk,n ≤ 1,

K∑

k=1

N∑

n=1

φk,n ≤ N, k ∈ K0, n ∈ N0

}. (1)

Here, Φ denotes the set of all possible subcarrier alloca-tion indicator matrices, andK0 = {1, 2, . . . ,K}, N0 ={1, 2, . . . , N} denote the set of all users and all subcarriers,respectively. The number of allocated subcarriers for thekth

user is denoted byNk, namely,Nk =N∑

n=1φk,n, and the

bandwidth allocated to thekth user is denoted byBk, i.e.,

Bk = NkB

N.

Each transmitter implements a first-in-first-out (FIFO)buffer, which prevents loss of packets that could occur whenthe source rate is higher than the service rate, at the expense ofincreasing the delay [13]. The upper-layer packets are dividedinto frames at the data-link layer and are stored at the transmitbuffer. The frames are then split into bit streams at the physicallayer. By utilizing perfect channel state information (CSI)knowledge fed back from the receiver and the predeterminedstatistical QoS constraint, adaptive modulation and coding(AMC) and adaptive power control policy are applied at thetransmitter side [17]. Then, the bit streams are read out ofthe buffer and are transmitted through the wireless fadingsubcarriers. At the receiver side, the reverse operations areperformed and the frames are recovered for further processing.We assume that each subcarrier experiences block fading, i.e.,the channel gains ofN subcarriers are invariant within a

Fig. 2: Queuing system model for each transmitter.

fading-block’s time durationTf , but independently varies fromone fading block to another. In addition, the length of eachfading-block,Tf , is considered to be an integer multiple ofthe symbol durationTs, and is assumed to be less than thefading coherence time [17].

For the kth user on thenth subcarrier at the fading-blockindex t, the subcarrier power gain is denoted byγk,n[t], k ∈K0, n ∈ N0. Also, each subcarrier is assumed to experiencei.i.d. additive white Gaussian noise (AWGN) with powerspectral density

η02

. Therefore, the instantaneous maximum

achievable rate of thekth user on thenth subcarrier at thetth

fading-block is given by

Rk,n [t] =B

NTf log2

(1 + Pk,n [t]

γk,n [t]

P kLη0(BN

))

(bits) , (2)

where P kL denotes the distance-based path-loss power andPk,n [t] is the nonnegative transmission power for thekth

user on thenth subcarrier, at thetth fading-block, i.e.,Pk,n [t] ≥ 0. Specifically, for the kth user, the sub-carrier power allocation vector is denoted asPk [t] =[Pk,1 [t] Pk,2 [t] ... Pk,N [t]

]2. The total achievable rate

over all allocated subcarriers for thekth user, which depends onthe subcarrier allocation indicator matrixφ and the subcarrierpower allocation vectorPk, can be denoted asRk (φ,Pk) =∑n∈Nk

φk,nRk,n, whereNk is the set of subcarriers allocated

to thekth user.

B. Multi-user Multi-carrier Effective Capacity and Link-layerEnergy Efficiency

For each transmitter, the FIFO buffer is assumed to be adynamic queueing system with stationary ergodic arrival andservice processes, depicted in Fig. 2 [22]. By using the largedeviation theory, the queue length processQ (t) converges indistribution to a steady-state queue lengthQ (∞) such that[22]

− limx→∞

ln (Pr{Q (∞) > x})

x= θ, (3)

wherePr{a > b} shows the probability thata > b holds. Thisdefinition implies that the probability of the queue length ex-ceeding a certain thresholdx decays exponentially fast asx in-creases [23]. Note that in (3), the parameterθ (θ > 0) indicatesthe exponential decay rate of the QoS violation probability. Asmaller value ofθ denotes a looser QoS requirement, whilelarger θ implies a lower probability of violating the queuelength and a more stringent delay constraint. Particularly, when

2Since the service rate process of thekth user on thenth subcarrier isconsidered to be stationary and ergodic [17], hereafter, the block index tcould be omitted for simplicity.

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θ → 0, which refers to a system with no delay constraint, theoptimum power allocation strategy is the traditional water-filling approach and the maximum achievable rate is ergodiccapacity. For a transmitter withθ → ∞, the optimum powerallocation is the channel inversion with fixed rate transmissiontechnique, under which the delay-limited capacity can beachieved. In other words, the ergodic capacity and the delay-limited capacity can be considered as two extreme cases ofthe effective capacity.

Taking the delay experienced by a source packet arrivingat time t, defined byD(t), into consideration, the probabilitythat the delay exceeds a maximum delay boundDmax, can beestimated as [13]

P outdelay = Pr{D(t) > Dmax} ≈ Pr{Q(t) > 0}e−θµDmax, (4)

where P outdelay presents the delay-outage probability,Dmax isin the unit of a symbol period,Pr{Q(t) > 0} denotes theprobability of a non-empty buffer at timet, and can beapproximated by the ratio of the constant arrival rate to theaverage service rate [17], [22], i.e.,Pr{Q(t) > 0} ≈

µ

E[R[t]].

Hence, in order to meet a target delay-outage probability limitP outdelay, a source needs to limit its data rate to the maximum ofµ, whereµ is the solution to (4).

Assume that the Gartner-Ellis theorem [24, Pages 34-36]is satisfied. For thekth user, the EC value, in b/s/Hz, over amulti-carrier transmission with a total bandwidthBk can beexpressed as [13]

Ekc (θk,φ,Pk) = −1

θkTfBkln(E

[e−θkRk(φ,Pk)

]), (5)

whereθk stands for the delay QoS exponent of thekth userwhich is associated with the statistical delay QoS requirementandE[·] indicates the expectation operator. Henceforth, EC ofthe kth user becomes a function ofθk, φ, andPk.

By expandingRk (φ,Pk) and inserting it into (5), EC ofthe kth user can be further expressed as

Ekc (θk,φ,Pk) = −1

θkTfBkln

(E

[e−θk

∑n∈Nk

φk,nRk,n])

.

(6)

For the multi-user OFDMA network, the overall EC valuecan be expressed as

Ec (θ,φ,P) =

∑Kk=1 NkE

kc (θk,φ,Pk)∑K

k=1 Nk(b/s/Hz) , (7)

where θ =[θ1 θ2 ... θK

]is the K × 1

vector of delay exponents for allK users. Pdenotes the transmission power allocation matrix,for all users over all subcarriers, i.e.,P ∈ P ,{[Pk,n]K×N ∈ R+ | Eγk

[∑Nn=1 φk,nPk,n

]≤ P kmax, k ∈ K0

}.

Here, P is all the possible power allocation matrices,Eγk [·] indicates the expectation over the PDF ofγk,where γk is the kth user’s subcarrier power gains, i.e.,γk =

[γk,1 γk,2 ... γk,Nk

]. P kmax represents the maximum

average power limit of thekth user.

Moreover, for thekth user, we define the link-layer EE as theratio of EC to the sum of its circuit powerP kc , and the average

transmission power scaled by the power amplifier efficiencyǫ, yielding

EEk (θk,φ,Pk) =Ekc (θk,φ,Pk)

P kc +1

ǫEγk

[∑

n∈Nk

φk,nPk,n

] . (8)

C. Problem Formulation

From a system point of view, the overall EC value needsto be maximized to achieve the best system performance.On the other hand, from the individual user point of view,each user has its own link-layer EE requirement, averagetransmission power limit and delay QoS constraint. Therefore,considering a multi-user multi-carrier network, the overallsystem throughput maximization problem, subject to eachuser’s resource constraints, can be formulated as

Q1 : maxφ∈Φ,P∈P

Ec (θ,φ,P) (9a)

subject to: EEk (θk,φ,Pk) ≥ ηkreq, ∀k, (9b)

Eγk

[N∑

n=1

φk,nPk,n

]≤ P kmax, ∀k, (9c)

K∑

k=1

φk,n ≤ 1, ∀n, (9d)

K∑

k=1

N∑

n=1

φk,n ≤ N, (9e)

φk,n ∈ {0, 1}, ∀k, ∀n, (9f)

Pk,n ≥ 0, ∀k, ∀n, (9g)

whereηkreq is thekth user’s required link-layer EE level, defined

by a certain ratio of its maximum achievable link-layer EEvalue, i.e.,ηkreq = χ

kEE × η

k,Nmax . Here, η

k,Nmax = EE

k∣∣∣ Nk=NPk=Pk

∗

EE

denotes thekth user’s maximum achievable EE value, whenall N subcarriers in the system are allocated to it.P k∗EE isthe operational average input power which achievesηk,Nmax .Further,χkEE ∈ [0, 1] is an adjustable EE requirement factor,which reveals the strictness of thekth user’s required EE leveland directly influences the system performance. In particular,χkEE = 0 indicates that thek

th user has no EE requirement,while χkEE = 1 means that userk requires an operational EEvalue atηk,Nmax . Sinceη

k,Nmax depends on the individual user’s

delay QoS exponent and its maximum averge power limit,its value is different for each user. Therefore, thekth user’srequired EE levelηkreq is different from the other users, evenwhen they have the same EE requirement factors.

Due to the conflicting property of the total system EC andeach user’s personal EE achievement, after introducingχkEE,the formulated problemQ1 becomes an adjustable tradeoffproblem. To be more specific, if the total system EC value hasa high priority, each user’s EE requirement factor value canberequired to be very low, which results in a low link-layer EElevel for each user. Correspondingly, if the total system ECvalue has a low priority, each user’s EE requirement factorcan be relatively high, so that each user will have a satisfiedhigh level of link-layer EE.

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III. O PTIMAL AND SUB-OPTIMAL SOLUTIONS

Since we assume that one subcarrier can be assigned toonly one user at a time, therefore there could beKN possiblesubcarrier assignments [25]. Hence, the complexity of theabove combinatorial integer programming problem in findingthe jointly optimal subcarrier and power allocation growsexponentially with the number of subcarriers. Furthermore,we note that it is very difficult to jointly obtain the optimalsubcarrier allocation sets and all power allocation valuesinevery frame, due to the reasons below. Firstly, from (6), wecan notice that the EC formulation of thekth user not onlyrequires the multiplication of two unknown parameters, i.e.,φk,n, and Rk,n, but also involves the expectation over thejoint PDF of all subcarriers’ channel power gains, i.e.,γk.Secondly, the expectation and the multiplication operationscannot be interchanged, even if all subcarriers are assumedto be i.i.d., and that is because the power allocation value oneach subcarrier is related to the other subcarriers.

Henceforth, in order to make the formulated problemQ1tractable, we divide the solving process into two steps: fre-quency provisioning which decides the number of subcarriersto be allocated to each user; and then optimal power allocationfor each user over all its allocated subcarriers. Specifically,the proposed frequency provisioning algorithms, which areindependent of the instantaneous CSI knowledge in eachframe, will be implemented only once within a period of time.On the other hand, for each user, the proposed optimal powerallocation strategy on each subcarrier, not only relies on theinstantaneous CSI of this subcarrier, but also depends on theother subcarriers’ CSI knowledge in each frame.

We start from introducing three frequency provisioningalgorithms: traditional exhaustive algorithm, fair-exhaustive al-gorithm and our proposed low-complexity heuristic frequencyallocation algorithm. After obtaining the subcarrier assign-ments, the optimal power allocation strategy for each single-user multi-carrier system will then be derived and obtainedinSection III-B.

A. Frequency Provisioning Algorithms

By applying frequency provisioning, we assume that allsubcarriers follows the same distribution. It is the numberof designated subcarriers which matters, regardless wherethose subcarriers are located in the frequency band [25]. Toreduce the problem complexity and the solving time, wefirst build a pre-calculated offline databaseD which storesall users’ maximum achievable link-layer EE values, i.e.,ηmax =

[η1max η

2max ... η

Kmax

]T, in terms of certain settings

of Pc, θ and N . Here ηkmax is a 1 × N vector of thekth

user’s maximum achievable EE values with different numberof allocated subcarriers, i.e.,ηkmax =

[ηk,1max η

k,2max ... η

k,Nmax

].

Defineηreq =[η1req η

2req ... η

Kreq

]Tas theK × 1 vector of

the EE requirement values for allK users, then we will trans-form ηreq to aK×1 vector which specifies all users’ requirednumber of subcarriers, i.e.,Sreq =

[S1req S

2req ... S

Kreq

]T.

Let us consider thekth user as an example. Its required link-layer EE value is denoted byηkreq, and correspondingly, itssubcarrier requirement value will be stored asSkreq.

Fig. 3: Transformηkreq to Sk

req.

To obtainSkreq, we provide a flowchart in Fig. 3 to compareηkreq with η

kmax. If the maximum achievable EE value obtained

with i subcarriers is larger than the required EE value, i.e.,ηk,imax ≥ ηkreq, then we can conclude that the minimum numberof subcarriers required to satisfy thekth user’s EE requirementηkreq, is i, i.e., S

kreq = i. Henceforth, allK users’ EE require-

ments inηreq can be transformed to the subcarrier requirementvectorSreq, by utilizing the flowchart in Fig. 3. In this way, thefeasibility of each user’s EE constraint can be easily checkedby comparing the number of allocated subcarriers with thenumber of required subcarriers.

1) Traditional Exhaustive Algorithm

The traditional way to solve an NP-hard problem, like theone we formulated in (9a)-(9g), is to carry out an exhaustivesearch, which systematically enumerates all possible combi-nations and finally locates the solution which optimizes theobjective function and satisfies all the problem constraints[25]. Specifically, for problemQ1, the set of feasible com-binations is found first. Then, the optimal power allocationstrategy proposed in the next section, will be applied to allthe feasible combinations. Finally, the feasible combinationwhich offers the maximum system throughput will be chosenas the optimal solution. Although exhaustive search is abletofind the optimal frequency provisioning solution, it also lacksuser fairness and has a high computational complexity whichexponentially grows with the size of the problem.

2) Fair-Exhaustive Algorithm

To further find the optimal frequency provisioning solutionwhich not only maximizes the total system EC value, whilesatisfying each user’s link-layer EE requirement, but alsoserves the maximum number of users that can be allowed,we propose a fair exhaustive algorithm. Firstly, the sum of allusers’ required subcarriers is compared with the total numberof subcarriersN to find the maximum number of users thatcan be served. For example, let us assumeN = 8, andthe subcarrier requirement vector for all users is[1, 2, 2, 4].Hence, the total available subcarriers can serve3 users at

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most. Secondly, the set of feasible subcarrier allocation vectorsis found, in which each allocation vector not only satisfiesall served users’ subcarrier requirements, but also servesthemaximum allowed number of users. Then, the optimal powerallocation strategy proposed in Section III-B will be appliedto all feasible allocation vectors to locate the fair and optimalsolution which outperforms the others.

Clearly, by enumerating all possible subcarrier allocationvectors which can serve the allowed maximum number ofusers, the above proposed algorithm exhaustively find theoptimal solution in a fair way. Although the fair-exhaustivealgorithm is less complex compared to the traditional exhaus-tive algorithm, but its computational complexity is still veryhigh, especially when the number of available subcarriersNis large. To further reduce algorithm complexity, we providethe following heuristic algorithm, which is simple, fair andclose-to-optimal.

3) Heuristic Algorithm

There are three steps included in the proposed heuristicfrequency provisioning algorithm, which are allocation pro-cess, calculation process and check process. Firstly, in orderto serve as many users as possible, in the allocation process,we start from the user which requires the minimum numberof subcarriers. Each served user will be allocated the exactnumber of its required subcarriers, so that all the allocatedusers can satisfy their EE requirements. The allocation willbe repeated until the remaining subcarriers run out, or thereare not enough subcarriers to satisfy the next user’s EErequirement, or all users’ subcarrier requirements have alreadybeen satisfied. Then, the calculation process starts, in whicheach served user operates the optimal power allocation strategydescribed in Table II to obtain its corresponding EC value. Inthe check process, we aim to maximize the system throughput,based on the strategy that the user with current minimumEC value has the allocation priority. Therefore, the remainingsubcarriers will be assigned one-by-one to the user who hasthe current minimum EC value, until all subcarriers run out.

Assume the final subcarrier allocation vector is denoted byN =

[N1 N2 ... NK

]. The Pseudocode of the proposed

heuristic algorithm is illustrated in Table I. We note that,theproposed algorithm only needs at mostK − 1 comparisonsper iteration, given that each user’s EC values with variousnumber of subcarriers, is pre-calculated off-line and is storedin a database. Therefore, the heuristic algorithm offers arelatively low computational complexity comparing to thetwo exhaustive algorithms whose complexity exponentiallyincrease with the number of subcarriers. On the other hand,later, in simulation results, we will demonstrate that theproposed low-complexity algorithm offers a close performancewith the fair-exhaustive algorithm.

Now, let us analyze and explain the strategies utilized in theproposed heuristic algorithm. Firstly, in the allocation process,the heuristic algorithm starts the allocation from the userwhichhas the minimum subcarrier requirement. Assume that userihas the relatively small number of required subcarriers,Sireq.By regardingθi and χiEE as the two influencing parameterson Sireq, a small value ofS

ireq may result from the following

two possibilities: 1) useri has a small delay QoS exponent

TABLE I: Heuristic Algorithm

Initialization:CalculateSreq, usingηreq and the pre-calculated

databaseD.DefineStol = N , H = Sreq.

Allocation Process:While Stol > 0

If H = 0Break;

EndFind Hi = min(H), andHi > 0;If Stol > Sireq

Ni = Sireq;

Stol = Stol − Sireq;Hi = 0;

ElseBreak;

EndEnd

Calculation Process:For each useri with Hi = 0, apply the optimal

power allocation process in Table II.Calculate theith user’s EC valueJi and define

J =[J1 J2 ... JK

].

Check Process:While Stol > 0

Find Ji = min(J), in which useri satisfiesHi = 0;

Ni = Ni + 1;Apply the optimal power allocation process

to useri and updateJi.End

Output: N ; Ec given in (7).

θi and the sameχiEE value, comparing to the other users; 2)useri has a small EE requirement factorχiEE, and the sameθivalue, comparing to the others. For the first situation, a smalldelay QoS exponentθi means a loose requirement on delayQoS, which will offer a bigger EC value, when the allocatednumber of subcarriers andχiEE are fixed. Meanwhile, for thesecond situation, a small value ofχiEE also provides a larger ECvalue, because now the EE requirement constraint is easy to besatisfied and the multi-carrier system will have more resourceand flexibility to maximize the EC performance. Consequently,the design idea of the allocation process not only makes surethat as many users as possible can be served, but also intendsto serve the user which can contribute a larger EC value.

On the other hand, the design strategy of the check process,i.e., the user with current minimum EC value has the allocationpriority, comes from Fig. 4, which describes the results ofmaximum EC versus delay QoS exponentθ, for various valuesof N , in a single-user multi-carrier system. Specifically, Fig.4 reveals that the user with current minimum EC value has ahigh possibility to offer the largest EC-increase, if givenonemore subcarrier. In more detail, from Fig. 4, we notice thatfor two users with the same values ofθ, if we allocate moresubcarriers to them, the user with current smaller EC value,i.e., the one which has smaller number of subcarriers, will geta larger EC-increase. Furthermore, for two users with the same

7

10−4

10−3

10−2

10−1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Delay QoS Exponent θ (1/bits)

Effe

ctiv

e C

apac

ity (

b/s/

Hz)

EC Maximization in a Single−user Multi−carrier System (BTf=200)

N=1N=2N=4N=8

Fig. 4: Effective capacity versus delay QoS exponentθ, for variousvalues ofN .

number of subcarriers, when we allocate each user two moresubcarriers, the user with relatively smaller EC value, namely,the one which has larger delay QoS exponent, will provide alarger EC-increase. Simulation results in Section IV confirmthe effectiveness of our design method, and inform that theproposed heuristic algorithm offers very close performancewith the fair-exhaustive algorithm.

B. Optimal Power Allocation For A Single-user Multi-carrierSystem

Given a subcarrier assignment matrixφ, the multi-userOFDMA system can be viewed as a frequency-division mul-tiple access (FDMA) system, where each user transmits datathrough a number of assigned subcarriers independently [26].Therefore, the original total EC maximization problem, subjectto each user’s link-layer EE requirement and maximum aver-age power limit, can be transformed into a link-layer EE-ECtradeoff problem for each single-user multi-carrier system.

Specifically, for thekth user, the problem can be expressedas

Q2 : maxPk,n≥0n∈Nk

Ekc (θk,Pk) (10a)

s.t. EEk (θk,Pk) ≥ ηkreq, (10b)

Eγk

[Nk∑

n=1

Pk,n

]≤ P kmax. (10c)

By recalling that the total bandwidth allocated to thekth

user isBk, the total instantaneous service rate of thekth useris given by

Rk =BkNk

Tf

Nk∑

n=1

log2

1+Pk,n

γk,n

P kLη0

(BkNk

)

(bits) . (11)

By inserting (11) into (5), we get the mathematical expressionof EC for thekth user. Correspondingly, the link-layer EE forthekth user, as the ratio of EC to the total power expenditure,

can be obtained. Therefore, problemQ2 can be expanded as

Q3 : maxP rk,n≥0

n∈Nk

−1

αklog2

Eγk

Nk∏

n=1

(1+NkP

rk,nγk,n

)−αkNk

(12a)

s.t.

−1

αklog2

Eγk

∏Nk

n=1

(1+NkP

rk,nγk,n

)− αkNk

Kkℓ

(P kcr+

1

ǫEγk

[Nk∑n=1

P rk,n

]) ≥ηkreq,

(12b)

Kkℓ Eγk

[Nk∑

n=1

P rk,n

]≤ P kmax, (12c)

whereαk ≡θkTfBkln (2)

, P rk,n =Pk,n

Kkℓ, andP kcr =

P kcKkℓ

. Here

Kkℓ = PkLη0Bk, which denotes the path loss factor, including

both AWGN power and path loss power. Setη̂kreq = Kkℓ η

kreq,

and P̂ kmax = Pkmax/K

kℓ . Then, K

kℓ in (12a)-(12c) can be

canceled to scale the system performance with respect to thepath loss factor.

From (12a)-(12c), one can notice that the EC expression ina single-user multi-carrier system is not a linear summation ofeach subcarrier’s achievable EC value. Hence, the concavityand monotonicity of the EC function in a single-subcarriersystem cannot be simply extended to the multi-carrier system.In order to find the joint energy and spectral efficient powerallocation strategy in a single-user multi-carrier system, westart from analyzing the proposed problemQ3.

By referring to the scaled multi-carrier transmit powervector asP rk =

[P rk,1 P

rk,2 ... P

rk,N

], we note that the

objective function (12a) is concave inP rk [18]. Then, thelink-layer EE, as the ratio of a concave function over a non-negative affine function inP rk, is a quasi-concave functionin subcarrier power allocations [18]. Therefore, its uppercontour set defined by (12b) is convex [27]. Hence, (12a)-(12c) is a concave optimization problem and the Karush-Kuhn-Tucker (KKT) conditions are both sufficient and necessary forthe global optimum value. Specifically, the proposed optimalpower allocation strategy for thekth user is related to the jointprobability density function (PDF) of the subcarrier powergainsγk, given byρ (γk).

To solve the concave optimization problem (12a)-(12c),we start from analyzing the power-unconstrained problem(12a)-(12b), which paves the way for the power-constrainedoptimization problem. By transforming (12b) to

−1

αklog2

Eγk

Nk∏

n=1

(1 +NkP

rk,nγk,n

)−αkNk

−η̂kreq

(P kcr +

1

ǫEγk

[Nk∑

n=1

P rk,n

])≥ 0, (13)

8

we get the Lagrangian function as follows

L (P rk, λ) =−1

αklog2

Eγk

Nk∏

n=1

(1+NkP

rk,nγk,n

)−αkNk

+ λ

− 1

αklog2

Eγk

Nk∏

n=1

(1 +NkP

rk,nγk,n

)−αkNk

−η̂kreq

(P kcr +

1

ǫEγk

[Nk∑

n=1

P rk,n

]))−

N∑

n=1

µnPrk,n, (14)

whereλ ∈ R is the Lagrange multiplier associated to (13)andµn is the Lagrange multiplier associated to the constraintP rk,n ≥ 0, ∀ n ∈ Nk.

At the optimal power allocation, we have

∂L (P rk, λ)

∂P rk= 0. (15)

Because of the complementary slackness condition [27], ifP rk,n > 0, then µn = 0, ∀ n ∈ Nk. On the other hand, ifP rk,n = 0, ∃ n ∈ Nk, then µn 6= 0. Thus, the followingtwo cases need to be considered to find the optimal powerallocation strategy.

1) Case 1: P rk,n > 0, ∀ n ∈ NkIn this case, allNk subcarriers are allocated non-zero

transmission power. Therefore, based on the complementaryslackness,{µn}

Nkn=1 = 0. Then, the KKT condition (15) can

be simplified as

Nk∏

i=1

(1 +NkP

rk,iγk,i

)−αkNk =

β

γk,n

(1 +NkP

rk,nγk,n

),

∀ n ∈ Nk, (16)

whereβ=λη̂kreq

ǫ(λ+ 1) log2 eEγk

∏Nk

n=1

(1+NkP

rk,nγk,n

)− αkNk

.

By multiplying the right and left-hand sides of theNkequations in (16), the optimal power allocation strategy canbe obtained as

P rk,n=1

Nk

1

β1

αk+1∏Nk

i=1 γαk

(αk+1)Nk

k,i

−1

γk,n

, ∀ n ∈ Nk. (17)

The derived power allocation strategy (17) is optimal onlywhen all subcarriers are assigned with positive powers. Ifthere are one or more subcarriers which are allocated non-positive powers, then the second case needs to be taken intoconsideration.

2) Case 2: P rk,j = 0, ∃ j ∈ NkIf there existsP rk,j ≤ 0, then the set of subcarriers, which

only positive powers should be assigned, needs to be found.

Firstly, we defineN̂k as

N̂k =

n∈Nk

∣∣∣∣∣1

Nk

1

β1

αk+1∏Nk

i=1 γαk

(αk+1)Nk

k,i

−1

γk,n

≥0

.

According to Lemma 1 in [17], the total power must beassigned to the subcarriers which belong tôNk, while the

subcarriersn 6∈ N̂k should not be allocated any power.Therefore, a new power-unconstrained optimization problemcould be expressed as

Q4 : maxP rk,n≥0

n∈N̂k

−1

αklog2

Eγk

N̂k∏

n=1

(1+NkP

rk,nγk,n

)−αkNk

(18a)

s.t.

−1

αklog2

Eγk

∏N̂k

n=1

(1+NkP

rk,nγk,n

)− αkNk

Kkℓ

(P kcr+

1

ǫEγk

[N̂k∑n=1

P rk,n

]) ≥ηkreq,

(18b)

whereN̂k = |N̂k| represents the cardinality of̂Nk.

Therefore, ifP rk,n > 0, ∀ n ∈ N̂k, then, the optimizationproblem can be solved exactly like Case 1. Otherwise, if thereare subcarriersn ∈ N̂k havingP rk,n = 0, then N̂k must befurther partitioned by recursively repeating the above processuntil a setN ∗k can be found, in which all subcarriers areallocated positive powers [18].

After obtaining N ∗k , the optimal power allocations arecomputed as

P rk,n=

1

Nk

1

βNk

Nk+αkN∗k

∏i∈N∗

kγ

αkNk+αkN

∗k

k,i

−1

γk,n

, n ∈ N ∗k

0, otherwise(19)

whereN∗k =| N∗k |.

The optimal value forβ, referred to asβ∗, is found whenthe kth user’s EE constraint is satisfied with equality, yielding

−1

αklog2

Eγk

Nk∏

n=1

(1 +NkP

rk,nγk,n

)−αkNk

−η̂kreq

(P kcr +

1

ǫEγk

[Nk∑

n=1

P rk,n

])= 0. (20)

Note that since EE versus EC is a bell shape curve, the requiredEE level, if possible, can be achieved at two different ECvalues, which means that there will be two solutions forβ, i.e.,β1 andβ2, to satisfy (20). Assume thatPk1 = Pk |β=β1 , and

Pk2 = Pk |β=β2 , wherePk stands forKkℓ Eγk

[∑Nkn=1 P

rk,n

].

Therefore, the feasible set of the average input power levelsat-isfying the EE constraint (12b) can be written as

[Pk1, Pk2

]3.

Considering our intention to maximize EC and the fact thatEC is a monotonically increasing function inPk [18], theoptimal average input power valueP ∗k , which solves thepower-unconstrained problem (12a)-(12b), is chosen as thelarger one which satisfies (20), i.e.,P ∗k = max

[Pk1, Pk2

].

Based on the assumption thatPk2 is larger thanPk1, thereforeP ∗k = Pk2, and correspondingly,β

∗ = β2. Here we completethe solving process of the optimal power allocation for the

3Without losing any generality, we assume thatPk2 is larger thanPk1.

9

TABLE II: Optimal Power Allocation Process

Input:[φ, θk, Tf, B,N,Nk, P

kc , ǫ,K

kℓ ,γk, P

kmax, η

kreq

]

Step 1:Have a initial guess ofβ.Repeat

Create (20), using (17) or (19), which appliesMonte Carlo method.

Updateβ using bisection method.Until find β∗ which solves (20).CalculateP rk,n, n ∈ Nk.

CalculateP ∗k = Kkℓ Eγk

[∑Nkn=1 P

rk,n

] ∣∣∣∣β=β∗

.

Step 2:If P kmax < P

∗k :

CreateP ∗k =Pkmaxand updateβ

∗, correspondingly.CalculateP rk,n, n ∈ Nk, in (17) or (19).

Step 3:Calculate the value ofEkc given in (6) and the

link-layer EEk value in (8).

Output:[P rk,n, P

∗k , E

kc ,EE

k]

power-unconstrained problem (12a)-(12b).

By utilizing the above proposed optimal power allocationstrategy, we start to analyze the optimization problem (12a)-(12c) with the average input power constraint. After the feasi-ble set of the average power value for the EE constraint (12b)is found, the power-constrained EC maximization problem forthe kth user, subject to a link-layer EE constraint, can besimplified to

Q5 : maxP rk,n≥0

n∈Nk

−1

αklog2

Eγk

Nk∏

n=1

(1+NkP

rk,nγk,n

)−αkNk

(21a)

s.t. Pk ∈[Pk1, Pk2

], (21b)

Pk ≤ Pkmax. (21c)

We note that EC is a monotonically increasing function inPk [18], therefore, the optimal average power value whichsolves the problem in (21a)-(21c) will be achieved at one of thethree endpoint values, i.e.,Pk1, Pk2, orP

kmax. In more detail, if

Pk2 ≤ Pkmax, the optimal power levelP

∗k , equals toPk2, and

the optimal power allocation strategy (17) will be achievedand operated. On the other hand, ifPk1 < P

kmax < Pk2, the

system has to operate atP kmax and the optimal power allocationto solve (12a)-(12b) is according to (17), wherein, optimalβ∗

is found such thatP ∗k |β=β∗= Pkmax. Moreover, ifP

kmax < Pk1,

the power-constrained problemQ5 has no feasible solution.For simplicity, we assume that each user’s maximum availablepower is always sufficient to support the feasibility of itsrequired EE value, i.e,P kmax ≥ Pk1. Otherwise, the proposedproblem will be infeasible.

To summarize, the Pseudocode of the optimal power alloca-tion process to solve the power-constrained link-layer EE-ECtradeoff problem for thekth user, through multiple subcarriers,is illustrated in Table II. After we obtain the optimal powerallocation strategy and optimal operational average power

for problem Q3, further analysis is needed to thoroughlyunderstand and investigate the impact of thekth user’s circuitpower value and the EE requirement factor on its link-layerEE-EC tradeoff performance. Hence, we provide the followinglemmas4.

C. The effects ofP kc andχkEE on thek

th user’s EE-EC tradeoffperformance

Lemma 1:The kth user’s tradeoff link-layer EE valueEE(P ∗k)

decreases withP kc .

Proof: The proof is provided in Appendix A.

Furthermore, the tradeoff optimal power value and thesystem performance can also be influenced by the introducedEE requirement factor. Specifically, whenχkEE increases, therequired link-layer EE level increases. Therefore, the finaloperational link-layer EE value which satisfies the EE require-ment equality increases. Since the proposed tradeoff averagepower operates at the EE-EC conflicting region, therefore thecorresponding EC value will decrease due to the increase inEE level. Hence, we can obtain the following lemma 2.

Lemma 2:The optimal average power valueP ∗k monotoni-cally decreases withχkEE, but the corresponding link-layer EEvalue EE

(P ∗k)

increases withχkEE.

Proof: The proof follows the above explanations and isomitted here due to page limit.

IV. SIMULATION RESULTS

In this section, we simulate the uplink transmission ina multi-user multi-subcarrier system, in which the fadingstatistics of the different subcarriers are considered to bei.i.d. Rayleigh distributed such that the subcarrier powergainsare realized as exponential random variables with unit mean.We numerically evaluate and compare the performance ofthe exhaustive algorithm, the fair-exhaustive algorithm,theheuristic algorithm and the algorithm proposed in [25], onthe total system EC-maximization problem, under the con-straints of each user’s link-layer EE requirement and averagetransmission power limit. To further analyze the problem andconfirm the lemmas proved in Section III-B, the impact ofthe delay QoS exponentθ, the EE requirement factorχEE andthe circuit-to-noise power ratioPcr on each user’s operationalEC value and the total system EC performance is simulatedand analyzed. In the following simulations, we assume thatB ·Tf = 200, the power amplifier efficiencyǫ = 1, each user’sindividual average transmission power limitPmax = 10dB,unless otherwise indicated.

In order to show the performance of the proposed heuristicalgorithm, Fig. 5 shows the results of the total system EC ver-sus the number of subcarriersN , for the heuristic algorithm,the exhaustive algorithm and the fair-exhaustive algorithm. Toget Fig. 5, the number of usersK is fixed, i.e.,K = 4, inwhich all users have the same settings of EE requirementfactor, i.e.,χEE = 0.7, and the circuit-to-noise power ratio, i.e.,Pcr = −10dB. The delay QoS exponent vectorθ is given by

4In these lemmas, we omit the influence ofP kmax, by assuming that it islarge enough to support the optimal power allocation strategy, i.e.,P kmax ≥ P

∗

k

10

4 6 8 10 12 140.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

The number of subcarriers N

Tot

al e

ffect

ive

capa

city

(b/

s/H

z)

Heuristic algorithmFair−exhaustive algorithmExhaustive algorithm

Fig. 5: The total effective capacity versus the number of subcarriersN , for heuristic algorithm, exhaustive algorithm and fair-exhaustive

algorithm.

4 6 8 10 12 140

1

2

3

4

5

The number of subcarriers N

The

num

ber

of s

erve

d us

ers

Heuristic algorithmExhaustive algorithmFair−exhaustive algorithm

Fig. 6: The number of served users versus the number ofsubcarriersN , for heuristic algorithm, exhaustive algorithm and

fair-exhaustive algorithm.

[0.001, 0.001, 0.01, 0.01]. For the exhaustive algorithm, whenthe number of subcarriersN increases, the total system ECvalue does not change very much, due to the loose delay QoSrequirements for all the users. For fair-exhaustive algorithmand heuristic algorithm, the total EC performance curves arevery close. This indicates that the proposed heuristic algorithmnot only has a low complexity and guarantees user fairness,but also offers close-to-optimal performance.

To further compare the three algorithms, the plots for thenumber of served users versus the number of subcarriersN areincluded in Fig. 6. Although the exhaustive algorithm offersthe best system performance in Fig. 5, Fig. 6 indicates that itserves the least number of users among all three algorithms.Especially, for the exhaustive algorithm, whenN ∈ [4, 8],it allocates all subcarriers to only one user, which shows alack of fairness. On the contrary, for the heuristic algorithmand the fair-exhaustive algorithm, the number of served usersshows an increasing trend until it equals to the total numberof usersK, i.e., 4. This happens because the increase ofN

10−4

10−3

10−2

10−1

100

12

14

16

18

20

22

24

26

Delay QoS exponent θk (1/bit)

Ave

rage

pow

er v

alue

(W

att)

Nk=2

Nk=4

Fig. 7: Average tradeoff optimal power value versus delay QoSexponentθk, for various values ofNk.

means more available frequency resources and the ability ofsupporting more users increases. We further note that normally,the traditional exhaustive algorithm prefers to choose thebest user and allocates all subcarriers to it. However, Fig.6indicates that whenN increases from 8 to 14, the number ofserved users for the exhaustive algorithm increases from 1 to3 and then stays stable. This means that we may not have onesingle best user, when all users’ delay QoS exponent valuesare different.

To understand the above phenomenon thoroughly, we con-sider thekth user’s multi-carrier system, and plot Fig. 7 andFig. 8 which respectively include the curves of the averagepower versusθk, and the tradeoff EC value versusθk, for twodifferent values ofNk, with χkEE = 0.2, andP

kcr = −10dB.

From Fig. 7, we note that with a fixedNk, whenθk increases,the average power value increases. To explain this, we firstrecall that, thekth user’s EE requirement value is defined as amultiplication ofχkEE andη

k,Nkmax , in which η

k,Nkmax is a function

of θk andNk. With the fixed values ofNk and χkEE, ηk,Nkmax

decreases withθk [18], and in turn, the EE requirement valuedecreases. Furthermore, the curve of link-layer EE versus aver-age power becomes wider when the user’s delay QoS exponentbecomes more stringent [15]. Therefore, whenθk increases,the optimal tradeoff average power obtained at a reduced EErequirement equality will become larger. Furthermore, Fig. 7indicates that with a fixed value ofθk, when Nk becomeslarger, the average power value reduces. This is due to the factthat when the values ofθk andχkEE are fixed,η

k,Nkmax increases

with Nk [18], as well as the required EE level. From Fig. 1 in[15], we note that a larger EE requirement will be satisfied ata smaller average power value. Hence, more available numberof subcarriers could lead to less average power consumption.

Fig. 8 shows the relationship between thekth user’s tradeoffEC value andθk for a single-user multi-carrier transmissionsystem. This figure reveals two important conflicting situationsand some insightful conclusions. Firstly, this figure indicatesthat one user’s operational EC value will not show a monotonictrend with its delay QoS exponent, when there is a link-layerEE constraint. This phenomenon violates the monotonic trend

11

10−4

10−3

10−2

10−1

100

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

X: 0.05125Y: 3.218

Delay QoS exponent θk (1/bit)

Effe

ctiv

e ca

paci

ty (

b/s/

Hz)

Nk=2

Nk=4

Fig. 8: Effective capacity versus delay QoS exponentθk, forvarious values ofNk.

of EC versus delay QoS exponent, in the EC-maximizationsituation provided in [17]. From [17], we note that for a fixeddelay QoS exponent, the EC value increases monotonicallywith the transmission power. Also, for a fixed transmissionpower, the EC value monotonically decreases with the delayQoS exponent. However, in our case, whenθk is small, thekth user’s link-layer EE requirement can be easily satisfiedwith a small value of transmission power. In contrast, whenθk becomes stringent, the required EE value has to be satisfiedwith a very large power value, like the trend indicated in Fig.7. In other words, the operational average power value willincrease withθk. But, the increase ofθk and the increase ofthe average power will have a conflicting influence on theuser’s operational EC value. Therefore, with the inconsistentinfluence of these two parameters, EC will not show a mono-tonic trend, which can be confirmed from Fig. 8. Clearly, whenθk is loose, the tradeoff EC value will be more influenced byθk. On the contrary, whenθk becomes stringent, the averagepower dominates the situation, therefore the operational ECvalue shows an increasing trend.

Secondly, Fig. 8 further reveals that one user’s tradeoff ECvalue achieved at a smaller number of subcarriers may behigher than the one obtained with relatively larger numberof subcarriers, when there is a link-layer EE constraint.Specifically, whenθk is loose, e.g.,θk ∈ [10−4, 0.05125], thetradeoff EC value with 4 subcarriers is higher than the oneobtained with 2 subcarriers. Whenθk becomes stringent, e.g.,θk ∈ [0.05125, 100], the tradeoff EC value achieved with 4subcarriers is lower than the one obtained with 2 subcarriers.This phenomenon also violates the monotonic trend of ECversus the number of subcarriers in EC-maximization situationanalyzed in [17]. This is due to the fact that with a link-layer EE requirement, whenNk increase, the average powervalue required to satisfy the EE constraint decreases. Sincethe increase ofNk and the corresponding decrease of theaverage power will have a conflicting influence on the user’soperational EC value, the EC will not show a monotonictrend. Apparently, Fig. 8 indicates that whenθk is loose, thetradeoff EC value will be more influenced byNk. When θkbecomes stringent, the average power dominates the situation,

2 3 4 5 6 7 8 9 100.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

The number of users K

The

tota

l effe

ctiv

e ca

paci

ty (

b/s/

Hz)

Heuristic algorithmFair−exhaustive algorithm

Fig. 9: The total effective capacity versus the number of users K,for heuristic algorithm and exhaustive algorithm.

therefore the operational EC value follows the same trend withthe average power. In conclusion, Fig. 7 and Fig. 8 indicatethat when there is a link-layer EE requirement, each user’soperational tradeoff EC value may not show a monotonic trendwith its delay QoS exponent value or its available numberof subcarriers. This reveals that we may have multiple bestusers, when all factors vary. Hence, in Fig. 6, the exhaustivealgorithm starts to serve more than one user whenN > 8.

To examine the effect of the number of users on a multi-usermulti-carrier system with limited resources, Fig. 9 includes theplots for the total EC value versus the number of usersK,for the heuristic algorithm and the fair-exhaustive algorithm.Specifically, the total number of available subcarriers is fixedat N = 10. All users are assumed to have the same circuit-to-noise power ratioPcr = −10dB, the same delay QoSexponentθ = 0.01, and the same EE requirement factorχEE = 0.7. When the number of usersK increases, the totalEC values calculated from the two algorithms decrease andthen stabilize whenK ≥ 6. This happens because, whenKincreases from 2 to 6, the number of served users increasesand correspondingly, the number of subcarriers allocated toeach served user decreases. Henceforth, the achievable EC foreach served user reduces and the total EC value calculatedfrom (7) decreases. WhenK ≥ 6, the number of served usersremains the same, due to the limited number of availablesubcarriers. Hence, the total EC value stays stable whenKbecomes greater than 6.

To indicate that the proposed heuristic algorithm performsbetter than the other state-of-the-art algorithms, Fig. 10plotsthe total EC versus the number of subcarriers for the proposedheuristic algorithm and the algorithm proposed in [25]. Forthe algorithm proposed in [25], an initial subcarrier allocationproportionate to each user’s EC requirement was set at thebeginning, and then the user with the highest slope of the ECversus signal-to-noise ratio (SNR) curve will keep releasingone subcarrier to the user with the lowest slope, until thetotal power consumption cannot be reduced any more. It isnoted that [25] does not consider the tradeoff between ECand the link-layer EE, and only the total power consumption

12

4 6 8 10 12 141

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

The number of subcarriers N

Tot

al e

ffect

ive

capa

city

(b/

s/H

z)

Heuristic algorithmAlgorithm in [24]

Fig. 10: The total effective capacity versus the number ofsubcarriersN , for heuristic algorithm and algorithm in [25] .

needs to be minimized. Since in our paper, we have link-layer EE requirement for each user, for the initial subcarrierallocation vector, we will make the elements proportionateto the corresponding users’ EE requirements5. The numberof users isK = 4, and all the users are assumed to havethe same delay QoS exponent, i.e.,θ = 0.001, and the sameEE requirement factor, i.e.,χEE = 0.7. When the number ofsubcarriers increases, the total EC values calculated fromthetwo algorithms first increase and then gradually stabilize.Thisindicates that when all users have loose delay QoS require-ments, increasing the number of available subcarriers willnotgreatly improve the total system EC value. Further, from Fig.10, it shows that our proposed heuristic algorithm performsbetter than the algorithm in [25]. Apart from this, the allocationalgorithm in [25] also has no guarantee of each user’s link-layer EE requirement, which can be confirmed from Fig. 14.Hence, we can conclude that, to solve the formulated total EC-maximization problem, subject to all users’ EE requirementconstraints, our proposed heuristic algorithm is more suitable,because it outperforms the algorithm in [25], and is alsosimple, fair and close-to-optimal.

Assume that allK users, having the sameθ value atθ = 0.01, and the same circuit-to-noise power ratio value atPcr = −10dB, are split into two groups. In group 1, allK1users are required to have the sameχEE, i.e., χEE = 0.1.Meanwhile, the EE requirement factor valueχEE for allK − K1 users in group 2 is 0.8. This indicates that theusers in group 1 have looser EE requirements compared tothe users in group 2. Set the total number of usersK = 6,and the total number of subcarriersN = 12. Fig. 11 includesthe plots for the results of the system total EC versus thenumber of usersK1 in group 1, for various values of circuit-to-noise power ratioPcr . With fixed Pcr , whenK1 increasesfrom 0 to 6, the total system EC value, in b/s/Hz, graduallyincreases. This is because whenK1 increases, the numberof users withχEE = 0.1 increases and correspondingly, thenumber of users withχEE = 0.8 reduces. We note that for

5Hence, we had to tweak the algorithm slightly to be able to provide theseresults.

0 1 2 3 4 5 60.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

The number of users K1 in group 1 with χ

EE = 0.1

Tot

al e

ffect

ive

capa

city

(b/

s/H

z)

Heuristic algorithm(P

cr

=−10dB)

Heuristic algorithm(Pc

r

=−5dB)

Fig. 11: The total effective capacity versus the number of usersK1in group 1, for various values ofPcr .

each user, a largeχEE value means that the user has a strictrequirement on its link-layer EE value and will end up with arelatively small EC value. Therefore, when the number of userswith χEE = 0.8 reduces, the system can save more resource tobenefit the total system EC value, rather than sacrifice thesystem performance to support the strict EE requirements.When K1 increases from 5 to 6, the number of users withχEE = 0.8 reduces from 1 to 0 and the total EC value growsdramatically. This is due to the fact that in the check processof heuristic algorithm, the user having current minimum ECvalue will get the priority, which, in this case, correspondsto the one withχEE = 0.8. Therefore, whenK1 = 5, theheuristic algorithm spends many resources on the user withχEE = 0.8. WhenK1 = 6, all users have the same loose EErequirements, i.e.,χEE = 0.1, therefore, the system resourcescan be arranged evenly, which results in a great growth inthe total EC value. Furthermore, from Fig. 11, we note thatwhenPcr becomes larger, the system total EC value increases.Since a bigger value ofPcr for all users will not change theirrelative difference, and correspondingly, will not changethesubcarrier assignment solution, this phenomenon indicates thatgiven a fixed subcarrier assignment, when one user’s circuitpower value increases, the system total EC value will increase,as well as its own EC value.

To analyze the impact of the circuit-to-noise power ratioP kcr and the EE requirement factorχ

kEE on thek

th user’s multi-carrier system, Fig. 12 plots the results of the link-layer EE(on the left hand side (LHS) y-Axis, in solid lines) and theoptimal tradeoff average power (on the right hand side (RHS)y-Axis, in dash lines) versusP kcr , for two different values ofχkEE, consideringNk = 4 andθk = 0.01. Whenχ

kEE is fixed,

the link-layer EE value decreases and the optimal averagepower increases withP kcr , which confirms the proved Lemma1. Furthermore, with a fixed value ofP kcr , whenχ

kEE becomes

larger, thekth user’s link-layer EE value increases, but theoptimal average power decreases, which confirms the proposedLemma 2 in Section III-B.

The plots of EE (on the RHS y-Axis, in dash lines) andEC (on the LHS y-Axis, in solid lines) versusχkEE, for two

13

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.2

0.4

0.6

0.8

1

Circuit−to−noise power ratio

Link

−la

yer

ener

gy e

ffici

ency

(b/

J/H

z)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

6

8

10

Opt

imal

ave

rage

pow

er (

Wat

t)

χkEE

=0.5

χkEE

=0.7

×1/Kl

Fig. 12: Link-layer energy efficiency and the optimal average powerversus circuit-to-noise power ratioP kcr , for various values ofχ

k

EE.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

10

χkEE

Effe

ctiv

e ca

paci

ty (

b/s/

Hz)

0 0.2 0.4 0.6 0.8 10

1

2

Link

−la

yer

ener

gy e

ffici

ency

(b/

J/H

z)

Nk=1, θ

k=0.001

Nk=1, θ

k=0.01

Nk=8, θ

k=0.001

Nk=8, θ

k=0.01

×1/Kl

Fig. 13: Effective capacity and link-layer energy efficiency versusχkEE, for various values ofθk andNk.

different values of delay QoS exponentθk and various valuesof Nk, are included in Fig. 13. From this figure, we note thatwith fixed number of subcarriersNk, whenχkEE increases, EEincreases. This confirms the proposed Lemma 2 in SectionIII-B. Furthermore, with a fixedNk, EC decreases withχkEE.This is due to the fact that the tradeoff system operates in theconflicting region of EE and EC, therefore the EE-increasesresult from EC-reductions. Moreover, with fixedχkEE andθk, asthe number of subcarriers increases, both EC and EE increase.With fixed Nk, when the delay QoS exponentθk increasesfrom 0.001 to 0.01, both EE and EC decrease. Especially,whenNk = 1, the decreases of EE and EC, as a result of theincrease inθk, are significant. However, whenNk is larger,e.g.,Nk = 8, the decreases of EE and EC are minor. Thisindicates that the multi-carrier communication system is morerobust against delay requirements, in comparison with single-carrier communication systems. In other words, when thesystem QoS requirement becomes more stringent, the multi-carrier system would sacrifice less EE and EC to guaranteethe requiredθk.

To further analyze and investigate the effect ofχEE on the

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.5

1

1.5

2

2.5

3

3.5

4

4.5

EE requirement factor χEE

Tot

al e

ffect

ive

capa

city

(b/

s/H

z)

Heuristic algorithm(θ=0.001)Heuristic algorithm(θ=0.1)Fair−exhaustive algorithm(θ=0.001)Fair−exhaustive algorithm(θ=0.1)Algorithm in [24] (θ=0.001)Algorithm in [24] (θ=0.1)

Fig. 14: The total effective capacity versus EE requirementfactorχEE for different values ofθ in heuristic algorithm, fair-exhaustive

algorithm and algorithm in [25].

0 1 2 3 4 5 6 7 8 90.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Per−user maximum available power value scaled by Kl (dB)

Tot

al e

ffect

ive

capa

city

(b/

s/H

z)

Heuristic algorithmFair−exhaustive algorithm

all θ =0.001

all θ = 0.1

Fig. 15: The total effective capacity versus maximum power value,for different values ofθ.

multi-user multi-carrier system, Fig. 14 includes the plots forthe total EC value versus EE requirement factorχEE, withtwo different values ofθ, for the heuristic algorithm, thefair-exhaustive algorithm and the algorithm in [25]. Assumethat the total number of usersK = 4 and the total numberof available subcarriersN = 8. Specifically, all K usershave the same settings of delay QoS exponentθ, χEE, withPcr = −10dB. WhenχEE increases, the total EC value of themulti-user multi-carrier system decreases in the heuristic al-gorithm and the fair-exhaustive algorithm. Furthermore, whenθ = 0.001, the EC curves of the two algorithms are exactlythe same. This indicates that for a system withK users havingloose delay requirements, the difference between the totalEC values calculated from the two algorithms is very little,even under different subcarrier assignment solutions. When θbecomes larger, the total EC values become smaller. Since alarger θ represents a more stringent delay QoS requirement,therefore each user’s maximum achievable arrival rate thatitcan support to maintain the target delay requirement, namely,its EC value, becomes small. Henceforth, the total systemEC value reduces, correspondingly. For the algorithm in [25],

14

10−4

10−3

10−2

10−1

10−5

10−4

10−3

10−2

10−1

100

Delay QoS exponent θk (1/bit)

Del

ay−

outa

ge p

roba

bilit

y

Nk = 8, χk

EE=0.9

Nk = 8, χk

EE=0.8

Fig. 16: Delay-outage probability versus delay QoS exponent θk,for different values ofχEE.

whenχEE increases, the total EC value stays stable. This isbecause, this algorithm proposed in [25] does not guaranteeall users’ EE requirements. Therefore, in this case,χEE has noinfluence on the total EC value.

Considering that the value of the maximum available powerfor each user can influence the optimal results, we include Fig.15 to show the performance of the heuristic algorithm andthe fair-exhaustive algorithm, when the maximum availablepower value varies. The total number of subcarriers is fixedat N = 6, and the number of users isK = 4. All users areassumed to have the same settings forχEE, i.e., χEE = 0.5,and the same circuit-to-noise power ratio, i.e.,Pcr = −10dB.In addition, two different scenarios of delay QoS exponentvector θ are included in Fig. 15, i.e., all elements inθ areeither 0.001 or 0.1. Firstly, we note that in both scenarios,the calculated final EC values from the two algorithms areclose, only with very little difference which makes the twocurves difficult to distinguish. This confirms that the proposedheuristic algorithm indeed guarantees a close-to-optimalper-formance. Furthermore, when all users have more stringentdelay QoS requirements, the total EC value reduces, whichmeans that the value of EC needs to be sacrificed in thissituation. More importantly, from Fig. 15, we can notice thatfor a fixedθ, the curves first increase, and then stabilize. Thisis because when the maximum available power value is toosmall to support the proposed optimal power value, the systemhas to operate atPmax. Therefore, the final calculated EC valueis smaller in this case, since it is obtained atPmax, rather than atthe optimal power value. On the other hand, when the value ofPmax becomes larger than the proposed optimal power value,then the system will operate at the optimal average power,which gives the maximum EC value, under each user’s EEconstraint. To find detailed analysis, please refer to SectionIII-B.

Fig. 16 plots the delay-outage probability for thekth user,P outdelay, versus delay QoS exponentθk, for various values ofχ

kEE

with a maximum tolerable delay thresholdDmax = 200 and thecircuit-to-noise power ratioP kcr = −10dB. This figure revealsthat for loose delay-constrained situations, e.g.,θk = 10−4,

the achievableP outdelay values stay the same with differentχkEE values. When the delay requirement is more stringent,e.g.,θk = 10−2, smallerχkEE ends up with less delay-outageprobability. This happens because smallerχkEE value meansmore sacrifices of EE from its maximum value, and in turn,results in more increases in its EC value. Therefore, theprobability that the buffer length exceedsDmax decreases,henceforth, the delay-outage probability reduces.

V. CONCLUSIONS

A close-to-optimal subcarrier assignment solution jointedwith an optimal power allocation strategy for end users tomaximize the system total EC value, subject to all users’ aver-age transmission power limits and link-layer EE constraints ina multi-user multi-carrier uplink network, were proposed anddeveloped in this paper. The traditional exhaustive algorithmwas introduced, followed by a fair-exhaustive algorithm, whichoffers the optimal frequency provisioning solution in a fairmanner. To reduce the computational complexity, we proposeda fair heuristic algorithm, which offers close-to-optimalsolu-tions, and has a low complexity linearly relating to the sizeof the problem. Given the subcarrier allocation matrix, thepower-constrained EC maximization problem in the single-user multi-carrier system, under each user’s individual link-layer requirement level, was solved. To thoroughly analyzethetradeoff problem, the effects of the circuit power and the EErequirement factor were proved and investigated. Simulationresults confirmed our design intentions, and further revealedthat when there is a link-layer EE constraint, each user’stradeoff EC value may not monotonically decrease with itsdelay QoS provisioning, and the tradeoff EC value obtainedwith less subcarriers may be higher than the one achieved withmore subcarriers.

APPENDIX A: PROOF OFLEMMA 1

Assume that the calculated tradeoff average power valuefor thekth user, obtained with a circuit power valueP kc,1, canbe denoted asP ∗k,1. Meanwhile, when the allocated numberof subcarriers isN , the maximum achievable EE value, i.e.,ηk,Nmax,1 = EE

k∣∣∣Nk=NPkc =P

kc,1

Pk=Pk∗

EE,1

, achieves atP k∗EE,1. If the kth user has

a higher circuit power, i.e,P kc,2 = Pkc,1 + ∆P

kc , ∆P

kc > 0,

the corresponding average power values which satisfy the EErequirement equality (20) andηk,Nmax,2 can be written asP

∗k,2

and P k∗EE,2, respectively. From (20), we have the followingequations:

EEk∣∣∣Pkc =Pkc,1Pk=P∗k,1

= χkEE × EEk∣∣∣Nk=NPkc =P

kc,1

Pk=Pk∗

EE,1

, (22a)

EEk∣∣∣Pkc =Pkc,2Pk=P∗k,2

= χkEE × EEk∣∣∣Nk=NPkc =P

kc,2

Pk=Pk∗

EE,2

. (22b)

In order to investigate the influence of circuit powerP kc onthe tradeoff EE value, we start from analyzing the effect ofP kcon the maximum EE valueηk,Nmax . For the system withP kc,1, if

15

we assume the operational average input power isP k∗EE,2, thecorresponding link-layer EE value can be calculated as

EEk∣∣∣Nk=NPkc =P

kc,1

Pk=Pk∗

EE,2

=

Ekc

∣∣∣Nk=NPk=Pk

∗

EE,2

P kc,1 +1

ǫP k

∗

EE,2

. (23)

Meanwhile, for the system withP kc,2, theηk,Nmax,2value is defined

as

EEk∣∣∣Nk=NPkc =P

kc,2

Pk=Pk∗

EE,2

=

Ekc

∣∣∣Nk=NPk=Pk

∗

EE,2

P kc,2 +1

ǫP k

∗

EE,2

. (24)

Apparently, we can notice that the link-layer EE value in (23)is larger than the one in (24), becauseP kc,1 < P

kc,2. Henceforth,

we can derive that the maximum achievable link-layer EEvalue ηk,Nmax,1 for the system with circuit powerP

kc,1 is larger

than the one obtaining at a larger circuit powerP kc,2. Thismeans that when one user’s circuit power becomes larger,its maximum achievable EE value reduces. From (22a)-(22b),we finally conclude that one user’s tradeoff EE level alsodecreases with its circuit power.

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