Hierarchical Resource Allocation: Balancing Throughput and Energy Efficiency in Wireless Systems
aa r X i v : . [ c s . I T ] F e b Hierarchical Resource Allocation:Balancing Throughput and Energy Efficiencyin Wireless Systems
Bho Matthiesen ∗ , Eduard A. Jorswieck † , and Petar Popovski ‡∗∗ University of Bremen, Department of Communications Engineering, Germany, email: [email protected] † TU Braunschweig, Department of Information Theory and Communication Systemes, Germany, email: [email protected] ‡ Aalborg University, Department of Electronic Systems, Denmark, email: [email protected]
Abstract —A main challenge of 5G and beyond wireless sys-tems is to efficiently utilize the available spectrum and si-multaneously reduce the energy consumption. From the radioresource allocation perspective, the solution to this problem isto maximize the energy efficiency instead of the throughput.This results in the optimal benefit-cost ratio between data rateand energy consumption. It also often leads to a considerablereduction in throughput and, hence, an underutilization of theavailable spectrum. Contemporary approaches to balance thesemetrics based on multi-objective programming theory often lackoperational meaning and finding the correct operating pointrequires careful experimentation and calibration. Instead, wepropose the novel concept of hierarchical resource allocationwhere conflicting objectives are ordered by their importance.This results in a resource allocation algorithm that strives tominimize the transmit power while keeping the data rate close themaximum achievable throughput. In a typical multi-cell scenario,this strategy is shown to reduces the transmit power consumptionby 65% at the cost of a 5% decrease in throughput. Moreover,this strategy also saves energy in scenarios where global energyefficiency maximization fails to achieve any gain over throughputmaximization.
Index Terms —multi-objective programming, global optimiza-tion, hierarchical optimization, mixed monotonic programming
I. M
OTIVATION AND P ROBLEM S TATEMENT
The goal of resource allocation in communication networksis to best utilize the available resources ensuring good Qualityof Service (QoS) to all users. While the QoS constraintsare mainly determined by the user’s requirements or networkslice configuration, the choice of a suitable utility functionis entirely up to the operator or system designer [1]–[3].Common choices are maximizing the throughput (TP) to bestutilize the available spectrum [4], minimizing the total transmitpower to save energy [5], or maximizing the energy efficiency(EE) to obtain a trade-off between these two [6], [7]. In general,these are conflicting metrics that can not be maximized simul-taneously. Indeed, the multi-objective optimization problem(MOP) max p ∈P (cid:2) f ( p ) , f ( p ) , . . . (cid:3) (1) The work of B. Matthiesen and P. Popovski is supported in part by theGerman Research Foundation (DFG) through Germany’s Excellence Strategyunder Grant EXC 2077 (University Allowance). The work of E. A. Jorswieckis supported in part by the DFG under Grant JO 801/24-1. with network utility functions f , f , . . . is known to possesan infinite number of noninferior solutions [8]. The MOP (1)is usually solved by transforming it into a scalar optimizationproblem, e.g., with the scalarization approach [9] where theweighted sum of the objectives is maximized, i.e., max p ∈P X i w i f i ( p ) , or by the utility profile approach [9] where the intersectionof a ray in the direction w and the outer boundary of theperformance region is computed, i.e., max t, p ∈P t s . t . ∀ i : tw i ≤ f i ( p ) . Both methods obtain Pareto optimal points but share the weak-ness that the weights w often have no operational meaning andneed to be chosen heuristically or by experimentation.For example, consider balancing the TP with the totaltransmit power. This problem is formally stated as ( max p , r (cid:2)P i r i , − P i p i (cid:3) s . t . r ∈ R ( p ) ∩ Q , ≤ p ≤ P (2) where P is the maximum transmit power, R ( p ) the achiev-able rate region, and Q contains the QoS constraints. Afterscalarization, the problem becomes max p , r w X i r i − w X i p i s . t . r ∈ R ( p ) ∩ Q , ≤ p ≤ P (3) with nonnegative weights w , w . By varying these weightssuch that w + w = 1 , the convex hull of the Paretoboundary is obtained. However, these weights do not havemuch operational meaning and there is no other guidancethan experience or experimentation to choose them for a givensystem. Another approach to balance TP and transmit poweris the notion of global energy efficiency (GEE), which isdefined as the benefit-cost ratio of system throughput and totaldissipated power, i.e., GEE = P i r i P i µ i p i + P c , where µ i ≥ and P c > are modeling constants reflecting the poweramplifier inefficiency and static circuit power consumptions.Maximizing the GEE results in a Pareto optimal solution of (2)[7, p. 241] and has a well defined operational meaning. With P max GEE max TP Power
GEE
PowerFig. 1. Typical solution of TP and GEE maximization. energy and spectrum being similarly scarce resources, the TPand GEE are considered to be the most important networkutility functions in 5G and beyond networks.A qualitative solution of TP and GEE maximization inwireless interference networks is displayed in Fig. 1. Whileleading to similar operating points in the low signal-to-noiseratio regime, it is characteristic for the GEE to saturate. Thelink and power budget in a wireless network often allowfor an operating point far in this saturation region. In sucha scenario, selecting the operating point by TP or GEEmaximization either results in poor EE or in low spectralefficiency. Thus, it has been proposed in [10] to balance TPand GEE with multi-objective programming theory. While theobtained performance region provides valuable insights forsystem design, the weights still have little operational meaning.A more straightforward method is to maximize the GEE underQoS constraints which is expected to provide the best rate-energy trade-off while still providing satisfactory service toall users.Taking the operator’s perspective, saving energy is just asecondary concern, while generating revenue from their costlyequipment and spectrum licenses is the primary goal. Thisrequires good service quality to outperform competitors andthereby ensure customer loyalty. Satisfying QoS constraintsand providing good connectivity is undoubtedly the foundationfor good service but from there it’s up to the operator tochoose an operating point in the resource allocation designspace. A viable strategy is to prioritize high service qualityand minimize energy consumption as a secondary objective toreduce operational expenditures and further increase revenue.This could be achieved by solving (3) with w ≫ w .A more rigorous approach is to use lexicographic ordering[11, §4.2], a recursive multi-objective programming techniquewhere objectives are strictly ordered by priority. In the contextof this paper and (2), a lexicographic ordering approach isto maximize the TP first and then select the solution withlowest transmit power, i.e., p ⋆ = min { P i p i | p ∈ T ⋆ } where T ⋆ is the set of throughput optimal power allocations, i.e., T ⋆ = arg max { P r i | r ∈ R ( p ) , ≤ p ≤ P } . When thesolution to the TP maximization problem is (almost) unique,i.e., the volume of T ⋆ is close to zero, the possible power re-duction due to this approach is negligible. However, significantgains are possible by slightly relaxing this strict ordering of theobjectives. For example, the goal could be to achieve at least95 % of the maximum TP instead of strictly maximizing it, i.e., p ⋆ = min { P i p i | P i r i ≥ . · r ⋆ Σ , r ∈ R ( p ) } , where r ⋆ Σ isthe optimal value of the TP maximization problem. Selectinga power allocation within this tight TP region leaves more freedom than lexicographic ordering, while still ensuring highservice quality.Leaving economical considerations aside, there are plentyof other technical motivations to strictly prioritize a highTP over other metrics. One application arises from cross-layer optimization where the queue of a base station (BS)needs to be stabilized. Regardless of the underlying queuingmodel, the total storage capacity is essentially limited bythe BS’s installed memory. The TP determines the maximumdeparture rate of this joint queue and, hence, maximizing theTP ultimately enlarges the stability region. Please refer to [4]for further application examples.The goal of this paper is to obtain a hierarchical Paretooptimal solution of (2) for wireless interference networks, andto evaluate the benefits of this approach over GEE maximiza-tion numerically. As the resulting optimization problem is NP-hard and numerically very challenging, this requires the carefuldesign of a solution algorithm. We show that, by reducing theTP by just 5 %, almost 65 % of transmit power can be savedin a typical wireless network. A. System Model
We consider a Gaussian interference network with powerallocation p = ( p , p , . . . ) and average power constraint P . The receive signal to interference plus noise ratio (SINR)is α i p i P j = i β ij p j + σ i and, under the assumption that interferenceis treated as noise, asymptotic error free communication ispossible at all rates r satisfying r i ≤ B log α i p i P j = i β ij p j + σ i ! for all i , where B is the communication bandwidth. In thissetting, α i is the effective channel gain of the direct channelfrom transmitter i to receiver i , β ij are the effective channelsfrom transmitter j to receiver i , and σ i is the variance ofcircularly-symmetric complex Gaussian noise.This adequately models the effective channel for multi-antenna transmission in 5G networks after precoder matrixselection [12, §11], for multi-cell networks with overlappingfrequencies, and for dense low earth orbit (LEO) satelliteconstellations [13]. Other applications include, e.g., massiveMIMO and relay-assisted CoMP networks [14].II. H IERARCHICAL O PTIMIZATION
Hierarchical optimization [11, §4.2.2], [15] is a solutionmethod for the MOP (1) where the objectives are arrangeda priori by their absolute importance. Without loss of general-ity, assume that f i is more important to the system designerthan f i +1 . The optimization is carried out recursively by firstmaximizing f and ignoring all other objectives f , f , . . . .Then, the next objective f is maximized with additionalconstraint that the value of f is close to the optimal value ofthe previous optimization. Mathematically, the i th optimizationproblem is max p ∈D i f i ( x ) with D i = { p ∈ D i − | f i − ( p ) ≥ ω i − f ⋆i − } for all i > and some initial feasible set D . Here, f ⋆i denotes the optimal value of the i th problem and ω , ω , . . . re so-called worsening factors. These are selected a priori bythe system designer and have, contrary to the weights in themulti-objective programming solution approaches discussedin Section I, a clearly defined operational meaning in manyengineering problems. Lexicographic ordering [11, §4.2] is aspecial case of this approach obtained by setting all worseningfactors to one. For a MOP with two objectives, the second(and final) optimization step is equivalent to the ε -constraintmethod [11, §3.2] and its solution is a strictly Pareto optimalpoint if it is unique [11, Thm. 3.2.4].Applying this approach to the MOP (2) and prioritizing theTP over the transmit power, we obtain two scalar optimizationproblems max p , r X i log α i p i P j = i β ij p j + σ i ! s . t . ∀ i : log α i p i P j = i β ij p j + σ i ! ≥ r i, min ≤ p ≤ P (4a)(4b)(4c) for minimum rate constraints r i, min ≥ , and min p , r X i p i s . t . X i log α i p i P j = i β ij p j + σ i ! ≥ ωr ⋆ Σ ∀ i : log α i p i P j = i β ij p j + σ i ! ≥ r i, min ≤ p ≤ P (5a)(5b)(5c)(5d) where r ⋆ Σ is the optimal value of (4) and ω ∈ [0 , is theworsening factor that determines the acceptable TP reduction.Clearly, it is necessary to solve (4) before (5).Both problems (4) and (5) are challenging global optimiza-tion problems due to the nonconvexity of the objective in(4) and constraint (5b). In particular, (4) is known to be NP-hard [16], and, hence, (5) is also NP-hard due to constraint(5b). While (4) can be solved efficiently using the mixedmonotonic programming (MMP) framework as discussed next,problem (5) needs a novel algorithm that is developed inSection III. A. Solution of Problem (4)MMP is a global optimization framework that exploitspartial monotonicity in the objective and constraints [17]. Itis much more versatile than classical monotonic optimization[18] and shows tremendous performance gains over state-of-the-art algorithms for global optimal power allocation ininterference networks and other scenarios [17, §IV].The concept of mixed monotonic (MM) functions general-izes differences of increasing functions. Let M be a box in R n , i.e., M = [ r , s ] = { x ∈ R n | ∀ i : r i ≤ x i ≤ s i } . Acontinuous function F : R n × R n → R is called MM functionif it satisfies F ( x , y ) ≤ F ( x ′ , y ) if x ≤ x ′ ,F ( x , y ) ≥ F ( x , y ′ ) if y ≤ y ′ . The constant B is inessential and moved into r i, min for notational clarity. for all x , x ′ , y , y ′ ∈ M and a continuous optimizationproblem max x ∈D f ( x ) with compact feasible set D ⊆ R n is called MMP problem if there exists an MM function F such that F ( x , x ) = f ( x ) for all x ∈ M , where M ⊇ D encloses D . The MMP framework [17] solves such a problemvery efficiently with global optimality using a branch andbound (BB) procedure.Applying the MMP framework requires MM representationsof the objective and constraint functions in (4). For theobjective, such a function is x , y P i R i ( x , y ) with [17,§IV-A] R i ( x , y ) = log α i x i P j = i β ij y j + σ i ! . (6) Likewise, the QoS constraints have MM representation x , y r i, min − R i ( x , y ) . Theoretically, such MM constraints lead toan algorithm without guaranteed finite convergence. This is,because for general MM constraints and some boxes M , itis impossible to determine whether M ∩ D contains feasiblepoints or not [17, §III-A]. However, in practise this is seldoma problem for typical minimum rate constraints as in (4b).The MMP framework is also applicable to (5). However, theminimum sum rate constraint in (5b) is very tight and leadsto a tiny feasible set compared to M = [ , P ] . This resultsin impractically slow convergence of the MMP procedure. Inthe next section, we develop an algorithm with much fasterand provably finite convergence.III. S UCCESSIVE I NCUMBENT T RANSCENDING S CHEME
The main challenge in solving (5) with the MMP frameworkis constraint (5b). An efficient solution to this problem is thesuccessive incumbent transcending (SIT) scheme developedin [19]. The main idea is to solve a sequence of easilyimplementable feasibility problems. Specifically, given a realnumber γ , the core problem of the SIT algorithm is to checkwhether (5) has a feasible solution p satisfying P i p i ≤ γ , or,else, establish that no such p exists. In this manner, a sequenceof feasible points (“incumbents”) with decreasing objectivevalue is generated until no point with lesser objective valuethan the current best solution γ exists.Consider the optimization problem min x ∈M f ( x ) s . t . g ( x ) ≤ (7) which generalizes (5) and assume that f is a nondecreasingfunction, g has an MM representation, and M is a box. Theoutlined SIT scheme for this problem is given in Algorithm 1. Algorithm 1
SIT Scheme [20, Sect. 7.5.1]
Step 0
Initialize ¯ x with the best known feasible solution and set γ = f (¯ x ) − η ; otherwise do not set ¯ x and choose some γ ≤ f ( x ) ∀ x ∈ M : g ( x ) ≤ . Step 1
Check if (5) has a feasible solution x satisfying f ( x ) ≥ γ ;otherwise, establish that no such feasible x exists and goto Step 3. Step 2
Update ¯ x ← x and γ ← f (¯ x ) − η . Go to Step 1. Step 3
Terminate: If ¯ x is set, it is an η -optimal solution; elseProblem (5) is infeasible. mplementing the feasibility check in Step 1 of Algorithm 1efficiently is crucial. Consider the optimization problem min x ∈M g ( x ) s . t . f ( x ) ≤ γ (8) which is dual to (7) in the sense that if the optimal value of(8) is greater than zero, the optimal value of (7) is greaterthan γ [20, Prop. 7.13]. Thus, any point x ′ in the feasible setof (8) with objective value less than zero is also a feasiblepoint in (7) with objective value less than γ . We can solve (7)sequentially by solving (8) with a BB method.At first, this approach seems to increase the computationalcomplexity significantly because if (7) is nonconvex, thenso is (8). However, given that f has favorable properties, problem (8) might be considerably easier to solve than (7).Moreover, the SIT scheme can be combined with the BBprocedure that solves (8). This eliminates the need to solve(8) multiple times.Exploiting the properties of MM functions, we can obtaina lower bound on the objective value of (8) over a box M =[ r , s ] from its MM representation G as min x ∈M : f ( x ) ≤ γ g ( x ) ≥ min x ∈M G ( x , x ) ≥ min x , y ∈M G ( x , y ) = G ( r , s ) . Together with an exhaustive rectangular subdivision [20], thisbound leads to a convergent BB procedure that can be incor-porated into the SIT scheme.The complete algorithm is stated in Algorithm 2. It involvesa parameter ε that is related to the concept of ε -essentialfeasibility explained in [21]. Its primary roles are to excludenumerically instable points from the feasible set and ensurefinite convergence of the algorithm. The latter is establishedin the theorem below. This is the first algorithm that combinesthe MMP approach with the SIT scheme. Theorem 1:
Algorithm 2 converges in finitely many steps tothe ( ε, η ) -optimal solution of (7) or establishes that no suchsolution exists. Proof sketch:
By virtue of [20, Prop. 7.14] a BBprocedure for solving (8) with pruning criterion G ( r , s ) > − ε and stopping criterion g ( r ) < or R k = ∅ implements Step 1in Algorithm 1. Thus, start with the MMP algorithm in [17,Alg. 1] for (8) and modify it according to the previous sentence.Establishing finite convergence is a minor modification of [17,Thm. 1]. Next, integrate the SIT scheme in Algorithm 1 intothis procedure: move the termination criterion g ( r ) < intothe incumbent update in Step 3 and update γ k if a box satisfiesthis criterion. It remains to show that continuing the procedureafter updating γ k preserves convergence. This part of the prooffollows along the lines of the proof of [21, Thm. 1].The purpose of the reduction in Step 2 is to speed upthe convergence. This is achieved by replacing the box underconsideration by a smaller one that still contains all candidatesolutions and, thereby, improves the quality of the computedbounds. One approach to determine this procedure for Algo-rithm 2 is to replace M by M ′ = [ r ′ , s ′ ] with r ′ i = min x ∈M : f ( x ) ≤ γ k x i , s ′ i = max x ∈M : f ( x ) ≤ γ k x i (10) Such favorable properties could be, e.g., linearity, convexity, or beingincreasing.
Algorithm 2
SIT Algorithm for (7)
Step 0 (Initialization)
Set ε, η > , Let k = 1 and R = {M } .If available, initialize ¯ x with the best known feasiblesolution and set γ k = f (¯ x ) − η . Otherwise, do not set ¯ x and choose γ ≥ f ( x ) for all feasible x . Step 1 (Branching)
Let M k = [ r k , s k ] be the oldest box in R k − . Bisect M k via ( v k , j k ) with j k ∈ arg max j s kj − r kj and v k = ( s k + r k ) , i.e., compute M − = { x | r kj ≤ x j ≤ v kj , r ki ≤ x i ≤ s ki ( i = j ) }M + = { x | v kj ≤ x j ≤ s kj , r ki ≤ x i ≤ s ki ( i = j ) } , and set P k = {M k − , M k + } . Step 2 (Reduction)
Replace each box in
M ∈ P k with some M ′ such that M ′ ⊆ M and min { g ( x ) | f ( x ) ≤ γ k , x ∈ M} = min { g ( x ) | f ( x ) ≤ γ k , x ∈ M ′ } (9) Step 3 (Incumbent)
Let I = { r | [ r , s ] ∈ P k , g ( r ) ≤ } . Ifnot empty, set r k = arg min r ∈ I f ( r ) . If ¯ x k − is not setor f ( r k ) < γ k − + η , set ¯ x = r k and γ k = f ( r k ) − η .In all other cases, set ¯ x k = ¯ x k − and γ k = γ k − . Step 4 (Pruning)
Delete every [ r , s ] ∈ P k with f ( r ) ≥ γ k or G ( r , s ) > − ε . Let P ′ k be the collection of remaining setsand set R k = P ′ k ∪ ( R k − \ {M k } ) . Step 5 (Termination)
Terminate if R = ∅ : If ¯ x k is not set, then(7) is ε -essential infeasible; else ¯ x k is an essential ( ε, η ) -optimal solution of (7). Otherwise, update k ← k + 1 andreturn to Step 1. for all i . For f nondecreasing, the solution to the first problemis always r i unless it is infeasible. For the upper bound in(10), recall that r minimizes f ( x ) over M . Thus, the optimalsolution to this optimization problem is to set x j = r j for all j = i . Then, the optimal x i = min { ˜ x i , s i } where ˜ x i satisfies f ( r + (˜ x i − r i ) e i ) = γ k . (11) Remark 1 (Branch selection):
Most BB procedures selectthe box with the largest bound for further partitioning. Therationale is that this choice leads to fastest convergence. Inpractice, when the number of boxes in R k grows very large,this selection rule might become the performance and memorybottleneck of the algorithm. First, it tends to store subop-timal boxes longer than necessary and therefore increasesmemory consumption. Second, inserting new boxes into R k has complexity O (log |R k | ) . Instead, with the oldest-first ruleemployed in Algorithm 2 inserting new boxes has constantcomplexity. Also, every box is visited after a fixed amount oftime and, thus, likely to be pruned much earlier than with thebest-first rule [17]. Since Algorithm 2 is essentially memorylimited, the oldest-first rule performs much better than thestandard best-first rule. Remark 2 (Other SIT applications):
Despite its tremendousnumerical advantages, the SIT approach is currently not widelyused. Besides the applications to DC and monotonic optimiza-tion problems in [19], [20], it is only employed in [21] whereit is applied to resource allocation problems with fractionalobjectives and partial convexity. The implementation mostclosely related to Algorithm 2 is the monotonic optimizationariant in [20, §11.3]. The key advantage of Algorithm 2over this procedure is that cumbersome transformations and anauxiliary variable are required to bring (5) into a suitable formfor [20, §11.3]. This leads to much slower convergence due tothe extra variable and much looser bounds on the constraints.
A. Solution of Problem (5)Identify M = [ , P ] and f ( p ) = P i p i . Note that f ( p ) isan increasing function. MM representations of (5b) and (5c)are x , y ωr ⋆ Σ − P i R i ( y , x ) and ∀ i : x , y r i, min − R i ( y , x ) , respectively, with R i ( x , y ) as in (6). They can bemerged into a single inequality constraint max i g i ( x ) ≤ withMM representation G ( x , y ) = max n ωr ⋆ Σ − X i R i ( y , x ) , max i (cid:8) r i, min − R i ( y , x ) (cid:9)o due to [17, Eq. (9)]. In the reduction step, the solution to (11)is ˜ x i = γ k − P j = i r j . Thus, every box M = [ r , s ] in Step 2can be replaced by [ r , s ′ ] with s ′ i = min { s i , γ k − P j = i r j } .With these choices, Algorithm 2 solves (5) in a finite numberof iterations. IV. N UMERICAL E VALUATION
We consider uplink transmission in a single-input single-output multi-cell system. User equipments (UEs) are placedrandomly in a rectangular area with edge length 1 km. Thisarea is divided into four equal sized cells with BSs locatedat the center of their cell. Path-loss is modeled accordingto the Hata-COST231 [22], [23] urban scenario with carrierfrequency 1.9 GHz, 30 m BS height and 8 dB log-normalshadow fading. Small scale effects are modeled as Rayleighfading. Each UE is associated to the BS with the best channel.Scenarios where more than one UE is associated to a BSare dropped. The receivers have noise spectral density N = −
174 dBm and noise figure F = 3 dB . The communicationbandwidth is B = 180 kHz and the noise power is calculatedas σ i = N F B . The UEs RF chains have a static powerconsumption P c = 400 mW and power amplifiers with anefficiency of 25 %. No cooperation between BSs is assumed,i.e., interference from other cells is treated as noise.TP and GEE are maximized using the MMP framework[17]. Algorithm 2 is used to solve (5) for ω = 0 . , i.e., theobtained resource allocation uses the minimum total transmitpower under the constraint that the system TP is not less than95 % of the maximum achievable system TP. We call thisresource allocation high throughput energy efficiency (HTEE)for reasons that will become apparent below. All algorithmsobtain the global optimal solution within an absolute toleranceof η = 0 . . In Algorithm 2, we set ε = 10 − . All resultsare averaged over 1000 independent and identically distributedchannel realizations.Figures 2 and 3 display TP and GEE, respectively, withvery typical behavior. With increasing transmit power budget,the maximum TP increases. Instead, the GEE saturates atsome point and the transmit power stays constant in the GEEoptimal resource allocation. Increasing the transmit powerbeyond the GEE saturation point, as is done in the TP optimal − −
10 0 10 20 3001234
Maximum Tx Power P [dBm] T h r oughpu t [ M b it/ s ] TPHTEE 95 %GEE
Fig. 2. Achievable throughput with different resource allocation approaches. − −
10 0 10 20 300246
Maximum Tx Power P [dBm] G EE [ M b it/ J ] TPHTEE 95 %GEE
Fig. 3. Global energy efficiency of the discussed resource allocation strategies. allocation, decreases the GEE. For a maximum transmit powerof 23 dBm, which corresponds to the typical UE power budget[12], [24], the GEE optimal allocation achieves 22.4 % or0.84 Mbit/s less TP than possible. Instead, the HTEE resourceallocation is within 95 % of the maximum achievable TP andachieves a 97 % higher GEE than the maximum TP allocationat 23 dBm. This corresponds to a gain of 1.8 Mbit/J at the costof 0.19 Mbit/s.However, the GEE is not the optimal metric to evaluatetransmit power savings. Consider a second operating point at−10 dBm, the median transmit power of 4G UEs in urban sce-narios [24]. The TP and GEE optimal strategies both achievealmost the same TP and GEE. Figure 4 displays the powerconsumption relative to the TP optimal resource allocation. Itcan be observed that the GEE strategy consumes almost asmuch transmit power as the TP strategy, and, thus, is unableto exploit the “rate reduction budget” of the system designer.Instead, the HTEE strategy uses almost 40 % less transmitpower at a TP cost of 50 kbit/s, which is less than the datarate of classical digital telephone line modem. Nevertheless,its GEE is worse than that of the other strategies, despite thetremendous transmit power reduction.Returning to our previous scenario with 23 dBm maximumtransmit power, it can be seen from Fig. 4 that the HTEEstrategy consumes only 35.7 % of the transmit power necessaryto achieve the maximum TP. Of course the GEE strategy saveseven more transmit power but at a much higher cost to thethroughput. This trade-off is illustrated in Fig. 5 where the −
10 0 10 20 300%50%100% po w e rr e du c ti on Maximum Tx Power P [dBm] R e l a ti v e T x P o w e r TPHTEE 95 %GEE
Fig. 4. Total power consumption relative to the throughput optimal strategy. po w e r – r a t e t r a d e - o ff Throughput [Mbit/s] R e l a ti v e T x P o w e r TPHTEE 95 %GEE
Fig. 5. Relative power consumption as a function of the achievable through-put. relative transmit power is plotted over the achievable data rate.It can be observed that a major advantage of the HTEE strategyover the GEE optimal power allocation is that the TP doesnot saturate and any data rate is achievable given a sufficienttransmit power budget. Thus, it results in an energy-efficientresource allocation while still ensuring high TP.Finally, to support the statement at the end of Section II-Athat (5) is hard to solve with a traditional BB method, wehave also employed the MMP framework to solve (5). Outof 1000 problem instances that ran on an Intel Xeon E5-2680v3 CPU with a memory usage limit of 21 GB, 483 probleminstances ran out of memory and 517 problem instances did notcomplete within 24 hours, i.e., not a single problem instanceof (5) could be solved by a traditional BB algorithm withreasonable usage of computational resources. In contrast, thesame problem instances could be solved with Algorithm 1using a maximum of 50 MB memory and not taking longerthan 752 ms to complete. The median computation time amongall problem instances was 1.75 ms.V. C
ONCLUSIONS
We have introduced the novel concept of hierarchical re-source allocation and applied it to minimize energy con-sumption while still ensuring high spectrum utilization. Thenumerical results show a transmit power reduction of 65 % ina multi-cell communication scenario at the cost of a 5 % drop in TP. Instead, state-of-the-art GEE maximization results ina TP reduction of almost 25 %. Moreover, this strategy alsosaves energy in scenarios where GEE optimization fails toprovide a gain over TP maximization. The developed algo-rithms solve the involved optimization problems with globaloptimality and, therefore, rigorously demonstrate the gains ofhierarchical resource allocation and high-throughput energyefficiency maximization over state-of-the-art approaches.R
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