Energy Efficiency and Delay in Wireless Systems:Is Their Relation Always a Tradeoff?

Changyang She and Chenyang Yang

Abstract—It is well-known that the average transmit powercan be traded off by average delay. This paper strives to studythe relation between the maximal energy efficiency (EE) andthe delay bound with a given violation probability for wirelesssystems serving randomly arrived traffic. We show that if theminimal average transmit and circuit powers consumed at abase station linearly increases with the required average servicerate, i.e., the power-rate relation is linear, then a non-tradeoffregion will appear in the EE-delay relation. By taking multi-input-multi-output system as an example, we show that thepower-rate relation will be linear if the transmit power andbandwidth are jointly allocated and if the bandwidth constraintis inactive to support a required delay bound. The impacts ofbandwidth constraint on the power-rate and EE-delay relationsare then analyzed. To study fundamental EE-delay relation, aqueue length dependent two-state policy is optimized. By furtherconsidering a compound Poisson arrival process in large numberof transmit antennas asymptotics, we find the boundary of thetradeoff and non-tradeoff regions, and provide a lower boundof the Pareto optimal EE-delay relation in the tradeoff region,all with closed-form expressions. Our results show that the non-tradeoff region increases with the maximal bandwidth and thenumber of transmit antennas.

Index Terms—EE-delay relation, power-rate relation, statisticalQoS requirement

I. INTRODUCTION

The fifth generation (5G) mobile networks are expected tosupport high throughputs for a wide variety of services withdiverse quality-of-service (QoS) requirements, ranging fromtactile internet with 1 ms latency to video streaming withmuch less stringent delay requirement [1, 2]. To support theever-growing traffic demands with satisfactory user experienceand to reduce the cost and global carbon dioxide emissions,energy efficiency (EE) has become one of the major designgoals for 5G systems. To meet the possibly conflicting per-formance metrics, several fundamental tradeoffs need carefulexamination [3], among which the EE/power-delay tradeoffhas drawn significant attention over the past decade. This isbecause delay is a representative QoS requirement that is morerelevant to characterize user experience than a minimal datarate requirement [4]. Such a metric is especially important fordelay sensitive traffic such as multimedia transmission [5, 6].

Manuscript received March 6, 2015; revised August 10, 2015, December15, 2015 and May 26, 2016; accepted August 3, 2016. The associate editorcoordinating the review of this paper and approving it for publication was M.C. Gursoy. The work is supported by China NSFC under Grant 61120106002and 973 Program under Grant 2012CB316003.

Changyang She and Chenyang Yang are with the School of Electronicsand Information Engineering, Beihang University, Beijing 100191, China(email:{cyshe,cyyang}@buaa.edu.cn).

EE/power-delay tradeoff has long been believed as an inher-ent property for wireless communications bounded by Shan-nons channel capacity. Since revealed in the pioneering workin [7], it has been widely accepted that the relation betweenEE/power and delay is a tradeoff, no matter if the delay metricis the average delay, strict deadline or the delay bound with asmall violation probability. Based on the fact that the power re-quired to transmit a bit reliably is a strictly convex function ofthe transmission rate for a given fading channel state, the studyin [7] shows that the average transmit power and the averagequeueing delay cannot be minimized at the same time, unlessboth the arrival rate and the channel are fixed for all time. ThePareto optimal power-delay tradeoff was characterized in [7]when the average delay approaches infinity and subsequentlyin [8] when the average delay approaches zero. When thedelay performance is modelled as a strict deadline, the power-delay tradeoff was also observed and then exploited to savepower by extending the delay deadline in [9, 10]. As pointedin [7], the average delay is relatively easy to analyze, but doesnot necessarily ensure the QoS required by delay sensitiveapplications. On the other hand, the strict delay deadline is tooexpensive to guarantee in wireless systems in terms of transmitpower. The statistical QoS requirement, defined as a delaybound and a delay violation probability, is more relevant forwireless multimedia transmission [11] but is more difficult toanalyze. In this context, the power-delay tradeoffs for severalorthogonal-frequency-division-multiplexing (OFDM) systemswere studied in [5], and the EE-delay tradeoffs were respec-tively observed for orthogonal frequency division multipleaccess (OFDMA) system in [12], narrow-band system in [13],and multi-input-multi-output (MIMO) systems in [14], all bysimulations.

While these research efforts have provided important in-sights, only the transmit power was assumed to consumeenergy implicitly despite that a fixed circuit power has beentaken into account in EE in several works, because onlytransmit power, modulation and coding schemes, and/or sub-carrier allocation were adjusted to save power, while fullbandwidth is used for transmission. In fact, this is the essentialreason that leads to the EE/power-delay tradeoff. In prevalentpractical systems, however, the energy consumed for runningthe circuits in a system is not negligible, hence other power-saving mechanisms become necessary [15] (and also viablein practice [16]), which not only reduce transmit power but

also reduce circuit power.1 For example, we can further adaptthe bandwidth or even turn a base station (BS) into idlemode to reduce the circuit power. While the importanceof reducing circuit power has been widely recognized ingreen communication literature, the impact of a rate-dependentcircuit power consumption model on the EE-delay relationis largely overlooked, which may change the characteristicsof the EE/power-delay relation. Recently, it was observed in[6] that the average power is not a monotonically decreasingfunction of the average sojourn time when BS idling strategiesare considered. Similar observation has been obtained in anearlier work [17] with transmission time as the delay metric.

A. Related Works

There have been a lot of works investigating transmit poweror energy efficiency under delay constraints. In general, theseworks can be divided into two major categories in terms ofgoal and hence methodologies.

The goal of the first category of works is to reveal thefundamental relation between average transmit power and av-erage delay requirement [7,8,18], which are from information-theoretic perspective. The fundamental studies aimed at find-ing the Pareto optimal power-delay tradeoff in [7, 8] implythat the power-delay tradeoff comes from the relationship be-tween the transmit power and transmission/service rate, whichis strictly convex according to the Shannon’s capacity. Anexceptional example in [18] demonstrates that if the averagetransmit power is a piecewise linear function of the requiredaverage rate, then the optimal power-delay tradeoff curve in[7] can be exceeded. To show how to achieve the Paretooptimal power-delay tradeoff, some transmission policies wereprovided in [7, 8, 18]. Since queueing delay is taken intoaccount, all the policies in these works depend on both channelstate information (CSI) and queue state information (QSI).

The goal of the second category of works is to design energyefficient policies with delay requirements, e.g., [5,6,9,10,12–14, 17, 19–25]. After optimizing a policy towards a specificobjective function, the EE/power-delay relation that can beachieved by the policy is either analyzed theoretically or dis-cussed via simulations, where the delay metric is hard deadlinein [9, 10, 19–22], delay bound with a violation probability in[5,12–14], and real average delay achieved by the policies in[6, 17, 23–25]. Except in [6, 17, 25], all these works either donot consider or only consider a fixed rate-independent circuitpower, and the power saving mechanism is power allocationthat only reduces the transmit power. For all the worksconsidering the statistical QoS requirement [5, 12–14], theoptimized policies only adapt to CSI. This may be owing to thefact that effective capacity is powerful in designing the policies

1It is worthy to note that the circuit power has been taken into account in[12, 13], which is a constant independent of the transmission rate. However,only the mathematical property of the optimization problem is affected bythe rate-independent circuit power, while the optimization variables (i.e.,the transmit power and/or subcarrier allocation) cannot reduce the circuitpower. Therefore, only transmit power is reduced, and the EE-delay relationis essentially the same as the power-delay relation.

with statistical QoS provision, which can translate a cross-layer problem related to both QSI and CSI into a physical-layer issue only related to CSI. As indicated in [7], wheneither source or channel is stochastic, the resource allocationshould depend on the QSI, otherwise the average power cannotbe minimized for a target average delay. This implies that theEE/power-delay relations observed in [5,12–14] are not Paretooptimal.

B. Differences from Related Works and Contributions

In this paper, we strive to investigate fundamental EE-delayrelation for wireless systems serving random sources withstatistical QoS requirement and connect it with the power-raterelation. By “ fundamental”, we mean that our study is frominformation-theoretic perspective, along the same line with thefirst category of research in [7, 8, 18]. Nonetheless, our studydiffers from these works in the following two aspects. First,we consider statistical QoS requirement, hence the tool weused for analysis differs from theirs. Second, we reduce bothtransmit and circuit powers. Though this seems nothing butonly more practical by accounting for the power consumedfor operating systems, it is the rate-dependent circuit powerconsumption model that leads to the fundamental differencebetween EE-delay relation and power-delay relation.

To this end, we first derive the EE limit, which is themaximal EE approachable by a system when serving a trafficwith any given average arrival rate and infinite delay bound,then find the optimal policy that is able to achieve the EElimit with finite delay bounds. This is in contrast to the secondcategory of research that aims to find a policy to maximizeor minimize a specific objective and then evaluates the EE-delay curve achieved by the policy. To maximize the EEof a system, we optimize the power-saving policy to reduceboth the circuit power and the transmit power, which differsfrom the previous studies in [5,7–10,12–14,18–24]. To reflectthe QoS requirement of delay sensitive traffic with variousdelay-bounded QoS provisioning, the delay metric is the delaybound with a small violation probability, which differs fromthe average delay bound in [7, 8, 18] and the real delay inserving a traffic in [6, 17, 23–25].2

The major contributions of this work are summarized asfollows.• We prove that the EE-delay relation can be divided into a

tradeoff region and a non-tradeoff region according to thedelay bound, which depends on the power-rate relation.We discover an important class of application scenariosreferred to as the linear case, where the minimal aver-age total power consumption linearly increases with theaverage service rate required to ensure a given QoS. Insuch a case, the EE-delay tradeoff vanishes even when

2This delay metric is more appropriate for elastic traffic, whose delay can betraded off for energy reduction. The delay bound we considered is the maximaldelay that a specific traffic can tolerate with a small violation probability inorder not to compromise user experience. Such a statistical QoS provision isappropriate for delay sensitive traffic such as video conference, where a datapacket becomes useless once its delay bound is violated, and the real delayto serve the traffic is not a concern.

both the arrival rate and the channel are random, whichdiffers from the results in [7, 8].

• We prove that the EE limit can be achieved by a QSIbased two-state policy when the required delay bound isfinite and lies in the non-tradeoff region of the EE-delayrelation, hence the obtained EE-delay relation in the non-tradeoff region is optimal in the sense of achieving theEE limit. This indicates that the EE of a system maynot reduce when supporting delay sensitive services withrespective to the delay tolerant services.

• To demonstrate whether and when the linear case existsin practice, we optimize a QSI dependent two-state policyby taking a downlink MIMO system as an example,where a BS serves multiple users with perfect CSI. Weprove that if transmit power and used bandwidth arejointly optimized to maximize EE, the linear case willoccur when the constraint on bandwidth is inactive toguarantee the delay bound required by a specific appli-cation. For a shorter delay bound, the required minimalbandwidth should be wider, which reflects an EE-delay-bandwidth tradeoff.

• By further taking the compound Poisson arrival processin large number of transmit antennas asymptotics as anexample, we find the boundary between the tradeoff andnon-tradeoff regions, and derive a lower bound of thePareto optimal EE-delay relation in the tradeoff region,all in closed-form.

The rest of the paper is organized as follows. Section II de-scribes queueing model and defines several notions. Section IIIproves that the EE-delay relation is determined by the power-rate relation. Section IV introduces the system and powerconsumption models for a MIMO system and formulates twoproblems to obtain the power-rate relation and the EE-delayrelation, respectively. Section V and VI analyze the EE-delayrelation and find the optimal two-state policies in single userand multi-user scenarios, respectively. Section VII providessimulation and numerical results to validate the analysis and toillustrate the EE-delay and power-rate relations. Section VIIIconcludes the paper.

II. QUEUEING MODEL AND DEFINITIONS

A. Queueing Model and Statistical QoS Requirement

Consider a downlink multiuser system, where a BS servesK users with delay-sensitive services. The statistical QoSrequirement of user k is defined as (Dmax

k , εDk), where Dmaxk

is the delay bound and εDk is the delay bound violationprobability allowed by the service. In this paper, we considerthe queuing delay and ignore the coding and transmissiondelay.

We consider a fluid queueing model [26,27], which is validwhen the time interval between arrived data packets and theinterval of updating transmit policy (called the transmit timeinterval (TTI) of the system) are much shorter than the delaybound required by the traffic. At time t, the data of the kthuser enters a first-in-first-out buffer of the BS at the arrival

rate ak(t), and is transmitted to the user at the departure ratebk(t). We assume that the data for the K users wait in Kqueues, and denote the queue length of the data for the kthuser as Qk(t). Then, the dynamics of the queue lengths canbe expressed as [27],

dQk (t)

dt= ak(t)− bk(t)

=

{max {ak (t)− sk (t) , 0} , Qk (t) = 0ak (t)− sk (t) , Qk (t) > 0

,

where k = 1, ...,K and sk(t) is the transmission rate (alsocalled service rate) of the kth user. Then, the throughput bk(t)is related with ak(t) and sk(t) as

bk (t) =

{min{ak (t) , sk(t)}, Qk (t) = 0sk (t) , Qk (t) > 0

. (1)

Under the assumption of infinite buffer size, Qk(t) = Ak(t)−Bk(t), where Ak(t) ,

∫ t0ak (τ) dτ and Bk(t) ,

∫ t0bk (τ) dτ

are the amounts of data arrived and departed in the intervalof [0, t], respectively. Assume that the queues of the users arein steady state during the interval of [0, t], and denote Q∞k asthe steady state queue length of the kth user.

Effective bandwidth [28] and effective capacity [29] arepowerful tools to design the systems with statistical QoSrequirement. Denote EBk(θk) and ECk(θk) as the effectivebandwidth of arrival process {ak(t), t > 0} and effectivecapacity of service process {sk(t), t > 0}, where θk is theQoS exponent reflecting the performance in terms of queuelength as [26]

limQmaxk →∞

− ln Pr (Q∞k > Qmaxk )

Qmaxk

= θk, (2)

where Qmaxk is the maximal queue length. A large value of

θk indicates a small value of Qmaxk with a given maximal

queue length violation probability, and vice versa. For a givenstationary arrival process with available effective bandwidthEBk(θk), the required QoS exponent θck can be obtained from(Dmax

k , εDk) as follows [30, 31],

Pr (D∞k ≥ Dmaxk ) ≈ exp [−θkEBk (θk)Dmax

k ] = εD, (3)

where D∞k is the steady state delay of the kth user. To guar-antee the QoS requirement characterized by (θc1, θ

c2, ..., θ

cK), a

transmit policy should satisfy [26]

ECk(θck) ≥ EBk(θck), k = 1, 2, ...,K. (4)

With given εDk , when Dmaxk → ∞ (i.e., θck → 0), (4)

degenerates into

Eh{sk(t)} ≥ E{ak(t)}, k = 1, 2, ...,K, (5)

where Ex{·} represents the expectation taken over x.

B. Two-state Transmit Policy

For randomly arrived process, the optimal transmit policythat minimizes the transmit power under delay constraintshould depend on both QSI and CSI [7, 8]. To study themaximal EE achieved by a system with statistical QoS require-

ments, we introduce a QSI-based two-state transmit policy.When Q∞k > 0, the policy allocates resources to the kth userdepending on the user’s own CSI, which is referred to as “ON”state of the user. When Q∞k = 0, no resource is allocated tothe kth user and hence sk(t) = 0, which is referred to as“OFF” state of the user.

Remark 1. If the TTI is much shorter than the delay bound(i.e., the system can switch between “OFF” and “ON” statesrapidly), then (Dmax

k , εDk) can be guaranteed with a two-state policy satisfying ECk(θck) ≥ EBk(θck), Q∞k > 0, k =1, 2, ...,K [14].

C. EE-Delay Relation, EE Limit, and Power-Rate Relaion

The “bits per Joule” EE metric is defined as the ratio of av-erage throughput3 to average total power consumed at the BS[32]. With the two-state policy, the average total power con-sumption is EQ∞,h {Ptot}, where Q∞ = (Q∞1 , Q

∞2 , ..., Q

∞K )

is QSI, and h is CSI. When the queue is in steady state,E{bk(t)} = E{ak(t)} [33, 34]. Then, for a system with any

given traffic load characterized byK∑k=1

E{ak(t)}, the EE is

EE ,

K∑k=1

E {bk (t)}

EQ∞,h {Ptot}=

K∑k=1

E {ak (t)}

EQ∞,h {Ptot}, (6)

where Ptot is the total power consumption including transmitand circuit power consumptions depending on the specificsystem, which can be controlled by the two-state transmitpolicy.

It is worthy to note that such a definition of EE differsfrom the “delay sensitive EE” in [12–14], which is defined asK∑k=1

ECk (θck)/Eh{Ptot}. Maximizing the delay sensitive EE

yields the policies only depending on CSI.

Definition 1: The EE-delay relation is denoted asEEmax(θc1, θ

c2, ..., θ

cK), which is defined as the maximal EE

achieved by a system with any given traffic load satisfyingthe QoS requirements (θc1, θ

c2, ..., θ

cK) ∈ RK+ , where RK+ is the

positive real space of K-dimension.Such an EE-delay relation is fundamental, which only

depends on the system and the service. When only the transmitpower is taken into account, the EE-delay relation degeneratesto the power-delay relation, which will be the same as thepower-delay relation in [7,8,18] if average delay requirementis considered. In [7,8,18], the relation is defined as the minimalaverage power required to ensure the average delay less thanD (i.e., average delay bound).

Definition 2: The EE limit is defined as EElim ,limθck→0,

k=1,2,...,K

EEmax(θc1, θc2, ..., θ

cK). The corresponding average

3The throughput is the number of bits actually transmitted per second in asystem for a given arrival process, rather than the maximal number of bits thesystem can transmit per second (i.e., capacity). Hence, the average throughput

isK∑

k=1E {bk (t)}.

total power consumption is referred to as the power limit,which is denoted as P lim

tot = limθck→0,

k=1,2,...,K

EQ∞,h{Ptot}.

The EE limit is an upper bound of EEmax(θc1, θc2, ..., θ

cK)

for arbitrary (θc1, θc2, ..., θ

cM ) ∈ RK+ , which is achievable when

Dmaxk →∞, k = 1, 2, ...,K.Definition 3: The power-rate relation is defined as

Eh{Pmintot }− (s1, s2, ..., sK), where Eh{Pmin

tot } is the minimalaverage total power consumption required to support theaverage service rates Eh{sk(t)} = sk, k = 1, 2, ...,K.

The characteristic of the EE-delay relation is determinedby the power-rate relation, as implied in [7, 8, 18]. This isbecause maximizing the EE is equivalent to minimizing theaverage total power consumption for any given traffic loadK∑k=1

E {ak (t)} according to the EE definition in (6). Besides,

the required delay bound has a close relation with the requiredaverage service rate. Intuitively, ensuring a stringent delaybound needs a high service rate.

The power-rate relation depends on the power-saving mech-anism and hence on whether circuit power can be reduced.In [7, 8, 18], the circuit power was not considered, thus theenergy is saved by adjusting transmit power and transmissionrate with adaptive modulation and coding. With the capacity-achieving coding, the power-rate relation, defined as the min-imal transmit power required to support a service rate s(t),is strictly convex, consequently the minimal average poweris a strictly decreasing function of the average delay boundD [7, 8]. For modern systems where circuit power is notnegligible but can be reduced by some new power-savingmechanisms, the power-rate relation and hence the resultingEE-delay relation behave quite differently. For example, forsingle antenna systems, we can adjust bandwidth together withtransmit power to reduce the total power consumption. Formulti-antenna systems, we can further adjust the number of an-tennas. This sharply differs from the analysis of only reducingtransmit power, where all the available bandwidth and antennasshould be used to save power. With more flexible power-savingmechanisms, it is possible that Eh{Pmin

tot } becomes a linearfunction of (s1, s2, ..., sK) as illustrated later, which is moreoptimistic than the strict convex power-rate relation.

III. EE-DELAY RELATION AND POWER-RATE RELATION

According to the distinguished properties of the EE-delayrelations, the power-rate relation can be divided into linear andnonlinear cases.

Linear case: This occurs for the scenario where the min-imum average total power required to support an average

service rate sk can be expressed as Eh{Pmintot } =

K∑k=1

cksk+c0,

where ck, k = 1, 2, ...,K and c0 are positive constants depend-ing on specific system.

As implied from the discussion in [8], a necessary conditionfor a transmit policy to minimize transmit power is not to serveempty buffer. This suggests that in order for a two-state policy

to achieve the EE-delay relation the following condition shouldbe satisfied

EQ∞,h{sk(t)} = E{ak(t)}. (7)

From (5) and (7), the EE limit can be derived as (see AppendixA),

EElim =

K∑k=1

E{ak(t)}/(K∑k=1

ckE{ak(t)}+ c0). (8)

In the linear case, the following proposition indicates thatfor any given arrival process with average rate E{ak(t)}, theEE limit is achievable by a simple QSI-based policy evenunder finite delay requirement, and the EE-delay tradeoffvanishes (proved in Appendix B).

Proposition 1. In the linear case, the EE-delay relation doesnot depend on the delay bounds Dmax

k , k = 1, ...,K, andEEmax(θc1, θ

c2, ..., θ

cK) = EElim can be achieved by a two-

state policy.

The proposition suggests that: (i) if the power-rate relationis linear, then the EE-delay relation is not a tradeoff, and themaximal EE of the non-tradeoff region equals to the EE-limit,(ii) a two-state policy can achieve the EE limit.

Furthermore, the following corollary indicates that the EE-delay tradeoff may vanish as well when the average delay isconsidered as the delay metric (proved in Appendix C).

Corollary 1. In the linear case, the EE does not depend onthe average delay bound D.

Nonlinear case: For other scenarios where Eh{Pmintot } is a

strictly convex function of sk, the tradeoff between averagetransmit power and average delay in single user systems hasbeen studied in large-delay and small-delay regimes [7, 8],and the result in large-delay regime has been extended intomulti-user systems [18]. In the context of statistical QoSrequirement, we will find the maximal EE achieved by thetwo-state policy. Since a QSI-based policy with more than twostates may achieve higher EE than the two-state policy, theEE achieved by the optimized two-state policy as a functionof Dmax

k is a lower bound of the Pareto optimal EE-delaytradeoff.

IV. MIMO: AN EXAMPLE SYSTEM

To show when the linear and non-linear cases happenin real-world applications, we consider a downlink MIMOsystem as a concrete example in the subsequent sections. Weemploy frequency division multiple access to avoid multi-userinterference, and maximum ratio transmission (MRT) for eachuser as precoding.

A. System Model

Consider a BS equipped with NT antennas serves K single-antenna users under block fading channels. For notationalsimplicity, we consider flat fading channel. For frequency-selective channel, there is no fundamental difference on theEE-delay relation, as shown by simulation later. The spatial

channel vector from the BS to the kth user during the lthchannel fading block is hk,l ∈ CNT×1, whose elementsare assumed as independent and identically distributed (i.i.d.)Gaussian variables with zero mean and variance µk. Assumethat the CSI is perfectly known at the BS. Then, the maximumachievable service/transmission rate of the kth user is

sk(t) = Wklog2

(1 +

µkpk,lgk,lN0Wk

), (l − 1)Tc < t ≤ lTc,

(9)

where Wk is the bandwidth allocated to the kth user, pk,l isthe transmit power allocated to the kth user during the lthchannel fading block, gk,l , 1

µkhHk,lhk,l is the instantaneous

channel power gain, Tc is the duration of each channel fadingblock, N0 is single-sided spectrum density of white Gaussiannoise, and (·)H is complex conjugate transpose.

Denote Wmax as the maximal available bandwidth of the

system. Then,K∑k=1

Wk ≤ Wmax. For i.i.d. block fading

channel, the effective capacity can be expressed as follows[29],

ECk(θk) = − 1

θckTclnE{exp [−θckTcsk(t)]}. (10)

To save energy consumed for transmitting data and oper-ating the system, the BS can adjust the service rate of eachuser by allocating transmit power and bandwidth.4 Since thebandwidth affects the circuit power, from the in-depth analysisof the powers consumed for various modules including radiofrequency transceivers and baseband processing in MIMOsystem [16], the total power consumption at the BS can be

modeled as Ptot = 1ρ

K∑k=1

pk,l+PCWK∑k=1

Wk+P0, where PCW

is the circuit power consumed for baseband processing perunit bandwidth, P0 is the rate-independent circuit power, andρ ∈ (0, 1] is the power amplifier efficiency. The values of PCWand P0 increase with the number of transmit antennas NT [16].Denote PTk = Eh{pk,l} as the average transmit power of thekth user. Then, the average total power consumption at the BSis

Eh{Ptot} =1

ρ

K∑k=1

PTk + PCW

K∑k=1

Wk + P0. (11)

B. Problem Formulation

In order to study the power-rate relation and the EE-delayrelation for the system, we formulate two problems.

According to Definition 3 and the power model in (11), thepower-rate relation can be found from the following problem,

minPTk ,Wk,k=1,...,K

K∑k=1

PTk + ρPCW

K∑k=1

Wk, (12)

4The number of active antennas can also be controlled by switching offsome unnecessary antennas to ensure QoS, which however leads to ratherinvolved optimization. To simplify the analysis, we fix NT in the followingderivations. The impact of further jointly optimizing NT on the EE-delayrelation will be illustrated with numerical results later.

s.t. Eh{sk(t)} = sk, k = 1, ...,K, (12a)K∑k=1

Wk ≤Wmax. (12b)

To provide the EE-limit achieving policy in the linear caseand find the lower bound of Pareto optimal EE-delay tradeoffin the nonlinear case, we need to find the EE-optimal two-statepolicy. According to (6), maximizing the EE is equivalent tominimizing the average total power consumption for any given

arrival process, whereK∑k=1

E{ak(t)} is fixed. Therefore, we

find the optimal two-state policy that minimizes the averagetotal power under the QoS constraint.

Denote ηk , Pr(Q∞k > 0), which is the non-emptyprobability of the buffer of the kth user. Then, the average totalpower consumed by the two-state policy can be expressed as,5

EQ∞,h{Ptot} =

K∑k=1

ηk

(P onTk

ρ+ PCWW

onk

)+ P0, (13)

where P onTk

and W onk are the average transmit power and the

bandwidth allocated to the kth user when Q∞k > 0 (i.e., in the“ON” state of the user), respectively.

Then, the EE-optimal two-state policy can be obtained fromthe following problem,

minP onTk,W on

k ,

k=1,...,K

K∑k=1

ηk

(P onTk

ρ+ PCWW

onk

)+ P0, (14)

s.t. ECk(θck) ≥ EBk(θk), ∀Q∞k > 0, k = 1, ...,K,(14a)

K∑k=1

W onk ≤Wmax. (14b)

V. EE-DELAY RELATIONSHIP IN SINGLE-USER SCENARIO

In this section, a single user scenario is considered, wherethe index k in problems (12) and (14) can be omitted for nota-tional simplicity. We first show that the minimal average totalpower consumption required to support an average service ratewill linearly grow with the average service rate requirement ifthe average transmit power and bandwidth are jointly allocatedand the constraint on Wmax is inactive. Then, we provide thetwo-state policy to achieve the EE limit in the linear case.Next, the impacts of the bandwidth constraint are analyzed,and the boundary between tradeoff and non-tradeoff regionsof the EE-delay relation are provided for a compound Poissonarrival process in large NT asymptotics. Finally, the lowerbound of the Pareto optimal EE-delay tradeoff is obtained inthe nonlinear case.

5Analogous to [7,8,18], when we study the power-rate relation, the averagetotal power consumption is Eh{Ptot}, and when we find the two-state policyto achieve the EE-delay relation, the average total power consumption isEQ∞,h{Ptot}.

A. EE-delay Relation in Linear Scenarios

We first find the relationship between Eh{Pmintot } and s for

the application scenario where the constraints on Wmax in(12b) and (14b) are not active. Then, we will discuss whenthe maximum bandwidth constraint is not active in SectionV.B.

1) Power-Rate Relation: The service rate s(t) in con-straint (12a) relies on the instantaneous transmit power plallocated in the lth fading block as shown from (9), whichdepends on the instantaneous power allocation policy withgiven average transmit power PT and bandwidth W , denotedas fp(PT ,W, gl). Then, problem (12) can be re-expressed as,

minPT ,W,

PT + ρPCWW, (15)

s.t. Eh{W log2

[1 +

µfp(PT ,W, gl)glN0W

]}= s. (15a)

To find the solution of problem (15), we first need to determinethe optimal form of the function fp(PT ,W, gl) that minimizesthe objective function in (15).

Finding the optimal form of a function belongs to the func-tional extreme problem, and the general method to solve sucha problem is very difficult. In order to obtain a closed-formsolution of fp(PT ,W, gl) for the succeeding optimization, inthe sequel we employ an alternative way. Specifically, we firstprove that the optimal instantaneous power allocation policyis water-filling, and then find the optimal solution of PT ,Wwith the optimized form of fp(PT ,W, gl).

As shown in [35], given the average transmit power PT andbandwidth W , the instantaneous power allocation policy thatcan maximize the average service rate is a water-filling policy,

pl = fwp(PT ,W, gl

),

{N0Wµ

(1gth − 1

gl

), gl ≥ gth,

0, gl < gth,(16)

where the water level gth can be obtained from∫ ∞gth

N0

µ

(1

gth− 1

g

)fh(g)dg =

PTW, (17)

and fh(g) is the distribution of channel gains, which followsthe Wishart distribution [36] as,

fh (g) =1

(Nt − 1)!gNt−1e−g, Nt > 1. (18)

The following Proposition shows that (16) is the optimalinstantaneous power allocation policy that minimizes theobjective function in (15) (the proof is omitted since thisproposition is a special case of Proposition 1 in [37]).

Proposition 2. Consider an arbitrary instantaneous power al-location policy fp(PT ,W, gl) differing from (16). The optimalsolutions of problem (15) with policies pl = fwp

(PT ,W, gl

)and pl = fp(PT ,W, gl) are denoted as {PwT ,Ww} and{ ˜PT , W}, respectively. Then,

PwT + ρPCWWw ≤ ˜PT + ρPCW W (19)

Note that problem (15) can be equivalently re-formulatedas minimizing the ratio of the objective function in (15) to theaverage service rate under the constraint in (15a). Then, withthe optimized form of fp(PT ,W, gl) in (16), problem (15) isequivalent to the following problem,

maxPT ,W,

PTW + ρPCW

Eh{

log2

[1 +

µfwp (PT ,W,gl)glN0W

]} , (20)

s.t. (15a).

Denote the average transmit power per unit bandwidth asPTW , PT

W .From (16) and (17), we can see that the value of

fwp (PT ,W,gl)W is determined by the value of PTW . Therefore,

the objective function in (20) is a function of PTW ratherthan a function of individual variables of PT and W . Denotethe average service rate per unit bandwidth as RW (PTW ) =

Eh{

log2

[1 +

µfwp (PT ,W,gl)glN0W

]}. Then, we can obtain the

following proposition.

Proposition 3. RW (PTW ) is strictly concave in PTW .Proof: When Nt = 1, this proposition is the same as

Proposition 3 in [37]. When Nt > 1, the distribution ofinstantaneous channel power gain in (18) is different fromthat when Nt = 1. It is not hard to see that fh (g) in (18) hasno impact on the concavity of RW (PTW ), thus RW (PTW )is strictly concave in PTW for Nt > 1. This completes theproof.

Since RW (PTW ) is strictly concave in PTW , the objectivefunction in (20) is strictly quasiconvex in PTW [38]. Thus,the value of PTW that maximizes the objective function isunique, which is denoted as P ∗TW . According to the definitionof P ∗TW , the optimal solution of problem (20), P ∗T and W ∗,should satisfy the following condition

P ∗TW ∗

= P ∗TW . (21)

Note that the value of P ∗TW is obtained without consideringthe average service rate requirement in (15a), hence it does notdepend on s. Further considering the service rate requirementin (15a), the optimal solution of problem (20) can be obtainedas follows,

P ∗T =P ∗TW

RW (P ∗TW )s and W ∗ =

1

RW (P ∗TW )s. (22)

Substituting (22) into (11), we can derive that

Eh{Pmintot } =

[P ∗TW

ρRW (P ∗TW )+

PCWRW (P ∗TW )

]s+ P0 = c1s+ c0,

(23)

where c1 =P∗TW

ρRW (P∗TW )+ PCW

RW (P∗TW )and c0 = P0. It is shown

that the minimal average transmit power linearly increaseswith the average service rate.

Remark 2. One may suppose that such a linear relation is an

artifact of the considered power consumption model, wherethe total power is a linear function of W . However, thisis not true. In fact, the linear relation is attributed to theflexible power-saving mechanisms in reducing both transmitand circuit powers, i.e., the joint transmit power and bandwidthallocation here. Later, we will use numerical result to showthat the power-rate relation is still linear when NT is jointlyoptimized with PT and W , where the total power consumptionis a non-linear function of NT .

2) EE-Delay Relation: In the linear case, according toProposition 1, the EE-delay relation EEmax(θc) equals to theEE limit EElim for all values of θc, i.e., there is no EE-delaytradeoff.

When θc → 0, from (8) and with the constants c1 and c0defined in (23), the EE limit and the corresponding power limitof the MIMO system can be respectively expressed as

EElim = E{a(t)}/Eh{P limtot } = E{a(t)}/(c1E{a(t)}+ c0),

(24)

Eh{P limtot } =

[P ∗TW

ρRW (P ∗TW )+

PCWRW (P ∗TW )

]E{a(t)}+ P0.

(25)

3) EE-Limit Achieving Policy: In what follows, we find thetwo-state policy to achieve the EE limit from a degeneratedversion of problem (14).

With a two-state policy, which is a special QSI-dependentpolicy, the necessary condition in (7) reduces to

E{a(t)} = EQ∞,h{s(t)} = Pr(Q∞ > 0)Eh {s (t) |Q∞ > 0}= ηEh {s (t) |Q∞ > 0} , (26)

from which we obtain the non-empty probability of the bufferas η = Pr(Q∞ > 0) = E{a(t)}

Eh{s(t)|Q∞>0} . Then, the average totalpower consumption of the two-state policy can be expressedas

EQ∞,h{Ptot} = ηP ontot + (1− η)P off

tot , (27)

where the average is taken over both queue length and in-stantaneous channel power gain, and P on

tot and P offtot are the

average total power consumptions in “ON” and “OFF” states,respectively.

Further considering (11), the average total power consump-tion can be obtained as,

EQ∞,h{Ptot} =E{a(t)}

Eh{s(t)|Q∞ > 0}

(P onT

ρ+ PCWW

on

)+ P0.

(28)

The optimal two-state policy that minimizes EQ∞,h{Ptot}under the statistical QoS constraint reflected by θc can befound from the degenerated version of problem (14) withoutthe bandwidth constraint, which is

minP onT ,W on

P onT

ρ + PCWWon

Eh{s(t)|Q∞ > 0}, (29)

s.t. EC(θc) ≥ EB(θc), ∀Q∞ > 0. (29a)

Again, the effective capacity EC(θc) for Q∞ > 0 depends onthe instantaneous power allocation policy with given averagetransmit power P on

T and bandwidth W on. This suggests thatthe optimal two-state policy also includes an instantaneouspower allocation implicitly except the average transmit powerand bandwidth allocation.

The objective function in (29) has the same form as theobjective function in (20). Thus, if the constraint in (29a)is not considered, the minimum of (29) can be achieved bythe instantaneous power policy in (16) that minimizes theobjective function in (20), and the average transmit power andbandwidth in the “ON” state similar to that in (22),

P on∗

T =P ∗TW

RW (P ∗TW )Eh{s(t)|Q∞ > 0}

W on∗ =1

RW (P ∗TW )Eh{s(t)|Q∞ > 0}. (30)

Substituting (30) into (28), we can derive that EQ∞,h{Ptot}is the same as the power limit in (25). This indicates that thepower limit can be achieved by using P on∗

T and W on∗ .Now we find the two-state policy that can achieve the power

limit under the statistical QoS requirement in (29a). To thisend, we need to derive the effective capacity. With the water-filling policy in the “ON” state, the effective capacity can beexpressed as follows,

EC (θc)

= −ln∫∞

0

[1 +

µfwp (P∗TWWon∗ ,W on∗ ,g)g

N0W on∗

]−βfh (g) dg

θckTc, (31)

where β =θckTcW

on∗

ln 2 . It is not hard to see (31) increases withW on∗ . Substituting (31) into EC (θc) = EB (θc), the minimalbandwidth required to ensure the statistical QoS requirement,min(W on∗), can be obtained numerically.

Remark 3. From EC (θc) = EB (θc), we can show thatmin(W on∗) increases with θc. In other words, in order toachieve the EE limit with a shorter delay bound Dmax

k , therequired minimal bandwidth in the “ON” state increases. Thisessentially reflects an EE-delay-bandwidth tradeoff.

From min(W on∗), we can obtain the minimal averagetransmit power required to achieve the power limit underthe statistical QoS requirement, which is min(P on∗

T ) =min(W on∗)P ∗TW . Then, the instantaneous power allocation toachieve the power limit under the QoS requirement is,

p∗l =

{N0 min(W on∗ )

µ

(1gth − 1

gl

), pon∗

l , gl ≥ gth, Q∞ > 0,

0, otherwise,(32)

which depends on both QSI and CSI, and pon∗

l is inthe form of water-filling, where gth can be obtained from∫∞gth

N0

µ

(1gth − 1

g

)fh(g)dg = P ∗TW .

Remark 4. pon∗

l in (32) is different from the optimal power

allocation in [14, 39] that maximizes effective capacity withgiven Eh{PT } and W (and hence can maximize the ratioof effective capacity to Eh{Ptot}). When using the powerallocation in [39], the average transmit power in the “ON”state of the queue is less than pon∗

l under the same constraintEC(θc) ≥ EB(θc). However, minimizing the average transmitpower in the “ON” state (i.e., Eh{PT }) is not equivalent tominimizing EQ∞,h{Ptot} (with expression in (28)). By usingthe optimal two-state policy (i.e., min(W on∗),min(P on∗

T ), p∗l ),EQ∞,h{Ptot} can be minimized. Given P on

T and W on, thewater-filling power allocation in the “ON” state pon∗

l can max-imize Eh{s(t)|Q∞ > 0}, and hence can minimize Pr(Q∞ >

0) = E{a(t)}Eh{s(t)|Q∞>0} with given E{a(t)}.

B. Boundary Between the Tradeoff and Non-tradeoff Regions

If min(W on∗) exceeds the maximal bandwidth Wmax,then the feasible solution of problem (29) cannot achievethe EE limit. This indicates that the required delay boundlies in the non-tradeoff region of the EE-delay relation whenmin(W on∗) ≤ Wmax. To obtain the closed-form expressionof the boundary between the tradeoff and non-tradeoff regions,we consider the large NT asymptotics in the rest of thissubsection.

When NT →∞, the service rate in (9) can be re-expressedas follows [40],

s(t) = W log2

(1 +

µNT PTN0W

), C, (33)

which can be achieved by the MRT when NT � K [40]. Dueto channel hardening, the transmit power pl = PT , and the ser-vice rate is constant when the transmit power and bandwidthare given. Then, EC(θ) = lim

t→∞− 1θct lnE {exp [−θctC]} =

C. Therefore, the constraint of problem (29) degenerates into

Con ≥ EB(θ), (34)

where Con , Eh{s(t)|Q∞ > 0} = W onlog2

(1 +

µNT PonT

N0W on

).

After substituting Eh{s(t)|Q∞ > 0} = Con, the powerlimit achieving average transmit power and bandwidth al-location policy in (30) can be simplified as P on∗

T =P ∗TWC

on/RW (P ∗TW ) and W on∗ = Con/RW (P ∗TW ). To en-sure (34), the minimal bandwidth can be expressed as

min(W on∗) = EB(θ)/RW (P ∗TW ). (35)

To achieve the EE limit, min(W on∗) ≤ Wmax should besatisfied. Thus, if the effective bandwidth of a certain arrivalprocess satisfies EB(θ)/RW (P ∗TW ) = min(W on∗) ≤ Wmax,then the EE limit can be achieved and the EE-delay tradeoffwill vanish.

To help understand the impacting factors on the boundary,in the sequel we provide a closed-form expression of theboundary by taking compound Poisson arrival process as anexample. The inter-arrival time interval between packets andthe packet size for the compound Poisson arrival process areexponential distributed with parameters λa and λu, respec-tively. The effective bandwidth of this arrival process can be

expressed as follows [41],

EB(θc) =λa

λu − θc, θc < λu. (36)

At the boundary of the non-tradeoff region, the QoSexponent and the related delay bound requirement are de-noted as θth and Dmax

th . Then, θth can be obtained fromEB(θ)/R∗W (P ∗TW ) = Wmax. From (3), the delay bound canbe obtained as Dmax

th = ln(1/εD)θthEB(θth)

. For compound Poissonprocess, we have

Dmaxth =

(λu − θth) ln(1/εD)

λaθth,

θth = λu − λa

Wmaxlog2

(1 + µNT

N0P ∗TW

) . (37)

As shown in (37), Dmaxth decreases with θth, and θth increases

with Wmax and NT . Therefore, by increasing Wmax orNT , the non-tradeoff region of the EE-delay relation canbe widened. Again, this comes from the EE-delay-bandwidthtradeoff (as mentioned in Remark 3), or an EE-delay-antennatradeoff. It suggests that the EE of a system will not reducewhen supporting delay sensitive services with respective to thedelay tolerant service if the bandwidth and/or the number ofantennas can be increased.

C. EE-delay Relation in the Strictly Convex Case

If the delay bound required by a service is stringentsuch that min(W on∗) ≥ Wmax, then in order to guaranteeEC(θc) ≥ EB(θc) the maximal bandwidth constraints in (14b)will be active, i.e., W on∗ = Wmax. With given bandwidth, theeffective capacity increases with average transmit power, andhence the minimal average transmit power min(P on∗

T ) can beobtained from EC(θc) = EB(θc) numerically.

To obtain closed-form results, we consider the large NTasymptotics again in this subsection. When Dmax < Dmax

th ,W on∗ = Wmax. As shown in what follows, Con lies in thestrictly convex region of the power-rate relation.

1) Power-rate Relation: Denote the service rate whenW = Wmax and PT = P ∗TWW

max as Cth ,

Wmaxlog2

(1 + µNT

N0P ∗TW

). To support higher service rate,

the system needs to increase transmit power. From the max-imum achievable rate in (33), we can obtain the minimaltransmit power to support a service rate C that is higher thanCth as P ∗T = N0W

max

µNT

(2C/W

max − 1). Substituting P ∗T and

W ∗ = Wmax into (11), the power-rate relation can be obtainedas

Pmintot (C) =

N0Wmax

ρµNT

(2C/W

max

− 1)

+ PCWWmax + P0,

(38)

which is strictly convex in C. With the strictly convex power-rate relation, the corresponding maximal EE increases with therequired delay bound, i.e., the EE can be traded off by delay.

2) Lower Bound of the Pareto Optimal EE-delay Tradeoff:In the following, we find the maximal EE achieved by atwo-state policy, which can serve as a lower bound of the

EE-delay relation. We first find the optimal two-state policyfrom problem (14) by setting K = 1. To gain useful insight,we again take the compound Poisson arrival process as anexample.

To obtain the two-state policy, we need the followingproposition (proved in Appendix D).

Proposition 4. Given the value of W on (or P onT ), the average

total power consumption in (28) first decreases and thenincreases with P on

T (or W on), and achieves a unique minimalvalue at P th

T (or W th). Moreover, P thT = P ∗TWW

on (orW th = P on

T /P ∗TW ).

From Proposition 4, we can obtain the following corollary(proved in Appendix E).

Corollary 2. If Dmax < Dmaxth , the optimal bandwidth in

“ON” state will be W on∗ = Wmax.

Since W on∗ = Wmax, we only need to solve P on∗

T . Asshown in Proposition 4, the average total power consumptionincreases with P on

T when P onT > P ∗TWW

max. Besides, Con

also increases with P onT . For compound Poisson process, to

minimize the average power consumption and satisfy theconstraint Con ≥ λa

λu−θc , P on∗

T can be obtained from Con =λa

λu−θc as

P on∗

T =N0W

max

µNT

[2

λa

(λu−θc)Wmax − 1]. (39)

Then, the optimal two-state policy employs the transmit powerin (39) and full bandwidth Wmax in “ON” state. Substituting(36) into (3), the required QoS exponent can be obtained asθc = λu ln(1/εD)

λaDmax+ln(1/εD) . Upon substituting θc into (39), andsubstituting (39) and W on∗ = Wmax into (28), the averagepower consumption of the two-state policy can be obtained asfollows,

EQ∞ {P ∗tot}

=

WmaxN0

ρµNT

{eλaDmax+ln(1/εD)WmaxλuDmax ln 2 − 1

}+ PCWW

max

1 + ln (1/εD) /(λaDmax)+ P0,

(40)

where E{a(t)} = λa

λu and Eh{s(t)|Q∞ > 0} = λa

λu−θcare applied. Considering the EE definition in (6), the lowerbound of the EE-delay relation can be obtained as EELB =

λa

λuEQ∞{P∗tot}.

VI. EE-DELAY RELATION IN MULTI-USER SCENARIO

As in the single user scenario, we first consider thecase without the bandwidth constraint and provide theEE limit that is an upper bound of the EE-delay re-lation EEmax(θc1, θ

c2, ..., θ

cK). Then, we consider the case

with the bandwidth constraint and study the maximal EEachieved by a two-state policy, which is a lower bound ofEEmax(θc1, θ

c2, ..., θ

cK). Finally, the boundary between EE-

delay tradeoff and non-tradeoff regions is briefly discussed.

A. EE Limit

When the required delay bounds are large such thatthe constraint on Wmax is not active, the transmitpolicy for each user does not affect the policies forothers. Then, the multi-user system can be decom-posed into K independent single user systems, and thepower limit can be directly extended from (25) as,

Eh{P limtot } =

K∑k=1

[P∗TWk

ρRWk (P∗TWk)

+ PCWRWk (P∗TWk

)

]E{ak(t)} +

P0, where P ∗TWkand RWk

(P ∗TWk) are the optimal average

transmit power and the average service rate per unit bandwidthof the kth user, respectively.

It is not hard to show that the power-rate relationis linear. According to Proposition 1, the EE-delay rela-tion always equals to the EE limit, which is EElim =M∑i=1

E{ak(t)}/Eh{P limtot }.

B. A Two-state Policy and the Boundary of Non-tradeoffRegion

In what follows, we first provide a two-state policy withthe bandwidth constraint, from which the lower bound of thePareto optimal EE-delay tradeoff can be obtained. Since thetransmit policy of each user has two QSI states, the policy ofthe system with K users has 2K states. To derive the averagepower consumption, we need to obtain the power consumedin each of the 2K states and the probability that the systemstays in each state. As a result, it is rather involved to findthe optimal two-state policy from problem (14) when K islarge. To tackle this difficulty, we find a policy by exploitingthe structure of EE limit-achieving policy.

As shown in (30), the EE limit can be achieved withP on∗

T = P ∗TWWon∗ . Recalling from Proposition 4 that the

average power consumption increases with∣∣P onT − P ∗TWW on

∣∣in large NT asymptotic. Intuitively, the average totalpower consumption in multi-user case will increase withK∑k=1

(P onTk− P ∗TWk

W onk

)2.6 Hence, we formulate the optimiza-

tion problem to find the two-state policy as follows,

minP onTk,W on

k

k=1,2,...,K

K∑k=1

(P onTk− P ∗TWk

W onk

)2(41)

s.t. (14a) and (14b).

The objective function in problem (41) is convex. From thediscussion in [12], it is not hard to know that ECk(θck) isjointly concave in average transmit power and bandwidth, andhence (14a) is convex. Furthermore, the constraint in (14b)are linear. Therefore, the problem is convex, whose optimalsolution can be solved with standard tools such as the interior-point method [38]. Due to the same reason as in the single user

6∣∣∣P on

Tk− P ∗

TWkW on

k

∣∣∣ is not differentiable. Therefore, we employK∑

k=1

(P onTk− P ∗

TWkW on

k

)2as the objective function for mathematical

tractability, which will not change the optimal solution.

case, the optimal two-state policy also implicitly includes aninstantaneous power allocation as in (32) except the averagetransmit power and bandwidth allocation.

When the constraint in (14b) is inactive, the problem can bedecoupled into several single-user problems, and the minimumof the objective function in (41) is zero, which can be achievedwith P on∗

Tk= P ∗TWk

W on∗

k , k = 1, 2, ...,K. This indicates thatthe resulting two-state policy can achieve EElim.

When the constraint in (14b) is active, the obtained two-statepolicy does not minimize the average power consumption, andhence the EE achieved by the policy is a lower bound of theEE-delay relation.

For a set of requirements (θc1, θc2, ..., θ

cK), if the solution

of problem (41) satisfiesK∑k=1

W on∗

k < Wmax, the bandwidth

constraint will not be active, and then EElim can be achievedby the two-state policy. This set of requirements are in thenon-tradeoff region.

For a set of requirements (θc1, θc2, ..., θ

cK), if the solution of

problem (41) is obtained when the constraint is active, i.e.,K∑k=1

W on∗

k = Wmax, then the objective function in (41) is

positive. Such a set of requirements are in the tradeoff region.

VII. SIMULATION AND NUMERICAL RESULTS

In this section, we first validate that the statistical QoSrequirement can be satisfied with the two-state policy in prac-tical systems by comparing simulation and numerical results.Then, we demonstrate the power-rate relation and the EE-delay curve, and show the EE-delay curve when the number oftransmit antennas are jointly adjusted with the transmit powerand bandwidth via numerical results.

We take the MIMO-OFDMA system as an example toillustrate the EE-delay relation, where the subcarrier separationis WS = 15 kHz. We consider K symmetric users servedby one BS, all have the same QoS requirement and the samedistance from the BS, since there is no fundamental differencefor more general multi-user scenarios. Frequency-selectivechannel is considered in the simulation with i.i.d. channelgains on all the subcarriers. Denote the number of subcarriersallocated to the kth user as NSk . Since a symmetric scenariois considered, the K-user problem can be decomposed intoK single-user problems with constraint on the number ofsubcarriers NSk ≤ Wmax/(WSK). The power-rate relationis obtained by solving problem (12), where Wk = WSNSk .The two-state policy and the EE achieved by it are obtained bysolving problem (14), where W on∗

k = WSNon∗

Sk. The methods

to solve problem (12) and problem (14) over frequency-selective channel are similar to that over flat fading channel.The only difference lies in the expression of sk(t), which is

sk(t) =NSk∑j=1

WS log2

(1 +

µkpk,l,jgk,l,jN0WS

), (l− 1)Tc < t ≤ lTc.

The packet arrival process of each user is a compound Poissonrandom process with average packet arrival rate λa and aver-age packet size 1/λu. The circuit power model and parametersin [16] are used, where the circuit power per unit bandwidth

PCW and the fixed circuit power consumption P0 increasewith the number of transmit antennas NT . The parameters usedin the following simulation and numerical results are listed inTable I. Unless otherwise specified, the above setup will beused in the sequel.

TABLE IPARAMETERS [11, 16]

Maximal available bandwidth Wmax 20 MHzEfficiency of power amplifier ρ 38 %Circuit power per unit bandwidthPCW

(72NT + N2T )

mW/MHzRate-independent circuit power P0 2NT + 1 WDistance between users to BS d 200 mPath loss model (dB) 35.3 + 37.6 log10 d

Power spectral density of noise N0 −174 dBm/Hz

0 1 2 3 4 5 6 7 8

x 104

10−2

10−1

100

Queue lenght (bits)

CC

DF

Pr(

Q∞>

q)

Numerical results, Dmax=10ms

Simulation results, Dmax=10ms

Numerical results, Dmax=2ms

Simulation results, Dmax=2ms

Qmax

εQ

Fig. 1. Complementary CDF of the queue length, K = 10, NT = 4,λa = 1000 packets/s, 1/λu = 5 kbits, and TTI =1 ms.

Figure 1 shows the complementary cumulative distributedfunction (CDF) of Q∞. As discussed in [42, 43], the QoSrequirement (Dmax, εD) is equivalent to (Qmax, εQ), whereQmax = EB(θc)Dmax and εQ = εD. The value of θc canbe obtained from exp [−θcEB(θc)Dmax] = εD [31]. Thesimulation results are obtained by computing the queue lengthin every TTI (which is 1 ms [11]) during the simulationtime of 100 s, where the compound Poisson arrival processis served by the two-state policy. The numerical results areobtained from Pr (Q∞ > q) ≤ exp (−θcq) [31], where η ≤ 1is applied. It is shown that the numerical results are upperbounds of simulation results. This indicates that the statisticalQoS requirement can be guaranteed by using the two-statepolicy, which validates Remark 1. Besides, the results showthat when Dmax = 2 ms, the maximum queue length, Qmax =EB(θc)Dmax, is large enough such that the exponential decayrate of Pr (Q∞ > q) is close to θc for the compound Poissonprocess.

Figure 2 shows the power-rate relation and the EE-delaycurves achieved by the two-state policy. For comparison, theEE-delay curves achieved by two existing QSI-independent

policies are also provided, which consider statistical QoSrequirement. The first one is the policy in [5], where the totaltransmit power of the K users is minimized under constraintsECk(θck) ≥ EBk(θck) and NSk ≤Wmax/(WSK) (with legend“power minimizing policy”). The second is the policy in [12],

where the ratio ofK∑k=1

ECk (θck) to the total power consumption

is maximized under the same constraints as in [5] (with legend“delay-sensitive EE policy”). Since the results in [13, 14] aresimilar to that in [12], we do not compare with [13, 14].

The power-rate relation in Fig 2(a) includes two segments,a linear region and a convex region. In the linear region,the optimal policy to support E{s(t)} = s satisfies P ∗Tk =P ∗TWWSN

∗Sk< P ∗TWW

max/K, and the bandwidth constraintis inactive. In the convex region, P ∗Tk > P ∗TWW

max/K andWSN

∗Sk

= Wmax/K. In the boundary of linear and convexregions, P ∗Tk = P ∗TWW

max/K and WSN∗Sk

= Wmax/K.Each EE-delay curve achieved by the two-state policy

in Fig. 2(b) includes a tradeoff region and a non-tradeoffregion. The vertical dash line is the boundary of these tworegions. In the non-tradeoff region, P on∗

Tk= P ∗TWWSN

on∗

Sk,

corresponding to the linear region in Fig. 2(a). In the tradeoffregion, P on∗

Tk> P ∗TWW

max/K and WSNon∗

Sk= Wmax/K,

corresponding to the convex region in Fig. 2(a). The non-tradeoff region exists when the delay bound is large, withinwhich supporting more stringent delay bound Dmax

k does notreduce the EE, in contrast to the traditional belief. In thetradeoff region, small additional Dmax

k leads to substantial EEincrease, which is consistent with the results in priori studies[5, 7–10, 18].

The EE achieved by the “power minimizing policy” alwaysincreases with the delay bound, which however is much lowerthan the EE achieved by the two-state policy. This is becausethe policy in [5] is QSI-independent (such that transmit powercannot be minimized) and only the transmission power andrate are optimized (such that the circuit power cannot be re-duced). The EE achieved by the “power minimizing policy” ishigher than the “delay-sensitive EE policy” when the requiredDmaxk is large, because the latter provides higher service rate

than required by the QoS (i.e., ECk(θk) > EBk(θk)) thatleads to a waste of energy. The EE achieved by the “delay-sensitive EE policy” also grows with the delay bound, thoughnot obvious in the figure.

From Fig. 2(b), we can also observe the impact of thenumber of users. With more users, non-tradeoff region shrinksbecause the bandwidth for each user decreases.

Figure 3 illustrates what happens when the number oftransmit antennas are jointly adjusted with the transmit powerand bandwidth. To obtain the EE-delay curve, we find the two-state policy with different values of NT , and then selectingNT that maximizes the EE. The achieved EE decreases withNT when the delay bound is large and increases with NTwhen the delay bound is short. We can see that when NTis jointly allocated with P on

Tkand Non

Sk, the EE-delay curve

also includes tradeoff and non-tradeoff regions, despite that thecircuit power consumption is not a linear function of NT as

0 50 100 150 2000

10

20

30

40

50

60

Average sum service rate of 10 users (Mbps)

Ave

rage

tota

l pow

er c

onsu

mpt

ion

(W)

Linear region

Convex region

(a) Power-rate relation, K = 10.

2 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4x 106

Delay bound (ms)

EE

(bi

ts/J

)

Two−state policy, K=10Power minimization policy, K=10Delay−sensitive EE policy, K=10Two−state policy, K=5

Non−tradeoff region

Non−tradeoff region

(b) EE-delay curves achieved by different policies for compound Poissonarrival process, εD = 0.01.

Fig. 2. Power-rate relation and EE-delay curves, NT = 4, λa = 500 pack-ets/s, 1/λu = 10 kbits.

shown in Table I. Numerical results show that the power-raterelation is linear when the resource constraints are inactive,which is not provided for conciseness.

VIII. CONCLUSION

In this paper, we studied fundamental EE-delay relation forwireless systems serving randomly arrived data with statisticalQoS requirements. Different from the widely accepted wisdomthat there is an inherent tradeoff between the EE and delay,we showed that the EE-delay relation includes tradeoff regionand non-tradeoff region. We illustrated that for a system withadjustable circuit power consumption, the required minimalaverage total power consumption may linearly increase withthe average service rate. We proved that in such a linearcase, the EE-delay tradeoff vanishes, and the EE-limit canbe achieved by a two-state policy. By taking MIMO systemas an example, we demonstrated when the linear case willoccur. Specifically, we proved that the power-rate relation willbe linear when the transmit power and bandwidth are jointlyoptimized in single-user scenario if the bandwidth constraint

2 5 10 15 200

1

2

3

4

5

6x 106

Delay bound (ms)

EE

(bi

ts/J

)

Non−tradeoff regionNT=1

NT=2

NT=8

NT=4

Tradeoff region

Optimized NT

Fig. 3. EE-delay curves achieved by the two-state policy with optimizedNT (the solid line) and given values of NT (the dash lines), K = 5, λa =500 packets/s, 1/λu = 10 kbits, and εD = 0.01.

is not active. We then obtained the EE-delay relation andthe EE-limit achieving policy in the linear case. Furtheringconsidering the compound Poisson arrival process in largenumber of transmit antennas asymptotics, the closed-formboundary between the tradeoff and non-tradeoff regions wasderived, and a lower bound of the Pareto optimal EE-delaytradeoff in the nonlinear case was obtained. Finally, theextension to the multi-user scenario was briefly addressed.Analytical and numerical results showed that increasing themaximal bandwidth or the number of transmit antennas canbroaden the non-tradeoff region, within which supporting morestringent delay bound does not reduce the EE. This comesfrom the EE-delay-bandwidth tradeoff or EE-delay-antennatradeoff. Besides, the EE-delay curve achieved by the two-state policy in the tradeoff region, i.e., the lower bound, ismuch higher than those achieved by existing policies that alsoconsider statistical QoS requirement.

APPENDIX APROOF OF EE LIMIT (8)

Proof: For any QSI based transmit policy that sat-isfies the necessary condition in (7), the average totalpower consumption in linear case can be expressed as

PLBtot = EQ∞{Eh{Pmintot }} = EQ∞{

K∑k=1

ckEh[sk(t)] + c0} =

K∑k=1

ckEQ∞,h{sk(t)}+ c0 =K∑k=1

ckE{ak(t)}+ c0. Because a

power limit achieving policy needs to further satisfy (5), P limtot

is no less than PLBtot . When Dmaxk →∞, k = 1, 2, ...,K, both

the QoS constraint (5) and the necessary condition in (7) canbe satisfied with a transmit policy that satisfies Eh{sk(t)} =E{ak(t)}, i.e., PLBtot is achievable. Therefore, P lim

tot = PLBtot ,

and EElim =K∑k=1

E{ak(t)}/(K∑k=1

ckE{ak(t)}+ c0).

APPENDIX BPROOF OF PROPOSITION 1

Proof: For a two-state policy, if Qk(t) = 0, then sk(t) =0. From (1), we have bk(t) = min{sk(t), ak(t)} = sk(t). IfQk(t) > 0, from (1), we also have bk(t) = sk(t). Hence, withthe two-state policy, bk(t) = sk(t),∀Qk(t). In steady state,E{bk(t)} = EQ∞k ,h{sk(t)}.

To guarantee the QoS requirement, ECk(θk) ≥ EBk(θk),Q∞k > 0 should be satisfied. The expectations of sk(t)under conditions Q∞k > 0 and Q∞k = 0 are denoted asEh{sk(t)|Q∞k > 0} and Eh{sk(t)|Q∞k = 0}, respectively.

Further considering that E{ak(t)} = E{bk(t)} in steadystate, we have

E{ak(t)} = E{bk(t)} = EQ∞k ,h{sk(t)}, k = 1, 2, ...,K.(B.1)

Denote EEts(θc1, θc2, ..., θ

cK) =

K∑k=1

E{ak(t)}/EQ∞{Eh(Pmintot )} as the EE achieved

by the two-state policy. In the linear case, i.e.,

Eh{Pmintot } =

K∑k=1

ckEh {sk (t)} + c0, the minimal average

power consumption achieved by the two-state policy can beobtained from the following expression,

EQ∞{Eh{Pmintot }}

= Pr (Q∞k > 0)EQ∞{Eh{Pmintot }|Q∞k > 0}

+ Pr (Q∞k = 0)EQ∞{Eh{Pmintot }|Q∞k = 0}

=

K∑k=1

ckPr (Q∞k > 0)Eh {sk (t) |Q∞k > 0}

+

K∑k=1

ckPr (Q∞k = 0)Eh {sk (t) |Q∞k = 0}+ c0

=

K∑k=1

ckEQ∞k ,h{sk(t)}+ c0 =

K∑k=1

ckE{ak(t)}+ c0. (B.2)

From (B.2), we can obtain that

EEts(θc1, θc2, ..., θ

cK) =

K∑k=1

E{ak(t)}/(K∑k=1

ckE{ak(t)}+ c0),

(B.3)

which is exactly the same as EElim in (8). This indicates thata two-state policy can achieve EElim for arbitrary set of delayrequirements (θc1, θ

c2, ..., θ

cK) ∈ RK+ . Further considering that

EEts(θc1, θc2, ..., θ

cK) ≤ EEmax(θc1, θ

c2, ..., θ

cK) ≤ EElim, the

proposition is proved.

APPENDIX CPROOF OF COROLLARY 1

Proof: For a finite QoS exponent θck, the tail probabilityof the steady state queue length of the kth user Q∞k can beapproximated as Pr(Q∞k > Qth) ≈ e−θ

ckQth [26]. Then, the

average of Q∞k is E{Q∞k } =∫∞

0qdPr (Q∞k ≤ q) ≈ 1

θck.

According to the Little’s Law, the average delay of the

kth user can be approximated as 1θckE{ak(t)} . Therefore, to

ensure the average delay D, a transmit policy should satisfyθck = 1

E{ak(t)}D . As shown in Proposition 1, the EE achievedby the two-state policy is independent of θck, and hence isindependent of D.

APPENDIX DPROOF OF PROPOSITION 4

Proof: Since E{a(t)} and P0 are fixed, minimizing theaverage power consumption (28) is equivalent to minimizingthe objective function in (29). In the sequel, we analyze thereciprocal of the objective function in (29), which is

ϕ(W on, P on

T

)=W onlog2

(1 +

µNT PonT

N0W on

)P onT

ρ + PCWW on, (E.1)

where (33) is used. In this appendix, we only prove the casewhen W on is given. For the other case, i.e., P on

T is given, theproof is similar and hence is omitted for conciseness.

To find the optimal P onT with given W on, we take the first

order derivative of (E.1) as

∂ϕ(W on, P on

T

)∂P on

T

=

W on

ρ ln 2ϕ1

(W on, P on

T

)(P onT

ρ + PCWW on)2 ,

where ϕ1

(W on, P on

T

)=

(PonT

Won +ρPCW

)µNTN0

1+µNT P

onT

N0Won

−

ln(

1 +µNT P

onT

N0W on

).

It is not hard to show that ϕ1 (W on, 0) > 0, ϕ1 (W on,∞) <0, and

∂ϕ1

(W on, P on

T

)∂P on

T

<

µNTN0W on(

1 +µNT P on

T

N0W on

)2 −µNTN0W on

1 +µNT P on

T

N0W on

< 0,

which means that ϕ1

(W on, P on

T

)strictly decreases with P on

T .Thus, the equation ϕ1

(W on, P on

T

)= 0 has a unique solution,

denoted as P thT .

If P onT < P th

T , then ϕ1

(W on, P on

T

)> 0, and hence

ϕ(W on, P on

T

)increases with P on

T . Since ϕ(W on, P on

T

)is the

reciprocal of the objective function in (29), the average powerconsumption decreases with P on

T . Similarly, if P onT > P th

T ,the average power consumption increases with P on

T . Therefore,P thT is the global optimal value. Moreover, the optimal average

transmit power per unite bandwidth is P ∗TW . Therefore, wehave P th

T = P ∗TWWon.

APPENDIX EPROOF OF COROLLARY 2

Proof: The delay requirement Dmax < Dmaxth can not

be guaranteed with any policy that satisfies W on ≤ Wmax

and P onT ≤ P ∗TWW

max (otherwise, Dmax lies in the non-tradeoff region, i.e., Dmax ≥ Dmax

th ). Then, to guarantee thedelay requirement, P on

T > P ∗TWWmax, and hence Wmax <

P onT /PTW

∗. According to Proposition 4, when W on ≤Wmax < P on

T /P ∗TW , the average total power consumption

decreases with W on. Hence for any given P onT > P ∗TWW

max,the average total power consumption is minimized withW on = Wmax. Therefore, W on∗ = Wmax.

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