4106 ieee transactions on wireless ...4106 ieee transactions on wireless communications, vol. 16,...

4106 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 6, JUNE 2017

Opportunistic Scheduling in Wireless PoweredCommunication Networks

Zoran Hadzi-Velkov, Senior Member, IEEE, Ivana Nikoloska, Student Member, IEEE,Hristina Chingoska, Student Member, IEEE, and Nikola Zlatanov, Member, IEEE

Abstract— In this paper, we apply the notion ofopportunistic scheduling in wireless-powered communicationnetworks (WPCNs). The considered WPCN model consists of abase station (BS) and multiple energy harvesting users (EHUs),where the BS broadcasts radio frequency energy to the EHUsover the downlink and receives information from the EHUs overthe uplink. We differentiate the WPCNs based upon the batterymanagement policy at the EHUs, i.e., whether an EHU spendsthe total amount of energy harvested in its battery for eachIT (WPCN type 1), or spends only a part of it for the currentIT and saves the other part for future ITs (WPCN type 2).We propose two opportunistic scheduling policies, referred to asthe harvest-then-select and harvest-or-select protocols, employedat the WPCN type 1 and WPCN type 2, respectively. Theseprotocols have significant practical advantages over the state-of-the-art schemes proposed for maximizing the WPCN sum-rate,because they introduce fairness in the resource utilization bythe EHUs, and require much lower amount of channel stateinformation. Both protocols achieve these benefits at the expenseof a minor rate degradation relative to the rates achieved bytheir counterpart protocols employing multiple access, denotedas the harvest-then-transmit and harvest-or-concurrently-transmitprotocols.

Index Terms— Energy harvesting, wireless information andenergy transfer, fair resource allocation, opportunistic scheduling.

I. INTRODUCTION

DUE to its capability to provide perpetual power sup-ply, radio-frequency (RF) energy harvesting (EH) has

emerged as a revolutionary technology for energy-constrainedwireless networks, such as sensor and ad-hoc networks [1]–[4].In theory, the maximum power available for RF energy har-vesting at free space distance of 40 meters is about 7μW and1μW at respective frequencies of 2.4GHz and 900MHz [5].In practice, the Powercast RF harvester operating at 900MHzcan harvest 3.5mW and 1μW of wireless power at respectivedistances of 0.6 and 11 meters [5].

Manuscript received February 10, 2016; revised October 28, 2016 andFebruary 1, 2017; accepted March 29, 2017. Date of publication April 12,2017; date of current version June 8, 2017. This work was supportedby the Reintegration Program of the Alexander von Humboldt Foun-dation. The associate editor coordinating the review of this paper andapproving it for publication was E. U. Biyikoglu. (Corresponding author:Zoran Hadzi-Velkov.)

Z. Hadzi-Velkov and H. Chingoska are with the Faculty of ElectricalEngineering and Information Technologies, Ss. Cyril and Methodius Uni-versity, 1000 Skopje, Macedonia (email: [email protected]; [email protected]).

I. Nikoloska and N. Zlatanov are with the Department of Electrical andComputer Systems Engineering, Monash University, Clayton, VIC 3800,Australia (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TWC.2017.2691785

The networks of communicating nodes with RF EHcapabilities are known as wireless-powered communicationnetworks (WPCNs) [6]–[9]. The WPCNs have been studiedunder various network setups, such as cellular network [6],random-access network [7], and multi-hop network [8]. In thispaper, we adopt the WPCN model proposed in [9], whichconsists of a base station (BS) and multiple energy harvestingusers (EHUs), where the BS broadcasts RF energy to the EHUsover the downlink and receives information from the EHUsover the uplink. In the following, we refer to this model ofnetwork simply as the WPCN. The primary resource sharingmechanism in the WPCN is the dynamic time-division multipleaccess (D-TDMA). In the D-TDMA, each TDMA frame isdivided into non-overlapping subintervals of variable durationsfor the energy transmission over the downlink and informationtransmission (IT) over the uplink in order to maximize thetotal number of bits delivered to the BS, i.e., the uplink sum-rate maximization [9]–[11]. The optimal durations of thesesubintervals with separated frequency channels for energybroadcast and ITs, has been studied in [9] and [10]. TheWPCNs with full-duplex EHUs and the BS equipped withtwo antennas has been studied in [11]. However, in [9]–[11],the BS broadcasts the same amount of energy in each TDMAframe (i.e., during each fading state), and power allocationover different fading states is not considered thereby. As aresult, the sum-rate is maximized only with respect to the dura-tion of the subintervals in the TDMA frame. The maximizationof the sum-rate with respect to both power and duration ofthe subintervals over multiple fading states (i.e., the ergodicsum-rate maximization) has been studied in [12].

Despite its spectral efficiency, a resource allocation strategybased upon the sum-rate maximization may unfairly distributethe system resources among EHUs at different distances fromthe BS. In particular, due to the large-scale fading (i.e., the pathloss), the EHUs at closer distances to the BS can transmitat much higher rates compared to the more distant EHUs,thus giving rise to the doubly near-far effect [9]. However,compared to the standard (non-EH) cellular systems, the near-far effect is manifested much more severely in the WPCNsbecause the closer EHUs are, they receive more energy fromthe BS but also achieve higher data rates over the uplink.Similarly to the non-EH systems [14]–[21], the doubly near-far effect in the WPCNs can be tackled by opportunisticscheduling, which implies policies for selecting a single userfor uplink transmission in each TDMA frame. In additionto the obvious practical advantages, opportunistic scheduling

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HADZI-VELKOV et al.: OPPORTUNISTIC SCHEDULING IN WPCNs 4107

policies can balance the tradeoff between the spectrum effi-ciency and the fairness among the EHUs. Furthermore, at theexpense of minor degradation of the sum-rate, the oppor-tunistic scheduling policies simplify the distribution and fairallocation of the network resources (e.g., power, time, and/orbandwidth) among the EHUs in the WPCNs. The fairness canbe controlled by associating a weight factor to each EHUs (ωk)whose value depends on the particular rate requirements ofall EHUs. One notable opportunistic scheduling policy thatassures fairness by dynamic adjustment of the weight vectoris the proportional fair (PF) scheduling, which maximizes theproduct of the achievable rates of all users. The PF schedulinghas been widely applied to classic (non-EH) communicationsystems, c.f. [16]–[18]. Recently, an offline resource allocationscheme that maximizes the rate in a proportionally fair wayover the downlink of an EH communication system has beenstudied in [23]. Furthermore, [24] proposes a PF schedulingpolicy for the WPCNs. However, the PF scheduling protocolsachieve only a single point in the achievable rate region,whereas in this paper we propose protocols that maximize theentire rate region.

We apply the notion of opportunistic scheduling to theuplink of two WPCN models, WPCN Type 1 and WPCNType 2. The two proposed opportunistic scheduling protocolsutilize a single user transmission per one TDMA frame, andare thus inherently suboptimal in terms of the average achiev-able rate, compared to the scheduling schemes that involvetransmissions from multiple users (either successively orconcurrently) per one TDMA frame. We also study twobenchmark protocols employing multiple EHUs’ transmissionsin each TDMA frame. The key contributions of this paper aresummarized as follows:

(1) We categorize WPCNs depending on the battery man-agement policy at the EHUs, i.e., whether an EHU spends thetotal amount of energy harvested in its battery for each IT(WPCN type 1), or spends only a part of it for the current ITand saves the other part for future ITs (WPCN type 2).

(2) We propose two opportunistic scheduling policies,the harvest-then-select protocol employed at the WPCNtype 1, and the harvest-or-select protocol employed at theWPCN type 2. These protocols propose allocations for the BStransmit power and the EH/IT time sharing, while requiringcausal channel state information (CSI) for their operation. Theproposed allocations maximize the achievable rate region overthe uplink of the WPCNs, while guaranteeing fair resourceallocation among the EHUs at various distances from the BS.

(3) For these two opportunistic protocols, we develop prac-tical online implementation algorithms (c.f., Algorithm 1 andAlgorithm 2).

(4) We propose two benchmark protocols employingtransmissions from multiple EHUs within a single TDMAframe, either in successive subintervals (the harvest-then-transmit protocol, employed at WPCN Type 1) or as con-current transmissions (the harvest-or-concurrently-transmitprotocol, employed at WPCN Type 2). The achievablerate regions of these benchmark protocols provide upperbounds to the rates achieved by the proposed opportunisticprotocols.

TABLE I

SUMMARY OF MOST IMPORTANT VARIABLES

The remainder of this paper is organized as follows:In Section II, we present the system and channel models, andalso categorize the WPCNs according to the EH and the energyconsumption policies employed by the EHUs. The followingtwo sections present the proposed opportunistic schedulingprotocols with optimal allocations of the BS transmit powerand the time-sharing among the EHUs, whereas Section IIIfocuses on the WPCN type 1 and Section IV focuses on theWPCN type 2. Section V presents numerical examples forthe achievable rate region of the proposed protocols of theconsidered WPCNs. Finally, section VI concludes the paper.For convenience, the most important variables used in thispaper are defined in Table I.

II. SYSTEM AND CHANNEL MODELS

The basic WPCN model consists of a half-duplex BS andK half-duplex EHUs, that operate in a random fading environ-ment. The EHUs are equipped with rechargeable EH batteriesthat store the RF energy received from the BS. We assume thattime is divided into epochs of equal duration T . Dependingon the opportunistic scheduling protocol, each epoch consistsof an EH and/or an IT phase. The EH phase is the time periodduring which the BS broadcasts RF energy to the EHUs, andthe IT phase is the time period during which the scheduledEHU transmits information to the BS.

The fading between the BS and EHU k (1 ≤ k ≤ K ) isassumed to be stationary and ergodic random process thatfollows the quasi-static block fading model (i.e., the channel isconstant during a single block but changes independently fromone block to the next). The duration of each fading block isequal to T , and thereby coincides with the duration of a singleepoch. In epoch i , let the fading power gain of the BS-EHUk

channel be denoted by x ′k(i). For convenience, the correspond-ing downlink (BS-EHUk) and uplink (EHUk-BS) channels areassumed to be reciprocal, although the generality of our resultsis unaffected by this assumption. These gains are normalizedby the additive white gaussian noise (AWGN) power, denoted


Fig. 1. Dynamic TDMA frame for harvest-then-select protocol.

by N0, yielding xk(i) = x ′k(i)/N0 whose average value is�k = E[x ′k(i)]/N0, where E[·] denotes expectation.

The BS is assumed to control and coordinate the uplinkand downlink transmissions. The BS schedules one EHU forIT over the uplink in each epoch based upon the specificopportunistic scheduling protocol. In order to do so, the BShas to have perfect CSI of all K fading links, {xk(i)}Kk=1,in each epoch. Before the start of each TDMA frame, the BS isassumed to broadcast information of log2(K ) bits over a low-capacity broadcast channel in order to notify the EHUs whichEHU is scheduled for IT in that epoch. On the other hand, eachEHU should have CSI available only of its own fading channel,i.e., EHU k should know only xk(i). The transmit power ofthe BS in epoch i is denoted by pi . We assume that the powerof the BS has to satisfy an average power constraint, Pavg, anda maximum power constraint, Pmax (i.e., 0 ≤ pi ≤ Pmax ).

A. Opportunistic Scheduling

Similar to the scheduling protocols in the classic (non-EH)systems (e.g., [22], [26], [27], [28]]), we utilize schedulingpolicies for a single EHU selection, based upon the weightedfeasible rates,

si = arg max1≤k≤K

ωkrk(i), (1)

where ωk is weight associated to EHU k and rk(i) is a feasiblerate of EHU k in epoch i .

Without loss of generality, the weights can be normalizedsuch that 0 ≤ ωk ≤ 1, 1 ≤ k ≤ K , and

∑Kk=1 ωk = 1. For the

proposed opportunistic scheduling policies, the TDMA frameis divided into two subintervals of variable duration (Fig. 1).Specifically, the TDMA frame in epoch i begins with the EHphase of duration τ0i T , which is followed by the IT phase ofduration (1 − τ0i )T . Hence, in epoch i , the BS broadcastsRF energy during τ0i T and then a single EHU transmitsinformation to the BS during (1− τ0i )T .

Given a fixed weight vector (ω1, ω2, . . . , ωK ), we aim atmaximizing the average uplink rate of the considered oppor-tunistic scheduling protocols by optimizing the transmit powerof the BS and the duration of EH phase, τ0i T,∀i . Note that,the vector (ω1, ω2, . . . , ωK ) can be seen as the vector whichdefines the operating point of the opportunistic schedulingprotocol, or, more specifically, one point of its achievable rateregion.

B. WPCN Types

Depending on the battery management policy employedby the EHUs, we distinguish between two typesof WPCNs:

1) WPCN Type 1: EHUs employ short-term EH beforeusing all of its harvested energy for IT. When an EHUtransmits information in a given epoch, it completely spendsall of the harvested energy in its battery during the previous BSbroadcast in that same epoch. To implement this policy, EHUsshould be equipped with rechargeable batteries that have lowenergy storage capacity and high discharge rate. For WPCNType 1, we propose an opportunistic scheduling policy referredto as harvest-then-select (HTS) protocol, with a TDMA frameshown in Fig. 1.

2) WPCN Type 2: EHUs transmit information by using theenergy harvested from the BS over multiple epochs. When anEHU transmits information in a given epoch, it spends eithera part or the total amount of the energy stored in its batterydepending on the channel fading conditions. To implement thispolicy, EHUs should be equipped with rechargeable batteriesthat have high energy storage capacity and low discharge rate.For this type of WPCN, we propose an opportunistic schedul-ing policy referred to as harvest-or-select (HOS) protocol.

Note that in the WPCN Type 2, the EHUs that are notselected for transmission can harvest energy from the BSduring the EH phase and store it in their battery for futureuse. On the contrary, in the WPCN Type 1, the EHUs that arenot selected for transmission in a given epoch can not storeenergy in their batteries during that epoch.

We adopt an idealized model for the power consumption bythe EHUs: The battery of the EHU k is drained only by theRF output power, Pk(i), whereas all other sources of powerconsumption (e.g., power consumed due to processing, non-ideal electronic circuitry, and channel estimation) are ignored.Therefore, the rates achieved by each of the consideredprotocols constitute information-theoretical upper bounds forreliable communication by the considered WPCNs assumingan ideal power consumption.

III. RATE MAXIMIZATION IN WPCN TYPE 1

For the HTS protocol developed for WPCN Type 1,we determine the optimal allocation of pi and τ0i so as tomaximize the average rate over the uplink, subject to thetwo power constraints and a fixed vector (ω1, ω2, . . . , ωK ).We note that a specific vector (ω1, ω2, . . . , ωK ) yields apoint in the achievable rate region of this protocol. Hence,by varying this vector, we can obtain all points of the rateregion.

A. Harvest-Then-Select Protocol

Let us assume the EHU si is the scheduled user for IT inepoch i . In this epoch, the EHU si harvests energy duringthe EH phase of duration τ0i T , and depletes this energy com-pletely to transmit information during the IT phase of duration(1−τ0i )T . The remaining K −1 EHUs are silent (i.e., neitherharvesting energy nor transmiting information). Assuming theBS broadcasts energy at power pi , the amount of harvestedenergy by EHU si during the period τ0i T is

Esi (i) = ηsi xsi (i)N0 piτ0i T, (2)

where ηsi is the energy harvesting efficiency of the scheduledEHU, and xsi (i) is the normalized fading power gain of the


channel between the scheduled EHU and the BS. Withoutloss of generality, we set ηsi = 1 (1 ≤ k ≤ K ). This isa safe assumption since ηk appears as the product ηk xk(i),and it can thus be incorporated into �k . During the IT phase,the EHU si spends all of its harvested energy, Esi (i), fortransmitting information to the BS. In particular, EHU si

transmits a complex-valued Gaussian codeword of duration(1− τ0i )T (comprised of infinitely many symbols that assurereliable information decoding) with an output power

Psi (i) =Esi (i)

(1− τ0i )T= N0 pi xsi (i)τ0i

1− τ0i(3)

and information rate

Rsi (i) = log(1+Psi (i)xsi (i)) = log

(

1+ pi asi (i)τ0i

1− τ0i

)

, (4)

where

asi (i) = N0x2si(i). (5)

If we adopt (4) as the feasible rates in (1), we arrive at thefollowing scheduling policy,


ωk log

(

1+ pi ak(i)τ0i

1− τ0i

)

. (6)

Since si in (6) depends on the optimization variables pi

and τ0i , the corresponding maximization problem would bedifficult (if not impossible) to solve. Instead, for the purposeof analytical tractability, we set pi = Pmax and τ0i = 1/2into (6) and propose the following opportunistic schedulingpolicy for the HTS protocol:


ωk log (1+ Pmaxak(i)) . (7)

Our goal is to determine pi and τ0i that maximize theachievable average rate over the uplink, i.e., to solve thefollowing maximization problem:

Maximizepi ,τ0i

1

M

M∑

i=1

(1− τ0i ) log

(

1+ pi τ0i asi (i)

1− τ0i

)

s.t. C1 : 1

M

∑M

i=1piτ0i ≤ Pavg

C2 : 0 ≤ pi ≤ Pmax , ∀iC3 : 0 < τ0i < 1, ∀i, (8)

where the index si is obtained from (7). The optimizationproblem (8) is the special case of [12, eq. (3)] with K = 1,a j i replaced by asi (i), and τ1i replaced by 1− τ0i . Therefore,the optimal allocations of the BS transmit power, p∗i , and thetime-sharing parameter, τ ∗0i , are obtained from [12, eqs. (4)and (5)], respectively, yielding

p∗i ={

Pmax , asi (i) > λ

0, otherwise,(9)

τ ∗0i = 1− asi (i)Pmax

asi (i)Pmax − 1

⎛

⎝1+ 1

W(

asi (i) Pmax−1

e1−λPmax

)

⎞

⎠

−1

,

(10)

where the constant λ is determined by setting C1 to equality,i.e., (1/M)

∑Mi=1 p∗i τ ∗0i = Pavg . Note, (10) is obtained by

applying definition of Lambert-W function to [12, eq. (5)].Hence, if scheduled in epoch i as per (7), EHU si should

first harvest energy for the time period τ ∗0i T and then transmitinformation at rate log

(1+ Esi (i) xsi (i)/((1− τ ∗0i )T )

)during

the time period (1 − τ ∗0i )T . To this end, the scheduled EHUshould be able to calculate τ ∗0i according to (10), whichrequires the CSI of its own channel, xsi (i). Additionally, in thebeginning of each epoch, the BS is assumed to broadcastlog2(K ) feedback bits in order to inform the EHUs whichEHU is scheduled for IT in that epoch.

The maximum achievable rate of EHU k under the HTSprotocol is determined by

R∗k = limM→∞

1

M

M∑

i=1

Ik(i)(1− τ ∗0i ) log

(

1+ ak(i)Pmaxτ∗0i

1− τ ∗0i

)

,

(11)

where Ik(i) is the indicator variable defined by

Ik(i) ={

1, if k = si

0, if k �= si .(12)

Using R∗k in (11), achievable rate region of the HTS protocolfor WPCN Type 1 can be defined as [27, Sec. V]

C1(Pavg, Pmax

) =⋃

F1

{(R1, R2, . . . , RK

) : Rk ≤ R∗k , ∀k}

,

(13)

where F1 denotes the set of all feasible allocations for τ0i andall feasible power allocations (P ) that satisfy the constrains onthe BS average available power and BS instantaneous transmitpower, i.e., F1 ≡ {(P , τ0i ) : E[piτ0i ] ≤ Pavg, 0 ≤ pi ≤Pmax , 0 < τ0i < 1}. A given weight vector (ω1, ω2, . . . , ωK )yields a specific value of the constant λ, and, therefore,a specific rate vector (R∗1 , R∗2 , . . . , R∗K ) that represents onepoint on the boundary of the achievable rate region of theHTS protocol, see [30, Lemma 3.10]).

1) HTS Protocol Implementation at the Base Station:Lemma 1 assumes an advanced knowledge of constant λ.However, its advanced calculation necessitates the knowledgeof the (cumulative) probability density functions (i.e., PDFsand CDFs) of all fading channels, which may be unknown inpractice. Instead, in practical implementations, λ can be esti-mated iteratively (online) by applying the stochastic gradientdescent method [25, Sec. III.C], as

λ(i) = λ(i − 1)+ β(

1

i

i∑

n=1

pn τ0n − Pavg

)

, (14)

where the average BS transmit power is estimated from theprevious i−1 epochs, and β is the step size. In (14), pn and τ0n

are calculated from (9) and (10), respectively, with λ replacedby its estimate in epoch i − 1, λ(i − 1). Due to the convexityof (8), the iterative calculation of pi from (9), τ0i from (10),and λ from (14) is guaranteed to converge to the optimalsolution of (8). The proper selection of β and the initial


Fig. 2. Dynamic TDMA frame for harvest-then-transmit protocol.

value λ(0) assures rapid convergence of λ(i) towards λ afterM ′ << M epochs, whereas λ(i) oscillates around λ fori > M ′. The online implementation of the HTS protocol atthe BS is summarized in Algorithm 1.

If we assume availability of the long-term statistics of thepower gain of the BS-EHUk channel, i.e., their PDF fXk (x)and CDF FXk (x), the time average in C1 of (8) can be replacedby a statistical average. In this case, C1 of (8) is transformed as

K∑

k=1

∫ ∞√λ/N0

Pmax τ0(x) vXk (x)dx = Pavg, (15)

where

vXk (x) = fXk (x)K∏

j=1j �=k

FX j

⎛

⎝

√(1+ Pmax N0x2

)ωk/ω j − 1

Pmax N0

⎞

⎠

(16)

whereas τ0(x) is obtained from (10) after replacing asi (i)by N0 x2. Derivation of (15) is similar to that of [28, Sec. III],but is omitted here for brevity. Based upon (15), the BS is ableto numerically calculate the constant λ in advance, in whichcase its iterative estimation is omitted from Algorithm 1.

Algorithm 1 HTS Protocol Implementation at the BS

1: Initialize λ, set β, and set i = 1;2: repeat3: Determine EHU si according to (7);4: if asi (i) > λ then5: Calculate τ0i from (10);6: Broadcast RF energy at pi = Pmax

during EH phase of duration τ0i T ;7: Schedule EHU si for uplink IT;8: else9: Set pi = 0 and τ0i = 0;

10: λ← λ+ β(

1i

∑in=1 pnτ0n − Pavg

);

11: i ← i + 1.12: until i ≤ M

B. Harvest-Then-Transmit Protocol

An upper bound of the achievable rate region C1 in (13)can be obtained by considering another protocol for WPCNType 1 which employs ITs of all EHUs in each epoch. To thisend, the K ITs are allocated in successive (non-overlapping)subintervals. In this case, the TDMA frame in epoch i isdivided into K + 1 subintervals of variable duration (Fig. 2):an EH phase of duration τ0i T , and the successive IT phases

of durations τki T (1 ≤ k ≤ K ), satisfying

K∑

k=0

τki = 1, ∀i, (17)

where τki are referred to as the time-sharing ratios. We notethat, this protocol corresponds to the well known harvest-then-transmit (HTT) protocol, which has been initially studiedin [9]. However, [9] considers a simplified version of theHTT protocol (referred to as the Benchmark HTT protocol),which assumes fixed BS transmit power (pi = const .,∀i )and only optimizes the time-sharing ratios τki for the uplinksum-rate maximization. We jointly optimize the BS transmitpower pi and the time-sharing ratios τki for weighted sum-ratemaximization.

For this protocol, during the EH phase, the amountof harvested power by each EHU is given by Ek(i) =ηk xk(i)N0 piτ0i T for 1 ≤ k ≤ K , where we again setηk = 1. For its IT, EHU k spends all of its harvested energy,Ek(i), for sending information to the BS. In particular, EHUk transmits a complex-valued Gaussian codeword of durationτki T (comprised of infinitely many symbols that assure reliableinformation decoding) with an output power

Pk(i) = Ek(i)

τki T= N0 pi xk(i) τ0i

τki, (18)

and information rate

Rk(i) = log

(

1+ pi ak(i)τ0i

τki

)

, (19)

where

ak(i) = N0x2k (i). (20)

The achievable rate region of the HTT protocol is definedas [27, Sec. V]

C2(Pavg, Pmax

) =⋃

F2

{(R1, R2, . . . , RK

) :

Rk ≤ limM→∞

1

M

M∑

i=1

τki Rk(i), ∀k}

,

(21)

where F2 denotes the set of all feasible power alloca-tion (P ) and time-sharing allocations (T ) satisfying the con-strains on the time-sharing ratios, the BS average availablepower and BS instantaneous transmit power, i.e., F2 ≡{(P , T ) : E[piτ0i ] ≤ Pavg, 0 ≤ pi ≤ Pmax ,

∑Kk=1 τki = 1

}.

Mathematically, each point of the boundary of the achiev-able rate region C2

(Pavg, Pmax

)can be obtained as the solu-

tion of the maximization problem [30, Lemma 3.10]:

MaximizeRk

K∑

k=1

ωk Rk s.t.(R1, R2, . . . , RK

)∈C2(Pavg, Pmax

),

(22)


which can be equivalently restated as a weighted sum-ratemaximization problem,

Maximizepi ,τ0i ,τki

1

M

M∑

i=1

K∑

k=1

ωk τki log

(

1+ piak(i)τ0i

τki

)

s.t. C1 : 1

M

∑M

i=1piτ0i ≤ Pavg

C2 : 0 ≤ pi ≤ Pmax , ∀iC3 :

∑K

k=0τki = 1, ∀i

C4 : τki > 0, ∀i, ∀k. (23)

Theorem 1: The optimal allocations of the BS transmitpower and the time-sharing ratios, found as the solutionof (23), are given by

p∗i =⎧⎨

⎩

Pmax ,∑K

k=1ωk ak(i)

1+ zk(i)> λ

0, otherwise,(24)

τ ∗0i =1

1+∑Kk=1

ak(i)Pmaxzk(i)

, (25)

τ ∗ki =ak(i)Pmax

zk(i)τ ∗0i , 1 ≤ k ≤ K , (26)

respectively, where the auxiliary variables zk(i) are calculatedas follows

zk(i) = −1−(

W0

(

−e−1−μi Pmax

ωk

))−1

, (27)

where W0(x) denotes the Lambert W function returning itsprincipal value (i.e., W0(x) ≥ −1). In (27), the parameterμi > 0 is determined as the root of the following transcen-dental equation,

λ+ μi +K∑

k=1

ak(i) ωk W0

(

− e−1−μi Pmax

ωk

)

= 0, (28)

where constant λ is found from (1/M)∑M

i=1 p∗i τ ∗0i = Pavg .Proof: Please refer to Appendix A.

The optimal pi and τki , ∀i, 1 ≤ k ≤ K , given inTheorem 1, yield to the achievable rate region of the proposedHTT protocol. Any point of C2

(Pavg, Pmax

)is determined by

the vector (ω1, ω2, . . . , ωK ), and the average achievable rateof EHU k in that point is given by

R∗k = limM→∞

1

M

M∑

i=1

τ ∗ki log

(

1+ Pmaxτ∗0i ak(i)

τ ∗ki

)

. (29)

In order to achieve rate (29), the EHU k in epoch ishould transmit during the time period τ ∗ki T at ratelog

(1+ Ek(i) xk(i)/(τ ∗ki T )

). In order to do so, EHU k should

be able to calculate its own τ ∗ki according to (26), whichrequires perfect CSI knowledge of all K fading links by allEHUs. The HTT protocol requires full CSI available at allthe nodes, as well as strict synchronization among the EHUs,which is not practical.

1) Sum-Rate Maximization: The vector (ω1, ω2, . . . ,ωK ) = (1, 1, . . . , 1) is a special point of the achievable rateregion where the sum-rate of the HTT protocol is maximized.In this case, Theorem 1 specializes to the following optimalallocations for the BS transmit power and time-sharing ratios,

p∗i ={

Pmax , b(i) > λ

0, otherwise,(30)

τ ∗0i = 1− b(i)Pmax

b(i)Pmax − 1

⎛

⎝1+ 1

W(

b(i)Pmax−1e1−λPmax

)

⎞

⎠

−1

, (31)

τ ∗ki =(1− τ ∗0i

)ak(i)

b(i), 1 ≤ k ≤ K , (32)

where b(i) = ∑Kn=1 an(i). The derivation for (30), (31)

and (32) is available in Appendix A. Note that this resultis already available in [12], where it is obtained by analternative derivation approach. Due to the doubly near-fareffect, the sum-rate maximization is unfair to EHUs at largerdistances to the BS since they receive much smaller portionof the average uplink rate compared to the closer EHUs.In addition, it also requires full CSI availability at all thenodes.

IV. RATE MAXIMIZATION IN WPCN TYPE 2

In WPCN type 1, the total amount of energy harvested bythe EHU during an EH phase is completely spent for IT in thesame epoch. Such battery management in WPCNs type 1 issimple but suboptimal. Actually, depending on the channelconditions, it may be better for a given EHU not to spend allof its harvested energy in the successive IT phase, but to saveit for ITs in the following epochs when the fading conditionsbecome more favorable. In addition, in WPCN Type 1, onlya single EHU harvests energy in a single epoch, although itwill be much more beneficial if all EHUs can harvest energyin each epoch. Such rationale gives rise to the WPCN type 2,whose EHUs harvest RF energy over multiple epochs beforetheir IT, which yields to higher rates then WPCN Type 1.

A. Harvest-or-Select Protocol

Let us assume a WPCN type 2, which employs an oppor-tunistic scheduling protocol and a TDMA frame structureaccording to Fig. 1. Let Ek(i − 1) denote the available energyin the battery of EHU k (1 ≤ k ≤ K ) in the beginning ofepoch i , and let battery’s storage capacity be unlimited (i.e.,Emax → ∞). Assuming the BS broadcasts with power pi inepoch i , the amount of harvested energy by EHU k during theEH phase is T N0 xk(i)piτ0i . Therefore, at the end of the EHphase of epoch i , the amount of stored energy in the batteryof EHU k is given by Ek(i − 1)+ T N0 xk(i)piτ0i .

Let the EHU si be the scheduled user in epoch i . Forthe HOS protocol, we propose the following opportunisticscheduling policy


ωk log (1+ Pk0 xk(i)) . (33)


According to (33), the scheduled EHU aims at transmit-ting a codeword with (fixed) desired power Psi 0 at ratelog

(1+ Psi 0 xsi (i)

).

Since the amount of available energy in the battery of thescheduled EHU is Esi (i − 1)+ T N0 xsi (i)piτ0i , its IT at thedesired output power Psi 0 may not necessarily be supplied bythe battery, and, in this case, the scheduled EHU transmits itscodeword with some lower power, Psi (i) ≤ Psi 0. In general,the output power of the scheduled EHU can be written as

Psi (i) = min

{Esi (i − 1)+ T N0xsi (i)piτ0i

T (1− τ0i ), Psi 0

}

. (34)

The output power adaptation in each epoch incur significantpower consumption due to processing, and therefore, maybe difficult to realize in practice due to the limited energyavailable at the EHUs. Therefore, we set the desired outputpowers of all EHUs to fixed values (P10, P20, . . . , PK 0). In thenumerical results section, the fixed desired output power ofEHU k is chosen so as to maximize the achievable rate of theWPCN Type 2 with a single EHU operating over the channelxk(i) (c.f. Appendix C).

Depending on whether a given EHU is the scheduleduser or one of the remaining K − 1 users, its output power isgenerally expressed as

Pk(i) ={

Psi (i), k = si

0, k �= si ,(35)

where Psi (i) is given by (34). Considering (35), the energyavailable in the battery of EHU k at the end of epoch i isgiven by

Ek(i) = Ek(i−1)+N0xk(i)piτ0i T−Pk(i)(1−τ0i)T . (36)

Thus, the maximization problem for the achievable averagerate over the uplink is written as

Maximizepi ,τ0i

1

M

M∑

i=1

(1− τ0i ) log

(

1+K∑

k=1

Pk(i) xk(i)

)

s.t. C1 : Pk(i) = Ik(i) ·min

{Bk(i − 1)

T (1− τ0i ), Pk0

}

, ∀i, ∀k

C2 : 1

M

∑M

i=1piτ0i ≤ Pavg

C3 : 0 ≤ pi ≤ Pmax , ∀iC4 : 0 ≤ τ0i ≤ 1, ∀iC5 : Ek(i) = Ek(i − 1)+ T N0xk(i) piτ0i

− T Pk(i)(1− τ0i ), ∀i, ∀k, (37)

where Ik(i) is the selection indicator variable defined by (12).For finite M , solving (37) is very difficult and may require

non-causal CSI knowledge. However, based upon [31], (37)can be significantly simplified when M → ∞, as shown bythe following lemma.

Lemma 1: For M →∞, (37) is equivalently written as

Maximizeui ,τ0i , ∀i

1

M

M∑

i=1

(1− τ0i ) log

(

1+K∑

k=1

xk(i)Ik(i)Pk0

)

s.t. C1 : Pk(i) = Pk0 Ik(i), ∀i, ∀k

C2 : 1

M

∑M

i=1piτ0i ≤ Pavg

C3 : 0 ≤ pi ≤ Pmax , ∀i

C4 : 0 ≤ τ0i ≤ 1, ∀i

C5 : T∑M

i=1Pk(i)(1− τ0i )

≤ T∑M

i=1N0xk(i)piτ0i , ∀i, ∀k, (38)

Proof: In [31], it is proven that if an EHU with unlimitedstorage capacity has a desired power allocation policy forwhich the average harvested energy in its battery is largerthan or equal to the average depleted energy from its battery,then this EHU can transmit with the desired transmit powerin almost all epochs, if the number of epochs M satisfiesM → ∞. In fact, in this case, the number of epochs inwhich the EHU transmits with a power which is smaller thanits desired power, due to insufficient energy in the battery,is negligible compared to M → ∞ and therefore theseepochs have negligible effect on the average rate. Now, forthe optimization problem in (37), note that all EHUs haveunlimited storage capacity and that M →∞. Moreover, notethat if we add constraint C5 in (38) as an additional constraintinto the optimization problem in (37), we will obtain that thefeasible power allocation policies of this new optimizationproblem are only those for which the average harvested energyin all the batteries of EHUs are larger than or equal to thecorresponding average depleted energies from their batteries.As a result, using the result from [26], we can replaceC1 in (37) by C1 in (38) since, according to [31], C1 in (38)now occurs in almost all epochs. Consequently, constraintC5 in (37) is not needed any more and can be removed.Thereby, the maximization problem in (37) becomes identicalto the maximization problem in (38). This completes theproof.

Note, the left hand side of the inequality in C5 is the totalenergy that is spent from the battery of EHU k, whereas itsright hand side is the total energy harvested in the batteryof EHU k. The solution of (38) is given by the followingTheorem.

Theorem 2: The optimal duration of EH phase is given by

τ ∗0i =

⎧⎪⎨

⎪⎩

0, d1(i) = 1 and d2(i) = 1

1, otherwise,(39)


whereas the optimal BS transmit power allocation is given by

p∗i =⎧⎨

⎩

Pmax , d1(i) = 0

0, otherwise.(40)

In (39) and (40),

d1(i) =

⎧⎪⎪⎨

⎪⎪⎩

1, λ0 >

K∑

k=1

N0λk xk(i)

0, otherwise,

(41)

and

d2(i) =⎧⎨

⎩

1, log(1+ Psi 0 xsi (i)

)> Psi 0λsi

0, otherwise,(42)

where si is the scheduled EHU in epoch i . The Lagrangemultipliers λ0 and {λk}Kk=1 are determined as the solutionof the system of equations resulting when the constraintsC1 and C2 of (38) are set to equality, i.e.,

⎧⎪⎪⎨

⎪⎪⎩

1

M

∑M

i=1p∗i τ ∗0i = Pavg

1

M

∑M

i=1Pk0 Ik(i)(1−τ ∗0i)=

1

M

∑M

j=1N0xk(i)p∗i τ ∗0i , ∀k.

(43)

Proof: Please refer to Appendix B.Note that, if τ0i = 0, the epoch i consists entirely of an

IT phase by the scheduled EHU (i.e., an uplink transmission),whereas, if τ0i = 1, the epoch i consists entirely of an energybroadcast by the BS (i.e., a downlink transmission).

Given (ω1, ω2, . . . , ωK ) and (P01, P02, . . . , P0K ),the achievable rate of EHU k under the proposed HOSprotocol for WPCN type 2 is given by

R∗k = limM→∞

1

M

M∑

i=1

Ik(i)d1(i)d2(i) log (1+Pk(i)xk(i)), (44)

where Pk(i) = min {Pk0, Ek(i − 1)/T }, and d1(i) and d2(i)are determined from (41) and (42), respectively. In order toachieve the rate (44), when scheduled, EHU k should transmitinformation at rate log (1+ Pk(i) xk(i)) for the entire epochduration T , which requires the perfect CSI knowledge only ofits own link, xk(i). Again, in the beginning of each epoch,the BS is assumed to broadcast log2(K ) feedback bits inorder to inform the EHUs of the scheduled EHU for thatepoch. The online implementation of the HOS protocol at theBS is summarized in Algorithm 2. Similarly to Algorithm 1,the gradient descent method is utilized for online estimationof λ0 and λk , with β0 and βk denoting the step sizes of eachof the constants.

If we assume availability of the long-term statistics for thefading channels between the BS and EHUs, the time averagesin the set (43) can be replaced by statistical averages. In this

case, (43) is transformed as⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Pmax · Pr

{∑K

n=1N0λn Xn > λ0

}

= Pavg

Pk0 · Pr

{∑K

n=1N0λn Xn < λ0, Xk >

ePk0λk − 1

Pk0,

Xn <(1+ Pk0 Xk)

ωk

ωi − 1

Pi0, n �= k

⎫⎪⎪⎬

⎪⎪⎭

= N0 · E[

Xk |∑K

n=1N0λn Xn > λ0

]

, ∀k,

(45)

where Pr {·} denotes probability, and E[· | ·] denotes condi-tional expectation. The probabilities and expectations in (45)can be expressed in closed form if some particular fadingdistributions (e.g., the Rayleigh fading) is assumed, but theseexpressions are omitted here for brevity. Based upon a closedform solution of (45), the BS is able to numerically calculatethe constants λ0 and {λk}Kk=1 in advance, in which case theiriterative estimations are omitted from Algorithm 2.

Algorithm 2 HOS Protocol Implementation at the BS

1: Initialize {λk}Kk=0; Set {βk}Kk=0 and i = 1;2: repeat3: if λ0 < N0

∑Kk=1 λk xk(i) then

4: Set τ0i = 1; Broadcast RF energy at pi = Pmax

for the entire epoch duration T ;5: else6: Set pi = 0 and τ0i = 1; Determine EHU si

according to (33);7: if log

(1+ Psi 0 xsi (i)

)> Psi 0λsi then

8: Set τ0i = 0; Schedule EHU si

for uplink IT for the entire epoch durationT ;

9: λ0 ← λ0 + β0

(1i

∑in=1 pnτ0n − Pavg

);

10: λk ← λk+ βki

∑in=1 Pk0 Ik(n)(1−τ0n)−N0 xk(n)pnτ0n ;

11: i ← i + 1.12: until i ≤ M

B. Harvest-or-Concurrently-Transmit Protocol

An upper bound of the achievable rate region of the HOSprotocol can be obtained by considering another protocol forWPCN type 2, which is not opportunistic since all EHUs cantransmit information in a multiple-access (MA) fashion duringa single epoch. We refer to this protocol to as the harvest-or-concurrently-transmit (HOCT) protocol, which has beenstudied in [32]. In this subsection, we briefly summarize themain results of [32] that lead to the achievable rate region ofthe HOCT protocol.

As in the HOS protocol, it turns out that, optimally, eachepoch of the HOCT protocol consists of either an uplink or adownlink transmission. Therefore, the appropriate definitionof the scheduling variable in this case is the following:

qi ={

0, epoch i is used for uplink transmission

1, epoch i is used for downlink transmission.(46)


Without loss in generality, let us order the EHUs indecreasing order of their corresponding weights, ω1 ≥ ω2 ≥· · · ≥ ωK . Then, the achievable rate of EHU k under theHOCT protocol for WPCN type 2 is given by [30, eq. (16)]

Rk= limM→∞

1

M

M∑

i=1

(1−qi

)log

(

1+ Pk(i) xk(i)

1+∑k−1j=1 Pj (i)x j (i)

)

,

(47)

which is achieved when epoch i is scheduled to be an uplinkepoch and EHU k transmits a codeword at rate

Rk(i) = log

(

1+ Pk(i) xk(i)

1+∑k−1j=1 Pj (i)x j (i)

)

. (48)

On the receiver side, the BS decodes the EHUs’ codewordsin the order K , K − 1, …, 1, by using successive interferencecancelation. In (47) and (48), the EHUs’ transmit powers,Pk(i), are obtained as

Pk(i) = min

{Ek(i − 1)

T, Pd,k(i)

}

, (49)

where Pd,k(i) is the desired output power of EHU k in epoch i .Note that, in this case, the EHUs’ desired output powers arenot fixed, but change from one epoch to the next, dependingon the channel conditions. The available amount of energy atthe end of epoch i is given by

Ek(i) = Ek(i−1)+qi N0xk(i)pi T−(1−qi)Pk(i)T . (50)

When M →∞, the optimal allocations of qi , pi and Pd,k(i)are given by [32, eq. (15)], [32, eq. (16)] and [32, eq. (17)],respectively. Similarly as the HTT protocol, the HOCT pro-tocol is impractical, since it requires CSI knowledge of all Kfading channels at all EHUs.

V. NUMERICAL RESULTS

In this section, we illustrate the achievable rates when eithera single EHU (K = 1) or two EHUs (K = 2) are present ina WPCN type 1 or a WPCN type 2. The achievable rates ofthe four proposed protocols are obtained from (11), (29), (44),and (47), respectively, by using Monte-Carlo simulations. Notethat our illustration is limited only to the cases of K = 1 andK = 2 because it is difficult to visualize the rate regions forK > 2. Nevertheless, the numerical results for the two-usercase presented in this section provide interesting insights thatwill carry over to the general K -user case, which is studiedin Sections III and IV.

Since Rayleigh fading is considered, x ′k(i) follows an expo-nential distribution. The deterministic path loss is calculatedas E[x ′k(i)] = 10−3 D−αk , where Dk is the distance of EHUk to the BS. The pathloss at a reference distance of 1m isset to 30dB, and the pathloss exponent is set to α = 3.We assume an AWGN power spectral density of −150dBm/Hzand a bandwidth of 1MHz, yielding N0 = 10−12 Watts. Thus,�k = 10−3 D−3

k /N0. In addition we consider that the EHUsare either at one of the following distances from the BS:(1) Dk = 10m yielding to �k = 106, (2) Dk = 12.5m yieldingto �k = 106/2, and (3) Dk = 20m yielding to �k = 106/8.

Fig. 3. Achievable rate in a WPCN with a single EHU (K = 1) at a distanceof 10 meters from the BS. The three dashed curves apply to the HOS protocoloperating over finite horizon: M = 102 (lowest rate curve), M = 103 andM = 104.

Fig. 4. Achievable rate in a WPCN with a single EHU (K = 1) at adistance of 12.5 meters from the BS. The three dashed curves apply to theHOS protocol operating over finite horizon: M = 102 (lowest rate curve),M = 103 and M = 104.

Figs. 3-5 present the achievable rates of each protocol as afunction of the BS average transmit power Pavg . The resultsare obtained for the first scenario (i.e., a point-to-point system)where the peak power is set according to Pmax = 5 Pavg.The Benchmark HTT protocol denotes the HTT protocolconsidered in [9], which maximizes the uplink sum-rate ofthe WPCN by optimizing the durations of the EH and ITphases, τki , for fixed BS output power in all epochs (i.e.,pi = Pavg). The protocol described by Theorem 1 can beseen as an extension of the Benchmark HTT protocol as itjointly adapts the BS output power and the durations of theEH and IT phases. In the case of the HOS protocol, variousnumber of epochs M is considered, including the asymptoticcase for M → ∞. For finite M , it can be observed that,the proposed HOS protocol is strictly suboptimal, comparedto M → ∞. Additionally, these figures show that HOSprotocol operates very close to the upper bound for M →∞.For the single-user scenario, the achievable rate of the HTSprotocol coincides with the upper bound achieved by the HTT


Fig. 5. Achievable rate in a WPCN with a single EHU (K = 1) at a distanceof 20 meters from the BS. The three dashed curves apply to the HOS protocoloperating over finite horizon: M = 102 (lowest rate curve), M = 103 andM = 104.

Fig. 6. Rate region of a WPCN with two EHUs (K = 2) at the samedistance from the BS (�1 = �2), where Pavg = 1W and Pmax = 5W . Thethree dashed curves apply to the HOS protocol operating over finite horizon:M = 102 (lowest rate curve), M = 103 and M = 104.

protocol. When the EHU is at a closer distance to the BS,Fig. 3 (D1 = 10m) and Fig. 4 (D2 = 12.5m) show that theHOS protocol is superior to the HTS protocol for any value ofPavg . However, at larger distances, Fig. 5 (D1 = 20m) showsthat the HTS protocol outperforms the HOS protocol, since thelatter protocol employs EHUs whose desired transmit powersare not adaptive but fixed to a single value in all epochs.Both the HOS and HTS protocols significantly outperform theBenchmark HTT protocol.

Figs. 6-8 present the trade-off in terms of rate in the twouser case scenario. In order to do so, we illustrate the capacityregion of each protocol. Each rate pair (R1, R2) correspondsto a pair of weights (ω1, ω2) where ω1 + ω2 = 1. The ratepairs are calculated according to (11), (29), (44) and (47) forHTS, HTT, HOS and HOCT, respectively. The BS averagetransmit power is set to 1W, thus Pmax = 5 Pavg = 5W. Here,the Benchmark HTT protocol, from [9] is represented by acircle in the rate region figures. Equivalently to Figs. 3-5,the HOS protocol is implemented for various number ofchannel realizations. The desired output powers of the twoEHUs, for the HOS protocol P10 and P20 are selected so

Fig. 7. Rate region of a WPCN with two EHUs (K = 2) at differentdistances from the BS (�1 = 2�2), where Pavg = 1W and Pmax = 5W .The three dashed curves apply to the HOS protocol operating over finitehorizon: M = 102 (lowest rate curve), M = 103 and M = 104.

Fig. 8. Rate region of a WPCN with two EHUs (K = 2) at differentdistances from the BS (�1 = 8�2), where Pavg = 1W and Pmax = 5W .The three dashed curves apply to the HOS protocol operating over finitehorizon: M = 102 (lowest rate curve), M = 103 and M = 104.

as to maximize the throughput of the respective point-to-point systems (c.f. Appendix C). Specifically, P10 and P20are selected according to (77). Our proposed opportunisticscheduling protocols result in a minor degradation of theachievable rate, compared to their respective upper bounds,while resulting in greater simplicity at the same time. TheHOCT protocol results in the largest rate region, except whenone EHU is far enough from the BS (e.g. Dk = 20m as in thissetup). In this case (Fig. 8), some operating points of HTT andHTS achieve larger rates for the further EHU. As expected,these proposed protocols again outperform Benchmark HTTprotocol.

VI. CONCLUSION

In analogy to the classic cellular systems, we have pro-posed opportunistic scheduling for fair resource allocation inWPCNs. The two proposed opportunistic scheduling schemes,HTS and HOS, can efficiently mitigate the doubly near-fareffect and accommodate fairness among the EHUs at variousdistances to the BS. The numerical examples show that theseadvantages can be achieved at the expense of only minor rate


degradation, compared to the rates of the benchmark protocolsthat employ transmissions from multiple EHUs within a singleTDMA frame (i.e., the HTT protocol for WPCN Type 1,and HOCT protocol for WPCN Type 2). Note that this workneglects the circuit power consumption and processing cost atthe EHUs, and thus arrives at upper bounds on the performanceof the two proposed opportunistic scheduling policies (HTSand HOS). However, regardless on how close these boundsare to the practical performance of WPCNs employing theseprotocols, the derived rate regions still clearly demonstrate theminor performance degradation of the two proposed protocolswith respect to the benchmark protocols. As a result, the use ofthese protocols is fully justified by their practical added ben-efits: (1) fair resource utilization, (2) lower CSI requirements,and (3) less stringent time synchronization.

APPENDIX APROOF OF THEOREM 1

We reformulate (23) by introducing the change of variables,ei � pi τ0i , which leads to

Maximizeei ,τ0i , τki

1

M

M∑

i=1

K∑

k=1

ωkτki log

(

1+ ak(i)ei

τki

)

s.t. C1 : 1

M

∑M

i=1ei ≤ Pavg

C2 : 0 ≤ ei ≤ Pmax τ0i ,∀iC3, and C4 as in (23). (51)

The objective function of (51) is jointly concave and theconstrains are linear in ei and τki . Thus, the Lagrangian isgiven by

L = 1

M

M∑

i=1

K∑

k=1

ωkτki log

(

1+ ak(i)ei

τki

)

− λ

M

M∑

i=1

ei

−ψi

(K∑

k=1

τki + τ0i − 1

)

+M∑

i=1

qi ei

−M∑

i=1

μi (ei − Pmaxτ0i ), (52)

where λ is the non-negative Lagrange multiplier associatedwith the constraint C1. Therefore,

∂L∂ei=

K∑

k=1

ωkak(i)

1+ ak(i)eiτki

− λ+ qi − μi = 0 (53)

∂L∂τki= ωk log

(

1+ ak(i)ei

τki

)

− ωkak(i)eiτki

1+ ak(i)eiτki

− ψi = 0, (54)

∂L∂τ0i= −ψi + μi Pmax = 0. (55)

The complementary slackness conditions are given byψi

(∑Kk=1 τki + τ0i − 1

)= qi ei = μi (ei − Pmaxτ0i ) =

0, ∀i, where ψi , qi , and μi are the non-negative Lagrangemultipliers.

Case 1: If τ0i = 0, then ei = 0 and no power is allocatedto epoch i , i.e., p∗i = 0.

Case 2: Let us assume 0 < τki < 1 and also ei = Pmaxτ0i .This case corresponds to p∗i = Pmax . Due to the slacknessconditions, qi = 0, ψi > 0, and μi > 0, leading to

μi =K∑

k=1

ωk ak(i)

1+ ak(i)Pmax τ0iτki

− λ, (56)

τ0i +K∑

k=1

τki = 1, (57)

ψi = μi Pmax = ωk log

(

1+ ak(i)Pmaxτ0i

τki

)

− ωkak(i)Pmax τ0i

τki

1+ ak(i)Pmax τ0iτki

. (58)

We now introduce the auxiliary variables

zk(i) = ak(i)Pmaxτ0i

τki, 1 ≤ k ≤ K . (59)

Combining (57) and (59), we obtain (25) and (26). Combin-ing (56), (58) and (59), we obtain the following two equationsrespectively, respectively leading to (27) and (28):

log (1+zk(i))− zk(i)

1+zk(i)= μi Pmax

ωk, 1 ≤ k ≤ K , (60)

μi =K∑

k=1

ωk ak(i)

1+ zk(i)− λ. (61)

A. Special Case: Sum-Rate Maximization

Setting ωk = 1 (1 ≤ k ≤ K ), (60) is transformed as

log (1+ zk(i))− zk(i)

1+ zk(i)= μi Pmax , 1 ≤ k ≤ K , (62)

which implies z1(i) = z2(i) = · · · = zK (i), i.e., τ1(i)a1(i)

=τ2(i)a2(i)= · · · = τK (i)

ak(i)= Ci . Applying (57), we obtain Ci =

1−τ0ib(i) , yielding z1(i) = z2(i) = · · · = zK (i) = b(i)Pmax τ0i

1−τ0i, and

μi = (1− τ0i )b(i)

1− τ0i + b(i)Pmaxτ0i− λ > 0. (63)

Thus, we obtain the condition b(i) > λ of (30). Introduc-ing (63) into (62) leads to

log

(

1+ b(i)Pmaxτ0i

1− τ0i

)

+λPmax = b(i)Pmax

1− τ0i + b(i)Pmaxτ0i,

(64)

which, after applying the definition of the Lambert-W func-tion, is finally transformed into (31).

APPENDIX BPROOF OF THEOREM 2

Introducing C1 into the objective function and the otherconstraints, and the change of variables ui = piτ0i , (38) is


transformed into a convex optimization problem,

Maximizeui ,τ0i , ∀i

1

M

∑M

i=1(1− τ0i ) log

(1+ Psi 0 xsi (i)

)

C1 : 1

M

∑M

i=1Pk0 Ik(i)(1− τ0i )

≤ 1

M

∑M

j=1N0xk(i)ui , ∀k

C2 : 1

M

∑M

i=1ui ≤ Pavg

C3 : 0 ≤ ui ≤ Pmaxτ0i

C4 : 0 ≤ τ0i ≤ 1 (65)

where the scheduled EHU si is determined from (33). TheLagrangian of (65) is given by

L = 1

M

M∑

i=1

(1− τ0i ) log(1+ Psi 0 xsi (i)

)

− 1

M

M∑

i=1

Psi 0λsi (1− τ0i )+ 1

M

M∑

i=1

K∑

k=1

N0λkui xk(i)

− λ0

(1

M

M∑

i=1

ui − Pavg

)

+M∑

i=1

ν2i (Pmaxτ0i − ui )

+M∑

i=1

ν1i ui +M∑

i=1

μ1i τ0i +M∑

i=1

μ2i (1− τ0i ), (66)

where {λk}Kk=1 and λ0 are the non-negative Lagrange multipli-ers associated to C1 and C2, respectively, whereas ν1i and ν2i

are associated to C3, and μ1i and μ2i are associated to C4.By differentiating (66) with respect to ui and τ0i , and settingboth derivatives to zero, we obtain

∂L∂ui=

K∑

k=1

N0λk xk(i)− λ0 + ν1i − ν2i = 0, (67)

∂L∂τ0i= − log

(1+ Psi 0 xsi (i)

)+ Psi 0 λsi

+ ν2i Pmax + μ1i − μ2i = 0, (68)

whereas the complementary slackness conditions are given by

ν1i ui=ν2i (Pmaxτ0i−ui)=μ1i τ0i =μ2i (1−τ0i)=0, ∀i.(69)

Case 1: Let 0 < ui < Pmaxτ0i and 0 < τ0i < 1.According to (69), ν1i = ν2i = μ1i = μ2i = 0, yielding∑K

k=1 N0λk xk(i) = λ0, and log(1+ Psi 0 xsi (i)

) = Psi 0 λsi .This case cannot occur, because the two latter equations cannotbe satisfied for arbitrary i .

Case 2: Let 0 < ui < Pmaxτ0i and τ0i = 0. Accordingto (69), ν1i = ν2i = μ2i = 0, whereas μ1i > 0. Introducingthe value of these multipliers into (67), we realize that this casecannot occur because (67) cannot be satisfied for arbitrary i .

Case 3: Let ui = 0 and 0 < τ0i < 1. According to (69),ν2i = μ1i = μ2i = 0, whereas ν1i > 0. Introducing the valueof these multipliers into (68), we realize that this case cannotoccur because (68) cannot be satisfied for arbitrary i .

Case 4: Let ui = 0 and τ0i = 1 (i.e., a downlinktransmission). Note, in this case, the BS is silent (i.e., pi = 0),and also all EHUs are silent (i.e., Pk = 0, 1 ≤ k ≤ K ).

Case 5: Let ui = Pmaxτ0i and 0 < τ0i < 1. Accord-ing to (69), ν1i = μ1i = μ2i = 0, whereas ν2i > 0.Introducing the value of these multipliers into (67), we obtainν2i = ∑K

k=1 N0λk xk(i) − λ0 > 0. Thus, Psi 0 λsi −log

(1+ Psi 0 xsi (i)

) = Pmax

(λ0 −∑K

k=1 N0λk xk(i))

, whichcannot be satisfied for arbitrary i , and this case thereforecannot occur.

Case 6: Let ui = Pmaxτ0i and τ0i = 1 (i.e., a downlinktransmission). Since all EHUs in this case are silent, we canassume Pk = 0, for 1 ≤ k ≤ K . According to (69), ν1i =μ1i = 0, whereas ν2i > 0 and μ2i > 0. Introducing the valueof these multipliers into (67) and (68), and setting Psi 0 = 0,we obtain

μ2i = Pmax ν2i = Pmax

(K∑

k=1

N0λk xk(i)− λ0

)

> 0, (70)

which yields (41) for the case d1(i) = 0.Case 7: Let ui = 0 and τ0i = 0 (i.e., an uplink transmis-

sion). According to (69), ν2i = μ2i = 0, whereas ν1i > 0 andμ1i > 0. Introducing the value of these multipliers into (67)and (68), we respectively obtain

ν1i = λ0 −K∑

k=1

N0λk xk(i) > 0, (71)

μ1i = log(1+ Psi 0 xsi (i)

)− Psi 0 λsi > 0. (72)

Note, (71) yields (41) for the case of d1(i) = 1, whereas (72)yields (42) for the case of d2(i) = 1.

APPENDIX CRATE MAXIMIZATION OF THE POINT-TO-POINT HOS

PROTOCOL IN RAYLEIGH FADING

Let us assume K = 1. Since the fading gain x1(i) isassumed stationary and ergodic random process, the equationset (43) can be solved numerically by using the PDF of channelx1, fX1(x). In Theorem 2, the conditions for d1(i) = 1and d2(i) = 1 occur when a1 < x1(i) < a2, where a1 =(exp(λ1 P10) − 1)/P10 and a2 = λ0/(N0λ1). Therefore, (43)attains the form

∫ a2

a1

P10 fX1(x)dx =∫ ∞

a2

N0 Pmax x fX1(x)dx . (73)∫ ∞

a2

Pmax fX1(x)dx = Pavg. (74)

In the case of Rayleigh fading, the solution of the above systemis given by

λ1 = 1

P10log

{

1− P10�1 log

[Pavg

Pmax

+ N0�1 Pavg

P10

(

1+ log

(Pmax

Pavg

))]}

, (75)

λ0 = λ1 N0�1 log

(Pmax

Pavg

)

. (76)


Using (75) and (76), we select P10 in order to maximize theaverage achievable rate of the point-to-point system, i.e.,

P∗10 = arg max∫ a2

a1

log(1+ P10x) fX1(x)dx . (77)

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Zoran Hadzi-Velkov (M’97–SM’11) received theDipl.Ing. degree (Hons.) in electrical engineering,the Magister Ing. degree (Hons.) in communicationsengineering, and the Ph.D. in technical sciencesfrom Ss. Cyril and Methodius University, Skopje,Macedonia, in 1996, 2000, and 2003, respectively.From 2001 to 2002, he was a Visiting Scholar withthe IBM Watson Research Center, NY, USA. From2012 to 2014, he was a Visiting Professor with theInstitute for Digital Communications, University ofErlangen-Nuremberg, Germany. He is currently a

Professor of Telecommunications with his alma mater. His research interestsare in the broad area of wireless communications, with particular emphasis onenergy harvesting communications, green communications, and cooperativecommunications. He received the Alexander von Humboldt Fellowship forexperienced researchers in 2012 and the Annual Best Scientist Award fromSs. Cyril and Methodius University in 2014. From 2012 to 2015, he wasthe Chair of the Macedonian Chapter of the IEEE Communications Society.He has served on the technical program committees of numerous internationalconferences, including ICC 2013, ICC 2014, ICC 2015, Globecom 2015,Globecom 2016, and ICC 2017. From 2012 to 2016, he has served as anEditor of the IEEE Communication Letters.

Ivana Nikoloska (S’15) received the bachelor’sand master’s degrees from Ss. Cyril and Method-ius University, Skopje, Macedonia, in 2014 and2016, respectively. She is currently pursuing thePh.D. degree with Monash University, Melbourne,Australia. From 2015 to 2016, she was a ResearchAssociate with the Faculty of Electrical Engi-neering and Information Technologies, Ss. Cyriland Methodius University, Skopje, where shewas involved in research projects funded by theAlexander von Humboldt Foundation. Her research

interests include green communications and information theory.


Hristina Chingoska (S’15) received the bachelor’sand master’s degrees from Ss. Cyril and Method-ius University, Skopje, Macedonia, in 2015 and2017, respectively. From 2015 to 2016, she was aResearch Associate with the Faculty of ElectricalEngineering and Information Technologies, Ss. Cyriland Methodius University, Skopje, where she wasinvolved in research projects funded by the Alexan-der von Humboldt Foundation. Her research interestsinclude energy harvesting communications.

Nikola Zlatanov (S’06–M’15) was born inMacedonia. He received the Dipl.Ing. and mas-ter’s degrees in electrical engineering from Ss.Cyril and Methodius University, Skopje, Macedonia,in 2007 and 2010, respectively, and the Ph.D. degreefrom the University of British Columbia (UBC),Vancouver, Canada, in 2015. He is currently aLecturer (Assistant Professor) with the Departmentof Electrical and Computer Systems Engineering,Monash University, Melbourne, Australia. His cur-rent research interests include wireless communica-

tions and information theory. He received several scholarships/awards for hiswork, including the UBC’s Four Year Doctoral Fellowship in 2010, the UBC’sKillam Doctoral Scholarship and Macedonia’s Young Scientist of the Yearin 2011, the Vanier Canada Graduate Scholarship in 2012, Best Journal PaperAward from the German Information Technology Society in 2014, and BestConference Paper Award at ICNC in 2016. He serves as an Editor of the IEEECommunications Letters. He has been a TPC member of various conferences,including Globecom, ICC, VTC, and ISWCS.

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