MITSUBISHI ELECTRIC RESEARCH LABORATORIEShttp://www.merl.com

QoS-Constrained Relay Control for Full-Duplex Relayingwith SWIPT

Liu, H.; Kim, K.J.; Kwak, K.S.; Poor, H.V.

TR2017-105 March 2017

AbstractThis study investigates relay control for simultaneous wireless information and power trans-fer in full-duplex relay networks under Nakagami-m fading channels. Unlike previous work,harvest-transmit (HT) and general harvest-transmit-store (HTS) models are respectively con-sidered to maximize average throughput subject to quality of service (QoS) constraints. Theend-to-end outage probability of the network in an HT model is presented in an exact integral-form. To prevent outage performance degradation in an HT model, time switching (TS) isdesigned to maximize average throughput subject to QoS constraints of minimizing outageprobability and maintaining a target outage probability, respectively. The optimal TS factorssubject to QoS constraints are presented for an HT model. In general, in an HTS model, en-ergy scheduling is performed across different transmission blocks and TS is performed withineach block. Compared with the block-based HTS model without TS, the proposed generalHTS model can greatly improve outage performance via greedy search (GS). By modelingthe relay’s energy levels as a Markov chain with a two-stage state transition, the outageprobability for GS implementation of the general HTS model is derived. To demonstrate thepractical significance of QoS-constrained relay control, numerical results are presented show-ing that the proposed relay control achieves substantial improvement of outage performanceand successful rate.

IEEE Transactions on Wireless Communications

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1

QoS-Constrained Relay Control for Full-DuplexRelaying with SWIPT

Hongwu Liu, Member, IEEE, Kyeong Jin Kim,Senior Member, IEEE,Kyung Sup Kwak,Member, IEEE, and H. Vincent Poor,Fellow, IEEE

Abstract—This study investigates relay control for simultane-ous wireless information and power transfer in full-duplex relaynetworks under Nakagami-m fading channels. Unlike previouswork, harvest-transmit (HT) and general harvest-transmit-store(HTS) models are respectively considered to maximize averagethroughput subject to quality of service (QoS) constraints.The end-to-end outage probability of the network in an HTmodel is presented in an exact integral-form. To prevent outageperformance degradation in an HT model, time switching (TS)is designed to maximize average throughput subject to QoSconstraints of minimizing outage probability and maintaininga target outage probability, respectively. The optimal TS factorssubject to QoS constraints are presented for an HT model. Ingeneral, in an HTS model, energy scheduling is performed acrossdifferent transmission blocks and TS is performed within eachblock. Compared with the block-based HTS model without TS,the proposed general HTS model can greatly improve outageperformance via greedy search (GS). By modeling the relay’senergy levels as a Markov chain with a two-stage state transition,the outage probability for GS implementation of the generalHTSmodel is derived. To demonstrate the practical significanceofQoS-constrained relay control, numerical results are presentedshowing that the proposed relay control achieves substantialimprovement of outage performance and successful rate.

Index Terms—Energy harvesting, wireless power transfer,amplify-and-forward relay, full-duplex relay, relay cont rol.

I. I NTRODUCTION

With capability to harvest energy from ambient radio-frequency (RF) signals, simultaneous wireless information andpower transfer (SWIPT) techniques provide a more promisingway for wireless communications to function in environmentswith physical or other limitations [1]–[3]. Based on two

Manuscript received December 29, 2015; revised June 21, 2016 and October18, 2016; accepted February 8, 2017. This work was supportedin part by SRFfor ROCS, SEM, Shandong Provincial Natural Science Foundation, China,under Grant 2014ZRB019XM, in part by the Ministry of Science, ICT andFuture Planning (MSIP), Korea, under the Information Technology ResearchCenter Support Program supervised by the Institute for Information andCommunications Technology Promotion under Grant IITP-2016-H8501-16-1019, in part by the National Research Foundation of Korea, Grant Funded bythe Korean Government, MSIP, under Grant NRF-2017R1A2B2012337, andin part by the U.S. National Science Foundation under Grant ECCS-1343210.The associate editor coordinating the review of this paper and approving itfor publication was Y. Jing.(Corresponding author: Kyung Sup Kwak.)

H. Liu is with Shandong Jiaotong University, Jinan 250357, China (e-mail:hong.w.liu@hotmail.com).

K. J. Kim is with Mitsubishi Electric Research Laboratories, Cambridge,MA 02139 USA (e-mail: kyeong.j.kim@hotmail.com).

K. S. Kwak (corresponding author) is with the Department of Informationand Communication Engineering, Inha University, Incheon 22212, SouthKorea (e-mail: kskwak@inha.ac.kr).

H. V. Poor is with the Department of Electrical Engineering,PrincetonUniversity, Princeton, NJ 08544 USA (e-mail: poor@princeton.edu).

practical receiver architectures, namely, time switching(TS)and power splitting (PS) receivers [1], SWIPT techniques havebeen widely applied in wireless networks [3, and referencestherein]. One line of research that has emerged is relay-assistedSWIPT [4]–[11]. It has shown that relay-assisted SWIPT notonly enables wireless communications over long distances oracross barriers, but also keeps energy-constrained relaysactivethrough RF energy harvesting (EH). In [4] and [5], TS andPS relaying protocols have been designed for amplify-and-forward (AF) and decode-and-forward (DF) relay networks,respectively. Several power allocation schemes for relay-assisted SWIPT networks with multiple source-destinationpairs were studied in [6]. Outage probability and diversityofrelay-assisted SWIPT networks with spatial randomly locatedrelays were investigated in [10] and distributed PS-basedSWIPT was investigated for interference relay networks in[11]. Moreover, multi-antenna technologies have been appliedin relay-assisted SWIPT networks in [7]–[9]. Nevertheless, allthese studies are limited to half-duplex relay (HDR) mode.Since the source-to-relay and relay-to-destination channels arekept orthogonal by either frequency division or time divisionmultiplexing, about 50% spectral efficiency (SE) loss occursin HDR mode. As a key technology for future relay networks,full-duplex relay (FDR) systems have drawn considerableattention [12]–[18]. Since FDR mode realizes an end-to-end (e2e) transmission via one channel utilization, significantimprovement of SE over HDR mode can be achieved.

A few studies have been conducted for FDR-assistedSWIPT networks. In [14] and [15], outage probability,throughput, and ergodic capacity have been analyzed forFDR-assisted SWIPT networks, in which a TS-based relayis operated cooperatively. In practice, since an FDR nodesuffers severe self-interference from its own transmit signal,FDR transmission is difficult to implement. To suppress self-interference, MIMO antennas have been employed at the relayto aid FDR-assisted SWIPT [19]. For conventional two-phaseAF HDR networks, a self-interference immunized FDR nodewas proposed by employing EH in the second time phase [20],so that the relay can transmit information and extract energysimultaneously via separated transmit and receive antennas.Note that all the above studies of FDR-assisted SWIPT areconducted to maximize network throughput without furtherconsidering quality of service (QoS) constraints, whereasit isenvisioned that SWIPT techniques will be required to supportvarious types of traffic having different QoS requirements[21]–[23]. For point-to-multipoint PS-based SWIPT networks,QoS constraints based on signal-to-interference-plus-noise ra-

2

tio (SINR) and mean-square-error (MSE) have been appliedto minimize transmission power in [21] and [24], respectively.A game-theoretic approach has been applied to optimizemultiple-pair SWIPT communications subject to SINR andEH constraints in [25]. For TS-based SWIPT, [23] studiesjoint TS and power control to maintain a given level ofthroughput. Notably, all the above control schemes are basedon estimation of instantaneous channel state information (CSI),which requires dedicated reverse-link training from the EHreceiver [26].

Motivated by these previous studies, in this paper weconsider the maximization of average throughput for a QoS-constrained FDR network with SWIPT. In the considered FDRnetwork, the source has a reliable power supply, whereasthe relay has to harvest energy from the source-emitted RFsignal via TS operation1. Since TS affects both SE and QoS,the relaying mode and corresponding TS have a complicatedrelationship in achieving the allowable maximum SE subjecttoQoS constraints. Compared with existing works, some distinctfeatures of our study are highlighted here.

• In [20], the effective information transmission time isthe same as that of HDR networks, so that the SEimprovement is achieved in HDR mode rather than FDRmode. In our study, we consider maximization of averagethroughput of FDR transmission, i.e., the relay receivesand forwards the source information to the destinationsimultaneously.

• Zhong et al. [14] and [15] investigated FDR-assistedSWIPT in the harvest-transmit (HT) model to improveaverage throughput with optimized TS. Unfortunately,outage performance seriously degrades in delay-limitedtransmissions when TS is optimized without a QoSconstraint. In our study, both outage probability mini-mization and target outage probability are adopted as QoSconstraints, under which TS is optimized to achieve theallowable maximum average throughput. Thus, seriousoutage performance degradation can be prevented andsuccessful rate can be maximized [27].

• The analysis of outage probability and throughput in[14] and [15] are conducted by modeling dual-hop andresidual self-interference (RSI) channels as Rayleigh fad-ing. However, SWIPT operates most efficiently within arelatively short range. In this situation, a line of sight(LoS) path exists with high probability and Nakagami-m fading can provides a better model [9], [28]. Fur-thermore, although a Rician fading model is appropriatefor the RSI channel in the RF-domain [29], the exactbehavior of the RSI channel in the digital-domain isstill unknown due to several complicated stages of activeinterference cancellation (IC) [30], [31]. Therefore, thisstudy investigates outage probability conditioned on theRSI channel power without modeling the RSI channelgain via a specific distribution and considers Nakagami-

1This setting has a number of potential applications in energy-limitedwireless networks, e.g., when the intermediate node is energy-selfish or lacksan energy-supply, or the direct link from the source to destination is blockedby a barrier while the relay has to be placed on a site without afixed powersupply.

m fading for dual-hop channels, so that our analyticalresults can be applied to a wide range of FDR-assistedSWIPT networks employing different IC schemes.

• Due to propagation loss and channel fluctuations, therelay-harvested energy within a transmission block canbe very limited and energy accumulation is needed toimprove the reliability of information transmission [32],[33]. Different from the block-based harvest-transmit-store (HTS) model [32], this study considers a generalHTS model by employing TS within each transmissionblock. With the aid of in-block TS, the general HTSmodel can greatly improve outage performance and in-cludes the block-based HTS model as a special case.

In this paper, the HT model and the general HTS are inves-tigated subject to QoS constraints. The corresponding controlschemes are developed and analytical results are presentedtoverify the QoS metrics. The contributions of the paper aresummarized as follows:

• By modeling dual-hop channels as Nakagami-m fading,we present the analytical e2e outage probability condi-tioned on the RSI channel power for the HT model.QoS constraints of minimizing outage probability andmaintaining a target outage probability are respectivelyconsidered in optimizing TS to achieve the allowablemaximum average throughput. The optimal TS factor thatmaximizes average throughput subject to minimizing out-age probability is presented in closed form. The optimalTS factor that maximizes average throughput subject toa target outage probability is also derived. Employingthe obtained optimal TS factors, the successful rate isachieved with the guaranteed outage performance.

• To accumulate energy for optimizing FDR transmission,a general HTS model is designed by employing TSwithin each block. In the general HTS model, the firstphase of each block is dedicated for EH. In the secondphase of each block, the relay node can switch to EHor FDR transmission depending on the relay’s residualenergy level and CSI. A greedy search (GS) policy isimplemented to realize the proposed general HTS model.By allocating a small portion of time for EH withineach block, the GS policy improves outage performancesignificantly over that of the block-based HTS model[32].

• The proposed general HTS model degenerates to theblock-based HTS model by setting a zero TS factor.Thus, the general HTS model and the block-based HTSmodel can be analyzed under a uniform framework. Withrespect to the time-switched two operational phases ineach block, the relay’s residual energy levels are modeledas a Markov chain (MC) with a two-stage state transition.Then, the e2e outage probability is derived for the GSimplementation of the general HTS mode including theblock-based HTS model as a special case.

The rest of this paper is organized as follows. Section II de-scribes the HT model and develops its e2e outage probability.The optimal TS that maximizes average throughput subjectto QoS constraints is also derived in Section II. Section III

3

presents the general HTS model and its GS implementation.The analytical e2e outage probability for the GS policy is alsoderived in Section III. Section IV presents numerical resultsand discusses the system performances of the proposed controlschemes. Finally, Section V summarizes the contributions ofour study.

Notation: ⌊·⌋ is the floor function,FX(·) andFX(·) denotethe cumulative distribution function (CDF) and the comple-mentary CDF (CCDF) of the random variableX , respectively,Γ(·) denotes the gamma function,Γu(·, ·) denotes the upperincomplete gamma function, andKn(·) is the n-th ordermodified Bessel function of the second kind [34, Eq. 8.432].

II. H ARVEST-TRANSMIT MODEL

In this paper, we consider a wireless dual-hop FDR network,in which a source node intends to transfer its information tothedestination node. Due to physical isolation or environmentallimitations between the source and destination, a cooperativerelay is employed to assist information transmission from thesource to the destination. The relay is assumed to be anenergy-selfish or energy-constrained device such that it needsto harvest energy from the source-emitted RF signal to forwardthe source information to the destination. For simplicity ofimplementation, the AF relaying scheme and TS architectureare chosen at the relay. The channels of the source-to-relayand relay-to-destination links are denoted byh1 =

√L1h1

and h2 =√L2h2, respectively, whereLi and hi (i = 1, 2)

are large-scale fading and small-scale fading of the dual-hopchannels, respectively. The large-scale fading is comprisedof the distance-dependent path loss as well as shadowingattenuation, i.e.,

Li ,AiL0Ls

(di/d0)ϕ, (1)

whereAi is the transmit antenna gain of thei-th hop link,L0

is the measured path loss at the reference distanced0, di is thedistance between the transmitter and receiver of thei-th hoplink, ϕ is the path loss exponent, andLs is the shadowingattenuation. To account for the LoS communication setting,the shadowing attenuation is set toLs = 1 in this study. Forthe sake of exposition, the channel gain ofhi is denoted bygi , |hi|2 for i ∈ {1, 2}. The small-scale channel magnitudeof the dual-hop links,|hi|, is modeled as Nakagami-m fadingwith unit mean such thatgi is distributed according to thegamma distribution with shape factormi and scale factorθi ,gimi

, where gi , E{|hi|2} is the average channel gain fori ∈ {1, 2}. Then, the CDF and CCDF ofgi (i = 1, 2) canbe respectively expressed as

Fgi(x) = 1− Γu(mi,x/θi)Γ(mi)

and Fgi(x) =Γu(mi,x/θi)

Γ(mi). (2)

For energy-constrained networks, the CSI can be estimatedvia dedicated reverse-link training from the EH receiver byemploying a two-phase training-transmission protocol [26] andhence, we assume that the relay can access perfect dual-hopCSI in this study.

According to the experimental results of [29], the RSIchannel incident on the receive antenna at a full-duplex nodecan be characterized as Rician. Nevertheless, RSI cannot be

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eliminated completely because of RF impairments [35] and thee2e detection performance of FDR networks is still limitedby RSI [36]. To the best of our knowledge, after severalcomplicated stages of active IC, the distribution of the RSIobserved in the digital-domain is not known in practice [30],[31], [36]. In this study, the RSI channel and RSI channelpower in the digital-domain after active IC are denoted byhr andgr , |hr|2, respectively. Furthermore, the normalizedtransmitted signals of the source and relay are denoted byxs(t) andxr(t), respectively, and the transmission powers atthe source and relay are denoted byps andpr, respectively.

In the HT model, each transmission block with a durationof T is divided into two phases for EH and FDR transmission,respectively. Denoting the TS factor byα (0 < α < 1),the first phase assigned with a duration ofαT is appliedfor power transfer from the source to the relay. The secondphase assigned with the remaining duration of(1−α)T isused for FDR transmission. The relay-received RF signals inthe two phases are sent to the EH receiver and informationprocessing (IP) receiver, respectively, as illustrated inFig. 1.With respect to the EH receivers’s sensitivitySmin [37], apiecewise behavior is assumed in the HT model, i.e., the EHand IP receivers at the relay are activated only when

psg1 ≥ Smin, (3)

otherwise the EH and IP receivers keep silent. When (3) issatisfied, the harvested energy at the relay can be expressedas

Eh = ηhpsg1αT, (4)

whereηh is the energy conversion efficiency depending on therectifier circuit [37]. By utilizing the harvested energy inthefirst phase, the relay transmission power in the second phaseis given by

pr =ηtEh

(1− α)T, κpsg1, (5)

whereηt ∈ (0, 1) is the energy utilization efficiency andκ ,αηhηt

1−α . For the sake of exposition, we assume a normalizedblock duration in the sequel and hence, we can use the termspower and energy interchangeably.

In the FDR mode, the relay concurrently receives the signalyr(t) and transmits the signalxr(t) on the same frequency,as depicted in Fig. 1, where⌣n

[r]

a (t) and ⌢n

[r]

a (t) are narrow-band Gaussian noises introduced by the receive and transmit

4

antennas, respectively. In addition,⌣n

[r]

c (t) and ⌢n

[r]

c (t) arebaseband additive white Gaussian noises (AWGNs) causedby down-conversion and up-conversion, respectively [1]. Forsimplicity, the equivalent baseband noise comprising both⌣n

[r]

a (t) and ⌢n

[r]

a (t) is modeled by the zero mean AWGNn[r]a (t) with varianceσ2

a, and the equivalent baseband noisecomprising both⌣

n[r]

c (t) and ⌢n

[r]

c (t) is modeled by the zeromean AWGN n[r]

c (t) with variance σ2c . Then, the overall

AWGN at the relay can be modeled as the zero mean AWGNnr(k) , n[r]

a (k) + n[r]c (k) with varianceσ2

r , σ2a + σ2

c . Thesampled baseband signal after some stages of IC is given by

yr(k) =√psh1xs(k) + hrxr(k) + nr(k), (6)

where k denotes the symbol index,xs(k) and xr(k) arethe sampled signals ofxs(t) and xr(t), respectively. Thetransmitted signal in (6) can be expressed as

xr(k) =√

βyr(k − τ), (7)

where τ ≥ 1 is the processing delay at the relay andβ =(psg1 + prgr + σ2

r )−1 is the normalization coefficient. The

sampled received signal at the destination is given by

yd(k) =√prh2xr(k) + nd(k), (8)

wherend(k) is the additive noise at the destination with zeromean and varianceσ2

d. In this network, the e2e SINR can beexpressed as

γe2e =γ

Rγ

D

γR+ γ

D+ 1

, (9)

whereγR ,psg1

prgr+σ2r

and γD ,prg2σ2d

are the SINRs at therelay and destination, respectively.

In each transmission block, an outage event occurs whenthe power of the ambient RF signal at the relay is less thanthe EH receiver sensitivity or when the e2e SINR is less thanthe required threshold for correct data detection given that EHis successful. Therefore, the e2e outage probability in theHTmodel can be expressed as

Pout , Pr {psg1 < Smin}+Pr {(psg1 ≥ Smin) ∩ (γe2e < γth)} , (10)

whereγth is the e2e SINR threshold for correct data detectionat the destination. Note that in conventional AF relay networkswithout EH, the information outage probability is defined as

P Iout , Pr{γe2e < γth}. (11)

Then, the average throughput can be expressed as

RHT

= (1− α)(1 − Pout)R, (12)

where R , log2(1 + γth) is the fixed transmission rate.The design goal is to maximize the average throughput byoptimizing TS subject to QoS constraints. The optimalα∗ canbe obtained by solving the following maximization problem:

α∗ = argmaxα

RHT

(α), s.t. 0 < α < 1 and Qi, (13)

where Qi ∈ {Q1, Q2} is a QoS constraint as we willexplain later. Notably, without considering a QoS constraint,the achievable maximum average throughput of (13) may

be obtained by a large1 − α at the cost of a largeP Iout,

which greatly degrades the system performance due to a largeamount of repetition transmissions [27], [38]. Therefore,theQoS requirements of decreasing outage probability [38] andmaintaining a target outage probability [27] are respectivelyconsidered, i.e.,

Q1 := {P Iout is minimized} and Q2 := {P I

out ≤ ε}, (14)

whereε is a given target information outage probability [27].To obtainα∗, the immediate task is to characterize the e2eoutage probability of the system. For the sake of mathematicaltractability, we focus on the RSI dominated scenario which isof practical interest [12], [14].

Proposition 1. Conditioned on gr, the e2e outage probabilityof the system in the HT model is given by

Pout = 1− (1− P Iout)Fg1

(

Smin

ps

)

, (15)

where

P Iout =

{

22−m1−m2

Γ(m1)Γ(m2)Dµ,ν(2

√ξ), 0 < α < 1

1+ηgrγth

1, 11+ηgrγth

≤ α < 1, (16)

ξ ,γthσ

2d(1 + κgr)

κpsθ1θ2(1− κγthgr), (17)

µ = m1+m2− 1, ν = m1−m2, Dµ,ν(y) =∫ y

0xµKν(x)dx,

and Fg1 (x) ,Γu(m1,x/θ1)

Γ(m1)is the CCDF of g1.

Proof. See Appendix A.

Proposition 1 shows that when 11+ηgrγth

≤ α < 1, Pout =

1. To avoidPout = 1, it is required to set0 < α < 11+ηgrγth

or equivalently to eliminate the RSI bygr < 1κγth

. Moreover,

asps → ∞, we haveFg1

(

Smin

ps

)

→ 1 and

Pout = P Iout →

{

0, 0 < α < 11+ηgrγth

1, 11+ηgrγth

≤ α < 1. (18)

Thus,P Iout → 0 and Pout → 0 can be achieved by setting

α ∈ (0, 11+ηgrγth

) asps → ∞. Furthermore, asps → ∞ andgr → 0, we can achieveP I

out → 0 andPout → 0 by settingα ∈ (0, 1). Recalling that the effective FDR transmission timeis 1 − α, α should be set as small as possible to achieve theallowable maximum average throughput. Thus, it can be shownthat the optimal solution of (13) with the QoS constraintQ1

is α∗ → 0 asps → ∞.With the obtainedP I

out of (16), maximizing1 − Pout isequivalent to maximizing1 − P I

out. Then, the maximizationproblem (13) can be simplified as

α∗ = argmaxα

(1− α)(1 − P Iout), (19a)

s.t. 0 < α <1

1 + ηgrγthand Qi. (19b)

Since outage probability and successful rate are two im-portant metrics in delay-limited transmissions [27], [39], wealso consider the successful rate for the FDR-assisted SWIPTnetwork. According to [27], the successful rate is defined as

5

the product of the fixed transmission rate and the success prob-ability. With respect to TS operation, we define the successfulrate for the considered FDR-assisted SWIPT network as

R(s)HT

,

{

(1 − α)(1 − ε)R, Pout ≤ ε0, otherwise,

(20)

where we have assumed that the physical-layer outage is fixedto the target outage probability [27]. Obviously, the successfulrate is a strictly QoS-dependent metric, which represents theaverage throughput subject to an explicitly defined outageconstraint. To achieve a non-zero successful rate, the obtainedP Iout subject to the outage constraint should be less than or

equal toε whenpsg1 ≥ Smin. In the following, the optimal TSfactors of (19) subject toQ1 andQ2 are respectively presented.

Proposition 2. The optimal TS factor that maximizes theaverage throughput subject to Q1 in the HT model is given by

α =1

1 + ηgr(√

γth(γth + 1) + γth)(21)

and the corresponding information outage probability is givenby

P Iout =

22−m1−m2

Γ(m1)Γ(m2)Dµ,ν

(

2

√

ξ

)

, (22)

where

ξ ,grγthσ

2d

(

2(

γth +√

γth(γth + 1))

+ 1)

psθ1θ2. (23)

Proof. See Appendix B.

Note thatα can be easily computed since it is independentof the CSI of the dual-hop channels. With the obtainedα andP Iout, the average throughput can be expressed as

RHT =ηgr(

√

γth(γth + 1) + γth)(1 − P Iout)Fg1

(

Smin

ps

)

R

1 + ηgr(√

γth(γth + 1) + γth).

(24)

As ps → ∞, we haveP Iout → 0, Fg1

(

Smin

ps

)

→ 1, and theasymptotic average throughput

R∞HT

→ ηgr(√

γth(γth + 1) + γth)R

1 + ηgr(√

γth(γth + 1) + γth). (25)

In such a case, the asymptotic average throughput is inde-pendent of the CSI of the dual-hop channels and is a mono-tonically increasing function ofgr. Thus, a larger asymptoticaverage throughput is obtained with a largergr, which canalleviate the IC burden in the highps region if α is applied.

When the QoS constraintQ2 is considered, we denote byF−1y (ε) the solution2

√ξ to the equationFy

(

2√ξ)

= ε [27],so thatξ = (F−1

y (ε))2/4. Note that we have applied the factP Iout = Fy

(

2√ξ)

from (A.6) to obtainξ; now we begin tocharacterize the optimal TS of (19) subject toQ2.

Proposition 3. For a given target information outage prob-ability ε, the optimal TS factor that maximizes the averagethroughput subject to Q2 in the HT model is given by

α =

{

κη+κ , gr ≤ gr

does not exsit, otherwise,(26)

where gr ,psθ1θ2(F

−1y (ε))2

4γthσ2d

(

2(

γth+√

γth(γth+1))

+1) and κ is given by

(27).

Proof. See Appendix C.

Proposition 3 shows thatα does not exist and the targetinformation outage probability cannot be achieved whengr >gr. In such a case, the corresponding FDR transmission cannotbe realized along with an e2e outage probability of 1. As canbe seen from Appendix C, the information outage probabilityachieved byα is P I

out = ε whengr ≤ gr. With the obtainedα andP I

out = ε, the average throughput given thatgr ≤ grcan be expressed as

RHT =(1− ε)ηRFg1

(

Smin

ps

)

η + κ. (28)

As ps → ∞, we haveκ → 0, Fg1

(

Smin

ps

)

→ 1, gr → ∞,andgr < gr. Thus, the asymptotic average throughput can beexpressed as

R∞HT

→ (1− ε)R. (29)

The expression in (29) implicitly shows thatα → 0 asps → ∞. In other words,α achieves an effective FDRtransmission time of a whole block asps → ∞, so that theasymptotic average throughput of (29) is the same as that ofa corresponding conventional FDR network.

Corollary 1. As ps → ∞, when gr < 1−ε

ηε(√

γth(γth+1)+γth

) ,

R∞HT

< R∞HT

; otherwise, R∞HT

≥ R∞HT

.

Proof. Comparing (25) and (29), it can be shown thatR∞HT

<

R∞HT

has an equivalencegr < 1−ε

ηε(√

γth(γth+1)+γth

) . Further-

more, it can be shown thatR∞HT

≥ R∞HT

has an equivalencegr ≥ 1−ε

ηε(√

γth(γth+1)+γth

) . This proves Corollary 1.

Being consistent with contemporary wireless systems wherean outage level near1% is typical [27], [39], it can beshown that 1−ε

ηε(√

γth(γth+1)+γth

) ≫ 1 with the substitution

of practical η and γth, whereasgr is less than 1 due toactive/passive IC. Thus, we havegr < 1−ε

ηε(√

γth(γth+1)+γth

)

in practice and the asymptotic average throughput achievedby α is larger than that ofα.

III. H ARVEST-TRANSMIT-STORE MODEL

In the HT model, all the harvested energy has been fullyutilized for FDR transmission within each block, without con-sidering energy accumulation and scheduling across channelrealizations. Although the HT model is easy to implement, itwould perform better if energy accumulation and schedulingwere allowed to store a part of the harvested energy for futureusage. Therefore, we propose a general HTS model with itsGS implementation for the considered FDR-assisted SWIPTnetwork. Note that a block-based HTS model has been pro-posed for HDR-assisted SWIPT networks in [32]. In contrast,our general HTS model achieves superior outage performanceover that of the block-based HTS model. Furthermore, under

6

κ =psθ1θ2(F

−1y (ε))2−4grγthσ

2d −

√

(psθ1θ2(F−1y (ε))2−4grγthσ2

d)2 − 16psθ1θ2grγ2

thσ2d(F

−1y (ε))2

2psθ1θ2grγth(F−1y (ε))2

. (27)

! "#$%&'()*+,**,-)

T ./ 0T !

! !

"#$%&'()*+,**,-)

!

./0 .10. 0 %()2% . 0h r

t t" " " "# #

./0 .10. 0 . 0h

t t" " "# #

.(0 3 %4 / $ $

.50 3 % 4 #

T

. 0r

t" "#

. 0h

t" "#

Fig. 2. Frame structure of the general HTS model: (a)0 < α < 1 and (b)α = 0.

our analysis framework, the block-based HTS model can betreated as the special caseα = 0.

The frame structure of the general HTS model is depictedin Fig. 2(a). In the first phase with a time durationαT , therelay switches to the EH modeµh. In the second phase witha time duration of(1 − α)T , the relay switches to the EHmodeµh or the FDR transmission modeµr depending on thebattery’s residual energy level and CSI. Note that whenα = 0,the frame structure of the general HTS model degenerates tothat of the block-based HTS model, as depicted in Fig. 2(b).For intermediate and high SINRs, the e2e SINR of (9) can beapproximated as [40]:

γe2e

≈ min{γR, γ

D}, (30)

whereγR= psg1

prgris the SINR at the relay in the considered

RSI dominated scenario. In order to decode the relaying signalreceived at the destination, it is required that the e2e SINRat least equals the target valueγth. Based on the aboveapproximation, the required relay transmission power thatensures signal detection can be simplified to

pr=

{

γthσ2d

g2, γ

R≥ γth

does not exist, otherwise,(31)

where γR

,psg1g2grγthσ2

d

denotes the SINR at the relay given

that pr =γthσ

2d

g2holds. Notably, the consumed energy for the

relay transmission is(1 − α)prT . In the following, we againassume the time normalization of each block, so that we canconsider energy and power interchangeably. Furthermore, weassume that a rechargeable battery has been employed at therelay with the battery sizepb = ρps (ρ > 0) . The battery isdiscretized intoL+ 2 energy levelsϕi , ipb/(L+ 1), wherei = 0, 1, . . . , L+1 [32], [41]. We definesi, i = 0, 1, . . . , L+1asL+ 2 energy states of the battery, so that the battery is inthe statesi when its stored energy equals toϕi.

Based on the considered discretized battery model, theenergy that can be harvested during the first phase is definedasϕh1 , ϕi∗

h1, where

i∗h1= arg max

i∈{0,...,L+1}{ϕi : ϕi < ϕh1} (32)

and ϕh1 , αηhpsg1. If the relay is operated in the EH modeµh in the second phase, the energy that can be harvested isdefined asϕh2 , ϕi∗

h2, where

i∗h2= arg max

i∈{0,...,L+1}{ϕi : ϕi < ϕh2} (33)

and ϕh2 , (1 − α)ηhpsg1. When the relay is operated in theFDR transmission modeµr in the second phase, the relay usesthe stored energy to power its transmission. Correspondingtopr of (31), the required energy level for transmission is givenby

ϕr ,

{

ϕi∗r , if (1−α)pr

ηt≤ pb

∞, otherwise,(34)

where i∗r = arg mini∈{1,...,L+1}

{

ϕi : ϕi ≥ (1−α)pr

ηt

}

. In each

block, an outage event occurs when the source signal cannotbe decoded at the destination or equivalently when the relayis operating in the EH modeµh in the second phase. Thus,the main optimization target is to minimize the number oftimes that the relay does not perform FDR transmission inthe second phase. To this end, the GS policy prioritizes theoperation modesµr in the second phase of each block. At thebeginning of the second phase, when the residual energy ofthe battery can support the required transmitted energy, the GSpolicy switches the relay node to FDR transmission; otherwiseit switches the relay node to EH. On denoting the battery’sresidual energy levels at the beginnings of the first and secondphases of thetth block byE1(t) ∈ {ϕi : 0 ≤ i ≤ L+ 1} andE2(t) ∈ {ϕi : 0 ≤ i ≤ L+1}, respectively, the GS policy canbe expressed asµ(1)(t) = µh for the first phase and

µ(2)(t)=

{

µr, (E2(k) ≥ ϕr) ∩ (γR ≥ γth)µh, ((E2(k)<ϕr)∩(γR

≥γth)) ∪ (γR<γth)

(35)

for the second phase, respectively. Furthermore, the battery’sresidual energy levels can be expressed as

E2(t) = min{pb, E1(t) + ϕh1} (36)

and

E1(t+1)=min{pb, E2(t) + wϕh2 + (1−w)ϕr}, (37)

where

w =

{

1, µ(2)(t) = µh

0, µ(2)(t) = µr.(38)

7

A. MC for the GS Policy

For the GS policy, a specific harvesting/relaying behavior ofthe relay’s battery can be modeled as a specific state-transitionof a finite-state MC, so that the energy level of the relay’sbattery at the beginning of each block represents a specificstate of the MC. Since the GS policy switches the relay to themodeµh in the first phase and to the modeµh or µr in thesecond phase, the MC model has a two-stage transition foreach block.

Assume that the initial, intermediate, and final states ofthe battery’s energy level in each block aresi, sk, and sj,respectively, the transitionssi → sk and sk → sj occur inthe first and second phases, respectively, and the transitionsi → sk → sj occurs throughout the whole block. In thefirst phase, we havek ≥ i due to the EH operation. In thesecond phase, we havek > j if the relay is operated in themodeµr or k ≤ j if the relay is operated in the modeµh.It can be shown that an outage event occurs whenk ≤ jfor si → sk → sj and a non-outage event occurs whenk > j for si → sk → sj . The transition matrix of the MCcan be denoted byP ∈ R

(L+2)×(L+2) with its ith-row andjth-column elementPi,j representing the probability of thetransition from the statesi to the statesj in a transmissionblock. Similarly, we defineP (1)

i,k as the transition probability

in the first phase and defineP (2)k,j and P

(2)k,j as the transition

probabilities in the second phase fork ≤ j and k > j,respectively. With respect to the two-stage state transition,Pi,j

can be written as

Pi,j ,∑

k≤j

P(1)i,k P

(2)k,j +

∑

k>j

P(1)i,k P

(2)k,j

, Pi,j + Pi,j , (39)

where Pi,j ,∑

k≤j P(1)i,k P

(2)k,j and Pi,j ,

∑

k>j P(1)i,k P

(2)k,j

are the transition probabilities corresponding to an outageevent and a non-outage event, respectively. Based on (39),the transition probabilities of the MC are determined in thefollowing:

1) The Empty Battery Remains Empty (s0 → sk → s0): Inthis case, an outage event occurs fors0 → s0 → s0 and anon-outage event occurs fors0 → sk → s0 with 1 ≤ k ≤L + 1. Thus, the transition probabilities corresponding to anoutage event and a non-outage event are respectively given byP0,0 = P

(1)0,0 P

(2)0,0 and P0,0 =

∑L+1k=1 P

(1)0,k P

(2)k,0 , where

P(1)0,k =

Pr{ϕh1 < ϕ1}, k = 0Pr{ϕh1 ≥ pb}, k = L+ 1

Pr{ϕk ≤ ϕh1 < ϕk+1}, otherwise

=

Fg1

(

ϕ1

αηhps

)

, k = 0

Fg1

(

pb

αηhps

)

, k = L+ 1

Fg1

(

ϕk+1

αηhps

)

−Fg1

(

ϕk

αηhps

)

, otherwise,

(40)

P(2)0,0 = Pr{ϕh2 < ϕ1} = Fg1

(

ϕ1

(1−α)ηhps

)

, (41)

and

P(2)k,0 =Pr{(ϕk−1 < ϕr ≤ ϕk) ∩ (γ

R≥ γth)}

= Fy

(

grγ2thσ

2d

psθ1θ2

)(

Fg2

(

(1−α)γthσ2d

ηtϕk−1

)

−Fg2

(

(1−α)γthσ2d

ηtϕk

))

. (42)

Then, the transition probability for this case is given byP0,0 =P0,0 + P0,0.

2) The Empty Battery Is Partially Charged (s0 → sk →sj: 0 < j < L + 1): In this case, an outage event occurswhen 0 ≤ k ≤ j and a non-outage event occurs whenj +1 ≤ k ≤ L + 1. The corresponding transition probabilitiesare respectively given byP0,j =

∑jk=0 P

(1)0,k P

(2)k,j and P0,j =

∑L+1k=j+1 P

(1)0,k P

(2)k,j , whereP (1)

0,k is given by (40),

P(2)k,j = Pr {(ϕj − ϕk ≤ ϕh2 < ϕj+1 − ϕk)∩

(((ϕk < ϕr) ∩ (γR≥ γth)) ∪ (γ

R< γth))}

=(

Fg1

(

ϕj+1−ϕk

(1−α)ηhps

)

− Fg1

(

ϕj−ϕk

(1−α)ηhps

))

(

Fg2

(

(1−α)γthσ2d

ηtϕk

)

Fy

(

grγ2thσ

2d

psθ1θ2

)

+Fy

(

grγ2thσ

2d

psθ1θ2

))

, (43)

and

P(2)k,j =Pr{(ϕk − ϕj+1 < ϕr ≤ ϕk − ϕj) ∩ (γ

R≥ γth)}

= Fy

(

grγ2thσ

2d

psθ1θ2

)(

Fg2

(

(1−α)γthσ2d

ηt(ϕk−ϕj+1)

)

−Fg2

(

(1−α)γthσ2d

ηt(ϕk−ϕj)

))

.(44)

Then, the transition probability for this case is given byP0,j =P0,j + P0,j .

3) The Empty Battery Is Fully Charged (s0 → sk → sL+1):This is the scenario in which the empty battery becomes fullycharged at the end of a transmission block. In such a case,the transition probability corresponding to an outage event isgiven by

P0,L+1 =L+1∑

k=0

P(1)0,k P

(2)k,L+1, (45)

whereP (1)0,k is given by (40) andP (2)

k,L+1 is given by

P(2)k,L+1 = Pr {(((ϕk < ϕr) ∩ (γ

R≥ γth)) ∪ (γ

R< γth))∩

(ϕh2 ≥ pb − ϕk)}=

(

Fg2

(

(1−α)γthσ2d

ηtϕk

)

Fy

(

grγ2thσ

2d

psθ1θ2

)

+Fy

(

grγ2thσ

2d

psθ1θ2

))

Fg1

(

pb−ϕk

(1−α)ηhps

)

. (46)

Furthermore, the transition probability corresponding toa non-outage event is given byP0,L+1 = 0 due to the absenceof discharging in the second phase. Then, the transitionprobability of this case can be expressed asP0,L+1 = P0,L+1.

4) The Battery Remains Full (sL+1 → sL+1 → sL+1):This case corresponds to the scenarios in which the batteryis fully charged at the beginning of the first phase, so thatthe battery cannot harvest more energy. In the second phase,a) the required transmitted energy is higher than the batterysize given thatγR ≥ γth or b) γR < γth. In such a case,PL+1,L+1 = P

(1)L+1,L+1P

(2)L+1,L+1, whereP (1)

L+1,L+1 = 1 and

P(2)L+1,L+1 can be obtained by substitutingk = L + 1 and

ϕL+1 = pb into (46). Thus, we have

PL+1,L+1 = Fg2

(

(1−α)γthσ2d

ηtpb

)

Fy

(

grγ2thσ

2d

psθ1θ2

)

+Fy

(

grγ2thσ

2d

psθ1θ2

)

.

(47)

8

Furthermore, we havePL+1,L+1 = 0 due to the absenceof discharging in the second phase. Therefore, the transitionprobability for this case can be expressed asPL+1,L+1 =

P(2)L+1,L+1.5) The Non-Empty and Non-Full Battery Remains Un-

changed (si → sk → si: 0 < i < L + 1): In this case,the transition probability corresponding to an outage event isPi,i = P

(1)i,i P

(2)i,i , where

P(1)i,i = Pr{ϕh1 < ϕ1} = Fg1

(

ϕ1

αηhps

)

(48)

andP (2)i,i is obtained from (43) with the substitutionk = j = i

in it, i.e.,

P(2)i,i = Fg1

(

ϕ1

(1−α)ηhps

)

(

Fg2

(

(1−α)γthσ2d

ηtϕi

)

Fy

(

grγ2thσ

2d

psθ1θ2

)

+Fy

(

grγ2thσ

2d

psθ1θ2

))

.(49)

The transition probability corresponding to a non-outage eventis Pi,i =

∑L+1k=i+1 P

(1)i,k P

(2)k,i , where

P(1)i,k =

{

Pr{ϕh1 ≥ pb − ϕi}, k = L+ 1Pr{ϕk − ϕi ≤ ϕh1 < ϕk+1 − ϕi}, otherwise

=

Fg1

(

pb−ϕi

αηhps

)

, k = L+ 1

Fg1

(

ϕk+1−ϕi

αηhps

)

− Fg1

(

ϕk−ϕi

αηhps

)

, otherwise(50)

andP (2)k,i is obtained from (44) with the substitutionj = i in it.

Then, the transition probability for this case can be expressedasPi,i = Pi,i + Pi,i.

6) The Non-Empty and Non-Full Battery Is Fully Charged(si → sk → sL+1: 0 < i < L + 1): In this case, we havePi,L+1 = 0 due to the absence of discharging. Then, thetransition probability is given by

Pi,L+1 = Pi,L+1 =

L+1∑

k=i

P(1)i,k P

(2)k,L+1, (51)

whereP (1)i,k is given by (50) andP (2)

k,L+1 is the same as thatof the case 3).

7) The Non-Empty and Non-Full Battery Is PartiallyCharged (si → sk → sj: 0 < i < j < L + 1): In thiscase, the transition probabilities corresponding to an outageevent and a non-outage event are respectively given by

Pi,j =

j∑

k=i

P(1)i,k P

(2)k,j and Pi,j =

L+1∑

k=j+1

P(1)i,k P

(2)k,j , (52)

whereP (1)i,k is given by (50),P (2)

k,j is given by (43), andP (2)k,j

is given by (44). Then, the transition probability of this casecan be expressed asPi,j = Pi,j + Pi,j .

8) The Non-Empty Battery Is Discharged (si → sk → sj:0 ≤ j < i ≤ L + 1): In this case, we havePi,j = 0 sincethe battery always discharges with respect toj < i. Thus, thetransition probability can be expressed as

Pi,j = Pi,j =L+1∑

k=i

P(1)i,k P

(2)k,j , (53)

whereP (1)i,k is given by (50) andP (2)

k,j is given by (44).

B. Stationary Distribution and System Performance

In order to explore how the relay’s battery is charged anddischarged according to the GS policy, the stationary distribu-tion of the MC needs to be derived in order to determine thee2e outage probability.

Proposition 4. The state transition matrix P = (Pi,j) of theMC that models the battery’s behavior is irreducible and rowstochastic.

Proof. The proof is similar to that of Proposition 1 of [32].

Since that the transition matrixP is irreducible and rowstochastic, the stationary distribution of the MC can be com-puted by [42]

π = (P T − I +B)−1b, (54)

where the element of the(L + 2) × (L + 2) matrix B inits i-th row andj-th column is given byBi,j = 1 ∀i, j andb = [1, 1, . . . , 1]T is an (L + 2) × 1 vector. With respect tothe two-stage transition in each block, an outage event occurswhen the battery does not discharge in the second phase ofa block, which transits the battery’s state to a non-decreasedenergy level in the second phase, i.e.,k ≤ j for si → sk → sj ,and the corresponding transition probability isPi,j . Therefore,the e2e outage probability achieved by the GS policy can beexpressed as

P(GS)out =

L+1∑

i=0

πi

L+1∑

j=i

Pi,j . (55)

The average throughput and the successful rate achieved bythe GS policy can be respectively expressed as

RGS

= (1 − α)(1 − PGSout)R (56)

and

R(s)GS

=

{

(1− α)(1 − ε)R, P(GS)out ≤ ε

0, otherwise.(57)

By comparing our proposed general HTS model with theblock-based HTS model proposed in [32], it can be shownthat the general HTS model degenerates to the block-basedHTS model whenα = 0. Thus, our analysis framework canbe directly applied to the block-based HTS model for the con-sidered FDR-assisted SWIPT network and the correspondinge2e outage probability can be computed by substitutingα = 0into the above analysis framework. Due to the complicatedcomputation of the stationary distribution, the optimal TSthat maximizes the average throughput subject to the QoSconstraints can hardly be obtained in closed form for thegeneral HTS model. Thus, in maximizing the QoS-constrainedaverage throughput (successful rate) for the general HTSmodel, we chooseα intuitively based on numerical resultsas discussed in the following section.

IV. N UMERICAL RESULTS

This section presents some numerical results to validatethe performance results of the developed schemes. In thesimulations, the EH receiver sensitivity is set asϕmin = −27

9

TABLE ISIMULATION PARAMETERS

No. Parameter Value1 Carrier frequency 868 MHz2 Fixed transmission rateR 3 bps/Hz3 Target information outage probabilityε 1%4 L at d0 = 1 m −30 dB5 Distances of dual-hop links:d1 andd2 8 and 18 m6 Path loss exponentϕ 2.57 Souce/relay transmit antenna gain 18/8 dB8 Additive noise power:σ2

r = σ2

d−90 dBm

9 Energy coefficients:ηh andηt 0.4 and 0.75

dBm [37] and the size of the relay’s battery is set aspb = ρps,where ρ = m1θ1. Consideringϕ1 ≥ ϕmin in practice, weset the actual number of energy levels of the relay’s batteryto L + 2, whereL , min{L, ⌊pb/ϕmin⌋ − 1}. For the sakeof simplicity, we use the terms general HTS model and theGS policy interchangeably in the following. Furthermore, theoptimal solution of (13) without any QoS constraint is denotedby α∗. Unless otherwise stated, the remaining parameters usedin the simulations are given in Table 1.

Fig. 3 investigates the e2e outage probability versusα ofthe considered schemes. In Fig. 3, we setps = 26 dBm,gr = −10 dB, m1 = 4, andm2 = 2. As observed in Fig.3, the e2e outage probability of the HT model first decreaseswith increasingα. After reaching a minimum, the e2e outageprobability of the HT model begins to increase with increasingα. Furthermore, the e2e outage probability exhibits a piecewisebehavior and approaches 1 in the highα region, as indicatedby Proposition 1. Note that the HT model suffers seriousoutage performance degradation in the low and highα regions.As expected, whenα is applied, the achieved e2e outageprobability is Pout = 0.005, which matches the minimumvalue of that of the HT model. Whenα is applied, Fig. 3shows that the target outage probabilityε = 0.01 has beenmaintained. For the general HTS model, Fig. 3 shows thatα = 0 achieves the worst outage performance over all valuesof α. Note thatα = 0 corresponds to the block-based HTSmodel. Asα increases from 0 to the highα region, it can beshown the e2e outage probability first decreases dramaticallyand then approaches an outage floor of 0.003. Notably, thegeneral HTS model achieves the best outage performance overα (note thatα and α are fixed).

The average throughput versusα of the considered schemesis depicted in Fig. 4, where the simulation parameters arethe same as those of Fig. 3. Fig. 4 clearly shows that themaximum average throughput of the HT model is achievedby α∗ = 0.17. Recalling the e2e outage performance in Fig.3, it can be shown that the maximum average throughputof the HT model is achieved withPout ≈ 0.1, so that theoutage performance seriously degrades compared toε = 0.01.Althoughα andα achieve a lower average throughput than themaximum point in the HT model, the corresponding outageperformance is acceptable, as depicted in Fig. 3. Furthermore,α achieves a higher average throughput than that ofα. For thegeneral HTS model, Fig. 4 shows that the maximum averagethroughput is achieved byα = 0. Unfortunately, we know

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

−5

10−4

10−3

10−2

10−1

100

α

Out

age

Pro

babi

lity

HT, anal.

HT, α, anal.

HT, α, anal.

GS, L=30, anal.

GS, L=150, anal.

HT, simul.

HT, α, simul.

HT, α, simul.

GS, L=30, simul.

GS, L=150, simul.

GS, α = 0

HT, fixed α = 0.498

HT, fixed α = 0.697

Fig. 3. Outage probability versusα.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1.0

1.5

2.0

2.5

3.0

α

Ave

rage

thro

ughp

ut (

bps/

Hz)

HT, anal.

HT, α, anal.

HT, α, anal.

GS, L=30, anal.

GS, L=150, anal.

HT, simul.

HT, α, simul.

HT, α, simul.

GS, L=30, simul.

GS, L=150, simul.

HT, fixed α = 0.498

HT, fixed α = 0.697

GS, α = 0

α∗ = 0.17

Fig. 4. Average throughput versusα.

that the corresponding outage performance is not acceptableby recalling the results of Fig. 3, so that the maximum averagethroughput achieved byα = 0 is not useful. Asα increasesfrom 0, the achieved average throughput first decreases slowlyand then maintains a fixed rate of decrease. Notably, the GSimplementation of the general HTS model always achieves ahigher average throughput than that of the HT model for allα.

Fig. 5 shows the successful rate versusα of the consideredschemes corresponding to Fig. 3 and Fig. 4. It is seen fromFig. 5 that the so-called optimalα∗ = 0.17 for the HTmodel achieves a zero successful rate due to the associatedpoor outage performance. Bothα and α achieve a non-zerosuccessful rate. The GS policy achieves the highest successfulrate forα. Furthermore, the non-zero successful rate achievedby the GS policy corresponds to a widerα region than thatof the HT model. Whenα = 0, the GS policy achieves azero successful rate. Therefore, it can be summarized that thegeneral HTS model can maximize the successful rate amongall the considered schemes by setting a non-zero TS factor.

The impact ofgr on the e2e outage probability is depictedin Fig. 6, where we setm1 = 4, m2 = 2, andps = 26 dBm.As observed, the e2e outage probabilities of all the consideredschemes increase with increasinggr. For the HT model, thenumerically optimizedα∗ achieves the worst outage perfor-mance in the low and middlegr regions. Fig. 6 also shows

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

α

Suc

cess

ful r

ate

(bps

/Hz)

HTHT, α = 0.498

HT, α = 0.697

GS, L=150

Fig. 5. Successful rate versusα.

−50 −45 −40 −35 −30 −25 −20 −15 −10 −5 010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

gr (dB)

Out

age

prob

abili

ty

HT, α∗, anal.

HT, α, anal.

HT, α, anal.

GS, α = 0, L=150, anal.

GS, α = 0.15, L=150, anal.

HT, α∗, simul.

HT, α, simul.

HT, α, simul.

GS, α = 0, L=150, simul.

GS, α = 0.15, L=150, simul.

Fig. 6. Outage probability versusgr.

that the e2e outage probability achieved byα experiences apiecewise behavior, as suggested by Proposition 3. For the HTmodel, the best e2e outage performance is achieved byα. Inthe low gr region,α results in an outage floor due to the factthat α → 1 asgr → 0. For the HTS model, the GS policy withα = 0 achieves a poor outage performance. Furthermore, inthe highpr region, the GS policy withα = 0.15 achieves thebest outage performance among all the considered schemes.

The average throughput versusgr is depicted in Fig. 7, inwhich the simulation parameters are the same as those of Fig.6. As observed in Fig. 7, except forα, the average throughputsachieved by all the schemes decrease with increasing ofgr. Forα, the obtained average throughput increases with increasinggr, which is consistent with the asymptotic analysis of theaverage throughput of Proposition 2. For the HT model,althoughα∗ achieves the highest average throughput, we knowthat it has poor outage performance by recalling the resultsofFig. 6. Fortunately, the GS policy achieves a higher averagethroughput than that of the HT model for allgr. Note that theaverage throughput achieved by the GS policy withα = 0 ishigher than that of the GS policy withα = 0.15, whereas thecorresponding outage performance is worse than that of thelatter.

Fig. 8 shows the successful rate versusgr of the consideredschemes corresponding to Fig. 6 and Fig. 7. Fig. 8 showsthat the so-called optimalα∗ for the HT model achieves a

−50 −45 −40 −35 −30 −25 −20 −15 −10 −5 00

0.5

1.0

1.5

2.0

2.5

3.0

gr (dB)

Ave

rage

thro

ughp

ut (

bps/

Hz)

HT, α∗, anal.

HT, α, anal.

HT, α, anal.

GS, α = 0, L=150, anal.

GS, α = 0.15, L=150, anal.

HT, α∗, simul.

HT, α, simul.

HT, α, simul.

GS, α = 0, L=150, simul.

GS, α = 0.15, L=150, simul.

Fig. 7. Average throughput versusgr.

−50 −40 −30 −20 −10 0−0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

gr (dB)

Suc

cess

ful r

ate

(bps

/Hz)

HT, α∗

HT, α

HT, α

GS, α= 0.15, L=150

Fig. 8. Successful rate versusgr.

zero successful rate. In the low and middlegr regions, thesuccessful rate achieved byα is higher than that ofα. In thehigh gr region, bothα and α achieve a zero successful rate.Notably, the GS policy withα = 0.15 achieves the highestsuccessful rate among all the considered schemes. Only whengr becomes larger than -2 dB, does the GS policy withα =0.15 achieve a zero successful rate.

Fig. 9 shows the e2e outage probability versusps. In Fig.9, we setm1 = 4, m2 = 2, andgr = −10 dB. As observed,the piecewise behavior occurs for the e2e outage probabilityachieved byα. For the HT model, the best outage performanceis achieved byα for the considered values ofps. For the HTSmodel, the GS policy withα = 0 approaches an outage floorwith increasingps. Moreover, the best outage performance isachieved by the GS policy withα = 0.15 in the middle andhigh ps regions among all the considered schemes.

The average throughput versusps is shown in Fig. 10,where the simulation parameters are the same as those of Fig.9. As observed in Fig. 10,α∗ achieves the highest averagethroughput among all the schemes in the HT model. However,the highest average throughput achieved byα∗ also results ina poor outage performance as depicted in Fig. 9. As expectedfrom Corollary 1, Fig. 10 shows thatα achieves a higheraverage throughput than that ofα in the highps region. Forthe general HTS model, Fig. 10 also shows that although theGS policy withα = 0 achieves the highest average throughput,

11

10 15 20 25 30 35 4010

−5

10−4

10−3

10−2

10−1

100

ps (dBm)

Out

age

prob

abili

ty

HT, α∗, anal.

HT, α, anal.

HT, α, anal.

GS, α = 0, L=150, anal.

GS, α = 0.15, L=150, anal.

HT, α∗, simul.

HT, α, simul.

HT, α, simul.

GS, α = 0, L=150, simul.

GS, α = 0.15, L=150, simul.

Fig. 9. Outage probability versusps.

10 15 20 25 30 35 40 420

0.5

1.0

1.5

2.0

2.5

3.0

ps (dBm)

Ave

rage

thro

ughp

ut (

bps/

Hz)

HT, α∗, anal.

HT, α, anal.

HT, α, anal.

GS, α = 0, L=150, anal.

GS, α = 0.15, L=150, anal.

HT, α∗, simul.

HT, α, simul.

HT, α, simul.

GS, α = 0, L=150, simul.

GS, α = 0.15, L=150, simul.

Fig. 10. Average throughput versusps.

the corresponding outage performance degenerates seriously,as depicted in Fig. 9. Notably, in the low and middlepsregions, the GS policy withα = 0.15 achieves a higheraverage throughput than all the schemes of the HT model.

Fig. 11 shows the successful rate versusps of the consideredschemes corresponding to Fig. 9 and Fig. 10. Fig. 11 showsthat the so-called optimalα∗ for the HT model achieves azero successful rate in most of the consideredps region. Forthe HT model,α always achieves a higher non-zero successfulrate than that ofα for the considered range ofps. Furthermore,the GS policy achieves a higher non-zero successful rate thanthat of α in the middleps region. However, in the highpsregion,α achieves a higher non-zero successful rate than thatof the GS policy. The reason for this phenomenon is that weset the fixedα = 0.15 for the GS policy intuitively, whereasα becomes smaller with increasingps.

V. CONCLUSION

This paper has studied QoS-constrained relay control for anFDR-assisted SWIPT network in terms of the e2e outage prob-ability, average throughput, and successful rate. Conditionedon the RSI channel power, the e2e outage probability hasbeen derived for the HT model. Subject to the QoS-constraintsof minimizing outage probability and maintaining a targetoutage probability, two optimal TS factors that maximize theaverage throughput have been respectively presented for the

10 15 20 25 30 35 40 42−0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

ps (dBm)

Suc

cess

ful r

ate

(bps

/Hz)

HT, α∗

HT, α

HT, α

GS, α= 0.15, L=150

Fig. 11. Successful rate versusps.

HT model. To enable energy accumulation and schedulingacross channel realizations, the general HTS model has beenproposed by employing TS within each transmission block. Toanalyze the e2e outage probability of the general HTS model,the residual energy levels of the relay’s battery have beenmodeled as an MC with a two-stage state transition. Basedon this uniform framework, the block-based HTS model canbe analyzed as a special case of the general HTS model. Thepractical significance of the proposed QoS-constrained controlschemes over the HT model without the QoS-constraints andthe block-based HTS model has been verified by numericalresults.

APPENDIX A: A PROOF OFPROPOSITION1

The e2e outage probability can be rewritten as

P(HT)out = Pr

{

g1 < Smin

ps

}

+Pr{

(g1 ≥ Smin

ps) ∩ (γe2e < γth)

}

. (A.1)

Sinceg1 follows the gamma distribution, we have

Pr{

g1 < Smin

ps

}

= Fg1

(

Smin

θ1

)

= 1−Γu

(

m1,Smin

psθ1

)

Γ(m1)(A.2)

and

Pr{

g1 ≥ Smin

ps

}

= Fg1

(

Smin

θ1

)

=Γu

(

m1,Smin

psθ1

)

Γ(m1). (A.3)

Then, the task is to evaluateP Iout = Pr{γ

e2e< γth}. By

substituting (9) intoP Iout = Pr {γ

e2e< γth}, conditioned on

the RSI channel power,P Iout can be expressed as

P Iout =

{

Pr{

g1g2 <γthσ

2d(1+κgr)

κps(1−κγthgr)

}

, gr < 1κγth

1, gr ≥ 1κγth

. (A.4)

Define xi , gi/θi for i = 1, 2 and y , x1x2. Notably, xi

(i = 1 and 2) has the standard gamma distribution andy isthe product of two independent gamma variables. By utilizingthe result of [43], the CDF ofy can be shown as

Fy(y) =22−m1−m2

Γ(m1)Γ(m2)Dµ,ν(2

√y), (A.5)

12

whereµ = m1 + m2 − 1, ν = m1 − m2, andDµ,ν(y) =∫ y

0 xµKν(x)dx. Then, conditioned ongr, P Iout can be further

expressed as

P Iout =

{

Pr{

y <γthσ

2d(1+κgr)

κpsθ1θ2(1−κγthgr)

}

, gr <1

κγth

1, gr ≥ 1κγth

=

{

Fy

(

γthσ2d(1+κgr)

κpsθ1θ2(1−κγthgr)

)

, 0 < α < 11+ηgrγth

1, 11+ηgrγth

≤ α < 1

=

{

22−m1−m2

Γ(m1)Γ(m2)Dµ,ν(2

√ξ), 0 < α < 1

1+ηgrγth

1, 11+ηgrγth

≤ α < 1,(A.6)

whereξ ,γthσ

2d(1+κgr)

κpsθ1θ2(1−κγthgr). By substituting (A.2), (A.3), and

(A.6) into Pout = Pr{

g1 < Smin

ps

}

+ Pr{

g1 ≥ Smin

ps

}

P Iout,

we arrive at (15).

APPENDIX B: A PROOF OFPROPOSITION2

From (A.6), we know thatP Iout = 1 when α ∈

[

11+ηgrγth

, 1)

. Whenα ∈(

0, 11+ηgrγth

)

, P Iout = Fy(2

√ξ),

which is a non-decreasing CDF ofy with respect to2√ξ,

whereξ is given by (17). Thus, minimizingP Iout is equivalent

to minimizingξ with respect toα ∈(

0, 11+ηgrγth

)

. Since there

exists a one-to-one mapping fromα ∈ (0, 1) to κ ∈ (0,∞),

minimizing ξ with respect toα ∈(

0, 11+ηgrγth

)

is equivalent

to minimizing ξ with respect toκ ∈(

0, 1grγth

)

.The second order derivative ofξ with respect toκ can be

expressed as

∂2ξ

∂κ2= −2γthσ

2d(1− 3grκγth + 3g2rκ

2γ2th + g3rκ

3γ2th)

κ3psθ1θ2(−1 + grκγth)3. (B.1)

Based on the fact thatκ ∈(

0, 1grγth

)

, we have−1+grκγth <

0 and 1 − 3grκγth + 3g2rκ2γ2

th + g3rκ3γ2

th > 0. Thus, itcan be shown that∂

2ξ∂κ2 > 0 and ξ is a convex function of

κ ∈(

0, 1grγth

)

. By solving ∂ξ∂κ = 0 with respect toκ, the

achievable minimumξ is obtained by

κ =1

grγth

(

√

γth(γth + 1)− γth

)

. (B.2)

Then, by substituting (B.2) intoα = κη+κ , we have

α =1

1 + ηgr(√

γth(γth + 1) + γth), (B.3)

which is the TS factor that achieves the minimum informationoutage probability. Due to the uniqueness ofα, it is alsothe TS factor that achieves the allowable maximum averagethroughput. By substituting (B.3) into (16), the achievedminimum information outage probability can be expressed as

P Iout =

22−m1−m2

Γ(m1)Γ(m2)Dµ,ν

(

2

√

ξ

)

, (B.4)

where

ξ ,grγthσ

2d

(

2(

γth +√

γth(γth + 1))

+ 1)

psθ1θ2. (B.5)

!

!

!"#

! "#

$ %

$ %d r

s r

g

p g

" #

$ $ "!

%

&'

!$ $ %%

&

yF (!

&

'

) *' !

(

Fig. C.1. An illustration ofξ versusκ.

APPENDIX C: A PROOF OFPROPOSITION3

Following the procedures at the beginning of AppendixB, we choose to optimizeκ ∈

(

0, 1grγth

)

with its one-to-

one mapping toα ∈(

0, 11+ηgrγth

)

to maximize the averagethroughput subject toQ2. Based on the results from (B.1)to (B.5), we know thatξ =

γthσ2d(1+κgr)

κpsθ1θ2(1−κγthgr)is a convex

function of κ ∈(

0, 1grγth

)

and the achievable minimumξ is

given by (B.5). Note thatQ2 := {P Iout = Fy(2

√ξ) ≤ ε} can

be rewritten asQ2 :={

ξ ≤ (F−1y (ε))2

4

}

, which can be satisfied

only whenξ ≤ (F−1y (ε))2

4 , i.e., gr ≤ gr, where

gr ,psθ1θ2(F

−1y (ε))2

4γthσ2d

(

2(

γth+√

γth(γth+1))

+1) . (C.1)

Moreover, whengr > gr, we haveξ >(F−1

y (ε))2

4 , so thatQ2

cannot be satisfied.As illustrated in Fig. C.1, whengr ≤ gr, there are

two cross-over pointsκ1 and κ2 betweenξ =(F−1

y (ε))2

4

and ξ =γthσ

2d(1+κgr)

κpsθ1θ2(1−κγthgr)due to the convexity ofξ =

γthσ2d(1+κgr)

κpsθ1θ2(1−κγthgr)with respect toκ. Without loss of generality,

we chooseκ1 ≤ κ2, where the equality holds whengr = gr, sothat the QoS constraintQ2 is satisfied byκ ∈ [κ1, κ2]. Whenthe target information outage probability can be achieved,theaverage throughput can be approximated as [27]

RHT = (1− α)(1 − ε)Fg1

(

Smin

ps

)

R. (C.2)

The above average throughput is maximized with the allow-able minimumα (or equivalently the allowable minimumκ)that satisfiesQ2. Thus, the average throughput is maximizedby κ of (27), which is obtained asκ = κ1 by solving

ξ =γthσ

2d(1+κgr)

κpsθ1θ2(1−κγthgr)=

(F−1y (ε))2

4 . Then, the TS factorthat maximizes the average throughput subject toQ2 can beexpressed as

α =

{

κη+κ , gr ≤ gr

does not exsit, otherwise.(C.3)

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Hongwu Liu received the Ph.D. degree from South-west Jiaotong University in 2008. From 2008 to2010, he was with Nanchang Hangkong Univer-sity. From 2010 to 2011, he was a Post-DoctoralFellow with the Shanghai Institute of Microsystemand Information Technology, Chinese Academy ofScience. From 2011 to 2013, he was a ResearchFellow with the UWB Wireless CommunicationsResearch Center, Inha University, South Korea. Heis currently an Associate Professor with ShandongJiaotong University. His current research interests in-

clude MIMO signal processing, cognitive radios, cooperative communications,wireless secrecy communications, and future Internet of Things.

14

Kyeong Jin Kim (SM’11) received the M.S. de-gree from Korea Advanced Institute of Science andTechnology, Daejeon, South Korea, in 1991, and theM.S. and Ph.D. degrees in electrical and computerengineering from the University of California, SantaBarbara, CA, USA in 2000. From 1991 to 1995, hewas a Research Engineer with the Video ResearchCenter of Daewoo Electronics, Ltd., South Korea. In1997, he joined the Data Transmission and Network-ing Laboratory, University of California at SantaBarbara. After receiving his degrees, he joined the

Nokia research center and Nokia Inc., Dallas, TX, USA, as a Senior ResearchEngineer, where he was an L1 specialist, from 2005 to 2009. From 2010 to2011, he was an Invited Professor with Inha University, Incheon, South Korea.Since 2012, he has been working as a Senior Principal Research Scientistwith the Mitsubishi Electric Research Laboratories, Cambridge, MA. Hiscurrent research interests include transceiver design, resource management,scheduling in the cooperative wireless communications systems, cooperativespectrum sharing system, physical layer secrecy system, and device-to-devicecommunications.

Dr. Kim currently serves as an Editor of the IEEE TRANSACTIONS

ON COMMUNICATIONS. He served as an Editor of the IEEE COMMU-NICATIONS LETTERS and INTERNATIONAL JOURNAL OF ANTENNAS AND

PROPAGATION. He also served as a Guest Editor of the EURASIP JOURNALON WIRELESS COMMUNICATIONS AND NETWORKING Special Issue onCooperative Cognitive Networks and IET COMMUNICATIONS Special Issueon Secure Physical Layer Communications.

Kyung Sup Kwak received the Ph.D. degree fromthe University of California at San Diego. He workedfor Hughes Network Systems, and IBM NetworkAnalysis Center, USA. Since then he has beenwith the School of Information and CommunicationEngineering, Inha University, South Korea, as a pro-fessor, and served the dean of the Graduate Schoolof Information Technology and Telecommunicationsand the director of UWB Wireless CommunicationsResearch Center, an IT research center, South Korea,since 2003. In 2006, he served as the President

of Korean Institute of Communication Sciences (KICS), and in 2009, thePresident of Korea Institute of Intelligent Transport Systems. He has authoredor co-authored over 200 peer-reviewed journal papers and served as TPCand Track chairs/organizing chairs for several IEEE related conferences. Hiscurrent research interests include multiple access communication systems,mobile and UWB radio systems, future Internet of Things, andwireless bodyarea network: nano network and Molecular Communications. In 1993, hereceived the Engineering College Achievement Award from Inha Universityand the Service Award from the Institute of Electronics Engineers of Korea, in1996 and 1999, respectively, and distinguished service awards from the KICS.He received the LG Paper Award in 1998 and the Motorola Paper Award in2000. He received official commendations for UWB radio technology researchand development from Minister of Information and Communication, PrimeMinister, and President of Korea, in 2005, 2006, and 2009. In2007, hereceived the Haedong Paper Award and in 2009, the Haedong Scientific Awardof research achievement. In 2008, he was elected an Inha Fellow Professorand he is currently an Inha Hanlim Fellow Professor.

H. Vincent Poor (S72-M77-SM82-F87) received thePh.D. degree in EECS from Princeton University in1977. From 1977 until 1990, he was on the facultyof the University of Illinois at Urbana-Champaign.Since 1990, he has been on the faculty at Princeton,where he is currently the Michael Henry StraterUniversity Professor of Electrical Engineering. From2006 to 2016, he served as the Dean of Princeton’sSchool of Engineering and Applied Science. Hisresearch interests are in the areas of informationtheory, statistical signal processing and stochastic

analysis, and their applications in wireless networks and related fields such assmart grid and social networks. Among his publications in these areas is thebook Mechanisms and Games for Dynamic Spectrum Allocation (CambridgeUniversity Press, 2014).

Dr. Poor is a member of the National Academy of Engineering andthe National Academy of Sciences, and is a foreign member of the RoyalSociety. He is also a fellow of the American Academy of Arts and Sciences,the National Academy of Inventors, and other national and internationalacademies. He received the Marconi and Armstrong Awards of the IEEECommunications Society in 2007 and 2009, respectively. Recent recognitionof his work includes the 2016 John Fritz Medal, the 2017 IEEE AlexanderGraham Bell Medal, Honorary Professorships at Peking University andTsinghua University, conferred in 2016, and a Doctor of Science honoriscausa from Syracuse University in 2017.