arooj mubashara siddiqui, leila musavian, sonia aissa and qiang nirepository.essex.ac.uk › 20516...

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL..., NO..., AUGUST 2017 1

Performance Analysis of Relaying Systemswith Fixed and Energy Harvesting Batteries

Arooj Mubashara Siddiqui, Leila Musavian, Sonia Aissa and Qiang Ni

Abstract—This paper focuses on the performance evaluation ofan energy harvesting (EH) equipped dual-hop relaying system forwhich the end-to-end signal-to-noise ratio (SNR) and the overallsystem throughput are analysed. The transmitter and the relaynodes are equipped with both fixed and EH batteries. The sourcefor harvesting at the transmitter is the solar energy, and at therelay node, the interference energy in the radio frequency is theharvesting source. Time switching scheme is used at the relayto switch between EH and decoding information. Harvest-useapproach is implemented, and we investigate the effects of theharvesting energy in enhancing the performance of the relayingsystem by deriving estimated closed-form expressions for thecumulative distribution function of each link’s individual SNRand of the end-to-end SNR. The analytical expression for theergodic capacity is also derived. These expressions are validatedthrough Monte-Carlo simulations. It is also shown that with theadditional EH at the transmitter (source and relay), a significantimprovement in the system throughput can be achieved whenfixed batteries are running on low powers.

Index Terms—EH, radio frequency EH, solar EH, relaying,time switching, cumulative distribution function.

I. INTRODUCTION

The continuously growing demand for higher data ratesand the increase in the number of mobile devices have ledto a rapid growth in the data traffic and communicationsinfrastructure. This excessive demand in communicating datarequires increasing energy consumption, which in turn, resultsin higher greenhouse gas emission, higher pollution and higherenergy costs [1]. On the other hand, while the wireless trafficis increasing rapidly, battery capacity is still limited. In fact,the battery advancement has been much slower than the needfor long-life batteries. Hence, the gap between increasing therate demands and the battery improvement had been widenedover the last few years [1]. 1

In addition to the battery limitation, according to climategroup SMART (2020) [2], higher greenhouse gas emissionshave increased the carbon footprint to 349Mt and the elec-trical energy consumption to 1700TWh, which could result in

A. M. Siddiqui is with Department of Electrical Engineering, COMSATSInstitute of Information Technology, Islamabad, 44000, Pakistan; Email: [email protected]. L. Musavian is with the School of Computer Scienceand Electronic Engineering, University of Essex, Colchester, CO4 3SQ; Email:[email protected]. S. Aissa is with the Institute National de laRecherche Scientifique (INRS-EMT), University of Quebec, Montreal, QC,Canada, H5A 1K6; Email: [email protected]. Q. Ni is with the School ofComputing and Communications, InfoLab 21, Lancaster University, UK, LA14WA; Email: [email protected].

1This work was supported by the UK EPSRC under Grant EP/N032268/1.

loosing at least 5% of the global gross domestic product (GDP)every year. Due to the increase in the carbon-dioxide emission,limited battery life advancements, and the operational costs,several projects started to look for solutions to reduce the highenergy consumption of communication networks [3], e.g., en-ergy aware radio and network technologies (EARTH) [4] andtowards real energy-efficient network design (TREND) [5].There is an increasing interest in the academics and relatedindustries to design and develop higher data rate devices whilelowering the device energy consumption. As discussed in [6],and references therein, there are four key trade-offs betweenenergy efficiency, spectrum efficiency, latency and deploymentcosts. In this regard, green wireless communication has apotential to provide possible solutions to the current energylimitation problems in the information and communicationtechnology (ICT) sector [7].

A. Motivation and Related Works

Referring to the famous Shannon capacity formula, whenthe transmission power increases, the received signal-to-noiseratio (SNR) improves, and in turn, the link capacity in-creases [8]. The capacity may also increase when the distancebetween the communicating nodes decreases, which will ef-fectively reduce the path loss [9]. Employing a relaying nodebetween the transmitter and its end receiver, which reduces thedistance between the communicating nodes, could improve thecapacity. On the other hand, to increase the transmission powerwithout spending more from the device fixed battery, one canemploy an additional energy harvesting (EH) battery [10]. Byimplementing EH at the source transmitter and at the relay, thetransmission rate can be improved while limiting the energyconsumption from the fixed batteries [11].

EH has indeed the potential to prolong the lifetime andimprove the performance of energy limited networks, e.g.,wireless sensor networks (WSNs) [12], [13]. This technologynot only promises to resolve the limited battery issues, butalso to reduce the carbon footprint of high data rate wirelessdevices, by reusing the energy from the surrounding environ-ment [13]. Energy can be harvested from natural sources suchas solar, thermal, wind and kinetic energy [1]. One of the mostcommonly used ambient energy sources is the solar energy,since light can be directly converted into electricity that runsa wide range of portable devices [14]. The solar energy is byfar the largest and the most widely available source among the




renewable energy sources. In a cellular network for example,the transmitter at the base station can have high solar energypanels [15].

Apart from conventional renewable sources, energy can alsobe harvested from ambient energy in the radio frequency(RF) [16]. The RF interference was used as a source of energyin many recent works [17]–[19]. In particular, the use of RFharvesting got popular due to the fact that it is autonomousand does not depend on dedicated energy sources [14], [16].Among modern technologies manufacturers, we note AmbientMicro and Maxim who have developed methods for combiningmultiple EH resources [20]. Also, Powercast powerharvesterdevices convert the radio waves into DC power in order toincrease the life-cycle of devices with low-power consump-tion [21].

On the other hand, due to practical circuit limitations, itis not possible to harvest energy and decode informationat the same time [15], [22]. From an architectural point ofview, there are two protocols for EH and wireless informationtransmission, e.g., time switching (TS) and power splitting(PS) [23]. In the former, the device switches over time to eitherharvest energy or to transmit information data. In PS, on theother hand, the receiver splits the received signal power intotwo parts, one for processing the information and the other forEH [23].

A Solar EH point-to-point communication system was con-sidered in [24] wherein the rate of the system was evaluated.A water-filling algorithm was proposed for improving thethroughput of the communication links with harvest-store-use EH limited battery devices in [10]. Directional water-filling algorithms to maximize throughput in EH networkswere later proposed in [25]. Optimum transmission policiesunder the constraints on energy causality condition of a fixedEH battery to maximise the throughput was studied in [26]. Acomprehensive receiver architecture and rate-energy tradeoffanalysis of an EH relaying system was presented in [27].

The combination of relaying and EH technology was shownuseful in increasing the battery ife of relays in [23], indeploying WSNs in remote areas [28], and in multiple-inputmultiple-output (MIMO) networks [29]. Also, optimal networkbeamforming in collaborative relaying systems that uses bothRF and solar EH was proposed in [35]. Optimal schedulingand power allocation for a two-hop relaying system withharvesting at the source and non-EH relays was developedin [36].

The received signal at the relay node is a combinationof unwanted signals (interference) and the signal from thesource [30]. We note that improving the signal-to-interference-plus-noise ratio (SINR) by decreasing the interference is animportant and challenging concern for most of the currentcommunication networks [18]. In this field, various methodsincluding multi-cell coordination or interference-alignmentwere used for interference cancellation [18]. Although inter-ference reduces the throughput of the system, if it is used as a

source of energy it may improve the system performance [30].In fact, interference emerges as an additional source of energythat can help the communication process by improving thesystem performance [17]–[19], [30].

The performance analysis of RF EH relaying network wasstudied in [22], which considered energy constraint on thetransmission power and transceiver hardware impairment formultiple EH relays. RF-powered WSNs with simultaneouswireless information and power transfer (SWIPT) was con-sidered in [31], wherein stochastic geometry was used toanalyze the performance on the downlink. Relay selectionwith residual impairments and multiple antennas devices wasconsidered in [32], wherein the best relay was chosen usingthe channel-state-information (CSI) at each hop and applyingmaximal ratio combining (MRC). Similar performance anal-ysis approach was used for non-orthogonal multiple access(NOMA) under Nakagami-m fading in [33]. Also, cooperativeNOMA with SWIPT, where the users close to the source actas EH relays, were considered in [34]. User selection schemeswere proposed to improve the system throughput and outageprobability in [34]. Outage probability and throughput of anamplify-and-forward relaying system using EH in Nakagami-m fading channel is derived in [35]. In addition to that, per-formance analysis of a dual-hop under-water communicationsystem subject to κ−µ shadowed fading channel with RF-EHwas analyzed in [36].

Despite the importance of energy constraint and its effectson the transceiver design issues in cooperative networks, mostof the research works in the literature focused on harvestingsources at the transmitter or at the relay with no fixed batteries.It is inevitable that a limited fixed battery is implementedwithin the communication system nodes [29]. Given the natureof the most types of EH sources, which is random and notnecessarily available all the times, a limited fixed battery canprovide extra flexibility and continuity in service. However, athorough study on how EH can improve the performance ofrelaying system with fixed, but limited power batteries, is yetto be done.

B. ContributionsIn this paper, a dual-hop relaying system, in which the

source transmitter and the relay are both equipped with fixed,as well as, EH batteries is considered. The harvesting at thetransmitter node is done from solar energy and the harvestingat the relay node is from RF interference. The TS scheme isused at the relay to harvest energy and to decode information.For the considered relaying system, we analyse and discuss theimpact of EH on the system performance. First, closed-formexpressions for the cumulative distribution function (CDF) ofthe end-to-end SNR, and also for each link SNRs are obtained.To do so, we analyse the randomness in the transmissionpower due to the RF interference energy at the relay, andwe evaluate the end-to-end SNR. Analytical expression forthe ergodic capacity is also derived. Numerical results for the




validation of the developed analysis are obtained using Monte-Carlo simulations. The effect of different parameters, such asthe EH power, the fixed batteries power levels, and the energyconservation coefficient, are discussed and analysed.

To summarize, the main contributions of this paper areenlisted below:• The benefits of having interference EH at the relay node

and solar EH at the source transmitter, along with limitedpower fixed batteries, on the performance of a dual-hop relaying system, are investigated. To the best of theauthors’ knowledge, this paper is the first to address this.

• Closed-form expressions for the CDF of the end-to-endSNR and the SNR at each link are derived. The closed-form expressions reduce the computational complexity,but are also challenging to derive due to the presenceof randomness in the interference at the relay node.Since, the channel between the relay and the sink is alsorandom, the presence of multiplication of several randomparameters in the received SNR, makes the analysisdifficult.2

• The analytical expression for the ergodic capacity, whichis an important performance metric for delay-insensitiveservices, of the relaying system is obtained.

• The impact of the additional energy sources to improvethe overall system performance is evaluated by usingmathematical closed-form expressions, which are vali-dated through Monte-Carlo simulations. Differences be-tween the numerical results corresponding to the analysisand the Monte-Carlo simulations are small, indicating thecorrectness of the closed-form expressions.

• Finally, the impact of solar and interference and also thefixed batteries power levels, on the system performance,is studied through numerical results.

C. Organization

The remainder of the paper is organized as follows. SectionII describes the system model and assumptions. Performanceanalysis of the EH relaying system is discussed in SectionIII. In Section IV, we derive the CDF of the first-hop SINR,that of the second-hop SNR, and the one of the end-to-end SNR. Results are expressed in closed-form. Also thissection includes the derivation of the ergodic capacity. Finally,numerical results and discussion are given in Section V,followed by the conclusion.

II. SYSTEM MODEL AND ASSUMPTIONS

We consider a dual-hop decode-and-forward (DF) relayingsystem, in which the transmitter T communicates with the sinkS through an EH relaying node R, as presented in Fig. 1. No

2The closed-form expressions developed here are of great help when itcomes to the system design in practice, e.g., when the impact of interference,solar energy and energy conversion coefficient on the system performanceneeds to be investigated.

Figure 1: Relaying system with fixed and energy-harvesting batteries.

Figure 2: Time-switching protocol for harvesting energy and infor-mation processing.

direct communication between transmitter T and the sink Sis possible, e.g., due to shadowing. Node T is equipped witha fixed battery B1 and a conventional solar EH battery. Therelay node is also energy constrained with a fixed battery B2

but can also harvest energy from RF signals in the form ofinterference. The reason behind this assumption is that manycurrent wireless networks, e.g., WSNs, operate on low andnon-replaceable batteries [37]. The source can be a transmitterat high levels, and can thus benefit from solar EH, whereas therelay can be considered as a sensor node that operates on lowand non-replaceable batteries. This system model is applicableto practical cases when we need to deploy WSNs for short-range communications. This type of networks can indeedbenefit from additional energy obtained through harvesting,even though this additional energy is not necessarily large.We note that, even if the fixed batteries are out of energy,the network may still be able to survive on EH batteries.The analysis and derivations of this paper can also apply tothe system scenario where the fixed batteries are completelydischarged, by simply replacing the value of the fixed batterylevel to zero in the paper’s derivations.

The knowledge of the instantaneous channel state informa-tion (CSI) is not required at the source transmitter, as we arenot considering adaptive power allocation at the source. Whenthe relay node operates as a receiver during the first timeslot, it is assumed to have the CSI of the link between the




Figure 3: Illustration of key parameters in time-switching protocolfor harvesting energy and information processing.

transmitter and the relay. However, instantaneous informationof the interference channels are not known at the relay node.When the relay operates as a transmitter, it does not haveknowledge of the instantaneous CSI of the link between therelay and the sink.

The power level of the transmitter and the relay fixedbatteries are given by P1 and P2, respectively. We also assumethat the arrival rate of the harvesting solar energy at thesource transmitter is constant, hence Qt has a fixed value.Similar assumption is used in [38] where harvesting energyis assumed to have a continuous fashion, which is summedup to a constant value. Also, solar energy is considered in[39] which introduces a practical sensor node prototype thatassumes solar energy to be constant for optimal event detectionprobability in the network. EH serving as the only source ofenergy, and modeled as a continuous function with fixed rate,is considered in [38] and a throughput maximization problemis solved in this paper.

The overall interference induced on the relay node originates

from N i.n.i.d. interferers [18], and is represented byN∑i=1

Ii,

where Ii denotes the interference power of the ith interferer. Itis assumed that the interference and the desired channels areindependent from each other. Here, the interference imposedon the other networks from the considered relaying system isnot analysed. However, the results of this paper can pave theway for analysing a system with both incoming and imposinginterference to the neighbouring networks. To do so, a furthercomplex variable can be introduced for inter-cell interference.By that, the analysis and derivations of this paper can befurther updated to be used for multi-cell applications.

We assume DF relaying is employed, hence, the messagereceived at the relay node is decoded and then is forwardedto the sink. Each node is assumed to have a single antennaand to work in a half duplex mode within the dual-hopcommunication system. TS approach is used at the relay forEH and processing information, as described in Fig. 2. LetTsym denote the time during which the overall communicationbetween the source transmitter and the sink takes place. τTsym,for which τ varies from 0 ≤ τ ≤ 1, is the fraction of timein which the relay harvests energy both from the receivedsignals and from the external interference. (1− τ)Tsym is then

divided in two equal parts and represents the fraction of timeduring which information is transmitted from the source to therelay and from the relay to the sink. An illustration of the keyparameters in the time allocation considered is provided in Fig.3. In the TS architecture demonstrated in Fig. 3, the sourcenode and the relay are assumed to have HU batteries, in whichharvesting energy cannot be stored and must be used immedi-ately when it becomes available for signal transmission. Whenthe source is transmitting, the relay uses part of this time tolisten and decode the source signal. When this phase finishes,the relay switches to harvesting. Since the relay cannot storethe harvesting energy, the harvesting period is considered tobe right before the relay signal transmission time. The relayharvests energy while the source is still transmitting. As soonas the harvesting energy becomes available at the relay, thelatter will transmit the signal, which has already been decoded,to the sink destination. When relay is transmitting, the sourcenode harvests energy, hence, no TS is required at the sourcetransmitter.

In the considered HU approach, energy collected throughharvesting is assumed available at the end of the harvestingtime [39]. As there is no buffer to store the harvestingenergy, the energy causality constraints are not applicable here.Rechargeable batteries, which consider energy causality con-straints for energy storage through harvesting, are discussedin [40]3. The EH circuit activation threshold, which definesthe minimum energy level required to activate the circuit forharvesting [48], is considered -10dBm for the relay node [37].Additive white Gaussian noise (AWGN) is considered at therelay and at the sink.

A. Channel Modeling Parameters

The channel modeling parameters considered for the systemmodel depend on different factors including path-loss, fadingmodel and asymmetric characteristics. We start with path-lossdescription which mainly depends on the distance betweenthe transmitter and the receiver, the operating frequency,and the environmental terrain [42]. We note that since thedistance between the transmitter and the relay, i.e., dR, isdifferent from the distance between the relay and the sink,i.e., dS, hence the path-loss factors of the first- and second-hop channels are different. Path-loss factor at first-hop is givenby PLR

= Kd−βR , with K indicating a frequency dependentconstant and β > 2 referring to the environmental/terraindependent path-loss exponent [42]. Similarly, PLS

representsthe path-loss factor for the second-hop, which is given byPLS

= Kd−βS . Note that the dual-hop relay considered in thispaper is asymmetric and the path-loss factors for the channelbetween the transmitter and the relay is not necessarily the

3More information on the HU approach is provided in [41]. In the presentwork, the solar EH battery level is considered fixed within a transmissioncycle, while the interference from the RF source is random given the natureof ambient energy. Similar to [39] and [41], no device equipment is dedicatedto store energy.




same as the one for the channel between the relay and thesink.

Channels follow the block flat-fading model, i.e., each of thechannel gains are invariant during each fading block, but varyindependently from one block to another. The length of eachfading block is denoted by Tb. The symbol duration is givenby Tsym = 1/B, where B is the system bandwidth. We assumethat the fading block duration is an integer multiplication ofthe symbol duration.4 Furthermore, the fading block durationof all channels are assumed to be same.

The first-hop and second-hop channels experience inde-pendent Rayleigh fading and their respective channel powergains are given by h and g such that h ∼ CN(0,Ωh) andg ∼ CN(0,Ωg). Parameters Ωh and Ωg denote the variance ofchannel power gains h and g, respectively.

III. PERFORMANCE ANALYSIS OF ENERGY-HARVESTINGRELAYING SYSTEM

In this section, we evaluate the end-to-end SNR and theergodic capacity of the relaying system with fixed and EHbatteries, as described in Section II. The end-to-end ergodiccapacity C (in b/s/Hz) is the average of the minimum betweenthe rate at the first hop (R1) and the one at the second hop(R2), represented by

C = E (min(R1, R2)) ,

where E(.) indicates the expectation operator and rates R1 andR2 are given by

R1 =(1− τ)

2log2 (1 + γTR) , (1)

R2 =(1− τ)

2log2 (1 + γRS) , (2)

where γTR denotes the received SINR at the relay, and γRS

indicates the received SNR at the sink. The effect of theinterference at the relay is considered as an AWGN whichis the worst effect of interference.

1) First-Hop (Transmitter-to-Relay): During this phase,node T transmits the information signal, consuming power P1

from its fixed battery and an additional power QT from its EHbattery. The SINR at the relay node can then be formulatedas

γTR =(P1 +QT)h

KR + PLR

N∑i=1

Ii

, (3)

where KR = PLRσ2

RB, with σ2R indicating the variance of the

AWGN and QT =(1− τ)Qt

1 + τ. The source transmitter-to-relay

SINR, γTR , is further simplified as

γTR =γh

1 + IR, (4)

4If Tb is smaller than Tsym, then one symbol will be divided into twofading blocks and will experience different fading. Hence, the derivations ofthe paper will not hold anymore.

where γh =(P1 +QT)h

KRand IR =

N∑i=1

Ii

σ2RB

. The received data

is assumed to be decoded correctly only when the SINR isgreater than a predefined threshold γ.

2) Second-Hop (Relay-to-Sink): For the second-hop com-munication, the total power at the relaying node is the sumof the power from its fixed battery and the RF harvestingpower from the interference and the source transmitter signal.Therefore, the power at the relay will be

PR = P2 +QR, (5)

where QR indicates the power harvested at the relay node5.The total energy that is harvested from the received informa-tion signal and from the interference signal for a duration ofτTsym at each block is given by

EH = η

[N∑i=1

Ii + (P1 +QT)h

]τTsym, (6)

with η indicating the energy conversion coefficient whichvaries from 0 to 1 [43]. The processing power at the relay,required by the transmit/receive circuitry is negligible com-pared to the power used for data transmission [30]. Therefore,we assume that the relay consumes the harvesting and thefixed battery energy for forwarding information to the sink.The transmission power at the relay can hence be written as

QR =EH

(1− τ)Tsym/2. (7)

Replacing the value of EH from (6) into (7) and substitutingit into (5), yields

PR = P2 +2τη

1− τ

[N∑i=1

Ii +(P1 +QT)h

PLR

]. (8)

The SNR at the sink S is hence given by

γRS =PRg

KS,

where KS = PLSσ2

SB, with σ2S indicating the variance of the

AWGN at the sink. Inserting the value of PR from (8) into γRS

yields

γRS =

[P2 +

2τη

1− τ

(N∑i=1

Ii +(P1 +QT)h

PLR

)]g

KS,

which can be further simplified into

γRS = γg + w(IR + γh), (9)

where γg =gP2

KSand w =

2ητσ2Rg

σ2SPLS

(1− τ). We note that w is

a random variable (RV) with the same distribution as of g butwith a different variance.

5 As previously mentioned, P2 is the power level of the relay’s fixed batteryB2 and is assumed constant.




IV. CLOSED-FORM DERIVATIONS

Here, we aim to derive a closed-form expression for theCDF of the transmitter-to-relay SINR, the relay-to-sink SNR,and also the end-to-end SNR for the considered EH DFrelaying system with TS approach. By definition, the CDFof SNR at a certain threshold, γ, shows the probability ofthe instantaneous SNR to be less than γ. This thresholdcan represent the criterion for a minimum quality-of-servicerequirement at each node. The CDF of the SINR at the first-hop is formulated as FγTR

(γ) = Pr(γTR ≤ γ), and the CDF ofthe SNR at the second-hop is given by FγRS

(γ) = Pr(γRS ≤ γ),where Pr(.) denotes the probability and Fx(x) stands for theCDF of RV X at x.

With the channels following independent Rayleigh fading,the probability distribution function (PDF) of γh is exponentialand is given by

fγh(x) =1

γhexp

(− x

γh

), x ≥ 0, (10)

where γh is the average SNR from the source transmitter

to the relay and is given by γh =(P1 +QT)Ωh

KR. Similarly,

the PDF of the interference signal, IR, which is the sum ofN statistically i.n.i.d. exponential random interferences, eachwith a mean of µi, can be written as [44]

fIR(y) =

v(A)∑i=1

τi(A)∑j=1

λij(A)µ−ji

(j − 1)!yj−1 exp

(−yµi

), y > 0,

(11)

where matrix A = diag(µ1, µ2, ...., µN ), with µ1 > µ2 >· · · > µv(A) being the diagonal elements in decreasing order,v(A) denotes the number of distinct diagonal elements ofA, τi(A) is the multiplicity of µi, and λij is the (i, j)thcharacteristic coefficient of A as discussed in [44].

A. CDF of the First-Hop SINR

The CDF of the first-hop received SINR, Pr(γTR ≤ γ), canbe expanded by inserting the value of γTR from (4) to get

Pr(γTR ≤ γ) = Pr (γh ≤ (1 + IR)γ) . (12)

Using the PDF of γh, given in (10), the CDF of the first-hopSINR for each value of interference IR can be expanded as

FγTR(γ) = EIR

(1− exp

(−γ(

1 + IR

γh

))), (13)

where EIR is the expectation operator with respect to RV IR.The expression in (13) can be expanded by using the definitionof expectation [45, p. 30], yielding

FγTR(γ) = 1−

∫ ∞0

exp

(−γ(1 + u)

γh

)fIR(u)du. (14)

We now replace the PDF of interference IR from (11) into (14)to get

FγTR(γ) = 1−

∫ ∞

0

exp

(−γ(1 + u)

γh

) v(A)∑i=1

τi(A)∑j=1

λij(A)

× µ−ji(j − 1)!

uj−1 exp

(−uµi

)du. (15)

In order to solve (15), we re-arrange the equation as

FγTR(γ) = 1−

v(A)∑i=1

τi(A)∑j=1

exp

(−γγh

)λij(A)

× µ−ji(j − 1)!

(1

µi

(1 +

γµiγh

)−1)

×∫ ∞0

uj−1

µi

(1 +

γµiγh

)exp

(−uµi

(1 +

γµiγh

))du. (16)

which can be solved in closed-form according to

FγTR(γ) = 1−

v(A)∑i=1

τi(A)∑j=1

exp

(−γγh

)λij(A)

µ−j+1i

(j − 1)!

×

((1 +

γµiγh

)−1(j − 1)!µj−1i

(1 +

γµiγh

)−j+1). (17)

Therefore, the CDF of the first-hop received SINR is obtainedby simplifying (17) to get

FγTR(γ) = 1−

v(A)∑i=1

τi(A)∑j=1

exp

(−γγh

)λij(A)

(1 +

γµiγh

)−j.

(18)

The closed-form expressions for the CDF of the first-hopSINR, which are given in (18), depends on the average SNRfrom the transmitter to the relay (γh) and on the mean valueof each interference signal at the relay, e.g., µi, i = 1, · · ·N .(18) shows that the CDF of the first-hop SINR is directlyrelated to µi, i = 1, · · ·N , whereas it is inversely related toγh. Hence, the CDF of the first-hop SINR decreases with theaverage SNR from the transmitter to the relay and increaseswith µi, i = 1, · · ·N .

B. CDF of the Second-Hop SNR

In this subsection, we obtain a closed-form expression forthe CDF of the second-hop received SNR, Pr(γRS ≤ γ). Theanalysis is difficult due to the presence of randomness inthe RF interference signals and in the fading channels. Thetransmit power at the relay follows a random exponentialdistribution according to (5). Since the channel between therelay and the sink is also random, the SNR will have theimpact of these two random parameters multiplied.

We start by defining z = γh + IR to simplify (9). Then, thetarget CDF becomes

FγRS(γ) = Pr(γg + wz ≤ γ)




= EzEγg

(1− exp

(−γ − γg

wz

)), (19)

where w =2τηΩgσ

2R

σ2SPLS

(1− τ), Ez and Eγg are the expectation

operators with respect to RVs z and γg, respectively. We notethat the PDF for γg is an exponential function given by

fγg(γg) =1

γgexp

(−γg

γg

), (20)

where γg =P2Ωg

KS. In order to solve (19), we first solve the

expectation with respect to γg, to get

Pr(γg + wz ≤ γ) = 1− 1

γgexp

(−γzw

)×∫ γ

0

exp

(−γg

γg+γg

wz

)dγg. (21)

Solving the integral in (21) and replacing it into (19) yields

Pr(γg + wz ≤ γ) =

Ez(∫ ∞

0

(1− wz

γg − wzexp

(− γ

γg

)+

wz

γg − wzexp

(−γzw

))f(z)dz

). (22)

Solving the expectation with respect to z in (22) is challengingbecause the outer expectations Ez and inner expectation Eγg

are multiplied and the result becomes computationally difficultto solve. Recall that z is defined as z = γh + IR, whichitself is a summation of two random variables with differentdistributions. Since γh and IR are independent, we get the jointdistribution, fγh,IR(x, y) = fγh(x)fIR(y). The PDF of z is thencalculated by inserting the individual PDFs, fγh(x) and fIR(y),from (10) and (11), and integrating according to [30, p. 6428,eq. 13], which yields

fz(z) =1

γhexp

(−zγh

) v(A)∑i=1

τi(A)∑j=1

λij(A)

(1− µi

γh

)−j

×

(1− exp (−aiz)

j−1∑k=0

akik!zk

), (23)

where ai ,1

µi− 1

γh. Then, by inserting the PDF of z from

(23) into (22), we obtain

FγRS(γ) = 1−

v(A)∑i=1

τi(A)∑j=1

λij(A)

(1− µi

γh

)−j

×

∫ ∞γ

wz

γh(γg − wz)exp

(−γzw− z

γh

)dz︸︷︷︸

J1

+1

γhexp

(− γ

γg

)∫ ∞γ

wz

γg − wzexp

(−zγh

)dz︸︷︷︸

J2

−j−1∑k=0

akik!

∫ ∞γ

1

γhexp

(− γ

wz− z

γh

)exp (−aiz) zkdz︸︷︷︸

J3

.

(24)

Since there is an additional fraction multiplied with the sameentity in the exponential function present in the integral in (24),obtaining closed-form espression for (24) is challenging. Westart by dividing (24) into three integrals, where the firstand the second integrals are solved by using the generalizedincomplete Gamma functions (given in [46]).Details are givenbelow.

1) Closed-Form Expression for J1: The integral J1 has afractional equation of z multiplied to an exponential function

of z. We start with simplifying the ratiowz

γg − wzin (24),

according towz

γg − wz= −

(γg − γg − wzγg − wz

)= −1 +

(1− wz

γg

)−1, (25)

The term(

1− wz

γg

)−1can be approximated to 1 +

wz

γgfor

wz

γg→ 0. Thus, we have

wz

γg − wz' wz

γg. (26)

Now, we recall the formulation for generalized incompleteGamma function given by [47, p. 372],

Γ(α, x; b) =

∫ ∞α

tα−1 exp(−t− bt−1

)dt, (27)

which will later be used for obtaining a closed-form expressionfor J1.

Lemma 1. Using the recurrence relation for the incompletegeneralized Gamma function in [48, p. 101], the followingequality can be obtained

Γ(α+ 1, x; b) = αΓ(α, x; b) + bΓ(α− 1, x; b)

+ xα exp (−x− bx−1), α > 0. (28)

Proof. The proof is provided in Appendix A.Using the results of Lemma 1 for α = 1 and (26), a solution

for the first integral J1 can be obtained according to

J1 'γg

γgγh

(Γ

(0,γ

γh;

γ

γhγgw

)−γ exp (−γ − bγ−1) + Γ

(1,γ

γh;

γ

γhγgw

)). (29)




2) Closed-Form Expression for J2: To obtain a closed-fromexpression for J2 (shown in (24)), we use the approximationresult in (26), yielding

J2 'w

γhγgexp

(− γ

γg

)∫ ∞γ

z exp

(−zγh

)dz,

which allows us to obtain a closed-form expression for J2 as

J2 ' −w

γhγg

(1− γ

γh

)exp

(−γ(

1

γg+

1

γh

)). (30)

3) Closed-Form Expression for J3: Using (27), we obtaina closed-form expression for J3, according to,

J3 =1

γhexp (ai)

j−1∑k=0

1

k!(−ai)k

×k∑

m=0

(k

m

)b−1i

(−bi)mΓ

(m+ 1, biγ;

biγ

w

), (31)

where bi =1

γh+

ai1 + γ

and ai is defined right after (23).

Finally, we obtain the closed-form solution for the CDF ofFγRS

(γ), by inserting (29), (30) and (31) into (24), yielding

FγRS(γ) ' 1−

v(A)∑i=1

τi(A)∑j=1

λij(A)

(1− µi

γh

)−j× w

γg

(−γ exp(−γ − bγ−1) + Γ

(1,γ

γh;

γ

γhγgw

)+Γ

(0,γ

γh;

γ

γhγgw

))− w

γgγh

(1− γ

γh

)× exp

(−γ(

1

γh+

1

γg

))− 1

γhexp (ai)

j−1∑k=0

1

k!(−ai)k

×k∑

m=0

(k

m

)b−1i

(−bi)mΓ

(m+ 1, biγ;

biγ

w

). (32)

C. CDF of the End-to-End SNR

The end-to-end SNR is defined by the probability thatthe instantaneous output SNR falls below a certain thresholdwhich is already defined as γ. Mathematically, it is written as

Pr(min(γTR , γRS) ≤ γ) = Pr((γg + wz)1 ≤ γ), (33)

where 1 symbolizes an indicator of RV, with 1 = 1 for γTR > γand 0 otherwise. This RV also indicates that the power at therelay is not only the power coming from its fixed battery B2,but also the random replenished energy from the interference.Let us introduce the RV z = (γh + IR)1. The PDF fz(z) isgiven by

fz(z) = 11

γhexp

(−zγh

) v(A)∑i=1

τi(A)∑j=1

λij(A)

(1− µi

γh

)−j

×

(1− exp

(−ai

z − γ1 + γ

) j−1∑k=0

akik!

(z − γ1 + γ

)k). (34)

Thus, by inserting (34) into (22), the PDF of the end-to-end SNR, γend-to-end, is obtained as Obtaining a closed-formsolution for (35) can be done by using a similar method as insubsection IV-B. It is noted that the solutions to the first andthe second integrals are already known from (29) and (30).In the third integral, we use the Taylor series expansion for(z−γ)k [49] and use the definition of generalized incompleteGamma function (given in [47, p. 372]). Details are givenbelow.

1) Closed-Form Expression for K3: The third integralK3 is solved by using direct substitution of a Taylor seriesexpansion of (z− γ)k with respect to z [30, p. 6430] and theidentity of the generalized incomplete Gamma function [46,Eq. (3.351.1), Eq.(3.334)]. This gives

K3 '1

γhexp

(aiγ

1 + γ

) j−1∑k=0

1

k!

(−aiγ1 + γ

)k×

k∑m=0

(k

m

)b−1i

(−biγ)mΓ

(m+ 1, biγ;

biγ

γgw

). (36)

We now obtain the closed-form solution for the CDF of theend-to-end SNR by inserting (29), (30) and (36), into (24),yielding

Fγend-to-end(γ) '

1−v(A)∑i=1

τi(A)∑j=1

λij(A)

(1− µi

γh

)−j× w

γg

(−γ exp(−γ − bγ−1) + Γ

(1,γ

γh;

γ

γhγgw

)+Γ

(0,γ

γh;

γ

γhγgw

))− w

γgγh

(1− γ

γh

)× exp

(−γ(

1

γh+

1

γg

))− 1

γhexp

(aiγ

1 + γ

)×j−1∑k=0

1

k!

(−aiγ1 + γ

)k k∑m=0

(k

m

)b−1i

(−biγ)m

× Γ

(m+ 1, biγ;

biγ

γgw

). (37)

We note that the transmission power at the relay, i.e., PR, doesnot only depend on the power level of the relay’s fixed battery,but also on the energy available through harvesting. We recallthat EH at the relay node is the combination of the energycollected from interference and the energy received from thesource’s signals. Hence, the distribution of the received SNR atthe sink node depends on the variations of the relay-to-sink, aswell as the transmitter-to-relay, channel power gains. One canshow that the second-hop SNR can be improved if the level ofinterference increases. However, the effect of the interferencepower on the end-to-end SNR, given in (37), cannot be easily




Fγend-to-end(γ) = 1−v(A)∑i=1

τi(A)∑j=1

λij(A)

(1− µi

γh

)−j∫ ∞γ

1

γh

wz

γg − wzexp

(−γzw− z

γh

)dz︸︷︷︸

K1

+1

γhexp

(− γ

γg

)∫ ∞γ

wz

γg − wzexp

(−zγh

)dz︸︷︷︸

K2

−j−1∑k=0

akik!

∫ ∞γ

1

γhexp

(− γ

wz− z

γh

)− exp

(−ai

z − γ1 + γ

)(z − γ1 + γ

)kdz︸︷︷︸

K3

. (35)

verified. This can also be intuitively explained since in theconsidered relaying system, the interference reduces the SINRof the first hop, whereas it increases the SNR of the secondhop.

D. Ergodic Capacity

In this subsection, we analytically derive the ergodic capac-ity of the relaying system which is calculated by taking theaverage of the minimum between the rate at the first-hop andthat at the second-hop [50], formulated in b/s/Hz as

C =

E(

min

((1− τ)

2log2 (1 + γTR) ,

(1− τ)

2log2 (1 + γRS)

))= E

(1− τ

2log2 (1 + min(γTR , γRS))

)=

1− τ2

∫ ∞0

log2(1 + γend-to-end)fγend-to-end(γend-to-end) dγend-to-end,

(38)

where fγend-to-end(.) is the PDF of the RV γend-to-end =min(γTR , γRS). We now solve (38) by using integration by partsas follows:

C =1− τ

2(log2 (1 + γend-to-end)

×(Fγend-to-end (γend-to-end)− 1

))∣∣∣∞0− 1− τ

2 ln 2

×∫ ∞0

Fγend-to-end(γend-to-end)− 1

1 + γend-to-enddγend-to-end

=1− τ2 ln 2

∫ ∞0

1− Fγend-to-end(γend-to-end)dγend-to-end

1 + γend-to-end.

where, Fγend-to-end(γend-to-end) is given in (37).

V. NUMERICAL RESULTS AND DISCUSSION

In this section, we numerically evaluate the effect of EH,the fixed batteries power levels and the energy conversion

Table I: Simulation parameters

Parameters Default value Parameters Default valueN 5 η 0.3 else statedγ 0.2 dB µi, i = 1, ..., 5 0.3τ 0.4 else stated σS

2B 1 else statedP 1 dB else stated KR 1 else statedΩh 1 else stated σ2

RB 1 else statedKS 1 else stated Ωg 1 else stated

coefficient on the ergodic capacity and the end-to-end SNRof the studied relaying system. In all the figures, we assumethat the power of the fixed battery at the source transmitterT is equal to the power of the fixed battery at the relayR, hence, P1 = P2 = P . In our simulation we have usedvalues directly for KR, e.g., , KR = 1, in which the effect ofdistance is included through the path-loss model. Hence wedid not choose any specific value of dR. In the simulations,we included the value for KR = KS = 1, else stated.The channels are assumed unit variance Rayleigh, hence,Ωh = Ωg = 1, unless otherwise indicated. The numberof interference signals at the relay is set to N = 5 withnormalized µi = 0.3, i = 1, ...5, and the TS parameter isset to τ = 0.4, unless otherwise indicated. Values for Qt andη are taken from studies shown in [51]. The noise variance atrelay and sink is assumed to be equal, hence σ2

R = σ2S = 1,

unless otherwise indicated.In Fig. 4, we start by plotting the CDF of the first-hop

SINR FγTR(γ), the CDF of the second-hop SNR FγRS

(γ), andthe CDF of the end-to-end SNR Fγend-to-end versus the powerconsumed from the fixed batteries P , at γ = 0.2, η = 0.3,KR = 0.5 and KS = 0.3 with Qt = 1dB. The figureincludes both analytical and Monte-Carlo simulation resultsto verify the correctness of the closed-form derivations. Fromthe graphs, we observe that the CDF of the received SNR atthe second hop is greater than the one of the SINR at the firsthop, for most of the time. It is noted that the CDF of theend-to-end SNR is similar to the CDF of the second hop SNR




0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P (dB)

Fγ T

R

(γ)

, F

γ RS

(γ)

, F

γ end−

to−

end (

γ)

Closed−form solution for Fγ

TR

(γ)

Monte−Carlo simulations for FγTR

(γ)

Closed−form solution for FγRS

(γ)

Monte−Carlo simulation for FγRS

(γ)

Closed−form solution for Fγend−to−end

(γ)

Monte−Carlo simulation for Fγend−to−end

(γ)

Figure 4: The CDF of γTR (first-hop), CDF of γRS (second-hop), andthe CDF of γend-to-end (end-to-end SNR) versus the power consumedfrom the fixed batteries P , when γ = 0.2, η = 0.3, KR = 0.5 andKS = 0.3 and Qt = 1dB.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P (dB)

Ave

rage

Rat

e (b

/s/H

z)

E(R1), No harvesting at source

E(R2), No harvesting at source

E(R1), Q

t = 1 dB

E(R2), Q

t = 1 dB

Figure 5: Average rates at the first and second hops (E(R1),E(R2))versus P with η = 0.3.

FγRS(γ). This is due to the fact that, for the settings used in

this figure, the first hop SINR is the minimum of the SINR atthe first and the SNR at the second hop for most of the time.Further, the plots confirm that the results obtained from theclosed-form expressions match to the ones obtained from theMonte-Carlo simulations, hence indicating that the obtainedclosed-form expressions yield accurate measures of the CDFof the individual links’ SNRs and the end-to-end SNR.

Fig. 5 shows the plots for E(R1) (the average rate at the firsthop) and E(R2) (the average rate at the second hop) versusthe power level of the fixed batteries P , at η = 0.3 and forvarious values of Qt. From the plots, we note that when P =0dB and no harvesting is available at the source, the averagerate of the second hop is slightly higher than the one at thefirst hop. The relay harvests energy both from source signaland RF interference and, hence, under the above-mentionedconditions, the second hop achieves a slightly higher rate thanthe first hop. When Qt = 1dB, and at the same time the energyconversion coefficient at the relay remains small (η = 0.3),the average rate at the first hop E(R1) dominates E(R2). This

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Qt (dB)

Fγ T

R

(γ),

F

γ RS

(γ)

FγRS

FγTR

Figure 6: FγTR(γ) and FγRS

(γ) versus Qt at γ = 0.2 with η = 0.3and P = 1dB.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

τ

Ave

rage

Rat

e (b

/s/H

z)

E(R1), Q

t= 1dB

E(R2), Q

t=1dB

C, Qt=1dB

E(R1), Q

t=0.5dB

E(R2), Q

t=0.5dB

C, Qt=0.5dB

Figure 7: Average rates at the first and second hops, (E(R1),E(R2)),versus τ , with η = 0.3 and various values of Qt.

happens because the solar energy at the transmitter is strongercompared to the RF energy level at the relay. We also observethat the difference between the two rates decreases at higherpower values P of the fixed batteries, indicating that EH isless beneficial in devices with higher powers.

Fig. 6 includes the plots for the CDF of the SINR at the first-hop γTR and the CDF of the SNR at the second hop γRS , versusQt, at η = 0.3, P = 1dB and γ = 0.2. From the graph, wenotice that not only the CDF of the first hop SINR (FγTR

(γ))decreases with the increase in the solar energy, but also theCDF of the second hop SNR (FγRS

(γ)) shows a similar trend.Fig. 7 illustrates the plots for the average rates E(R1),

E(R2) and C versus the TS parameter τ with η = 0.3,µ = 0.3, P = 1dB, and various values of Qt. It is noted thatwhen Qt = 0.5dB, E(R2) shows bell curve while E(R1) ismonotonically decreasing. This is due to the fact that althoughwith increasing τ , more energy will be available at the relaynode through harvesting, but at the same time, less time is




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

η

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

Ave

rage

Rat

e (b

/s/H

z)

E (R1) , No harvesting at source transmitter

E (R2) , No harvesting at source transmitter

E(R1), Q

t= 1dB

E(R2), Q

t= 1dB

Figure 8: Average rates (first-hop and second-hop) versus the energyconversion coefficient η, with P = 1dB and various values of Qt.

available for signal information transmission. This conflictingeffect, makes the shape of E(R2) with respect to τ bell-shape.When solar energy increases, e.g., Qt = 1dB, the two curvescoincide each other at smaller values of τ = 0.5. This happensbecause with higher Qt, E(R1) is bigger. Hence, the amount oftime which will be used for harvesting, instead of informationtransmission reduces E(R1) quicker and hence, the two curvecoincide at a smaller value of τ , when compared to the casewith Qt = 0.5dB. Generally, the trend shows the higher thesolar energy is, the less time should be allocated for harvestingin the second hop.

In Fig. 8, we present the plots for the average rate at thefirst-hop and the average rate at the second-hop versus energyconversion coefficient η at P = 1dB with various values ofQt. At small values of η, i.e., η < 0.1, with Qt = 1dB, theaverage rate at the first-hop is slightly higher than that at thesecond-hop, due to the small energy conversion coefficient atthe relay harvesting battery. As η increases, i.e., η > 0.1, theaverage rate at the second hop dominates the average rate atthe first hop.

Fig. 9 shows the plots for the average rates versus Qt withη = 0.3 and P = 1dB. The figure shows three different rateplots defined by the average rate at the first hop, the averagerate at the second hop and the average end-to-end ergodicrate C. We notice that E(R1) increases rapidly with Qt, whilethe average rate at the second hop increases slowly with Qt.When Qt = 0dB, we observe that the E(R2) is higher thanE(R1). As Qt increases, e.g., Qt > 1.5dB, the average rate atthe first hop becomes larger than that at the second hop. Thishappens due to the fact that in E(R1) the transmitter node hasa direct access to solar energy, whereas in case of E(R2), theharvested power gets affected by the Rayleigh fading channelh. Furthermore, the total end-to-end capacity C increases withQt, showing that the higher the solar energy is, the better theend-to-end ergodic rate is. The numerical results validate themathematical expression for the ergodic capacity in (38).

0 1 2 3 4 5 6 70.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Qt (dB)

Ave

rage

Rat

e (b

/s/H

z)

E(R1) : first−hop ergodic rate

E(R2) : second−hop ergodic rate

C : end−to−end ergodic rate

Figure 9: Average rates (first hop, second hop and end-to-end) versusQt at η = 0.3 with P = 1dB.

0 1 2 3 4 5 6 7

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Qt (dB)

Ave

rage

SN

R/S

INR

γTR

: Average SINR at the first hop

γRS

: Average SNR at the second hop

Figure 10: Average SNR/SINR versus Qt at η = 0.3 with P = 1dB.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

E(IR

)

Ave

rage

Rat

e (b

/s/H

z)

E(R1) : first−hop ergodic rate

E(R2) : second−hop ergodic rate

C : end−to−end ergodic rate

Figure 11: Average rates (first hop, second hop and end-to-end) versusaverage interference at the relay E(IR), with τ = 0.4, Qt = 1dB andvarying µ.

Fig. 10 shows the plots for the average SINR at the first-hop and average SNR at the second-hop, versus Qt, at η = 0.3and P = 1dB. As observed, both values increase when solarenergy increases. Meanwhile, at Qt = 0dB, the average SNR at




the second hop is higher than the average SINR at the first hop.This is due to the fact that while Qt is small, i.e., Qt = 0dB,the relay still has access to harvesting power from the RFinterference signal. At Qt = 1.5dB, the two plots coincideeach other. Overall, the first-hop SINR increases more rapidlywith Qt compared to the second hop SNR.

Finally, Fig. 11 demonstrates the plots for the average rateat the first hop, the average rate at the second hop, andthe average end-to-end ergodic rate C, versus the averageinterference power at the relay E(IR), with η = 0.3, P = 1dBand Qt = 1dB. When Qt = 1dB and the average interferencepower is small, i.e., E(IR) < 0.2, the first-hop rate is slightlyhigher than the one at the second hop. The average interferenceadversely affects the rate of the first hop, therefore E(R1) isalways decreasing with E(IR); whereas it improves the rateof the second hop through EH. When the interference signalis weak, the rate of the first hop is better than the rate of thesecond hop. With an increase in the average interference at therelay, E(R2) gives higher values since the solar transmitter hasaccess to solar energy. Furthermore, the average end-to-endcapacity C shows a monotonic decrease with E(IR), revealingthat the more the average interference is, the less the end-to-end ergodic rate will be.

VI. CONCLUSION

In this paper, we investigated the system performance of adual-hop DF relaying system, in which the transmitter and therelay are equipped with fixed, as well as, EH batteries. Har-vesting at the transmitter is done through solar energy source,whereas interference from the RF is used as the EH sourceat the relay. TS is used for EH and information transmissionat the relay. We showed that EH at the source transmitter andat the relay can improve the end-to-end SNR. Closed-formexpressions were derived for the CDF of the SNR at each hop,and also for the end-to-end SNR. The analytical expression forthe end-to-end ergodic capacity was also obtained. Numericalresults were provided to verify the correctness of the closed-from derivations. The effect of energy conversion efficiencyand the fixed batteries power levels on the CDF of the SNR forthe individual links were also investigated. The results furtherdemonstrated that with the addition of EH at the source andthe relay, significant improvement in the system throughputcan be achieved. Future work includes consideration of inputpower constraints at the fixed batteries and catering inter-cellinterference and hardware impairments.

APPENDIX A

Here, we aim to prove the equation in Lemma 1, givenin (1). We start by referring to the below recurrence equa-tion [48, p. 101]

d

dxxα exp

(−x− bx−1

)= αxα−1 exp(−x− bx−1)

+ bxα−2 exp(−x− bx−1)− xα exp(−x− bx−1), (39)

For α = 1, the expression simplifies to

d(x exp(−x− bx−1)

)dx

=(1 + bx−1 − x)

× exp(−x− bx−1). (40)

Re-arranging (40) and taking integration on both sides gives,∫ ∞γ

x exp(−x− bx−1)dx =∫ ∞γ

bx−1 exp(−x− bx−1) + x exp(−x− bx−1)∣∣∣∞γ

+

∫ ∞γ

exp(−x− bx−1)dx. (41)

Different parts of (41) can be re-written into generalizedincomplete Gamma function [47] as∫ ∞

γ

x exp(−x− bx−1) = Γ(2, x; b), (42)∫ ∞γ

bx−1 exp(−x− bx−1) = Γ(0, x; b), (43)∫ ∞γ

exp(−x− bx−1) = Γ(1, x; b). (44)

Hence, replacing (42)-(44) into (41) yields,

Γ(2, x; b) = Γ(0, x; b) + x exp(−x− bx−1)∣∣∣∞γ

+ Γ(1, x; b). (45)

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Arooj Mubashara Siddiqui received her B.Sc. de-gree in Electrical (Telecommunication) Engineering,from COMSATS Institute of Information Technol-ogy, Pakistan, in 2009 and the M.Sc. degree in Mo-bile Broadband Communications and the PhD de-gree in Communication Systems from the School ofComputing and Communications, InfoLab21, Lan-caster University, Lancaster, UK, in 2013 and 2017respectively. She is currently working as AssistantProfessor in Department of Electrical Engineering,COMSATS Institute of Information Technology, Is-

lamabad 44000, Pakistan. Her research interests include Energy Harvesting,energy- efficient transmission techniques and optimization towards 5G Greencommunications.

Leila Musavian (S’05-M’07) received her Ph.D.degree in Telecommunications from Kings CollegeLondon, UK, in 2006. She is currently working asa Reader at the School of Computer Science andElectrical Engineering, University of Essex. Priorto that, she was a Lecturer at InfoLab21, LancasterUniversity (2012-2016). She was a ResearchAssociate at McGill University (2011-2012) anda post-doctoral fellow at INRS-EMT, Canada,from 2006 to 2008. Her research interests lie inRadio Resource Management for next generation

wireless networks, CRNs, Energy Harvesting, Green Communication,energy-efficient transmission techniques, cross-layer design for delay QoSprovisioning and 5G systems. She is an editor of IEEE Transactions onWireless Communications, Executive Editor of Transactions on EmergingTelecommunications Technologies and Associate Editor of Wiley’s InternetTechnology Letters.

Sonia Aïssa (S’93-M’00-SM’03) received her Ph.D.degree in Electrical and Computer Engineering fromMcGill University, Montreal, QC, Canada, in 1998.Since then, she has been with the Institut Nationalde la Recherche Scientifique-Energy, Materials andTelecommunications Center (INRS-EMT), Univer-sity of Quebec, Montreal, QC, Canada, where sheis a Full Professor.

From 1996 to 1997, she was a Researcher withthe Department of Electronics and Communicationsof Kyoto University, and with the Wireless Systems

Laboratories of NTT, Japan. From 1998 to 2000, she was a Research Associateat INRS-EMT. In 2000-2002, while she was an Assistant Professor, shewas a Principal Investigator in the major program of personal and mobilecommunications of the Canadian Institute for Telecommunications Research,leading research in radio resource management for wireless networks. From2004 to 2007, she was an Adjunct Professor with Concordia University,Montreal. She was Visiting Invited Professor at Kyoto University, Japan, in2006, and Universiti Sains Malaysia, in 2015. Her research interests includethe modeling, design and performance analysis of wireless communicationsystems and networks.

Dr. Aïssa is the Founding Chair of the IEEE Women in EngineeringAffinity Group in Montreal, 2004-2007; acted as TPC Symposium Chair orCochair at IEEE ICC ’06 ’09 ’11 ’12; Program Cochair at IEEE WCNC2007; TPC Cochair of IEEE VTC-spring 2013; and TPC Symposia Chairof IEEE Globecom 2014; and serves as TPC Vice-Chair of IEEE Globecom2018. Her main editorial activities include: Editor, IEEE TRANSACTIONS ONWIRELESS COMMUNICATIONS, 2004-2012; Associate Editor and TechnicalEditor, IEEE COMMUNICATIONS MAGAZINE, 2004-2015; Technical Editor,IEEE WIRELESS COMMUNICATIONS MAGAZINE, 2006-2010; and AssociateEditor, Wiley Security and Communication Networks Journal, 2007-2012. Shecurrently serves as Area Editor for the IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS. Awards to her credit include the NSERC UniversityFaculty Award in 1999; the Quebec Government FRQNT Strategic FacultyFellowship in 2001-2006; the INRS-EMT Performance Award multiple timessince 2004, for outstanding achievements in research, teaching and service;and the Technical Community Service Award from the FRQNT Centre forAdvanced Systems and Technologies in Communications, 2007. She is co-recipient of five IEEE Best Paper Awards and of the 2012 IEICE BestPaper Award; and recipient of NSERC Discovery Accelerator SupplementAward. She served as Distinguished Lecturer of the IEEE CommunicationsSociety and Member of its Board of Governors in 2013-2016 and 2014-2016, respectively. Professor Aïssa is a Fellow of the Canadian Academyof Engineering.

Qiang Ni (S’93-M’00-SM’03) received B.Sc.,M.Sc., and Ph.D. degrees from Huazhong Universityof Science and Technology (HUST), China, all inengineering. He is a Professor and the Head ofCommunication Systems Research Group, School ofComputing and Communications, InfoLab21, Lan-caster University, U.K. His research interests liein the area of future generation communicationsand networking, including green communications,5G, energy harvesting,cognitive radio, ultra-densenetworks, SDN, IoTs and vehicular networks. He

has published 180+ papers in these areas. He is a Voting Member of IEEE1932.1 standard. He was an IEEE 802.11 Wireless Standard Working GroupVoting Member and a Contributor to various IEEE Wireless Standards.

arooj mubashara siddiqui, leila musavian, sonia aissa and qiang nirepository.essex.ac.uk › 20516...

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