research article characterization of energy availability...

Research ArticleCharacterization of Energy Availability in RF EnergyHarvesting Networks

Daniela Oliveira1 and Rodolfo Oliveira12

1 Instituto de Telecomunicacoes (IT) Avenida Rovisco Pais 1 1049-001 Lisboa Portugal2Dep de Eng Electrotecnica Faculdade de Ciencias e Tecnologia (FCT) Universidade Nova de Lisboa2829-516 Costa da Caparica Portugal

Correspondence should be addressed to Rodolfo Oliveira radofctunlpt

Received 12 June 2016 Accepted 8 September 2016

Academic Editor Babak Shotorban

Copyright copy 2016 D Oliveira and R Oliveira This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The multiple nodes forming a Radio Frequency (RF) Energy Harvesting Network (RF-EHN) have the capability of convertingreceived electromagnetic RF signals in energy that can be used to power a network device (the energy harvester) Traditionally theRF signals are provided by high power transmitters (eg base stations) operating in the neighborhood of the harvesters Admittingthat the transmitters are spatially distributed according to a spatial Poisson process we start by characterizing the distribution of theRF power received by an energy harvester node Considering Gamma shadowing and Rayleigh fading we show that the receivedRF power can be approximated by the sum of multiple Gamma distributions with different scale and shape parameters Using thedistribution of the received RF power we derive the probability of a node having enough energy to transmit a packet after a givenamount of charging time The RF power distribution and the probability of a harvester having enough energy to transmit a packetare validated through simulationThe numerical results obtained with the proposed analysis are close to the ones obtained throughsimulation which confirms the accuracy of the proposed analysis

1 Introduction

The nodes forming a Radio Frequency (RF) Energy Har-vesting Network (RF-EHN) have the capability of convertingreceived electromagnetic RF signals in energyThe RF energyis converted by an energy harvester device which is com-posed of an RF antenna a band pass filter parametrized tothe RF signals and a rectifying circuit able to convert RF toDCpower [1] In this way the converted RF signals are used tocharge a battery (usually a supercapacitor) with finite capacity[2]Theharvested energy accumulated in the battery can thenbe used to transmit a packet However the transmission isonly possible if the level of accumulated energy is higher thana given threshold representing the minimum level of energyrequired to complete a packet transmission

Recently RF energy harvesting has attracted muchattention and many efforts are being dedicated to developinnovative RF energy harvesting technologies as well asto investigate the performance of the networks formed by

the harvesting devices The RF energy harvesting literaturededicated to the efficient design of RF harvesting devices(see [3ndash7] for a few examples) is mainly focused on theminimization of the loss effects due to the RF-to-DC con-version and battery charging process A different focus isalso found in the literature where the main goal is thestudy and characterization of RF-EHNs Adopting a genericmodel for the RF energy harvesting devices the goals areusually related with the scheduling of the harvesting devicesin order to maximize the utilization of the RF energy and thefrequency band constrained by specific throughput fairnesspolicies [8] the optimization of the harvester communicationtask to deal with the multiple tradeoffs associated with thephysical and MAC layers [9 10] the characterization of theRF-EHN performance (throughput) and stability when RFenergy harvesting is adopted [11] Reference [12] investigatesthe performance (throughput) of a slotted Aloha randomaccess wireless network consisting of two types of nodes withunlimited energy supply and solely powered by an RF energy

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7849175 9 pageshttpdxdoiorg10115520167849175

2 Mathematical Problems in Engineering

harvesting circuit To illustrate the design considerations ofRF-based harvesting networks [13] points out the primarychallenges of implementing and operating such networksincluding nondeterministic energy arrival patterns energyharvesting mode selection and energy-aware cooperationamong base stations Reference [14] adopts a stochasticgeometry framework based on the Ginibre model to ana-lyze the performance of self-sustainable communicationsover cellular networks with general fading channels Theexpectation of the RF energy harvesting rate the energyoutage probability and the transmission outage probabilityare evaluated over Nakagami-m fading channels

RF-EHNs may also act as cognitive radio networks(CRNs) that is using the spectrum in an opportunistic waywithout being licensed Several works have explored thesekinds of networks Reference [15] provides an overview of theRF-EHNs including system architecture RF energy harvest-ing techniques and existing applications The authors alsoexplore various key design issues in the development of RF-EHNs including cognitive radio networks The work in [16]provides a comprehensive overview of recent developmentand challenges regarding the operation of cognitive radionetworks powered by RF energy Spectrum efficiency andenergy efficiency are two critical issues in designing cognitiveradio RF-EHNs Reference [17] provides an overview of theRF-powered CRNs and discusses the challenges that arisefor dynamic spectrum access in these networks Focusingon the trade-off among spectrum sensing data transmissionand RF energy harvesting the authors discuss the dynamicchannel selection problem in a multichannel RF-poweredCRN Reference [18] proposes a novel method for wirelessnetworks coexistingwhere low-powermobiles in a secondarynetwork harvest ambient RF energy from transmissions bynearby active transmitters while opportunistically accessingthe spectrum licensed to the primary network The authorsanalyze the transmission probability of harvesting terminalsand the resulting spatial throughput The optimal trans-mission power and terminalsrsquo density are also derived formaximizing the throughput The work in [19] considers anRF-powered green cognitive radio network where a centralnode harvests energy from ambient sources and wirelesslydelivers random harvested energy to cognitive users Thework evaluates the performance of such a network showingthe feasibility of the behavior if the energy transmissionrate is below a certain threshold Reference [20] considersa network where the unlicensed users can perform channelaccess to transmit a packet or to harvest RF energy whenthe selected channel is idle or occupied by the primaryuser respectively The work is mainly focused on finding thechannel access policy that maximizes the throughput of thesecondary user Reference [21] analyzes an energy harvesting-based cognitive radio system to find the optimal spectrumsensing time which maximizes the harvested energy Thework in [22] analyzes a cognitive and energy harvesting-based device-to-device (D2D) communication in cellularnetworksThe authors employ tools from stochastic geometryto evaluate the performance of the proposed communicationsystem model with general path-loss exponent in termsof outage probability for D2D and cellular users One of

the work conclusions is that energy harvesting can be areliable alternative to power cognitive D2D transmitterswhile achieving acceptable performance

In this work we are particularly focused on the charac-terization of the RF power received by each harvester and itsimpact in terms of the probability of accumulating enoughenergy to transmit a packet A generalized radio propagationenvironment is considered Assuming that the sources ofhigh power RF signals (eg base stations) are distributedaccording to a spatial Poisson process we characterize thedistribution of the received RF power from the multipletransmitters Path loss shadowing and fading effects areconsidered The distribution of the RF power is then usedto derive the probability of a harvester node having enoughenergy to transmit a packet after a given period of time Asoft computational model (Gaussian approach) and a morecomplex model (non-Gaussian approach) are presented tocompute the probability of a harvester node having enoughenergy to transmit These are the main contributions ofthe paper Considering multiple spatial and propagationscenarios we validate the distribution of the RF powerand the probability of a harvester have enough energy totransmit a packet The numerical results obtained with theproposed analysis are close to the ones obtained throughsimulation which confirms the accuracy of the proposedanalysis In this way we provide a characterization of thebattery charging time considering innovative assumptionsincluding the spatial distribution of the RF transmitters thepropagation effects and the losses associated with the RF-to-DC conversion and battery charging process The proposedmodel can thus be adopted to determine the probability of aharvester accumulating enough energy after a given periodof time which is a determinant condition to compute thethroughput of RF-EHNs As far as we known this is thefirst work to derive such a probability when the multiple RFsignals received by the harvesters are differently affected bymultiple propagation effects

The rest of the paper is organized as follows The systemdescription is presented in Section 2 The characterizationof the received RF power is characterized in Section 3Section 4 derives the probability of accumulating enoughenergy to transmit a packet through the Gaussian and non-Gaussian approaches Finally validation results are presentedin Section 5 and conclusions are drawn in Section 6

2 System Description

This work considers a RF energy harvesting network whereeach node accumulates energy from the base stations andother RF transmitters located in the neighborhood A har-vester node is able to initiate a packet transmission wheneverthe level of accumulated energy is above a transmissionthreshold

21 Spatial Distribution of the RF Transmitters We considerthe scenario illustrated in Figure 1 where the node 119873119867 (theharvester node) accumulates energy from the transmittersthat might be located in the area 119860 = 120587((119877119871119900 )2 minus (1198771119894 )2) The

Mathematical Problems in Engineering 3

middot middot middotNH

R1i

R1o equiv R2

i

R2o equiv R3

i

RLi

RLo

Figure 1 The harvester node 119873119867 receives RF power from thetransmitters located in the area 119860 = 120587((119877119871119900 )2 minus (1198771119894 )2)

area 119860 can be obtained via calculus by dividing the annulusup into an infinite number of annuli of infinitesimal width 119889120594and area 2120587120594 119889120594 and then integrating from 120594 = 1198771119894 to 120594 = 119877119871119900 that is 119860 = int119877119871119900

11987711198942120587120594 119889120594 Using the Riemann sum 119860 can be

approximated by the sum of the area of a finite number (119871) ofannuli of width 120588

119860 asymp 119871sum119897=1

119860 119897 (1)

where 119860 119897 = 120587((119877119897119900)2 minus (119877119897119894)2) denotes the area of the annulus119897 119877119897119900 = (1198771119894 + 119897120588) and 119877119897119894 = (1198771119894 + (119897 minus 1)120588) represent the radiusof the larger and smaller circles of the annulus 119897 respectively

The number of transmitters located in a specific annulus119897 isin 1 119871 represented by the random variable (RV)119883119897 isapproximated by a Poisson process being its ProbabilityMassFunction (PMF) for a finite domain given by [23]

119875 (119883119897 = 119896) = ((120573119897119860 119897)119896 119896) 119890minus120573119897119860119897sum119899119894=0 ((120573119897119860 119897)119894 119894) 119890minus120573119897119860119897

119896 = 0 1 119899(2)

where 120573119897 is the spatial density of the RF nodes transmitting inthe annulus and 119899 is the total number of mobile nodes

22 Propagation Assumptions We consider that the RFpower 119868119894 received by the harvester119873119867 from the RF transmit-ter 119894 is given by

119868119894 = 119875119879119909120595119894119903minus120572119897 (3)

where 119875119879119909 is the transmitted power level of the 119894th RFtransmitter (119875119879119909 = 20 times 103mW is assumed for each node)and 120595119894 is an instant value of the fading and shadowing gainobserved in the channel between the receiver 119873119867 and thetransmitter node 119894 119903119897 represents the distance between the 119894thtransmitter and the receiver The values 119903119897 and 120595119894 representinstant values of the random variables 119877119897 andΨ119894 respectively120572 represents the path-loss coefficient

The PDF of 119877119897 can be written as the ratio between theperimeter of the circle with radius 119909 and the total area 119860 119897being represented as follows

119891119877119897 (119909) = 2120587119909119860 119897

119877119897119894 lt 119909 lt 1198771198971199000 otherwise (4)

To characterize the distribution of Ψ119894 the small-scalefading (fast fading) and shadowing (slow fading) effectsmust be considered The amplitude of the small-scale fadingeffect is assumed to be distributed according to a Rayleighdistribution which is represented by

119891120577 (119909) = 1199091205902120577

119890minus119909221205902120577 (5)

where 119909 is the envelope amplitude of the received signal 21205902120577is the mean power of the multipath received signal 21205902120577 = 1is adopted in this work to consider the case of normalizedpower

Regarding the shadowing effect we have assumed that itfollows a log-normal distribution

119891120585 (119909) = 1radic2120587120590120585119909119890minus(ln(119909)minus120583)221205902120585 (6)

where 120590120585 is the shadow standard deviation when 120583 = 0The standard deviation is usually expressed in decibels andis given by 120590120585dB = 10120590120585 ln(10) For 120590120585 rarr 0 no shadow-ing results Although (6) appears to be a simple expres-sion it is often inconvenient when further analyses arerequired Consequently [24] has shown that the log-normaldistribution can be accurately approximated by a Gammadistribution defined by

119891120585 (119909) asymp 1Γ (120599) ( 120599120596119904)120599 119909120599minus1119890minus119909(120599120596119904) (7)

where 120599 is equal to 1(1198901205902120585 minus1) and120596119904 is equal to 119890120583radic(120599 + 1)120599Γ(sdot) represents the Gamma functionThe probability density function ofΨ119894 is thus represented

by

119891Ψ119894 (119909) asymp 1198911205772 (119909) sdot 119891120585 (119909)asymp 2Γ (120599) ( 120599120596119904)

(120599+1)2 119909(120599minus1)2119870120599minus1(radic4120599119909120596119904 ) (8)

which is the Generalized-119870 distribution where119870120599minus1(sdot) is themodified Bessel function of the second kind

Due to the analytical difficulties of the Generalized-119870 distribution an approximation of the PDF (8) by amore tractable PDF is needed Reference [25] provides anapproximation of theGeneralized-119870distribution by using themoment matching method to determine the parameters ofthe approximated Gamma distribution With this method


[25] shows that the scale (120579120595) and shape (119896120595) parameters ofthe Gamma distribution are given by

120579120595 = (2 (120599 + 1)120599 minus 1)120596119904 (9)

119896120595 = 12 (120599 + 1) 120599 minus 1 (10)

respectively

3 RF Received Power

31 RF Received Power due to the Transmitters Located withinthe Annulus 119897 The amount of RF power received by theharvester node 119873119867 located in the centre of an annulus 119897 isgiven by

Σ = 119899119897sum119894=1

119868119894 (11)

where 119868119894 is the RF power received from the 119894th transmitter and119899119897 is the total number of transmitters in the annulus 119897Let 119872119894

119868(119904) represent the MGF of the 119894th transmitterlocated within the annulus (119894 = 1 119899119897) given by

119872119894119868 (119904) = E119868119894 [119890119904119868119894] = EΨ119894 [E119877119897 [119890119904119868119894]] (12)

Using the PDF of the distance given in (4) and the PDF of thesmall-scale fading and shadowing effects in (8) the MGF ofthe power received by the node 119873119867 from the 119894th transmitterin (12) can be written as follows

119872119894119868 (119904) = int+infin

0int119877119897119900119877119897119894

119890119904119868119894119891119877119897 (119903119897) 119891Ψ119894 (120595119894) 119889119903119897 119889120595119894 (13)

which using (3) (9) (10) and (4) can be simplified to

119872119894119868 (119904) = 2120587

119860 119897 (2 + 119896120595120572) (119875119879119909120579120595119904)119896120595sdot ((119877119897119900)2+119896120595120572 984858 (119877119897119900) minus (119877119897119894)2+119896120595120572 984858 (119877119897119894))

(14)

where 984858(119909) = 2F1(119896120595 119896120595+2120572 1+119896120595+2120572 minus119909120572119875119879119909120579120595119904) and2F1 represents the Gauss Hypergeometric function [26]

Departing from the fact that the individual power 119868119894 isindependent and identically distributed when compared tothe other transmitters the PDF of the aggregate RF power 119868given a total of 119896 active transmitters is the convolution of thePDFs of each 119868119894 Following this rationale theMGFof 119868 is givenby

119872119868119896 (119904) = 1198721119868 (119904) times 1198722

119868 (119904) times sdot sdot sdot times 119872119896119868 (119904)

= (119872119894119868 (119904))119896 (15)

Using the law of total probability the PDFof theRF power119868 can be written as

119891119868 (119895) = 119899sum119896=0

119891119868 (119895 | 119883119897 = 119896)P (119883119897 = 119896) (16)

leading to the MGF of the aggregate power 119868 which can bewritten as

E [119890119904119868] = 119899sum119896=0

P (119883119897 = 119896)int+infinminusinfin

119890119904119895119891119868 (119895 | 119883119897 = 119896) 119889119895= 119899sum

119896=0

P (119883119897 = 119896)119872119868119896 (119904) (17)

Using (15) the MGF of 119868 is given as follows

E [119890119904119868] = 119899sum119896=0

P (119883119897 = 119896) 119890119896 ln(119872119894119868(119904)) (18)

Using the MGF of the Poisson distribution in (18) theMGF of 119868 is finally given by

E [119890119904119868] = 119890120573119897119860119897(119872119894119868(119904)minus1) (19)

The first- and second-order statistics of the aggregate RFpower received by 119873119867 from the transmitters located withinthe annulus 119897 are an important tool E[119868] the expected valueof the aggregate RF power can be determined by using theLaw of Total Expectation It can be shown that

E [119868] = E [E [119868 | 119883119897]]= 2120587120573119897119875119879119909radic1198901205902120585 ((119877119897119900)

2minus120572 minus (119877119897119894)2minus1205722 minus 120572 ) (20)

Making similar use of the Law of Total Variance the varianceof the aggregate RF power can be described as

Var [119868] = Var [119868119894]E [119883119897] + E [119868119894]2 Var [119883119897] (21)

Since119883119897 is given by a Poisson distribution (with mean 120573119897119860 119897)the variance of the RF power is given as follows

Var [119868] = 120573119897119860 119897 (1205972119872119894119868 (0)1205971199042 )

= 12058712057311989711987521198791199091198961205951205792120595 (1 + 119896120595)((119877119897119900)2minus2120572 minus (119877119897119894)2minus21205721 minus 120572 )

(22)

The first and second moments can be matched with therespectivemoments of a given distribution to obtain a closed-form approximation for the aggregate received RF powerAs shown in [27] the aggregate RF power due to path lossfast fading and shadowing effect can be approximated by aGamma distribution Consequently the shape and the scaleparameters of the Gamma distribution denoted by 119896119897 and 120579119897are respectively given by

119896119897 asymp E [119868]2Var [119868] (23)

120579119897 asymp Var [119868]E [119868] (24)


32 RF Received Power due to the Transmitters Located within119871 Annuli As shown in the previous subsection the RFpower 119868 received from the transmitters located within the 119897thannulus is approximated by aGammadistribution withMGF119872119897

119868(119904) = (1minus120579119897119904)minus119896119897 Since the annulus of width119877119871119900 minus1198771119894 wherethe transmitters are located can be expressed as a summationof 119871 annuli of width 120588 the MGF of the aggregate powerreceived from the transmitters located within the 119871 annuli isgiven by

119872119868agg(119904) = 119871prod

119897=1

(1 minus 120579119897119904)minus119896119897 (25)

Finally the expectation of the aggregate RF power can becomputed as follows

E [119868agg] = 120597119872119868agg(0)120597119904 (26)

33 Distribution of the Aggregate RF Power The aggregate RFpower may be stated as being the summation of the 119871 indi-vidual aggregated RF powers received from the transmitterslocated within each annulus Expressions for the PDF and theCDF of the summation of 119871 independent Gamma randomvariables were initially derived by Mathai in [28] Those weresimplified in [29] in order to be computed more efficiently

Let 119885119897119871119897=1 be independent but not necessarily identicallydistributed Gamma variables with parameters 119896119897 (shape) and120579119897 (scale) The PDF of the aggregate RF power is written as119868agg = sum119871

119897=1 119885119897 which can be approximated by [29]

119891119868agg (119909)asymp 119871prod

119897=1

(1205791120579119897 )119896119897 +infinsum119908=0

120575119908119909(sum119871119897=1 119896119897+119908minus1) exp (minus1199091205791)120579(sum119871119897=1 119896119897+119908)1 Γ (sum119871119897=1 119896119897 + 119908)

(27)

where 1205791 = min119897120579119897120575119908 coefficients are computed recursively120575119908+1 = (1(119908+1))sum119908+1119894=1 [sum119871

119897=1 119896119897(1minus1205791120579119897)119894]120575119908+1minus119894 and 1205750 = 1Γ(sdot) is theGamma function Finally the CDF of 119868agg119865119868agg(119909) =int119909minusinfin119891119868agg(119911)119889119911 is computed as follows [29]

119865119868agg (119909) asymp 119871prod119897=1

(1205791120579119897 )119896119897 infinsum119908=0

120575119908120579sum119871119897=1 119896119897+1199081 Γ (sum119871119897=1 119896119897 + 119908)

times int1199090119911sum119871119897=1 119896119897+119908minus1 exp(minus 1199111205791)119889119911

(28)

4 Probability of Transmission

41 Gaussian Approach Departing from the fact that the RFpower received from the transmitters located in the annulus119897 can be approximated by a Gamma distribution

Gamma (119896119897 120579119897) (29)

with 119896119897 and 120579119897 given by (23) and (24) respectively theenvelope signal (amplitude) received from the transmittersis given by the square root of a Gamma distributed randomvariable which is given by aGeneralizedGammadistributionwith the following parameters

119860119897119868 = GG(radic120579119897 2119896119897 2) (30)

Since a Gamma distribution with shape 119896119897 and scale 120579119897 is thesum of 119896 Exponential (1120579119897) distributions using the CentralLimit Theorem (CLT) when 119896119897 is large the GeneralizedGamma distribution can be approximated by a normaldistribution [30] In these conditions the amplitude of theaggregate signals received by the harvester 119873119867 from thetransmitters located in the annulus 119897 can be also approximatedby a normal distribution represented by

119860119897119868 asympN (120583119860119897119868 1205902119860119897119868) asympN(120578radic120579119897 Γ (119896119897 + 12)Γ (119896119897) 1205782120579119897(Γ (119896119897 + 1)Γ (119896119897) minus Γ (119896119897 + 12)2Γ (119896119897)2 ))

(31)

where the loss factor 0 lt 120578 lt 1 represents the losses asso-ciated with the RF-to-DC conversion and battery chargingefficiency

During the battery charging period the received power(119860119897119868)2 is accumulated in a discrete period of time Δ 119905 Theamount of energy stored in the battery of the harvester nodeduring 119899119905 time intervals is given by

120598119897 = Δ 119905

119899119905sum119899=1

10038161003816100381610038161003816119860119897119868100381610038161003816100381610038162 (32)

Considering the unit variance random variable 1205981015840119897 =sum119899119905119899=1 |119860119897119868120590119860119897119868 |2 with

120598119897 = 1205981015840119897 times 1205902119860119897119868 (33)

and considering Δ 119905 = 1 for the sake of simplicity 1205981015840119897 followsa noncentral Chi-squared distribution with noncentralityparameter 120582119897 = sum119899119905

119896=1(120583119860119897119868120590119860119897119868)2 When 119899119905 is large enough itis possible to use the Central Limit Theorem to approximatethe Chi-square distribution to a Gaussian distribution [31]and the following approximation holds

1205981015840119897 asympN (119899119905 + 120582119897 2 (119899119905 + 2120582119897)) (34)

Using (33) the energy accumulated in the battery of theharvester node 119873119867 due to the transmitters located in theannulus 119897 follows the following Gaussian distribution

120598119897 asympN (120583119897 1205902119897 )asympN (1205902119860119897119868 (119899119905 + 120582119897) 1205904119860119897119868 [2 (119899119905 + 2120582119897)])

(35)


Because 119871 annuli are considered the energy accumulated inthe battery of the harvester node119873119867 due to the transmitterslocated in the 119871 annuli is given by

120598 = 119871sum119897=1

120598119897 (36)

and 120598 follows the following distribution120598 asympN( 119871sum

119897=1

120583119897 119871sum119897=1

1205902119897 ) asympN (120583Σ119871 1205902Σ119871) (37)

Therefore denoting 120574 as the level of battery charge(accumulated energy) required to transmit a packet theprobability of reaching a 120574 level of accumulated energy after119899119905 units of time is given by

119875119862 = Q(120574 minus 120583Σ1198711205902Σ119871 ) (38)

where Q(119909) = (1radic2120587) intinfin119909119890minus11990622119889119906 is the complementary

distribution function of the standard normal

42 Non-Gaussian Approach In the last subsection the CLTwas used to approximate the amplitude of the aggregatesignals received by the harvester 119873119867 from the transmitterslocated in the annulus 119897 119860119897119868 as represented in (31) Howeverwhen 119896119897 is small theCLTdoes not hold and consequently theGaussian approach is not valid In what follows we present aformulation when CLT does not hold While the formulationis more computationally complex it exhibits higher accuracyfor low 119896119897 values

Departing from the MGF in (25) the CharacteristicFunction (CF) of the RF power received from the annulus 119897is written as

120593119897 (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (39)

where 0 lt 120578 lt 1 is the loss factor When the aggregate powerfrom the 119871 annuli is considered the CF is written as follows

120593119868agg (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (40)

Because the amount of energy received in 119899119905 samples isexpressed as

120598 = 119899119905sum119899=1

119868agg (119905) (41)

the CF of 120598 is as follows120593120598 (119905) = 119899119905prod

119887=1

(120593119868agg (119905)) = 119899119905prod119887=1

( 119871prod119897=1

(1 minus 120578120579119897119894119905)120581119897)= 119871prod

119897=1

[(1 minus 120578120579119897119894119905)120581119897]119899119905 (42)

Table 1 Parameters adopted in the validation and simulations

120588 400 200 100 4m119871 2 4 100120573119897 1 2 3 4 times 10minus7 nodesm2

120572 2120578 05Δ 119905 1 minute1198771119894 10m119877119871119900 410m119875119879119909 20W120590120585dB 45 dB120574 10mAh119899119905 20

Using the Fourier Transform the PDF of 120598 is written as

119891120598 (119909) = 12120587 int+infin

minusinfin119890minus119894119905119909120593120598 (119905) 119889119905 (43)

Again the probability of reaching a 120574 level of accumulatedenergy after 119899119905 units of time is given by

119875119862 (120598 gt 120574) = 1 minus 119875 (120598 le 120574) = 1 minus int120574minusinfin119891120598 (119909) 119889119909 (44)

5 Validation Results

This section describes a set of simulations and numericalresults to validate the analyticalmethodology proposed in thepaper The simulated scenario considered a spatial circulararea 119860 as described in Section 2 with 1198771119894 = 10m and119877119871119900 = 410m The multiple nodes were spread over the area 119860according to the spatial Poisson process and 4 different spatialdensities were simulated 120573119897 = 1 2 3 4 times 10minus4 nodesm2In each simulation the RF propagation scenario described inSection 2 was parametrized with 120572 = 2 and 120590120585dB = 45 dBFinally we have considered the battery operation voltageequal to 1 V the loss factor 120578 = 05 and the required energythreshold to transmit a packet (120574) equal to 10mAh Theparameters used in the validation are summarized in Table 1

The first results presented in Figure 2 compare the CDFof the RF received power computed with (28) The simulatedresults were obtained for the spatial density value 120573119897 = 1 times10minus4 In the model a different number (119871 = 2 4 100) ofannuli were adopted to compute the model and compare theaccuracy of the model for different number of annuli Ascan be seen the accuracy of the model increases with thenumber of annuli considered in the model This is becauseas more annuli are considered for the same circular area119860 = 120587((119877119871119900 )2 minus (1198771119894 )2) the width of each annulus 120588 decreasesleading to a more accurate value of the mean and variance(E[119868] and Var[119868] resp) of the RF power received from thetransmitters located in a single annuli This fact increases theaccuracy of the conditions in (23) and (24) leading to a moreaccurate characterization of the distribution of the receivedRF power From the results plotted in Figure 2 we observethat for 119871 = 100 the numerical results are close to the results

Mathematical Problems in Engineering 7CD

F

0

01

02

03

04

05

06

07

08

09

1

Simulation

Iagg (mW)101 102

Model L = 4

Model L = 2 Model L = 100

Figure 2 CDF of the RF power received by the node 119873119867 from thetransmitters located in the area 119860 = 120587((119877119871119900 )2 minus (1198771119894 )2)

CDF

0

01

02

03

04

05

06

07

08

09

1

Iagg (mW)101 102

1205731 = 1e minus 4

1205731 = 2e minus 4

1205731 = 3e minus 4

1205731 = 4e minus 4

Figure 3 CDFof theRFpower received by the node119873119867 consideringdifferent densities of transmitters (120573119897 in nodes per square meter)located in the area 119860 = 120587((119877119871119900 )2 minus (1198771119894 )2)obtained through simulation confirming the accuracy of theproposed model

The numerical results presented in Figure 3 comparethe CDF of the RF received power (computed with (28))for different spatial density values (120573119897 = 1 times 10minus4 2 times10minus4 3times10minus4 4times10minus4)The numerical results were obtainedconsidering 119871 = 100 (consequently 120588 = 4) As expectedthe results confirm that the average of the RF power receivedby the harvester increases with the spatial density of thetransmitters

In Figure 4 we compare the probability of having enoughenergy accumulated in the harvester battery to transmit a

Time (minutes)0 2 4 6 8 10 12 14 16 18 20

0

01

02

03

04

05

06

07

08

09

1

Simulation

PC

Model L = 1


Figure 4 Gaussian approach model probability of having enoughenergy accumulated in the harvester battery to transmit a packet(119875119862) for different density 120573 = 4 times 10minus4 of transmitters (120573119897 in nodesper square meter) located in the area 119860 = 120587((119877119871119900 )2 minus (1198771119894 )2)

packet (119875119862) The probability was computed using (38) thatis the Gaussian approach was adopted The simulated valueswere obtained for the spatial density value 120573119897 = 4 times 10minus4nodesm2 The charging threshold 120574 was defined to 10mAhand we have considered a battery voltage of 1 V The modelwas computed for 119871 = 1 (120588 = 400) 119871 = 2 (120588 = 200) and 119871 =40 (120588 = 10) As can be observed the results computed withthemodel do notmatchwith the ones obtained by simulationThis fact was intentionally exploited to show that while fordifferent parameterizations the model and the simulationresults match for the specific parameterization adopted inthe validation scenario the CLT does not hold because the119896119897 values are too small Consequently (30) is not an accurateapproximation and a large deviation of the 119875119862rsquos model isobserved In this case it would be better to adopt the non-Gaussian approach because it leads to more accurate modelresults

In Figure 5 we compare the probability of having enoughenergy accumulated in the harvester battery to transmit apacket (119875119862) The probability was computed using (44) thatis the non-Gaussian approach for 119871 = 2 (120588 = 200) 119871 = 4(120588 = 100) and 119871 = 40 (120588 = 10) The simulated values are thesame as depicted in Figure 4 that is the considered scenariois the same As can be observed the results computedwith themodel do not match with the ones obtained by simulation for119871 = 2 and 119871 = 4 However if more annuli are used the modelaccurately characterizes 119875119862 as is the case for 119871 = 40 in thefigureThis fact is due to the approximation of 119896119897 and 120579119897 in (23)and (24) respectively As the number of annuli (119871) increasesE[119868] and Var[119868] in (23) and (24) become more accurate

As can be observed the results computed with the modelfor 119871 = 40 are close to the ones obtained by simulationMore-over the probability of reaching a battery charging level equal


0

01

02

03

04

05

06

07

08

09

1

Time (minutes)0 2 4 6 8 10 12 14 16 18 20

Simulation

PC

Model L = 4


Figure 5 Non-Gaussian approach model probability of havingenough energy accumulated in the harvester battery to transmit apacket (119875119862) for different density 120573 = 4 times 10minus4 of transmitters (120573119897 innodes per square meter) located in the area 119860 = 120587((119877119871119900 )2 minus (1198771119894 )2)

to the 120574 threshold increases over time as expectedThe resultsconfirm the accuracy of the proposed characterization whichmay be easily adopted to evaluate the probability of chargingover time

Finally we highlight that the mean aggregate RF power(119868agg) considered in the validation scenario is low to showthe error of the Gaussian approach For higher 119868agg valuesthe error of the Gaussian approach becomes smaller and themodel becomes more accurate The non-Gaussian approachis generally a better solution (because it does not dependon the CLT) however it exhibits a higher computationalcomplexity

6 Final Remarks

In this paper we have characterized the battery chargingtime of a harvester node that accumulates the received RFenergy in a battery Admitting that the transmitters arespatially distributed according to a spatial Poisson processwe use the distribution of the received RF power frommultiple transmitters to derive the probability of a harvesterhaving enough energy to transmit a packet after a givenamount of charging time The distribution of the RF powerand the probability of a harvester node having enoughenergy to transmit a packet are validated through simulationThe numerical results obtained with the proposed analysisare close to the ones obtained through simulation whichconfirms the accuracy of the proposed analysis

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the Portuguese Scienceand Technology Foundation (FCTMEC) under the projectUIDEEA500082013

References

[1] T Soyata L Copeland and W Heinzelman ldquoRF energyharvesting for embedded systems a survey of tradeoffs andmethodologyrdquo IEEE Circuits and Systems Magazine vol 16 no1 pp 22ndash57 2016

[2] X Lu P Wang D Niyato D I Kim and Z Han ldquoWirelessnetworks with RF energy harvesting a contemporary surveyrdquoIEEE Communications Surveys and Tutorials vol 17 no 2 pp757ndash789 2015

[3] Y Yao J Wu Y Shi and F F Dai ldquoA fully integrated 900-MHzpassive RFID transponder front end with novel zero-thresholdRF-DC rectifierrdquo IEEE Transactions on Industrial Electronicsvol 56 no 7 pp 2317ndash2325 2009

[4] T Salter K Choi M Peckerar GMetze andN Goldsman ldquoRFenergy scavenging system utilising switched capacitor DC-DCconverterrdquo Electronics Letters vol 45 no 7 pp 374ndash376 2009

[5] G Papotto F Carrara and G Palmisano ldquoA 90-nm CMOSthreshold-compensated RF energy harvesterrdquo IEEE Journal ofSolid-State Circuits vol 46 no 9 pp 1985ndash1997 2011

[6] S Scorcioni L Larcher A Bertacchini L Vincetti and MMaini ldquoAn integrated RF energy harvester for UHF wirelesspowering applicationsrdquo in Proceedings of the IEEE WirelessPower Transfer (WPT rsquo13) pp 92ndash95 IEEE Perugia Italy May2013

[7] B R Franciscatto V Freitas J M Duchamp C Defay andT P Vuong ldquoHigh-efficiency rectifier circuit at 245GHz forlow-input-power RF energy harvestingrdquo in Proceedings of theEuropeanMicrowave Conference (EuMC rsquo13) pp 507ndash510 IEEENuremberg Germany October 2013

[8] H Ju and R Zhang ldquoThroughput maximization in wire-less powered communication networksrdquo IEEE Transactions onWireless Communications vol 13 no 1 pp 418ndash428 2014

[9] R Zhang and C K Ho ldquoMIMO broadcasting for simultaneouswireless information and power transferrdquo IEEE Transactions onWireless Communications vol 12 no 5 pp 1989ndash2001 2013

[10] X Zhou R Zhang and C K Ho ldquoWireless information andpower transfer architecture design and rate-energy tradeoffrdquoIEEE Transactions on Communications vol 61 no 11 pp 4754ndash4767 2013

[11] N Pappas J Jeon A Ephremides and A Traganitis ldquoOptimalutilization of a cognitive shared channel with a rechargeable pri-mary source noderdquo Journal of Communications and Networksvol 14 no 2 pp 162ndash168 2012

[12] A M Ibrahim O Ercetin and T ElBatt ldquoStability analysis ofslotted aloha with opportunistic RF energy harvestingrdquo IEEEJournal on Selected Areas in Communications vol 34 no 5 pp1477ndash1490 2016

[13] A Ghazanfari H Tabassum and E Hossain ldquoAmbient RFenergy harvesting in ultra-dense small cell networks perfor-mance and trade-offsrdquo IEEE Wireless Communications vol 23no 2 pp 38ndash45 2016

[14] X Lu I Flint D Niyato N Privault and P Wang ldquoSelf-sustainable communicationswith RF energy harvesting ginibrepoint process modeling and analysisrdquo IEEE Journal on SelectedAreas in Communications vol 34 no 5 pp 1518ndash1535 2016


[15] X Lu P Wang D Niyato D I Kim and Z Han ldquoWirelessnetworks with rf energy harvesting a contemporary surveyrdquoIEEE Communications Surveys and Tutorials vol 17 no 2 pp757ndash789 2015

[16] L Mohjazi M Dianati G K Karagiannidis S Muhaidatand M Al-Qutayri ldquoRF-powered cognitive radio networkstechnical challenges and limitationsrdquo IEEE CommunicationsMagazine vol 53 no 4 pp 94ndash100 2015

[17] X Lu P Wang D Niyato and E Hossain ldquoDynamic spectrumaccess in cognitive radio networks with RF energy harvestingrdquoIEEE Wireless Communications vol 21 no 3 pp 102ndash110 2014

[18] S Lee R Zhang and K Huang ldquoOpportunistic wireless energyharvesting in cognitive radio networksrdquo IEEE Transactions onWireless Communications vol 12 no 9 pp 4788ndash4799 2013

[19] S Khoshabi Nobar K Adli Mehr and J Musevi Niya ldquoRF-powered green cognitive radio networks architecture andperformance analysisrdquo IEEE Communications Letters vol 20no 2 pp 296ndash299 2016

[20] D T Hoang D Niyato P Wang and D I Kim ldquoOpportunisticchannel access and RF energy harvesting in cognitive radionetworksrdquo IEEE Journal on Selected Areas in Communicationsvol 32 no 11 pp 2039ndash2052 2014

[21] A Bhowmick S D Roy and S Kundu ldquoThroughput ofa cognitive radio network with energy-harvesting based onprimary user signalrdquo IEEE Wireless Communications Lettersvol 5 no 2 pp 136ndash139 2016

[22] A H Sakr and E Hossain ldquoCognitive and energy harvesting-based D2D communication in cellular networks stochasticgeometry modeling and analysisrdquo IEEE Transactions on Com-munications vol 63 no 5 pp 1867ndash1880 2015

[23] A Papoulis and S U Pillai Probability Random Variables andStochastic Processes McGraw-Hill New York NY USA 4thedition 2001

[24] A Abdi and M Kaveh ldquoOn the utility of gamma pdf in mod-eling shadow fading (slow fading)rdquo in Proceedings of the IEEE49th Vehicular Technology Conference vol 3 IEEE HoustonTex USA May 1999

[25] S Al-Ahmadi and H Yanikomeroglu ldquoOn the approximationof the generalized-120581 distribution by a gamma distribution formodeling composite fading channelsrdquo IEEE Transactions onWireless Communications vol 9 no 2 pp 706ndash713 2010

[26] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1965

[27] M Haenggi and R K Ganti Interference in Large WirelessNetworks Now Publishers Inc 2009

[28] A M Mathai ldquoStorage capacity of a dam with gamma typeinputsrdquoAnnals of the Institute of Statistical Mathematics vol 34no 1 pp 591ndash597 1982

[29] P G Moschopoulos ldquoThe distribution of the sum of inde-pendent gamma random variablesrdquo Annals of the Institute ofStatistical Mathematics vol 37 no 1 pp 541ndash544 1985

[30] N L Johnson and S Kotz Continuous Univariate DistributionsDistributions in Statistics vol 1 Houghton Mifflin 1970

[31] H Tang ldquoSome physical layer issues of wide-band cognitiveradio systemsrdquo in Proceedings of the 1st IEEE InternationalSymposium on New Frontiers in Dynamic Spectrum AccessNetworks (DySPAN rsquo05) pp 151ndash159 IEEE BaltimoreMdUSANovember 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


harvesting circuit To illustrate the design considerations ofRF-based harvesting networks [13] points out the primarychallenges of implementing and operating such networksincluding nondeterministic energy arrival patterns energyharvesting mode selection and energy-aware cooperationamong base stations Reference [14] adopts a stochasticgeometry framework based on the Ginibre model to ana-lyze the performance of self-sustainable communicationsover cellular networks with general fading channels Theexpectation of the RF energy harvesting rate the energyoutage probability and the transmission outage probabilityare evaluated over Nakagami-m fading channels

RF-EHNs may also act as cognitive radio networks(CRNs) that is using the spectrum in an opportunistic waywithout being licensed Several works have explored thesekinds of networks Reference [15] provides an overview of theRF-EHNs including system architecture RF energy harvest-ing techniques and existing applications The authors alsoexplore various key design issues in the development of RF-EHNs including cognitive radio networks The work in [16]provides a comprehensive overview of recent developmentand challenges regarding the operation of cognitive radionetworks powered by RF energy Spectrum efficiency andenergy efficiency are two critical issues in designing cognitiveradio RF-EHNs Reference [17] provides an overview of theRF-powered CRNs and discusses the challenges that arisefor dynamic spectrum access in these networks Focusingon the trade-off among spectrum sensing data transmissionand RF energy harvesting the authors discuss the dynamicchannel selection problem in a multichannel RF-poweredCRN Reference [18] proposes a novel method for wirelessnetworks coexistingwhere low-powermobiles in a secondarynetwork harvest ambient RF energy from transmissions bynearby active transmitters while opportunistically accessingthe spectrum licensed to the primary network The authorsanalyze the transmission probability of harvesting terminalsand the resulting spatial throughput The optimal trans-mission power and terminalsrsquo density are also derived formaximizing the throughput The work in [19] considers anRF-powered green cognitive radio network where a centralnode harvests energy from ambient sources and wirelesslydelivers random harvested energy to cognitive users Thework evaluates the performance of such a network showingthe feasibility of the behavior if the energy transmissionrate is below a certain threshold Reference [20] considersa network where the unlicensed users can perform channelaccess to transmit a packet or to harvest RF energy whenthe selected channel is idle or occupied by the primaryuser respectively The work is mainly focused on finding thechannel access policy that maximizes the throughput of thesecondary user Reference [21] analyzes an energy harvesting-based cognitive radio system to find the optimal spectrumsensing time which maximizes the harvested energy Thework in [22] analyzes a cognitive and energy harvesting-based device-to-device (D2D) communication in cellularnetworksThe authors employ tools from stochastic geometryto evaluate the performance of the proposed communicationsystem model with general path-loss exponent in termsof outage probability for D2D and cellular users One of

the work conclusions is that energy harvesting can be areliable alternative to power cognitive D2D transmitterswhile achieving acceptable performance

In this work we are particularly focused on the charac-terization of the RF power received by each harvester and itsimpact in terms of the probability of accumulating enoughenergy to transmit a packet A generalized radio propagationenvironment is considered Assuming that the sources ofhigh power RF signals (eg base stations) are distributedaccording to a spatial Poisson process we characterize thedistribution of the received RF power from the multipletransmitters Path loss shadowing and fading effects areconsidered The distribution of the RF power is then usedto derive the probability of a harvester node having enoughenergy to transmit a packet after a given period of time Asoft computational model (Gaussian approach) and a morecomplex model (non-Gaussian approach) are presented tocompute the probability of a harvester node having enoughenergy to transmit These are the main contributions ofthe paper Considering multiple spatial and propagationscenarios we validate the distribution of the RF powerand the probability of a harvester have enough energy totransmit a packet The numerical results obtained with theproposed analysis are close to the ones obtained throughsimulation which confirms the accuracy of the proposedanalysis In this way we provide a characterization of thebattery charging time considering innovative assumptionsincluding the spatial distribution of the RF transmitters thepropagation effects and the losses associated with the RF-to-DC conversion and battery charging process The proposedmodel can thus be adopted to determine the probability of aharvester accumulating enough energy after a given periodof time which is a determinant condition to compute thethroughput of RF-EHNs As far as we known this is thefirst work to derive such a probability when the multiple RFsignals received by the harvesters are differently affected bymultiple propagation effects

The rest of the paper is organized as follows The systemdescription is presented in Section 2 The characterizationof the received RF power is characterized in Section 3Section 4 derives the probability of accumulating enoughenergy to transmit a packet through the Gaussian and non-Gaussian approaches Finally validation results are presentedin Section 5 and conclusions are drawn in Section 6

2 System Description

This work considers a RF energy harvesting network whereeach node accumulates energy from the base stations andother RF transmitters located in the neighborhood A har-vester node is able to initiate a packet transmission wheneverthe level of accumulated energy is above a transmissionthreshold

21 Spatial Distribution of the RF Transmitters We considerthe scenario illustrated in Figure 1 where the node 119873119867 (theharvester node) accumulates energy from the transmittersthat might be located in the area 119860 = 120587((119877119871119900 )2 minus (1198771119894 )2) The



R1i

R1o equiv R2

i

R2o equiv R3

i

RLi

RLo





119860 asymp 119871sum119897=1

119860 119897 (1)



119875 (119883119897 = 119896) = ((120573119897119860 119897)119896 119896) 119890minus120573119897119860119897sum119899119894=0 ((120573119897119860 119897)119894 119894) 119890minus120573119897119860119897

119896 = 0 1 119899(2)



119868119894 = 119875119879119909120595119894119903minus120572119897 (3)



119891119877119897 (119909) = 2120587119909119860 119897

119877119897119894 lt 119909 lt 1198771198971199000 otherwise (4)


119891120577 (119909) = 1199091205902120577

119890minus119909221205902120577 (5)







by







120579120595 = (2 (120599 + 1)120599 minus 1)120596119904 (9)

119896120595 = 12 (120599 + 1) 120599 minus 1 (10)

respectively

3 RF Received Power


Σ = 119899119897sum119894=1

119868119894 (11)



119872119894119868 (119904) = E119868119894 [119890119904119868119894] = EΨ119894 [E119877119897 [119890119904119868119894]] (12)


119872119894119868 (119904) = int+infin

0int119877119897119900119877119897119894

119890119904119868119894119891119877119897 (119903119897) 119891Ψ119894 (120595119894) 119889119903119897 119889120595119894 (13)


119872119894119868 (119904) = 2120587

119860 119897 (2 + 119896120595120572) (119875119879119909120579120595119904)119896120595sdot ((119877119897119900)2+119896120595120572 984858 (119877119897119900) minus (119877119897119894)2+119896120595120572 984858 (119877119897119894))

(14)



119872119868119896 (119904) = 1198721119868 (119904) times 1198722


= (119872119894119868 (119904))119896 (15)


119891119868 (119895) = 119899sum119896=0

119891119868 (119895 | 119883119897 = 119896)P (119883119897 = 119896) (16)


E [119890119904119868] = 119899sum119896=0


119890119904119895119891119868 (119895 | 119883119897 = 119896) 119889119895= 119899sum

119896=0

P (119883119897 = 119896)119872119868119896 (119904) (17)


E [119890119904119868] = 119899sum119896=0

P (119883119897 = 119896) 119890119896 ln(119872119894119868(119904)) (18)


E [119890119904119868] = 119890120573119897119860119897(119872119894119868(119904)minus1) (19)


E [119868] = E [E [119868 | 119883119897]]= 2120587120573119897119875119879119909radic1198901205902120585 ((119877119897119900)



Var [119868] = Var [119868119894]E [119883119897] + E [119868119894]2 Var [119883119897] (21)


Var [119868] = 120573119897119860 119897 (1205972119872119894119868 (0)1205971199042 )


(22)


119896119897 asymp E [119868]2Var [119868] (23)

120579119897 asymp Var [119868]E [119868] (24)




119872119868agg(119904) = 119871prod

119897=1

(1 minus 120579119897119904)minus119896119897 (25)


E [119868agg] = 120597119872119868agg(0)120597119904 (26)




119891119868agg (119909)asymp 119871prod

119897=1

(1205791120579119897 )119896119897 +infinsum119908=0


(27)



119865119868agg (119909) asymp 119871prod119897=1

(1205791120579119897 )119896119897 infinsum119908=0

120575119908120579sum119871119897=1 119896119897+1199081 Γ (sum119871119897=1 119896119897 + 119908)


(28)



Gamma (119896119897 120579119897) (29)


119860119897119868 = GG(radic120579119897 2119896119897 2) (30)



(31)



120598119897 = Δ 119905

119899119905sum119899=1

10038161003816100381610038161003816119860119897119868100381610038161003816100381610038162 (32)


120598119897 = 1205981015840119897 times 1205902119860119897119868 (33)



1205981015840119897 asympN (119899119905 + 120582119897 2 (119899119905 + 2120582119897)) (34)


120598119897 asympN (120583119897 1205902119897 )asympN (1205902119860119897119868 (119899119905 + 120582119897) 1205904119860119897119868 [2 (119899119905 + 2120582119897)])

(35)



120598 = 119871sum119897=1

120598119897 (36)


119897=1

120583119897 119871sum119897=1

1205902119897 ) asympN (120583Σ119871 1205902Σ119871) (37)


119875119862 = Q(120574 minus 120583Σ1198711205902Σ119871 ) (38)





120593119897 (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (39)


120593119868agg (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (40)


120598 = 119899119905sum119899=1

119868agg (119905) (41)


119887=1

(120593119868agg (119905)) = 119899119905prod119887=1

( 119871prod119897=1

(1 minus 120578120579119897119894119905)120581119897)= 119871prod

119897=1

[(1 minus 120578120579119897119894119905)120581119897]119899119905 (42)





119891120598 (119909) = 12120587 int+infin








F

0

01

02

03

04

05

06

07

08

09

1

Simulation

Iagg (mW)101 102

Model L = 4



CDF

0

01

02

03

04

05

06

07

08

09

1

Iagg (mW)101 102

1205731 = 1e minus 4

1205731 = 2e minus 4

1205731 = 3e minus 4

1205731 = 4e minus 4




Time (minutes)0 2 4 6 8 10 12 14 16 18 20

0

01

02

03

04

05

06

07

08

09

1

Simulation

PC

Model L = 1







0

01

02

03

04

05

06

07

08

09

1

Time (minutes)0 2 4 6 8 10 12 14 16 18 20

Simulation

PC

Model L = 4





6 Final Remarks


Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











R1i

R1o equiv R2

i

R2o equiv R3

i

RLi

RLo





119860 asymp 119871sum119897=1

119860 119897 (1)



119875 (119883119897 = 119896) = ((120573119897119860 119897)119896 119896) 119890minus120573119897119860119897sum119899119894=0 ((120573119897119860 119897)119894 119894) 119890minus120573119897119860119897

119896 = 0 1 119899(2)



119868119894 = 119875119879119909120595119894119903minus120572119897 (3)



119891119877119897 (119909) = 2120587119909119860 119897

119877119897119894 lt 119909 lt 1198771198971199000 otherwise (4)


119891120577 (119909) = 1199091205902120577

119890minus119909221205902120577 (5)







by







120579120595 = (2 (120599 + 1)120599 minus 1)120596119904 (9)

119896120595 = 12 (120599 + 1) 120599 minus 1 (10)

respectively

3 RF Received Power


Σ = 119899119897sum119894=1

119868119894 (11)



119872119894119868 (119904) = E119868119894 [119890119904119868119894] = EΨ119894 [E119877119897 [119890119904119868119894]] (12)


119872119894119868 (119904) = int+infin

0int119877119897119900119877119897119894

119890119904119868119894119891119877119897 (119903119897) 119891Ψ119894 (120595119894) 119889119903119897 119889120595119894 (13)


119872119894119868 (119904) = 2120587

119860 119897 (2 + 119896120595120572) (119875119879119909120579120595119904)119896120595sdot ((119877119897119900)2+119896120595120572 984858 (119877119897119900) minus (119877119897119894)2+119896120595120572 984858 (119877119897119894))

(14)



119872119868119896 (119904) = 1198721119868 (119904) times 1198722


= (119872119894119868 (119904))119896 (15)


119891119868 (119895) = 119899sum119896=0

119891119868 (119895 | 119883119897 = 119896)P (119883119897 = 119896) (16)


E [119890119904119868] = 119899sum119896=0


119890119904119895119891119868 (119895 | 119883119897 = 119896) 119889119895= 119899sum

119896=0

P (119883119897 = 119896)119872119868119896 (119904) (17)


E [119890119904119868] = 119899sum119896=0

P (119883119897 = 119896) 119890119896 ln(119872119894119868(119904)) (18)


E [119890119904119868] = 119890120573119897119860119897(119872119894119868(119904)minus1) (19)


E [119868] = E [E [119868 | 119883119897]]= 2120587120573119897119875119879119909radic1198901205902120585 ((119877119897119900)



Var [119868] = Var [119868119894]E [119883119897] + E [119868119894]2 Var [119883119897] (21)


Var [119868] = 120573119897119860 119897 (1205972119872119894119868 (0)1205971199042 )


(22)


119896119897 asymp E [119868]2Var [119868] (23)

120579119897 asymp Var [119868]E [119868] (24)




119872119868agg(119904) = 119871prod

119897=1

(1 minus 120579119897119904)minus119896119897 (25)


E [119868agg] = 120597119872119868agg(0)120597119904 (26)




119891119868agg (119909)asymp 119871prod

119897=1

(1205791120579119897 )119896119897 +infinsum119908=0


(27)



119865119868agg (119909) asymp 119871prod119897=1

(1205791120579119897 )119896119897 infinsum119908=0

120575119908120579sum119871119897=1 119896119897+1199081 Γ (sum119871119897=1 119896119897 + 119908)


(28)



Gamma (119896119897 120579119897) (29)


119860119897119868 = GG(radic120579119897 2119896119897 2) (30)



(31)



120598119897 = Δ 119905

119899119905sum119899=1

10038161003816100381610038161003816119860119897119868100381610038161003816100381610038162 (32)


120598119897 = 1205981015840119897 times 1205902119860119897119868 (33)



1205981015840119897 asympN (119899119905 + 120582119897 2 (119899119905 + 2120582119897)) (34)


120598119897 asympN (120583119897 1205902119897 )asympN (1205902119860119897119868 (119899119905 + 120582119897) 1205904119860119897119868 [2 (119899119905 + 2120582119897)])

(35)



120598 = 119871sum119897=1

120598119897 (36)


119897=1

120583119897 119871sum119897=1

1205902119897 ) asympN (120583Σ119871 1205902Σ119871) (37)


119875119862 = Q(120574 minus 120583Σ1198711205902Σ119871 ) (38)





120593119897 (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (39)


120593119868agg (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (40)


120598 = 119899119905sum119899=1

119868agg (119905) (41)


119887=1

(120593119868agg (119905)) = 119899119905prod119887=1

( 119871prod119897=1

(1 minus 120578120579119897119894119905)120581119897)= 119871prod

119897=1

[(1 minus 120578120579119897119894119905)120581119897]119899119905 (42)





119891120598 (119909) = 12120587 int+infin








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Simulation

Iagg (mW)101 102

Model L = 4



CDF

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Iagg (mW)101 102

1205731 = 1e minus 4

1205731 = 2e minus 4

1205731 = 3e minus 4

1205731 = 4e minus 4




Time (minutes)0 2 4 6 8 10 12 14 16 18 20

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PC

Model L = 1







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PC

Model L = 4





6 Final Remarks


Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120579120595 = (2 (120599 + 1)120599 minus 1)120596119904 (9)

119896120595 = 12 (120599 + 1) 120599 minus 1 (10)

respectively

3 RF Received Power


Σ = 119899119897sum119894=1

119868119894 (11)



119872119894119868 (119904) = E119868119894 [119890119904119868119894] = EΨ119894 [E119877119897 [119890119904119868119894]] (12)


119872119894119868 (119904) = int+infin

0int119877119897119900119877119897119894

119890119904119868119894119891119877119897 (119903119897) 119891Ψ119894 (120595119894) 119889119903119897 119889120595119894 (13)


119872119894119868 (119904) = 2120587

119860 119897 (2 + 119896120595120572) (119875119879119909120579120595119904)119896120595sdot ((119877119897119900)2+119896120595120572 984858 (119877119897119900) minus (119877119897119894)2+119896120595120572 984858 (119877119897119894))

(14)



119872119868119896 (119904) = 1198721119868 (119904) times 1198722


= (119872119894119868 (119904))119896 (15)


119891119868 (119895) = 119899sum119896=0

119891119868 (119895 | 119883119897 = 119896)P (119883119897 = 119896) (16)


E [119890119904119868] = 119899sum119896=0


119890119904119895119891119868 (119895 | 119883119897 = 119896) 119889119895= 119899sum

119896=0

P (119883119897 = 119896)119872119868119896 (119904) (17)


E [119890119904119868] = 119899sum119896=0

P (119883119897 = 119896) 119890119896 ln(119872119894119868(119904)) (18)


E [119890119904119868] = 119890120573119897119860119897(119872119894119868(119904)minus1) (19)


E [119868] = E [E [119868 | 119883119897]]= 2120587120573119897119875119879119909radic1198901205902120585 ((119877119897119900)



Var [119868] = Var [119868119894]E [119883119897] + E [119868119894]2 Var [119883119897] (21)


Var [119868] = 120573119897119860 119897 (1205972119872119894119868 (0)1205971199042 )


(22)


119896119897 asymp E [119868]2Var [119868] (23)

120579119897 asymp Var [119868]E [119868] (24)




119872119868agg(119904) = 119871prod

119897=1

(1 minus 120579119897119904)minus119896119897 (25)


E [119868agg] = 120597119872119868agg(0)120597119904 (26)




119891119868agg (119909)asymp 119871prod

119897=1

(1205791120579119897 )119896119897 +infinsum119908=0


(27)



119865119868agg (119909) asymp 119871prod119897=1

(1205791120579119897 )119896119897 infinsum119908=0

120575119908120579sum119871119897=1 119896119897+1199081 Γ (sum119871119897=1 119896119897 + 119908)


(28)



Gamma (119896119897 120579119897) (29)


119860119897119868 = GG(radic120579119897 2119896119897 2) (30)



(31)



120598119897 = Δ 119905

119899119905sum119899=1

10038161003816100381610038161003816119860119897119868100381610038161003816100381610038162 (32)


120598119897 = 1205981015840119897 times 1205902119860119897119868 (33)



1205981015840119897 asympN (119899119905 + 120582119897 2 (119899119905 + 2120582119897)) (34)


120598119897 asympN (120583119897 1205902119897 )asympN (1205902119860119897119868 (119899119905 + 120582119897) 1205904119860119897119868 [2 (119899119905 + 2120582119897)])

(35)



120598 = 119871sum119897=1

120598119897 (36)


119897=1

120583119897 119871sum119897=1

1205902119897 ) asympN (120583Σ119871 1205902Σ119871) (37)


119875119862 = Q(120574 minus 120583Σ1198711205902Σ119871 ) (38)





120593119897 (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (39)


120593119868agg (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (40)


120598 = 119899119905sum119899=1

119868agg (119905) (41)


119887=1

(120593119868agg (119905)) = 119899119905prod119887=1

( 119871prod119897=1

(1 minus 120578120579119897119894119905)120581119897)= 119871prod

119897=1

[(1 minus 120578120579119897119894119905)120581119897]119899119905 (42)





119891120598 (119909) = 12120587 int+infin








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Simulation

Iagg (mW)101 102

Model L = 4



CDF

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Iagg (mW)101 102

1205731 = 1e minus 4

1205731 = 2e minus 4

1205731 = 3e minus 4

1205731 = 4e minus 4




Time (minutes)0 2 4 6 8 10 12 14 16 18 20

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PC

Model L = 1







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Time (minutes)0 2 4 6 8 10 12 14 16 18 20

Simulation

PC

Model L = 4





6 Final Remarks


Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119872119868agg(119904) = 119871prod

119897=1

(1 minus 120579119897119904)minus119896119897 (25)


E [119868agg] = 120597119872119868agg(0)120597119904 (26)




119891119868agg (119909)asymp 119871prod

119897=1

(1205791120579119897 )119896119897 +infinsum119908=0


(27)



119865119868agg (119909) asymp 119871prod119897=1

(1205791120579119897 )119896119897 infinsum119908=0

120575119908120579sum119871119897=1 119896119897+1199081 Γ (sum119871119897=1 119896119897 + 119908)


(28)



Gamma (119896119897 120579119897) (29)


119860119897119868 = GG(radic120579119897 2119896119897 2) (30)



(31)



120598119897 = Δ 119905

119899119905sum119899=1

10038161003816100381610038161003816119860119897119868100381610038161003816100381610038162 (32)


120598119897 = 1205981015840119897 times 1205902119860119897119868 (33)



1205981015840119897 asympN (119899119905 + 120582119897 2 (119899119905 + 2120582119897)) (34)


120598119897 asympN (120583119897 1205902119897 )asympN (1205902119860119897119868 (119899119905 + 120582119897) 1205904119860119897119868 [2 (119899119905 + 2120582119897)])

(35)



120598 = 119871sum119897=1

120598119897 (36)


119897=1

120583119897 119871sum119897=1

1205902119897 ) asympN (120583Σ119871 1205902Σ119871) (37)


119875119862 = Q(120574 minus 120583Σ1198711205902Σ119871 ) (38)





120593119897 (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (39)


120593119868agg (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (40)


120598 = 119899119905sum119899=1

119868agg (119905) (41)


119887=1

(120593119868agg (119905)) = 119899119905prod119887=1

( 119871prod119897=1

(1 minus 120578120579119897119894119905)120581119897)= 119871prod

119897=1

[(1 minus 120578120579119897119894119905)120581119897]119899119905 (42)





119891120598 (119909) = 12120587 int+infin








F

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Simulation

Iagg (mW)101 102

Model L = 4



CDF

0

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1

Iagg (mW)101 102

1205731 = 1e minus 4

1205731 = 2e minus 4

1205731 = 3e minus 4

1205731 = 4e minus 4




Time (minutes)0 2 4 6 8 10 12 14 16 18 20

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PC

Model L = 1







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Time (minutes)0 2 4 6 8 10 12 14 16 18 20

Simulation

PC

Model L = 4





6 Final Remarks


Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120598 = 119871sum119897=1

120598119897 (36)


119897=1

120583119897 119871sum119897=1

1205902119897 ) asympN (120583Σ119871 1205902Σ119871) (37)


119875119862 = Q(120574 minus 120583Σ1198711205902Σ119871 ) (38)





120593119897 (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (39)


120593119868agg (119905) = 119871prod119897=1

(1 minus 120578120579119897119894119905)119896119897 (40)


120598 = 119899119905sum119899=1

119868agg (119905) (41)


119887=1

(120593119868agg (119905)) = 119899119905prod119887=1

( 119871prod119897=1

(1 minus 120578120579119897119894119905)120581119897)= 119871prod

119897=1

[(1 minus 120578120579119897119894119905)120581119897]119899119905 (42)





119891120598 (119909) = 12120587 int+infin








F

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Simulation

Iagg (mW)101 102

Model L = 4



CDF

0

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1

Iagg (mW)101 102

1205731 = 1e minus 4

1205731 = 2e minus 4

1205731 = 3e minus 4

1205731 = 4e minus 4




Time (minutes)0 2 4 6 8 10 12 14 16 18 20

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Simulation

PC

Model L = 1







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Time (minutes)0 2 4 6 8 10 12 14 16 18 20

Simulation

PC

Model L = 4





6 Final Remarks


Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










F

0

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Simulation

Iagg (mW)101 102

Model L = 4



CDF

0

01

02

03

04

05

06

07

08

09

1

Iagg (mW)101 102

1205731 = 1e minus 4

1205731 = 2e minus 4

1205731 = 3e minus 4

1205731 = 4e minus 4




Time (minutes)0 2 4 6 8 10 12 14 16 18 20

0

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Simulation

PC

Model L = 1







0

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Time (minutes)0 2 4 6 8 10 12 14 16 18 20

Simulation

PC

Model L = 4





6 Final Remarks


Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

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Time (minutes)0 2 4 6 8 10 12 14 16 18 20

Simulation

PC

Model L = 4





6 Final Remarks


Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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