Information Exchange in Randomly Deployed Dense WSNs with Wireless Energy Harvesting Capabilities

1.1 MB·

by  Prodromos-Vasileios Mekikis

Information Exchange in Randomly Deployed
Dense WSNs with Wireless Energy Harvesting
Capabilities
Prodromos-Vasileios Mekikis, Student Member, IEEE, Angelos Antonopoulos, Elli Kartsakli, Senior Member,
IEEE, Aris S. Lalos, Member, IEEE, Luis Alonso, Christos Verikoukis, Senior Member, IEEE
Abstract—As large-scale dense and often randomly deployed to consider the use of distributed algorithms that encourage
wireless sensor networks (WSNs) become widespread, local in- data processing on the node side. Distributed estimation [4],
6
formationexchangebetweenco-locatedsetsofnodesmayplaya
1 distributed clustering [5] and distributed data storage [6] are
significantroleinhandlingtheexcessivetrafficvolume.Moreover,
0 among the applications that support local data exchange and
to account for the limited life-span of the wireless devices,
2 harvesting the energy of the network transmissions provides processingtoimprovethenetworkperformanceandtheenergy

n significant benefits to the lifetime of such networks. In this efficiency. Hence, the design of effective schemes that enable
a paper, we study the performance of communication in dense neighboringnodestoexchangemessagesandapplydistributed
J networkswithwirelessenergyharvesting(WEH)-enabledsensor
algorithmslocallyisbecomingofconsiderableimportance[7].

2 nodes. In particular, we examine two different communication
Given the dense deployment, it is very probable that sur-
scenarios (direct and cooperative) for data exchange and we
] provide theoretical expressions for the probability of successful rounding nodes overhear the transmissions of the network
T communication. Then, considering the importance of lifetime in and are willing to assist the communication by acting as
I WSNs, we employ state-of-the-art WEH techniques and realistic relays. This concept, known as cooperation [8], can provide
s. energy converters, quantifying the potential energy gains that noteworthy gains in the communication and was initially
c canbeachievedinthenetwork.Ouranalyticalderivations,which
studied in small-scale networks where the relays are deployed
[ arevalidatedbyextensiveMonte-Carlosimulations,highlightthe
importanceofWEHindensenetworksandidentifythetrade-offs infavorablepositions(e.g.,inbetweenthetransmittingnodes)
1
between the direct and cooperative communication scenarios. [9]. However, in large-scale networks, i) relay selection needs
v
considerable overhead and signaling [10] and ii) it is hard to
2 Index Terms—Wireless Sensor Networks, Cooperative Net-
5 works, Wireless Energy Harvesting, Dynamic Power Splitting, maintainafavorablepositionoftherelaysforeverypairofthe
1 Realistic RF-to-DC Conversion Efficiency, Stochastic Geometry randomlydeployedtransmittingnodes.Nevertheless,although
0 cooperation cannot always guarantee notable performance
0 I. INTRODUCTION gains in large-scale dense networks [11], it is possible to
.
1 DESPITEtheirlimitedprocessingandenergycapabilities, achieve diversity gains that increase the network reliability.
0 Besides, dueto thelimited humanintervention forpractical
6 wirelesssensornetworks(WSNs)applyinanincreasing
matters (e.g., replacing batteries), energy efficient communi-
1 number of domains, such as environmental monitoring [1],
cation becomes an essential concern in the design of large-
: mobile healthcare [2] and intelligent transportation systems
v scale networks. Although cooperation can improve the energy
i [3].Withtheintroductionofnewparadigms,suchasMachine-
X efficiency of a WSN, there are more effective ways to extend
to-MachinecommunicationandInternetofThings,thenumber
the network lifetime, which is a key parameter of a WSN and
r of wireless nodes in WSNs increases constantly, creating
a strongly depends on the limited-capacity batteries. Currently,
large-scale and dense randomly deployed networks. In such
a popular and drastic way to prolong the network operation
networks, the interference and the excessive traffic can signif-
is by harvesting energy from the environment to either power
icantly affect the quality of service (QoS) and, consequently,
entirelythesensornodesor extendthelifetimeoftheexisting
the network lifetime. Therefore, although in typical WSN
batteries [12]–[14]. In this new paradigm, which is broadly
scenariostheinformationcollectedbythesensorsisforwarded
known as energy harvesting (EH), the most typical energy
throughthenetworktoacentralcontrolstation(sink)forcen-
sources are solar, thermal, wind and kinetic energy. Recently,
tralized handling and decision-making, recent applications in
wireless energy harvesting (WEH) [15] has emerged as an al-
dense networks drive the need for local data exchange among
ternativeapproachtoharvesttheenergyoftheelectromagnetic
the nodes. To that end, many works have been motivated
radiation (EMR) from the network transmissions without the
This work was supported by the projects Cellfive (TEC2014-60130-P), need of expensive hardware systems. WEH can be adopted
AGAUR(2014-SGR-1551)andESEE(324284). even in cases where the aforementioned energy sources are
P.-V. Mekikis, E. Kartsakli, A. S. Lalos and L. Alonso are with the
scarceorunstableduetotheirdependenceonstochasticevents
Department of Signal Theory and Communications (TSC), Technical Uni-
versity of Catalonia (UPC), Spain, (e-mail: {vmekikis, ellik, aristeidis.lalos, liketheweatherconditions.Thisconstitutesitareasonableand
luisg}@tsc.upc.edu). straightforward method to extend the lifetime of the wireless
A. Antonopoulos and C. Verikoukis are with the Telecommunications
nodes and, consequently, of the whole network.
Technological Centre of Catalonia (CTTC), Spain, (e-mail: {aantonopoulos,
cveri}@cttc.es). Due to the dependence of the energy conversion efficiency
2
oftheharvesterontheamountofreceivedEMR[16],[17],the gains from cooperation are not always guaranteed in dense
benefitsfromWEHaremarginalforsmall-scalenetworkappli- networks,itisinterestingtoinvestigatethepotentialbenefitsof
cations, but interestingly high for large-scale dense networks. cooperationinaWEH-enableddensenetwork.Inaddition,we
Ideally,withWEH,itwouldbepossibletoimprovevastlythe employ a realistic model for the WEH conversion efficiency
network performance by simultaneously transferring informa- of the receivers [17]. Our contribution can be summarized in
tion and harvesting all the power. However, since the reuse of the following points:
the whole received signal both from the energy harvester and • We analytically derive the probability of successful data
the information receiver is not yet possible, various methods exchange, while taking into account DPS.
have been proposed in order to facilitate WEH [18]. In the • In order to demonstrate the potential energy gains of
class of these techniques, dynamic power splitting (DPS) [19] WEH, we analytically estimate the network lifetime with
has been proved to be among the most efficient approaches and without WEH. We assume a variable and, thus,
that facilitates simultaneous message decoding and energy realistic energy conversion efficiency for the harvester to
harvesting. Using DPS, it is possible to dynamically share comply with state-of-the-art rectennas.
the received energy between the information decoder and the • We provide theoretical expressions for a well-established
energy harvester, according to the channel condition that is end-to-end QoS performance metric, namely the spatial
assumed to be known at the receiver. throughput, and derive theoretically the optimal intensity
To that end, several studies that consider large-scale net- that maximizes the network lifetime.
works with WEH have lately appeared in the literature [20]– • We conduct an extensive performance assessment for
[23]. In his pioneer work [20], Huang studies the network the two schemes (direct and cooperative), which reveals
throughput in a basic mobile ad hoc scenario, where the intriguing trade-offs that provide useful insights for the
communication between the transmitter and the receiver is design of WSNs with WEH.
conductedthroughanidealwirelesschannel(i.e.,nopathloss
The rest of the paper is organized as follows. Section II
is assumed in the link). It is worth noting that, although some
describes the system model and the communication scenarios.
of the potential benefits of the WEH technology are identified
Section III presents the analysis for the probability of suc-
in [20], the results cannot be generalized for cooperative
cessful message exchange. Section IV includes the theoretical
communications. Particularly, in cooperative scenarios, the
expressions of the average network lifetime for the different
existenceofrelaynodesimplyavolatileandcomplexenviron-
scenarios, while, in Section V, we present useful performance
ment that requires a dedicated study. Similarly, in [21], Guo
metrics. Section VI presents the model validation and the
andWangstudytheeffectsofWEHinadirectcommunication
numerical results. Finally, Section VII concludes the paper.
scenario.Nevertheless,theanalysisisbasedonspecificphysi-
callayerconfigurations,sincetheauthorsprovideclosed-form
II. SYSTEMMODEL
expressions for the QoS metrics only for specific path loss
A. Network and Channel Model
conditions, i.e., a particular value for the path loss exponent.
However, the range of values that the path loss exponent can We consider a large-scale network consisting of two sets of
haveindifferentenvironmentsstressestheneedfortheoretical source nodes S ={s ,...,s }, S ={s ,...,s } and a
1 11 1i 2 21 2j
expressionsthatprovidegeneralandenvironment-independent setofambientnodesactingasrelaysR={r ,...,r }intwo
1 k
solutions.Recently,aninterestingapproachhasbeenpresented different communication scenarios: i) direct, where the sets of
in [22] by Krikidis, where the coverage of a large-scale source nodes exchange messages directly, and ii) cooperative,
network is studied, while the receivers employ a technique where the randomly deployed relays R assist S and S to
1 2
for simultaneous information and energy transfer. The author the message exchange. In cases where it is convenient, a
provides incentives for cooperation, highlighting the possible set of sources will be denoted as S , ϕ ∈ {1,2} while S
ϕ ϕˆ
benefits,howevertheproposedmodelconsidersfixeddistances will denote the complementary set (i.e., when S = S then
ϕ 2
between preassigned nodes. In addition, the model assumes S = S and vice versa). The relays are assumed to be
ϕˆ 1
a constant energy conversion efficiency for the harvester, other sensor nodes or other type of devices (e.g., smartphones
althoughinrealisticimplementationstheefficiencydependson with dedicated interface for relaying). The different sets of
the input power. In the same context, the work in [23] studies sources measure different phenomena and broadcast their
a bidirectional scenario with nodes that harvest EMR with measurements. More specifically, each individual source node
a constant energy conversion efficiency. The authors provide receives a local measurement, either directly or cooperatively
important insights into the probability of data exchange in from the nearest node of the other type (i.e., nearest-neighbor
such scenarios, but there is no analysis with regard to the model [24]). Consequently, each node is required to be aware
end-to-end network performance, which is essential for the of the location of itself and of its neighbors, via localization
evaluation of the proposed model. In addition, the possibility schemes that act in higher network layers [25].
ofdirectcommunicationamongtherandomlydeployednodes All nodes are identical and assumed to be moving on
is neglected, as only cooperative operation is considered. the same Euclidean plane. They are represented by three
In this paper, we study the impact of WEH using DPS on independent homogeneous PPPs, a reasonable approach for
theinformationexchangeinlarge-scalenetworks.Weconsider WSNs according to [26]. The locations of the sources S are
1
two sets of sources that exchange their messages either di- described by the PPP Φ = {x ,...,x } with intensity λ ,
S1 1 i 1
rectly or via randomly deployed relay nodes. As performance where x , ∀i ∈ N, denotes the location of the source s on
i 1i
Tx
3
EHS
PR PT, PR %) 100
RF-to-DC Information PI υ(ψ)·PI Information PI υ(ψ)·PI InRfoercmeiavteiorn ency ( 80
Conversion Receiver Power Receiver Power Effici 60
Splitting C 40
P =ε(P )P Splitting Energy −D
EH R R o 20
Energy (1-υ(ψ))·PI Harvester F−t
(1-υ(ψ))·P Harvester R 0
I −20 −15 −10 −5 0 5 10 15 20
Input Power (dBm)
Fig. 1: Schematic of a node at reception mode. The received
Source
Battery power is dynamically split based on the rule given in (1). Fig. 2: Behavior of the RF to DC efficiency of a rectenna.
the plane R2. Similarly, the location of the sources S2 on R2 the informationreceiver, thena fractionof the receivedpower
aλr2e,rwephreerseenytje,d∀bjy∈thNe PdePnPotΦesS2th=e {loyc1a,t.io..n,oyfj}thweitshouinrcteenss2itjy. ewqiuthaoluttod(e1te−rioψhra)ti∈ng[0t,h1e]ciosmbmeinugnicfeadtiotnoptheerfoenrmeragnyceh.aArvtetshteisr
For the modeling of the relay nodes, there is an additional point, we should mention that the employed DPS technique
PPP ΦR = {z1,...,zk} with intensity λR, which represents does not necessarily provide optimal performance in terms of
the location zk, ∀k ∈N, of the relay rk. harvestedenergyforourinterference-limitedsystem.However,
For our analysis, without loss of generality, we assume that itisanoveltechniquethatconsiderstheimpactoffadingand,
the respective receiving node in each slot is located at the thus, avoids compromising the communication performance.
origin (Slyvnyak’s theorem [27]). The received power PR at Furthermore, the conversion efficiency of the radio fre-
a node located in a distance d from the transmitting node quency (RF) energy into direct current electricity is denoted
is PR = Pthd−α, where Pt is the transmission power of by (cid:15). As the conversion efficiency of a rectenna depends on
EHS the nodes, α > 2 is the path loss exponent and h is the the received power [16], [17], we adopt a variable conversion
square of the amplitude fading coefficient (i.e., the power efficiency (cid:15) modeled as a quadratic polynomial that captures
fadingcoefficient)thatisassociatedwiththechannelbetween the behavior of state-of-the-art rectennas [16], [17], as in Fig.
P the nodes. We also assume that the fading coefficients are 2, given by
Rx
independent and identically distributed (i.i.d.). Moreover, the
P√P
amplitude fading Tx, hRixs Rayleigh with a scale parameter (cid:15)(P )=a P3+a P2+a P +a , (2)
I 3 I 2 I 1 I 0
CPU P RF-to-DC Peh=εPRx Power σInf=orm1,athioenn ce h is exponentially distributed with mean value
et
Conversion Conditioning µ.RTecheeivcehrannelisassumedtoremainconstantinonetimeslot where P in Watts is the input power or the total received
I
λ (i.e., a time perioNdoiinsewhich a transmission takes place). power, which consists of the received signal and the inter-
e
Solar panel Pc Noise All nodes are powered by a battery with initial energy ference, while a3,a2,a1,a0 are the coefficients of the cubic
level BI and in every time slot consume energy to com- polynomial.
municate (i.e., Pt power is consumed for transmission and After taking into account DPS, a message is considered
Power conditioning Buffer Battery Source BaPttrerfyor reception). Also, they are capable of WEH using a to be successfully decoded at a receiver when the signal-to-
power splitter that dynamically adjusts the power ratio that is interference-plus-noise ratio (SINR) from its nearest transmit-
allocated to the information receiver and the energy harvester, ter, denoted as γ, is higher than a threshold γ∗; otherwise the
i.e., DPS [19]. A simplified illustration of a node is provided message is dropped [28]. The SINR of a mobile node located
in Fig. 1, where the various parts of the node are shown. A at the origin at a distance d from its nearest transmitter is
node is able to recharge its battery by harvesting the EMR
energyfromthetransmissionsofthesourcesandtherelaysin v(ψ)P hd−α
γ = t , (3)
the network. According to DPS, the splitting depends on the v(ψ)I +N
d
channel condition and it is described by the following rule:
where I is the aggregated interference caused by the trans-
(cid:40) 1, if h<ψ d
mitter’s PPP, defined as I = (cid:80) P h x−α and N is the
v(ψ)= ψ d x∈Φ t x
, if h≥ψ (1) additive white Gaussian noise power, modeled as a constant
h
zero mean Gaussian Random Variable (RV).
wherehisthepowerfadingcoefficientofthechannelbetween
the receiver and the nearest transmitter and ψ is a parameter
that defines the amount of power that is split between the
B. Communication Model
energy harvester and the information receiver. Later in this
paper, we provide an empirical method to choose the value The time is divided into time slots of fixed duration t ,
s
of the ψ parameter. In addition, it is assumed that h is in which the transmission of one packet can take place. The
known at the receiving node, but unknown to the transmitter. time needed for the two sets of sources to exchange messages
According to (1), when the channel conditions are poor, all of is called communication period (CP). Each CP consists of g
thereceivedsignalisbeingfedtotheinformationreceiver.On time slots, depending on the communication scheme, as we
the contrary, when the channel conditions are satisfactory for will describe in detail next.
2 1 R 2R Stelnei to 1
To lamvanoun to
2 1 R 2R
2 kai to R
Stelnoun ta 2 kai
2 1 R 2R
2R
2 1 R 2R Stelnei to 1
1 1
R
1 2
1
I
1
h
R1
I
R 2
1 2
Failed Transmission Successful Transmission
Interference Target Transmission
13 23 11, 13
11, 1231 11 13 23 21 1121
21
21, 23 13 R1
21, 23 13 R1
R2 22
13 R2 22
12 Range 24 13
22
12
22
Failed Transmission Successful Transmission
11
11 21
21 22
12
22 2312 23 21
12
Interference Target Transmission 1x Data from node 1x 2y Data from node 2y Interference Target Transmission 1x Message from node 1x 2y Message from node 2y through relay
11 21 Time slot #1 11 21 Time slot #2 13 23 21 11 13 23 11 21 11
22 11 21
1 1 1 1
1 2 1 2
12 21 12
12 22 23 12 22 23 13 R1 13 23 13 R11321
R2 12 22 13 122R22 2213
22
Interference Target Transmission 12 Time slot #1 12 Time slot #2
11 Time slot #1
21
13 23 21 11 23 1313 21 11
11 12 11 2121 1113 2121
12
12 22 23 2321 13 R1 13 R1 1321
1x Message fro1m1 node 1x 2y MeTssaimge efr osml onot d#e 22y 1321 2321
21
21 22 1222
11 12 12 2222 R2 2213 R2 1222 221313
12 22 23 12 Time slot #3 12 2222 Time slot #4
4
ITnatregrfeet rTenracnesmission 13 23 21 11 13 23 11 21 11
11 21
1 Node x from set S
x 1
13
2 Node y from set S
y 2 1 R 1 R
3 1 3 1
Rz Relay z from set R 23 13 21
1x Message received from node 1x
22yy MMeessssaaggee rreecceeiivveedd ffrroomm nnooddee 22yy R2 12 22 13 22 R222 12 22 13
through a relay 1 1
2 CC Time slot #1 2 CC Time slot #2
11 2 DC Time slot #1
1 13 2233 2211 1111 2233 13 13 21 11
1 1 11 21 21 11 13 21 21
1 2
12
12 22 23 23 21 1133 RR11 13 R1 13 21
11 2 DC Time slot #2 13 21 23 21
1
22 12 22
2111 12 12 22 22 RR22 2222 13 RR22 12 22 22 13 13
12 22 23 1122 CC Time slot #3 1122 22 22 CC Time slot #4
(a) DC scenario phases (b) CC scenario phases
Fig. 3: Communication phases. (a) DC scenario phases: i) Slot 1 (S →S ), ii) Slot 2 (S →S ), (b) CC scenario phases: i)
1 2 2 1
Slot 1 (S →S ,R), ii) Slot 2 (S →S ,R), iii) Slot 3 with active relay (S ←R), iv) Slot 4 with active relay (S ←R)
1 2 2 1 1 2
1) Direct communication scenario (DC): In the DC sce- on the plane that attempt to decode the messages from their
nario, illustrated in Fig. 3a, the CP consists of two time slots nearest source nodes to assist the communication. Therefore,
(i.e., g = 2) of duration t . In the first time slot, each S in the following two time slots, the relays are consecutively
DC s 1
sourceisbroadcastingitsmessageandeachS sourceattempts broadcastingthemessagesoftheirnearestS andS node.In
2 1 2
to decode the message of its nearest S source. The rest this way, there is a diversity gain, since the sources have two
1
transmissions of the S sources are considered as interference possible ways of receiving a message from a source of the
1
for the S source. However, when the circumstances allow it other type. At the fourth time slot in Fig. 3b, we notice that
2
(i.e., h ≥ ψ), this interference is beneficial for the network, mostsourcenodeshavereceivedthesamemessagetwice.This
because a part of it is harvested. In the second time slot, means that these nodes have higher probability to decode this
the system follows a similar procedure and each S source message. However, depending on the random topology, there
1
attempts to decode a message from its nearest S source. In is a chance that some source nodes will receive two different
2
the end of the CP, all source nodes have attempted to decode messages, as it happens in nodes 1 and 2 and, thus, deduce
3 1
a message from their nearest node of the other type, as it is moreinformationabouttheirenvironment.Moreover,ifarelay
depicted in Fig. 3a (i.e., small rectangular next to each node). fails to decode the messages in the first two time slots, then it
Inthesecondtimeslotofthisfigure,itcanbenoticedthatnode transmits power to the sources to cooperate only in terms of
2 has attempted to decode the message from its nearest node energy.
3
1 ,althoughthelatterhasattemptedtodecodethemessageof
2
its nearest S node, i.e., 2 . Therefore, there are not always
2 2
certain pairs in the network, as it happens with nodes 1 and
1
21. In this way, all nodes manage to receive a message from III. SUCCESSFULMESSAGEEXCHANGEPROBABILITY
their nearest neighbor, which is the goal in such scenarios.
2) Cooperative communication scenario (CC): In the CC In this section, we present the probability of successful
scenario, illustrated in Fig. 3b, the CP consists of four time message exchange between the two types of sources in one
slots (i.e., g = 4). Similar to the DC scenario, in the first CP for the DC and CC scenarios. The successful message
CC
two slots, the S and S sources are attempting to decode exchange probability is an important QoS metric, defined
1 2
the message from their nearest neighbor of the other type. as the probability of both S and S sources to decode
1 2
However, in this scenario, there are also relays distributed successfully the received messages within a CP.
5
√ √ (cid:113)
A. Direct Communication Scenario γ∗(π/2 − arccot( γ∗))), ζ(r,γ∗) = Ptγ∗ and
Pt−r4γ∗N
In the first time slot of the DC scenario, all S2 source ω(γ∗)=µγ∗N/Pt.
nodes decode successfully a direct message from their nearest
Proof. TheproofofLemma1isprovidedinAppendixB.
S neighbor with a probability denoted as p . Similarly,
1 DC1
with p we denote the probability that all S source nodes
DC2 1
decode successfully a direct message from their nearest S B. Cooperative Communication Scenario
2
neighbor in the second time slot. These probabilities (i.e.,
In the case of the cooperative scenario, the two sets of
p and p ) are independent and have common network
DC1 DC2 sources exchange their messages either directly or with the
parameters except for the intensity λ and λ , respectively.
1 2 assistanceofrelaynodes.Therefore,theoverallprobabilityof
Therefore, the probability p =f(λ ) is a function of the
DCϕ ϕ successful exchange in the cooperative case, denoted as pCC,
intensity and the probability p that all source nodes have
DC depends both on the probabilities p and p derived in
successfully decoded a message from the nearest neighbor of DC1 DC2
SectionIII-Aandontheprobabilityp ,whichdenotesthe
the other type is given by CCRϕ
probability that relay has decoded a message from its nearest
2 type ϕ source and a type ϕˆ source node has successfully
(cid:89)
pDC =pDC1pDC2 = pDCϕ (4) decodedthismessagethroughthisrelay.Hence,therearethree
ϕ=1 events for successful exchange in the cooperative scenario: i)
Tothatend,toderivep wehavetocalculatetheprobability both directly and through a relay, ii) only directly, or iii) only
DC
p . Moreover, in order to account for the power splitting througharelay.Sincetheseeventsaremutuallyexclusive,the
DCϕ
process described by (1), we have to differentiate between the probability of successful exchange in the cooperative case is
cases of h < ψ and h ≥ ψ. Therefore, the probability of given by the following lemma1.
successful message exchange for the direct scenario is given
Lemma 2. Theprobabilityofsuccessfulmessageexchangein
by the following theorem.
one CP for the cooperative scenario is given by
Theorem 1. The probability of successful message decoding
(cid:0) (cid:1)(cid:0) (cid:1)
p = p +p (1−p ) p +p (1−p ) .
in one time slot for the DC scenario is given by (5), where CC DC1 CCR1 DC1 DC2 CCR2 DC2
F (a,b;c;z)=(cid:80)∞ (a)n(b)nzn isthehypergeometricfunc- (11)
2 1 n=0 (c)n n!
tion. Proof. TheproofofLemma2isprovidedinAppendixC.
Proof. By taking into account (1) and (3), the probability
Remark 1. In interference-limited systems, thermal noise
p is given by
DCϕ is not an important consideration that results in a weak
p =Pr(γ >γ∗∩h<ψ)+Pr(γ >γ∗∩h≥ψ). (6) dependenceoftheprobabilityofsuccessfultransmissionp
DCϕ DCϕ
with the node intensity [29]. To that end, it follows that
CdeofinndiittiioonnionfgcoonndthiteiovnaallueproofbathbeiliRtiVes,hwuesinogbttahine Kolmogorov pDC1 (cid:39) pDC2 and, thus, pCC (cid:39) (cid:0)pDCϕ +p2DCϕ −p3DCϕ(cid:1).
Fromthelatter,itcanbeeasilyproventhatp ≥p holds
pDCϕ =Pr(h<ψ)Pr(γ>γ∗|h<ψ)+Pr(h≥ψ)Pr(γ>γ∗|h≥ψ). always. Still, although it is always more proCbCable toDCachieve
(7)
Since h is exponentially distributed with rate µ, (7) can be a successful message exchange in the CC scenario, this result
written as does not imply higher performance of the CC scenario in the
(cid:18) (cid:19) end-to-end performance of the network. Consequently, in the
1 1
pDCϕ= 1−eµψ Pr(γ >γ∗|h<ψ)+eµψPr(γ >γ∗|h≥ψ). following,weperformananalysisonthenetworklifetimeand
other end-to-end performance metrics (e.g., spatial through-
(8)
In (8), the probability Pr(γ > γ∗|h < ψ) can be easily put) to identify trade-offs between the two scenarios.
calculated using guidelines from [29] and it is given as
Pr(γ>γ∗|h<ψ)= (9) IV. NETWORKLIFETIME
(cid:90) ∞ (cid:20) (cid:18) (cid:90) ∞ γ∗2/α (cid:19) µγ∗N (cid:21)
=πλϕ 0 exp −πλϕr 1+ γ∗−α2 1+ua/2du − Ptr−α/2 dr, One of the most important metrics for a WSN is its operat-
ing lifetime. In this section, the analysis for the derivation of
whereas the proof for the probability Pr(γ > γ∗|h ≥ ψ) is
the network lifetime and the average harvested power is given
provided in Appendix A. Replacing Pr(γ > γ∗|h < ψ) and
forallscenarios.Inthisway,itbecomespossibletodetermine
Pr(γ >γ∗|h≥ψ) in (8), concludes the proof.
the gains of WEH using DPS.
Lemma 1. For the special but common case when the path
loss exponent is α=4, Theorem 1 is simplified into
A. Direct Communication Scenario
pDCϕ=πλϕ(1−e−µψ)(cid:114)ω(πγ∗)exp(cid:18)χ(4λωϕ(,γγ∗∗))2(cid:19)Q(cid:18)χ(cid:112)(λ2ωϕ(,γγ∗∗))(cid:19)+ After wd ∈ N0 communication periods and without taking
+2πλϕe−µψ(cid:90) ∞exp(cid:20)−πλϕr2(cid:16)1+ζ(r,γ∗)arctan(cid:2)ζ(r,γ∗)(cid:3)(cid:17)(cid:21)rdr, EH into account, the average battery level of a source node
0
(10) 1It should be noted that, although the interference at the relay and
where Q(x) = √1 (cid:82)∞exp(−q2/2)dq is the tail probability destination in the two first time slots comes from the same set of nodes,
2π x theimpactoffadingminimizesthecorrelationand,therefore,theeventscan
of the standard normal distribution, χ(λϕ,γ∗) = πλϕ(1 + beconsideredindependent.
6
(cid:90) ∞ (cid:20) (cid:18) (cid:90) ∞ 1 (cid:19) µγ∗N (cid:21)
p =πλ (1−e−µψ) exp −πλ r 1+γ∗2/α du − rα/2 dr+
DCϕ ϕ ϕ 1+ua/2 ψP
0 γ∗−2/α t
(5)
(cid:90) ∞ (cid:20) F (cid:0)1,α−2;2α−2; γ∗Ptψ (cid:1)(cid:0) 1 − N (cid:1)−α2(cid:0) γ∗Ptψ (cid:1)αα−2 (cid:21)
+2πλ e−µψ exp −2πλ 2 1 α α Ptψ−rαγ∗N γ∗rα Ptψ Ptψ−rαγ∗N −πλ r2 rdr
ϕ ϕ α−2 ϕ
0
in the DC scenario is defined by the amount of energy E Remark 3. At this point, it should be mentioned that the
con
consumed per CP and it is given by average lifetime L¯ EH is limited by the set of sources with
d
the least average harvested power. Thus, it holds that
B¯ (w )=B −w E =B −w t (P +P ), (12)
d d I d con I d s r t
L¯ EH =B (cid:14)[t (P +P −min{P¯ EH,P¯ EH})] . (17)
where B is the initial energy level, t is the duration of a d I s r t d1 d2 +
I s
timeslot,Pr isthepowerconsumptionatthereceptionmode, This happens because when a set of sources consumes all of
and Pt is the power consumption at the transmission mode. its energy, then we assume that the system has reached its
In the case that the source nodes have EH capabilities, their lifetime.
battery level is increased in each CP by the average harvested
power per CP denoted as P¯ EH. Thus,
d B. Cooperative Communication Scenario
B¯dEH(wdEH)=BI−wdEHts(Pr+Pt)+wdEHtsP¯dEH. (13) In the cooperative communication scenario, a set of relay
Therootsof(12)and(13)(i.e.,thevaluesofw andwEH that nodes assists the source nodes to exchange their messages.
d d
Therefore, without taking EH into account, the battery level
the battery is discharged) provide the source node’s average
lifetime in CPs L¯ and L¯ EH, respectively of a node after wc ∈ N0 CPs in the cooperative scenario is
d d
defined by the initial battery level and the amount of energy
B
L¯d = t (P +I P ) (14) Econ consumed per CP and it is given by
s r t B¯ (w )=B −w E =B −w t (cid:16)2P +P (cid:0)1+1 (cid:1)(cid:17),
and c c I c con I c s r t R
L¯ EH = BI , (15) (18)
d [ts(Pr+Pt−P¯dEH)]+ where 1R is the indicator function that determines whether
(18) represents the battery level of a relay node or a source
where [ξ] =max(ξ,0).
+ and it is described by
Remark 2. In the extreme case that the denominator of (15) (cid:26) 1, Relay node. (19)
is equal to zero, the consumed power is lower or equal than 1 =
R
0, Source node.
the average harvested power and, hence, the network lifetime
becomes infinite (i.e., the perpetual network operation). Similarly,inthecasethatthenodeshaveEHcapabilities,their
battery level at any CP wEH is
In the following theorem, the average harvested power c
P¯dEH of a source node is provided, in order to complete the B¯cEH(wcEH)=BI−wcEHts(cid:16)2Pr+Pt(cid:0)1+1R(cid:1)(cid:17)+wcEHtsP¯cEH,
derivation of the average network lifetime with EH in the DC (20)
scenario L¯dEH, given in (15). where P¯cEH is the average harvested power in one CP. The
Theorem 2. The average harvested power in one CP of a roots of (18) and (20) provide the node’s average lifetime in
type Sϕ source node at the DC scenario, while taking into CPs for each case, respectively
account DPS and before the RF-to-DC conversion efficiency
is described by B
L¯ = I (21)
P¯DPSdϕ =Pte−µψ(cid:34)µ(παα−λϕˆ2)+ψe−µψEi[−µψ](cid:20)παα−λϕ2ˆ−E(cid:8)rc−ϕˆα(cid:9)(cid:21)(cid:35), c ts(cid:16)2Pr+Pt(cid:0)1+1R(cid:1)(cid:17)
and
whereas the actual average harvested power after applying
B
the RF-to-DC conversion efficiency is given by L¯EH = I . (22)
P¯ EH =P¯ (cid:104)a (cid:0)P¯ (cid:1)3+a (cid:0)P¯ (cid:1)2+a (cid:0)P¯ (cid:1)+a (cid:105), c [ts(cid:16)2Pr+Pt(cid:0)1+1R(cid:1)(cid:17)−tsP¯cEH]+
dϕ DPSdϕ 3 log 2 log 1 log 0
(16) As in the DC scenario, the average harvested power P¯EH
where P¯ = 10log P¯DPSdϕ, Ei[x] = −(cid:82)∞ e−tdt for must be derived to complete the calculation of the netwcork
nonzero lvoaglues of x 1d0eno1tmeWs the exponential −inxtegtral and lifetime with EH in the CC scenario L¯EH, given in (22).
c
E(cid:8)r−α(cid:9) denotes the expected value of the path loss to the
cϕˆ Lemma 3. The average harvested power of a type S source
nearest type S transmitter for different path loss exponent ϕ
ϕˆ P¯ orarelaynodeP¯ forthecooperativescenario,
values α>2, given within the proof. DPScϕ DPScR
while taking into account DPS and before the RF-to-DC con-
Proof. The proof of Theorem 2 is provided in Appendix D. version efficiency, is the sum of the average power harvested
by the transmissions of the other two sets. Hence, we obtain
7
(cid:34) (cid:20) (cid:21)(cid:35)
P¯ =P e−µψ πα(λR+λϕˆ) +ψe−µψEi[−µψ] πα(λR+λϕˆ) −E(cid:8)r−α(cid:9)−E(cid:8)r−α(cid:9) (23)
DPScϕ t µ(α−2) α−2 cϕˆ cR
(cid:34) (cid:20) (cid:21)(cid:35)
P¯ =P e−µψ πα(λϕ+λϕˆ) +ψe−µψEi[−µψ] πα(λϕ+λϕˆ) −E(cid:8)r−α(cid:9)−E(cid:8)r−α(cid:9) (24)
DPScR t µ(α−2) α−2 cϕˆ cϕ
(23) and (24), where Ei[x] is the exponential integral of x A. Optimal Intensity
and E(cid:8)r−α(cid:9) denotes the expected value of the path loss to
cR In previous works with WEH networks that do not take
the nearest relay.
into account the RF-to-DC conversion efficiency, the network
The actual average harvested power after applying the RF-
intensity is a monotonic function of the average harvested
to-DC conversion efficiency P¯cιEH, ι∈{ϕ,R} is given by power. However, in a more realistic approach where the
antennas are not ideal, as the network intensity and, thus, the
P¯cιEH =P¯DPScι(cid:104)a3(cid:0)P¯clog(cid:1)3+a2(cid:0)P¯clog(cid:1)2+a1(cid:0)P¯clog(cid:1)+a0(cid:105), interference increases, the average harvested power rises to
(25) a local maximum and then decreases due to the low RF-to-
where P¯clog =10log10 P¯D1mPWScι. DC conversion efficiency. Therefore, it is important to know
the network topology characteristics such as the intensity
Proof. The same line of thought is followed for this proof as
of the transmitting set of nodes that achieves the maximum
in Theorem 2. However, for the cooperative case, the sources
average harvested power for the receiving set of nodes. The
are assisted by a set of relays. Therefore, each source node
optimization problem considered can be described as
receivesonaverageenergyfromtwosets(i.e.,inonetimeslot
from the relay transmissions and in another timeslot from max P¯ EH(λ )
dϕ ϕˆ
the transmissions of the other set of sources). Moreover, the λϕˆ
(28)
relays are receiving the energy from the transmissions of the s.t. λϕˆ ≥0
two source sets. Thus, the average harvested power while 0≤(cid:15)(P )≤1
I
taking into account DPS and before the RF-to-DC conversion
and a solution of this problem is given in the following
efficiency of an S source is
ϕ
Lemma.
P¯ =P¯ +P¯ , (26) Lemma 4. The optimal intensity λ to achieve maximum
DPScϕ DPSdϕ DPSdR opt
lifetime in a network with DPS and RF-to-DC conversion
where P¯ can be derived from P¯ using λ as the efficiency described by (1) and (2), respectively, is given by
DPSdR DPSdϕ R
intensity. For aP¯relay nod=eP¯the avera+geP¯harves.ted power is(27) λopt=µ1(0α3P−tπ2α)1e0−−µ3ψ0α+α213 exp(cid:20)2620/α33f + ln210(α222−−4/33α610αα33)f+900α23(cid:21),
DPScR DPSd1 DPSd2 where
Substituting (26) or (27) to (25) and following a procedure as (cid:114)
(cid:113)
in Theorem 2, yields the respective actual average harvested f = 3 −ρ+ ρ2−4(cid:0)ln210(α2−3α α )+900α2(cid:1)3
2 1 3 3
power after applying the RF-to-DC conversion efficiency,
which concludes the proof. and
ρ=ln310(cid:0)27α α2−9α α α +2α3(cid:1)+54000α3
0 3 1 2 3 2 3
Thus, by combining (25) with (22), the maximum lifetime
of a node with EH in the cooperative scenario can be derived. Proof. TheproofofLemma4isprovidedinAppendixE.
Similar to Remark 3, the average lifetime in the CC scenario
is defined by the minimum between P¯ EH and P¯ EH. Remark 4. It should be noted that the optimal intensity of
c1 d2
the S source nodes calculated using Lemma 4 maximizes the
1
lifetime of the S set of nodes. Similarly, the optimal intensity
2
V. OPTIMALINTENSITYANDPERFORMANCEMETRICS of the S2 set of nodes maximizes the lifetime of the S1 set.
In this section, we will introduce the optimal intensity,
B. ST and TME
whichprovidesanaccurateestimationofthenumberofnodes
per unit area needed to achieve the highest possible lifetime The probability of successful exchange derived in Section
forthenetwork,andtwometricsthatareusefulforevaluating III for all scenarios is a throughput metric for the link
the performance of the network, i.e., the spatial throughput under examination. In order to have a complete picture of
that indicates the average number of messages exchanged per the network performance, we employ the metric of spatial
unit area and the total messages exchanged on average. throughput [27, 5.3.1], [34], which provides an average of the
8
throughputoverallthelinksinthenetwork.Hence,thespatial g
throughpuSt(m=es(sλag1e+s/sλ/2u)npits-car(ema)esosfatghees/nse/utwniot-rakreisa)d,efine(d2a9s) ability of sful decodin00000.....056666.912336 1% reduction SAinmaulylatictiaoln
sc gscts Probcces00..5578
u −10 −8 −6 −4 −2 0 2 4 6 8 10
where sc = {DC,CC}, p and g denote the successful s
erthexsrcFophiuenacgnathlgilvpeyeu,ltpya.rnioosbthatbehrielmitayevterairncadgthetahsttceoctananulmbmebeesdscresdaougfceessdloeutxsscinphgearntghseecdesnpiaanrtiioaal, Average Power Harvested (mW)02468 ~150% increase SAinmaulylatictiaoln
−10 −8 −6 −4 −2 0 2 4 6 8 10
lifetimeperunitarea(TME),whichisgivenbymultiplyingthe ψ−parameter in dB
spatial throughput with the network lifetime and the number Fig. 4: Comparison of probability p and average harvested
DC
of slots per CP for each scenario. TME can be written as power P¯EH versus the ψ-parameter.
d
TME =S w g (messages/unit-area), (30)
sc sc sc sc
1
where w denotes the network lifetime for the various sce-
sc ng 0.8
nariosderivedinSectionIV.Inthefollowingsection,wewill of odi
pthraetsehnatveanbdeevnalpidreasteenttheed nsoumfaerr.ical results of all the metrics bability sful dec 00..46 Simulation DC
VI. ANALYTICALANDSIMULATIONRESULTS Prosucces 0.2 SAAinnmaaullyylattiicctiaaolln DC CCCC
In this section, we validate the proposed theoretical frame- −03 0 −25 −20 −15 −10 −5 0 5 10 15 20
work via extensive simulations and provide useful insights on SINR Threshold γ* (dB)
the use of WEH by comparing the metrics of interest for the Fig. 5: Probability of successful message exchange vs. decod-
different communication scenarios. ing threshold γ∗ for the direct and cooperative scenarios.
A. Simulation Setup
power). It can be observed that by sacrificing only 1% in
We compare the two proposed scenarios, direct and simple
the probability of successful decoding, the average harvested
cooperative without EH (DC and CC, respectively) and with
powerisincreasedby∼150%.Thisisduetothefactthatthe
EH(DC-EHandCC-EH,respectively).Forhighaccuracy,we
probabilityofexchangedropswithalowrateasψ isreduced,
create 10.000 realizations of a 500 m by 500 m area with
while the average harvested power rises with a much higher
intensities varying from 0.01 to 0.5 devices per m2 (i.e., the
rate. Therefore, in our experiments, the ψ-parameter has been
numberofdevicesperrealizationisfrom3.000upto150.000).
fixed at −10 dB or ψ =0.1.
The time slot duration is denoted as t and depends on the
s
applicationscenarioandthechosenbitrate.Thetransmitpower
B. Model Validation and Performance Evaluation
is P = 75 mW, while the power for the reception mode is
t
P =100 mW [35] and the initial level of a node’s battery is Inthissection,wevalidatethebasicmetrics(i.e.,probability
r
L =1000J.Additionally,thepathlossexponentischosento ofsuccessfulmessageexchangeandaverageharvestedpower)
I
be α=4, although it is possible to use any value α>2. For ofouranalysis,thatareusedforthederivationsoftheend-to-
the model validation, the channel fading gain is set to µ=1 endQoSandlifetimemetrics.InFig.5,weplottheprobability
and the noise power to N =−124 dBm for 100 kHz system ofsuccessfulmessageexchangeforthedirectandcooperative
bandwidth for all scenarios, unless otherwise stated, while we communicationscenariosversusthedecodingthresholdγ∗.As
vary the values of decoding threshold γ∗ and intensity λ in we can see, the probability p matches perfectly with the
DC
ordertopresenttheperformanceofthesystemunderdifferent simulations and, thus, Theorem 1 is validated. Furthermore,
conditions. In addition, if not explicitly stated otherwise, the the probability p becomes lower as the decoding threshold
DC
decoding threshold is fixed at γ∗ = 0 dB and the intensity λ increases. This result can be justified by the fact that, for
of the PPPs is set to λ =0.1, λ =0.5 and λ =0.25. higher decoding thresholds, the received signal must be much
1 2 R
Moreover, in all the experiments, the coefficients for the stronger than the interference plus noise. Similar conclusions
RF-to-DC conversion efficiency (cid:15) given in (2) are α = canbederivedintheresultforthecooperativecommunication
3
−4.6·10−5, α = −7.8·10−4, α = 0.03 and α = 0.62, scenario. As we can see, Lemma 2 is validated and the
2 1 0
according to [17] for the case of 940 MHz. Regarding the ψ- probability p decreases for higher decoding thresholds.
CC
parameter in (1), since it defines the amount of power that is By comparing the two curves, we can also notice that the
splitbetweentheharvesterandtheinformationreceiver,itcan probabilityofsuccessfulexchangeishigherinthecooperative
bechoseninawaythatincreasestheaverageharvestedpower communication case compared to the direct one for the same
without affecting the probability of successful exchange. In decoding thresholds. This has been already proven in small-
Fig. 4, we provide the relation of ψ with the two metrics scale networks and with our study we extend this result even
(i.e.,probabilityofsuccessfuldecodingandaverageharvested forlarge-scalenetworkswithrandomrelaydeployment.Thus,
9
(a) µ=0.5 (b) µ=1 x 104 (a) DC scenario x 104 (b) CC scenario
30 30 3 3
m) m) Analytical w/o EH
B25 B25 Simulation w/o EH
d d
wer (20 wer (20 e (t)s2.5 ASnimaulylatictiaoln f ofor rµ µ==00.5.5 e (t)s2.5
Harvested Po11055 AAnnaallyyttiiccaall DDCC ww/itoh ccoonnvv.. eeffff.. Harvested Po11055 AAnnaallyyttiiccaall DDCC ww/itoh ccoonnvv.. eeffff.. erage Lifetim 2 ASnimaulylatictiaoln f ofor rµ µ==11 erage Lifetim 2 69%
age 0 SAinmaulylatictiaoln C DCC w w/oit hc ocnovn.v e. fef.ff. age 0 SAinmaulylatictiaoln C DCC w w/oit hc ocnovn.v e. fef.ff. Av 1.5 63% Av1.5
er−5 Analytical CC with conv. eff. er−5 Analytical CC with conv. eff.
Av Simulation CC with conv. eff. Av Simulation CC with conv. eff.
−10 0.1 0.2 0.3 0.4 0.5 −10 0.1 0.2 0.3 0.4 0.5 1 0.1 0.2 0.3 0.4 0.5 1 0.1 0.2 0.3 0.4 0.5
Intensity λ Intensity λ Intensity λ Intensity λ
1 1
Fig.6:Averageharvestedpowervs.Intensity.(a)µ=0.5,(b) Fig.7:AverageLifetimevs.Intensity:(a)Comparisonbetween
µ=1. DC and DC-EH, (b) Comparison between CC and CC-EH.
techxoacmInnhkmaFsnuigngteoi.c6wad,tiiivwlolenertsafpiaktliyeol,stp.ttlhhaeecreeavveiisraaagreeplharyoarbvnaeobsditleeitdsy,petovhweantertihbfeythmaeesdossiuraregcceet al Throughput ages/t/unit−area)s00..12 DCCC
in one CP versus the node intensity, considering two different patiess 0
Sm 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
casesforthechannelconditions,a)favorablewithµ=0.5and ( Intensity λ
1
b) moderate with µ=1. One first straightforward observation ea3000
fcreormtainboptohinfit,gtuhreesnoidsetshhaat,rvaesstthmeorientpeonwsiteyr,idnuceretaosethseuhnigtihlear unit−ar2000
interference. Also, compared to Fig. 6a, the results in Fig. 6b per 1000
need higher intensity to achieve the same average harvested ME DC DC−EH CC CC−EH
power, because the fading conditions attenuate the received T 00 .1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Intensity λ
power and, thus, the average harvested power. However, it is 1
very interesting to see that, after a peak value, the average Fig.8:(a)Spatialthroughputvs.Intensityand(b)Successfully
harvested power is decreasing. This can be seen clearly in exchangedmessagesinalifetimevs.Intensityforthedifferent
Fig.6aanditstemsfromthefactthattheRF-to-DCconversion communication scenarios.
efficiency of the rectennas, given in (2) and shown in Fig. 2,
decreases as the received power increases over a certain
point. Indeed, to highlight the difference between the average cooperative scenario, where the lifetime gains can reach up
harvested power with and without RF-to-DC conversion, we to 69%, compared to a gain of 63% in Fig. 7a. The gains
also plot in the same figure the cases without the conversion, are higher for the CC scenario, because relays contribute to
which show the significant amount of energy that is lost due the average harvested energy during each CP compared to
to the conversion (e.g., for µ = 0.5 and λ = 0.2 in the DC the DC scenario. Additionally, it can be noticed that, in the
scenario,thedifferencebetweenthetwocasesisover3dBm). CC case, there is a limit in the average lifetime from the
This is a very important insight which implies that i) adding intensities between 0.2 and 0.3. This stems from the fact that
more nodes in the network does not necessarily increase the S sources cannot achieve higher lifetime than this limit (i.e.,
2
lifetime of the network and ii) there is a unique maximum of λ =0.3), which limits the lifetime of the whole network, as
2
theaverageharvestedpoweraccordingtotheconditionsofthe it is explained in Remark 3.
system.Inaddition,bycomparingthedifferentcommunication Having validated the analysis, we now present a perfor-
scenarios in both figures, we notice that the cooperative mance evaluation for the two communication scenarios in
scenarioprovidesthehighestamountofharvestedpower.This Fig. 8. In this figure, the simulation results appear as markers
is due to the fact that, in this scenario, there are also relays while the lines represent the analytical results. As depicted
that provide more energy to the system in one CP. in Fig. 8a, the spatial throughput increases with the intensity,
In Fig. 7, we present the average network lifetime with sincemorenodesexchangemessagesperunitarea.Moreover,
and without EH for both scenarios versus the intensity λ it is interesting to notice that, although the probability of
1
of the S source nodes. For the DC scenario (Fig. 7a), we message exchange is always higher in the cooperative com-
1
assume that the intensity λ is equal to the optimal intensity munication(seealsoRemark1),thespatialthroughputforthe
2
calculated using Lemma 4 (i.e., λ (cid:39) 0.25 for µ = 0.5 and cooperative scenario presents lower performance than the DC
2
λ (cid:39) 0.5 for µ = 1). Similarly, for the CC scenario (Fig. scenario. This can be justified by considering the randomness
2
7b), we assume that the intensity of the relays is equal to in the deployment of the relays and the longer CPs in the
the optimal (λ = 0.25) and we set λ = 0.3. As expected, cooperative scenario. To clarify, although the performance
R 2
EH increases the lifetime of the network, especially for the gains from cooperation are obvious in a scenario where the
10
(a) λ/λ=2.5, λ=λ =λ counter-intuitive insight that the DC scenario presents better
1 2 1 R opt
2500 communicationperformancethantheCCscenarioinrandomly
a
e DC
−ar2000 CC −25% messages/unit−area deployed dense networks. Nevertheless, thanks to its higher
nit1500 lifetime,theCCscenariocouldbeprovedidealforapplications
u
er 1000 suchasincaseswherethenodesareembeddedinbuildingsor
ME p 500 +lif4e0ti%m ein bodieswithouteasyaccess,wherelongevityismoreimportant
T 0 than high data rates.
0 0.5 1 1.5 2 2.5 3
Time (ts) x 104
(b) λ/λ=1, λ=λ =λ
1 2 1 R opt
ea2500 DC −25% messages/unit−area VII. CONCLUSION
−ar2000 CC This paper has studied the impact of WEH on the infor-
nit1500
er u1000 +38% in mation exchange in large-scale networks. The purpose of the
E p 500 lifetime randomly deployed WSN nodes is to exchange successfully
M
T their messages locally with their neighbors, either directly
0
0 0.5 1 1.5 2 2.5 3 (direct communication scenario) or through a relay node
Time (ts) x 104 (cooperativecommunicationscenario).Thedifferentscenarios
Fig. 9: Successfully exchanged messages per unit-area vs. werecomparedintermsofmessageexchangeprobability,spa-
Time for the different scenarios. tial throughput and network lifetime. The theoretical deriva-
tions were validated by extensive Monte-Carlo simulations.
Finally, the comparison of the two scenarios highlighted the
relays are located in between the source nodes, this is not the importance of WEH in large-scale networks and revealed that
case for randomly deployed networks. In such networks, it the direct communication scenario presents better communi-
is possible for a direct link to provide better communication cation performance than the cooperative scenario in randomly
than a cooperative link, whereas the performance of the deployeddensenetworks.However,thecooperativescenariois
cooperative scenario is limited and depends on the random more advisable in applications where longevity matters, since
relaydeployment.ThisfactinconjunctionwiththelongerCPs it is superior in terms of lifetime.
in the cooperative scenario are the reasons that the message
exchange rate of CC drops in comparison to the DC scenario.
Moreover, in Fig. 8b, we combine the two metrics given APPENDIXA
in Fig. 7 and Fig. 8a and estimate the number of successfully PROOFOFPR(γ >γ∗|h≥ψ)INTHEOREM1
decodedmessagesduringthenetworklifetimeperunitareaas In this section, we will derive the probability Pr(γ >
a function of the intensity. From this figure, it is evident that γ∗|h≥ψ). Conditioning on the nearest transmitting source at
the CC-EH scenario presents lower performance compared to a distance r, the probability of successful message reception
the direct scenario with EH, which shows that the additional given that h≥ψ is given by
time slots in the CC scenario drop the performance. However,
in Fig. 8b, it is worth noting that the performance of the Pr(γ >γ∗|h≥ψ)=E [Pr(γ >γ∗|h≥ψ,r)]=
r
network through time is not taken into account. Since the (cid:90) ∞
battery capacity of the CC scenario is decreased through time = Pr(γ >γ∗|h≥ψ,r)fr(r)dr =
0
withalowerratethanintheDCscenario,wecouldidentifythe (cid:90) ∞ (cid:18)P hr−αv(ψ) (cid:12) (cid:19)
trade-offsbetweenthetwoscenarioswhiletakingintoaccount = Pr t >γ∗(cid:12)r f (r)dr =
v(ψ)I +N (cid:12) r
the total exchanged messages and the average lifetime. 0 r
(cid:90) ∞ (cid:18) γ∗rαI (cid:12) (cid:19)
Finally, in Fig. 9, we present the average exchanged mes- = Pr h> r >γ∗(cid:12)r f (r)dr,
sages per unit area versus time for two different intensity 0 1−φγ∗rα (cid:12) r
combinations (i.e., in Fig. 9a, λ = λ = λ and λ =
1 R opt 2 wheref (r)denotestheprobabilitydensityfunction(PDF)of
λopt/2.5 and in Fig. 9b, λ1 =λ2 =λR =λopt). We observe r, givenrin [27, 2.9.1] and φ=N/(P ψ). Since h follows an
t
that, in Fig. 9a, the network has lower lifetime compared exponential distribution, we have
to Fig. 9b, because the network lifetime is limited by the
lhnoaewntwdero,rwiknhtleeinnfestiaitlmyleosefitssthmheaavSxe2imtshiezeteodop.ftMismoouarlrecoienvtenerno,sdiiettys.ca(OFningb.teh9ecbl)oe,athtrhleyer P=r(γ(cid:90)0>∞γE∗I|rh(cid:20)≥exψp)(cid:18)=−(cid:90)01∞−µEγφ∗Iγrr∗(cid:20)αrPαr(cid:18)Irh(cid:19)>(cid:12)(cid:12)(cid:12)(cid:12)r(cid:21)1fγ−r∗(φrrαγ)dI∗rrrα=>γ∗(cid:12)(cid:12)(cid:12)r(cid:19)fr(r)dr=
seen that the communication scenarios present different trade- (cid:90) ∞ (cid:18) µγ∗rα (cid:19)
offs. For instance, in Fig. 9a, the DC scenario has higher = 0 LIr 1−φγ∗rα fr(r)dr, (31)
number of exchanged messages but lower lifetime, while
where L (s) defines the Laplace transform of the interfer-
the CC scenario demonstrates higher lifetime (+40%) with ence. WeIraim to calculate the Laplace transform by applying
fewer exchanged messages (−25%). Similarly, in Fig. 9b, the the following steps:
CC scenario demonstrates higher lifetime (+38%) with again
fewTeortehxacthaenngde,dthmeesresasugletss(i−n2F5%ig.).8 and Fig. 9 reveal the LIr(cid:18)1−µγφ∗γr∗αrα(cid:19)=Ee−(cid:0)1−µγφ∗γr∗αrα(cid:1)Ir =E(cid:20) (cid:89) e−(cid:0)1−µγφ∗γr∗αrα(cid:1)rhα(cid:21),
i∈Φ/x