throughput analysis of cooperative communication...
TRANSCRIPT
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Throughput Analysis of CooperativeCommunication in Wireless Ad Hoc Networks with
Frequency ReuseYong Zhou, Student Member, IEEE, and Weihua Zhuang, Fellow, IEEE
Abstract— In this paper, we investigate the network throughputachieved by both spatial diversity and spatial frequency reuse ina wireless ad hoc network with randomly positioned single-hopsource-destination pairs and relays. Compared to conventionaldirect transmissions, cooperative communication can enhancesingle-link transmission reliability, but reduce network-wide spa-tial frequency reuse due to relay transmissions. To study thetradeoff between these two competing effects, we construct ageographically constrained region for relay selection based onchannel state information. The network throughput, defined asthe product of the success probability of each link and theexpected number of concurrent transmissions, is derived as afunction of the total number of links, relay density, size of relayselection region, and distance between the source and destination.The performance analysis is carried out for both selection com-bining and maximum ratio combining at the destination. Suchanalytical results can evaluate the effectiveness of cooperativecommunication and provide useful insights on the design of large-scale networks. Finally, extensive simulations are conducted tovalidate the performance analysis.
Index Terms—Cooperative communication, spatial diversity,spatial frequency reuse, constrained relay selection region, Pois-son point process, selection combining, maximum ratio combin-ing.
I. INTRODUCTION
Wireless ad hoc networks have received extensive attentionsfrom both academia and industry because of their low costsand wide applications [2]. To meet the ever increasing trafficdemand and quality-of-service (QoS) requirement, the maincauses of performance degradation (e.g., channel fading andtransmission interference) should be addressed. As a result,cooperative communication as an effective technique for real-izing spatial diversity has been proposed [3]. Via coordinatingmultiple nearby nodes to work together and form a virtual an-tenna array, cooperative communication can improve channelcapacity and enhance transmission reliability [4]–[7].
Due to the scarcity of the radio spectrum, it is almostimpossible to allocate an exclusive channel for each source-destination pair, especially in a wireless ad hoc network. Byseparating the concurrent transmissions in space using thesame radio channel, spatial frequency reuse is an efficientmethod to enhance spectrum utilization [8]. Hence, two types
This work was supported by a research grant from the Natural Science andEngineering Research Council (NSERC) of Canada. This work was presentedin part at IEEE INFOCOM 2013 (mini conference) [1].
The authors are with the Department of Electrical and Computer Engineer-ing, University of Waterloo, 200 University Avenue West, Waterloo, Ontario,Canada, N2L 3G1 (E-mail: {y233zhou, wzhuang}@uwaterloo.ca).
S D
R
AB
IR of S IR of D
IR of R
Fig. 1: IR enlargement due to relay transmission under the protocol interfer-ence model.
of gains can be achieved by utilizing spatial resources, namelyspatial diversity gain and spatial reuse gain. Existing worksmainly focus on how to exploit either maximal spatial diversitygain or maximal spatial reuse gain [9], [10]. Maximizing onetype of gain, however, does not necessarily maximize theother. We take a network scenario with two source-destinationpairs, as shown in Fig. 1, as an example. Under the protocolinterference model, while relay R can enhance the transmis-sion reliability of the concerned link S − D by achievinga spatial diversity gain, it can also enlarge the interferenceregion (IR) of the concerned link to block the transmission ofits neighboring link B − A. Generally, cooperation occupiesmore spatial resources to achieve the spatial diversity gain,at a potential cost of reducing the spatial reuse gain due torelay transmissions. There exists a tradeoff between single-linkcooperation gain and network-wide reduced spatial frequencyreuse. Hence, the effectiveness of cooperation in a wireless adhoc network should be evaluated from a perspective of overallnetwork performance [1], rather than the performance of asingle source-destination pair.
In a wireless ad hoc network, node locations are dynamicdue to node mobility. Such characteristics pose challengeson analyzing the overall network performance achieved byboth spatial diversity and spatial frequency reuse. Firstly,the single-link cooperation gain depends on relay selection,which should take account of both the spatial distribution ofrelays and time-varying channel fading. Secondly, the locationrandomness of both source-destination pairs and relays makes
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it difficult to analyze the reduction in spatial frequency reuseof the whole network. Without network-wide information,a locally beneficial cooperation decision is not guaranteedto be network-wide beneficial. Although there have beensignificant efforts in exploiting the benefits and demonstratingthe effectiveness of cooperation for a single source-destinationpair, the performance analysis for cooperation in a wirelessad hoc network is very limited. Such analytical results canprovide insights for the network design. Hence, it is desirableto fully understand the benefits and limitations of cooperationin a wireless ad hoc network, which motivates this work.
In this paper, we analyze the network throughput of co-operative transmissions in a wireless ad hoc network withrandomly positioned single-hop source-destination pairs andrelays, to study the tradeoff between the performance achievedby spatial diversity and spatial frequency reuse. Under theprotocol interference model, we construct a diamond-shapedrelay selection region to restrict the detrimental effect of theenlarged IR due to relay transmissions. Based on stochasticgeometry, we model the relay locations by a homogeneousPoisson point process (PPP). Within a relay selection region,the relay with the best channel to the destination that receiveda packet from the source is selected as the best relay. Oversuch a network model, we derive the outage probabilities forboth selection combining (SC) and maximum ratio combining(MRC) at the destination. The single-link cooperation gain ischaracterized by comparing the success probabilities of thecooperative and direct transmissions. On the other hand, wemodel the source locations by a binomial point process (BPP)and approximate the link interference region by an ellipticalregion. Based on a randomized scheduling scheme, we derivethe expected number of concurrent transmissions that canbe accommodated within the network coverage area. Thenetwork-wide reduced spatial frequency reuse is characterizedby comparing the expected numbers of concurrent cooperativeand direct transmissions. As a result, we can evaluate theeffectiveness of cooperation from a perspective of overallnetwork performance, by checking whether or not the achievedsingle-link cooperation gain can compensate for the reductionin network-wide spatial frequency reuse.
The main contributions of this paper are three-fold:i) We develop a theoretical performance analysis framework
for cooperation in a wireless ad hoc network with randomlypositioned single-hop source-destination pairs and relays. Itcan be extended to analyze more complicated relay selectionschemes. The effectiveness of cooperation is evaluated froma perspective of overall network performance by consideringboth single-link cooperation gain and network-wide reducedspatial frequency reuse;ii) We construct a diamond-shaped relay selection region
to balance the tradeoff between spatial diversity and spatialfrequency reuse. The amount of spatial resources allocated toachieve each type of gain is controlled by the size of relayselection region. Such a relay selection region covers the bestrelay locations by introducing a small enlarged IR;iii) The network throughput is derived for both SC and
MRC at the destination in terms of important network andprotocol parameters. Extensive simulations are conducted to
validate the performance analysis. The analytical results canbe used to evaluate the network performance and provideguidance on the network design.
The rest of this paper is organized as follows. The relatedwork is reviewed in Section II. In Section III, the system modelunder consideration is described. We present the proposedcooperation scheme and relay selection strategy in SectionIV. In Section V, we characterize the single-link cooperationgain. The network-wide reduced spatial frequency reuse isanalyzed in Section VI. Numerical results are given in SectionVII. Finally, Section VIII concludes this work. The importantsymbols used in this paper are summarized in Table I, and theproofs of three propositions are given in Appendices.
II. RELATED WORK
Two categories of performance analysis for cooperativecommunication can be distinguished in the literature. In thefirst category, the relay locations are known and fixed; whilethe second category includes the scenario with random relaylocations.
The scenario with fixed relay locations is studied in [11]–[15]. Without power control, employing more relays leads toa higher spatial diversity gain, but incurs a larger IR for eachcooperative link. In contrast, a single-relay cooperative schemeis easier to implement and achieves full-order spatial diversityby selecting the best relay [11]. Because of its simplicity andefficiency, the single-relay cooperative scheme is used in manyexisting studies. Opportunistic relaying [11] and selectioncooperation [12] are two representative single relay selectionapproaches. In opportunistic relaying, the relay that maximizesthe minimum signal-to-noise ratio (SNR) of the source-relayand relay-destination links is selected. The outage probabilitiesof opportunistic amplify-and-forward (AF) and decode-and-forward (DF) relaying under different fading channels arederived in [13] and [14], respectively. On the other hand,in selection cooperation, the relay with the best channel tothe destination is selected. The outage probability of selectioncooperation is derived in closed form in [15]. Although thesestudies demonstrate the potential benefits of the single relayselection approach, they do not take account of the spatialdistribution of relays and cannot be directly extended to awireless ad hoc network.
The scenario with randomly positioned relays is studiedin [16]–[19]. The stochastic geometry [20], [21] as an effec-tive mathematical tool is used to deal with random networktopologies by treating node locations in a probabilistic manner.Via modeling the relay locations by a homogeneous PPP, theoutage probabilities of both opportunistic relaying and selec-tion cooperation are analyzed for Rayleigh fading channels in[16]. The performance analysis is extended for general fadingchannels in [17]. An uncoordinated cooperation scheme isproposed in [18], where each relay contends to cooperate witha specific probability calculated based on both local channelstate information (CSI) and spatial distribution of relays.Moreover, the authors in [19] introduce a QoS region withinwhich any relay can be selected to satisfy a specified QoSconstraint. These studies focus on the performance analysis
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TABLE I. Summary of Important Symbols
Symbol DefinitionADR0(ω) Area of DR0(ω)dxy Euclidean distance between nodes x and yDR0(ω) Diamond-shaped relay selection region with angle ω for link L0
G Network throughput gain in using cooperative transmissionsHxy Random distance-independent fading coefficient between nodes x and yK0 Number of potential relays for link L0
N Total number of links within the network coverage areaN(D/C)T Expected number of concurrent direct/cooperative transmissionsP (k, n) Probability that k links can be scheduled after checking the first n linksq
(S/MR)CCT Outage probability of the cooperative transmission with SC/MRCqDT Outage probability of the direct transmissionQ(D/C)T Interference-free probability between any two direct/cooperative transmissionsR0b Best relay of link L0 for the cooperative transmission with SC
R0b′ Best relay of link L0 for the cooperative transmission with MRCRI(D/C)T Interference range of the direct/cooperative transmissionT(D/C)T Network throughput of the direct/cooperative transmissionZi Location of relay Riα Path loss exponentβr Required reception threshold for the packets transmitted at rate rγxy SNR of the link between nodes x and yλR Relay densityΦ0 Set of qualified relays for link L0
ΦR PPP formed by relay nodes
for a single source-destination pair, which although providinguseful insights on the potential benefits of cooperation, cannotcharacterize the performance of cooperation in a wireless adhoc network.
A study on cooperation in a wireless ad hoc network ispresented in [22], which employs a unit disk graph model toanalyze the penalty of the enlarged IR. This method suffersfrom three limitations: First, the link density is not considered,which is an influential factor for the reduction in network-widespatial frequency reuse; Second, the unit disk graph modelcannot accurately characterize the IRs for both direct andcooperative links; Third, the relay selection is not considered,which determines the single-link cooperation gain.
In this paper, we characterize both single-link cooperationgain and network-wide reduced spatial frequency reuse ina wireless ad hoc network, while taking into account thespatial distributions of sources and relays, constrained relayselection region, and time-varying channel fading. We evaluatethe effectiveness of cooperation from a perspective of overallnetwork performance, in terms of the total number of links,relay density, size of relay selection region, and distancebetween the source and destination.
III. SYSTEM MODEL
A. Network Topology
Consider a wireless ad hoc network with nodes randomlylocated in the network coverage area. The locations of sourcesat any time instant can be specified by a BPP. Specifically,
N sources are independently and uniformly distributed withinthe network coverage area. To explicitly illustrate the impactof the distance between the source and destination on theeffectiveness of cooperative communication, we assume thateach source (Si) has an associated destination (Di) locatedat a fixed distance dSD away with a random direction [22]–[24]. Assuming that the relays do not have their own packetsto transmit and they are always willing to forward packetsfrom the sources [11]–[19]. At any time instant, they form ahomogeneous PPP ΦR with density λR (the number of nodesper unit area). Let Zi denote the location of relay Ri.
B. Propagation ChannelThe channel between any pair of nodes is characterized
by both Rayleigh fading and path loss. The channel impactbetween nodes x and y on the received signal power is repre-sented by Hxyd
−αxy , where Hxy denotes the random distance-
independent fading coefficient, dxy denotes the Euclideandistance between the nodes, and α ≥ 2 denotes the pathloss exponent. All the fading coefficients are independentand identically distributed (i.i.d.) exponential random variableswith unit mean. All nodes transmit with the same power. Fora packet transmission from source S0 to destination D0, thereceived SNR is given by
γS0D0 =PtHS0D0
d−αSDW
(1)
where Pt and W denote the transmission power and noisepower, respectively.
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C. Interference Model
We study the impact of interference on spatial reuse underthe protocol interference model. The interference range definesa region within which the transmission from an interfererinterrupts a packet reception [25]. The impact of interferenceis binary with respect to the interference range. The inter-ference from an interferer within the interference range of areceiver is intolerable, while the interference from an interfereroutside the interference range is negligible. Thus, a packettransmission from S0 to D0 is successful only if the followingconditions are satisfied: γS0D0
≥ βr and dSkD0> RIDT , for
every Sk, k > 0, transmitting concurrently, where βr denotesthe required reception threshold for the packets transmitted atrate r (in bit/s), and RIDT denotes the interference range forthe direct transmission. Define the required reception thresholdβr ≡ 2r/B − 1 so that r = B · log2(1 + βr), based onShannon’s formula, where B denotes the channel bandwidth inHz. In two-hop cooperative transmission, rate 2r is utilized inboth hops to achieve the target rate r between the source anddestination. The required reception threshold and interferencerange for the cooperative transmission are denoted as β2r andRICT , respectively.
D. Packet Transmission
We consider a time-slotted packet transmission scheme, inwhich the time-slot duration is a constant and all sources aresynchronized in time. The fading coefficients and node loca-tions remain invariant during one time-slot. There is only onecommunication channel and the concurrent transmissions areenabled across different locations. Before a packet transmis-sion, the coordination signaling among a source, its intendeddestination, and neighboring relays is required. Each sourcealways has a packet for transmission [26] and it transmitsto its intended destination via either direct or cooperativetransmission. For both direct and cooperative transmissions,all packets have equal length and each packet is transmitted inexactly one time-slot. Each node has a single omni-directionalantenna and operates in half-duplex mode. As the main focusof this paper is to study the tradeoff between spatial diversityand spatial frequency reuse, the protocol overhead incurred bycoordination signaling and relay selection is not considered.
IV. COOPERATIVE TRANSMISSION AND RELAYSELECTION
A single relay is considered for the cooperative transmis-sion; hence, each source and its best relay share one time-slotin transmitting the same packet. The DF scheme is adopted bythe best relay. A time-slot is partitioned equally to two sub-time-slots [27]. Each source transmits a packet at rate 2r inthe first sub-time-slot. Due to the broadcast nature of wirelesscommunications, the intended destination and neighboringrelays can successfully receive the packet, depending on bothinstantaneous channel fading and path loss. A CSI-based relayselection strategy is employed to select the best relay, whichforwards the packet to the intended destination at rate 2rin the second sub-time-slot. Finally, the destination decodesthe packet using either SC or MRC. In SC, the destination
S Dx
( )DR ω
ω ω( )0,0
tan2
SDd ω
2SDd
y
2SDd−
O
Reliable
Outage
Qualified relays
Unqualified relays
R3
R2
R1
R4
Fig. 2: A diamond-shaped constrained relay selection region under therectangular coordinate system. Only the relays within the constrained relayselection region (e.g., R2, R3, and R4) and successfully receive the packetfrom the source (e.g., R2 and R3) are qualified relays (e.g., R2 and R3).
selects only one link from the direct and forward link forpacket decoding. On the other hand, in MRC, the destinationcombines the signals from both the direct and forward linkfor packet decoding. Hence, two cooperative transmissionschemes are considered, that is CSI-based relay selection withSC and MRC, respectively. Studying both SC and MRC canprovide insights for the tradeoff between performance andcomplexity.
At the link layer, as each transmission requires handshaking,each node of one link becomes both a transmitter and receiverduring the transmission of one packet. The IR of a direct(or cooperative) link is the combination of the IRs of thesource and destination (and relays). As more nodes takepart in the transmission of one packet, a cooperative linkoccupies more spatial resources, which reduces the spatialfrequency reuse. As shown in Fig. 1, the size of the IRoccupied by a cooperative link is determined by the relaylocation. Because all relays are randomly distributed acrossthe network, the size of the IR of each cooperative link israndom and it may be large enough to significantly reduce thespatial frequency reuse. To restrict the size of the IR occupiedby each cooperative link, a simple way is to construct aconstrained geographical region for relay selection. The relayswithin the relay selection region are called potential relays andonly the potential relays can contend to be the best relay. Inorder to select the best relay and in turn achieve full-orderspatial diversity, the packet collision at each potential relayshould be avoided and the signaling among the potential relaysfor relay selection should be collision-free. As a result, allpotential relays should be protected from interference. Thesize of the IR of a cooperative link is determined by therelative location of the furthermost potential relay, which canbe controlled by adjusting the size of relay selection region.Such a relay selection region establishes a connection betweensingle-link cooperation gain and network-wide reduced spatialfrequency reuse. A larger relay selection region leads to ahigher cooperation gain by incorporating more potential relays,at the cost of reducing the network-wide spatial frequencyreuse.
We consider a diamond-shaped relay selection region foreach cooperative link, as it covers the best relay locations by
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introducing a small enlarged IR and only has one parameter.As shown in Fig. 2, a source and its intended destination locateat two endpoints of one diagonal. Such a diamond region (DR)is characterized by an angle ω, which determines the size ofthe constrained relay selection region. The potential relays thatsuccessfully receive the packet from source S0 in the firstsub-time-slot are referred to as qualified relays, which form adecoding set, Φ0. Mathematically, the decoding set Φ0 can beexpressed as
Φ0 = {Zi ∈ ΦR ∩DR0(ω), γS0Ri ≥ β2r} (2)
where DR0(ω) denotes the constrained relay selection regionwith angle ω for link L0.
We assume that each qualified relay knows its qualificationstatus and has instantaneous SNR information of the channelbetween itself and its intended destination. A back-off schemecan be used to select the best relay in a distributed manner,which requires only local CSI [28]. When the decoding setΦ0 is not empty, a qualified relay with the best channel to thedestination obtains the shortest back-off duration and contendsto be the best relay first. Other qualified relays quit contentionas soon as they receive the signaling from the best relay, R0
b .The location of the best relay is given by
ZR0b
= arg maxZi∈Φ0
{HRiD0d−αRiD0
}. (3)
In addition to spatial diversity, spatial frequency reuse isanother way to enhance spectrum efficiency. As all links arerandomly distributed across the network coverage area andinteract with each other, the optimal scheduling problem isshown to be NP-hard in [29]. For simplicity, a randomizedscheduling scheme proposed in [30] is employed to activatenon-interfering links for concurrent transmissions. The mainidea of the randomized scheduling scheme is to check alllinks in a random order and remove a new link if it interfereswith existing ones. The remaining links can be activated con-currently without interrupting each other. Two links interferewith each other when any node of one link locates within theinterference range of any node of the other link.
The network throughput, defined as the product of thesuccess probability of each link and the expected numberof concurrent transmissions, measures the expected numberof concurrent successful transmissions within the networkcoverage area, given the total number of links and relaydensity. The network throughput for the direct and cooperativetransmissions can be, respectively, expressed as
TDT = (1− qDT ) ·NDTTCT = (1− qCT ) ·NCT
(4)
where qDT and qCT represent the outage probabilities of thedirect and cooperative transmissions respectively, NDT andNCT represent the expected numbers of concurrent direct andcooperative transmissions respectively. Due to the enlarged IR,NCT is not larger than NDT .
The expected number of concurrent direct transmissionsis affected by the total number of links (i.e., N ), while theexpected number of concurrent cooperative transmissions isfurther affected by the size of relay selection region (i.e., ω)
and relay density (i.e., λR). The values of parameters ω andλR can be set to balance the tradeoff between spatial diversityand spatial frequency reuse. Specifically, with an increase ofω or λR, the single-link cooperation gain is enhanced as morepotential relays are available and the probability of selectinga better relay increases. On the other hand, with an increaseof ω or λR, the IR of a cooperative link is enlarged and thenetwork-wide spatial frequency reuse is reduced, which willbe discussed in Section VI.
From a perspective of overall network performance, thenetwork throughput gain in using the cooperative transmissionover the direct transmission can be expressed as
G =(1− qCT ) ·NCT(1− qDT ) ·NDT
= ρ · η (5)
where ρ = 1−qCT1−qDT represents the single-link cooperation gain
and η = NCTNDT
represents the network-wide reduced spatialfrequency reuse.
According to (5), cooperation in a wireless ad hoc networkis beneficial only when the achieved single-link cooperationgain can compensate for the reduction in network-wide spatialfrequency reuse, that is, ρ·η > 1. To evaluate the effectivenessof cooperation, we derive the single-link cooperation gain andnetwork-wide reduced spatial frequency reuse in Sections Vand VI, respectively.
V. SINGLE-LINK COOPERATION GAIN ρ
In this section, we characterize the single-link cooperationgain by comparing the success probabilities of the cooperativeand direct transmissions. Based on stochastic geometry, wederive the outage probability of the CSI-based relay selectionstrategy for both SC and MRC, while taking into accountthe spatial distribution of relays, constrained relay selectionregion, and time-varying channel fading.
A. CSI-based Relay Selection with SCWith SC at the destination, an outage occurs when both
the direct and forward links cannot support the requiredtransmission rate. Specifically, as a source transmits a packetat rate 2r in the first sub-time-slot, the direct link fails whenthe SNR at the destination is smaller than β2r. On the otherhand, the forward link fails when one of the following eventsoccurs: 1) There are no potential relays within DR0(ω); 2)Event E1: there are no qualified relays when there exists atleast one potential relay; 3) Event E2: destination D0 fails todecode the packet from the best relay in the second sub-time-slot when decoding set Φ0 is not empty. Hence, the outageprobability, denoted as qSCCT , is given by
qSCCT = P (γS0D0< β2r)
×
[P (K0 = 0) +
∞∑k=1
P (K0 = k) · P (E1 ∪ E2 |K0 = k )
](6)
where K0 represents the number of potential relays withinDR0(ω), and outage events E1 and E2 can be expressed as
E1 = {Φ0 = ∅,K0 > 0}
E2 ={
Φ0 6= ∅, γR0bD0
< β2r
}.
(7)
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qSCCT = [1− exp (−MdαSD)]× exp
[−2λR
∫ dSD2 tanω
0
∫ dSD2 −
ytanω
ytanω−
dSD2
exp(−M
(dαS0R + dαRD0
))dxdy
]. (8)
qSCCT = [1− exp (−MdαSD)]× exp
[−√
2π√MλR
∫ dSD2 tanω
0
exp
(−2My2 − Md2
SD
2
)erf
(√2M
(dSD
2− y))
dy
]
= [1− exp (−MdαSD)]× exp
[−∞∑k=0
λR22k+2Mk
(2k + 1)!! exp (Md2SD)B(
4t2 − 2dSDt,dSD
2(1− tanω),
dSD2
)].
(9)
The outage probability can be evaluated in terms of the relaydensity, size of relay selection region, and distance between thesource and destination, as stated in the following proposition.
Proposition 1. Given a diamond-shaped relay selection regionDR0(ω), as shown in Fig. 2, the outage probability of theCSI-based relay selection strategy with SC at the destina-tion receiver, given by (6), can be written as (8), whereM = β2rW
Pt, dS0R =
√(x+ dSD
2 )2 + y2, and dRD0=√
(x− dSD2 )2 + y2.
The outage probability in (8) decreases exponentially as therelay density increases, and it can be calculated numerically inMAPLE. For the special case of α = 2, the integration can bereplaced by the series summation in the following corollary,which can be calculated more efficiently [31].
Corollary 1. For the special case of α = 2, the outageprobability given in (8) can be simplified as (9), whereerf(x) = 2√
π
∫ x0e−t
2
dt is the error function and
B(g(t),m, n) =
∫ n
m
exp [−Mg (t)]t2k+1dt. (10)
B. CSI-based Relay Selection with MRCWith MRC at the destination receiver, in the following
analysis, the source is treated equivalently as a qualified relayand it transmits the packet in the second sub-time-slot onlywhen it is selected as the best relay, i.e., all qualified relayskeep silent. Let R0
b′ denote the best relay in this cooperationscheme. The instantaneous SNR of the channel between R0
b′
and D0 is given by
γR0b′D0
= max{γR0
bD0, γS0D0
}. (11)
An outage occurs when the destination fails to decode thepacket after combining the signals transmitted by the sourceand the best relay in the first and second sub-time-slots,respectively. Hence, the outage probability, denoted as qMRC
CT ,is given by
qMRCCT = P
(γS0D0 + γR0
b′D0< β2r
). (12)
Proposition 2. Given a diamond-shaped relay selection regionDR0(ω), as shown in Fig. 2, the outage probability of the CSI-based relay selection strategy with MRC at the destinationreceiver, given by (12), can be written as (13).
Similarly, the outage probability in (13) decreases exponen-tially as the relay density increases, and it can be numericallyevaluated in MAPLE.
In the direct transmission, a source utilizes the whole time-slot to transmit a packet at rate r. An outage occurs whenthe SNR at the destination is smaller than βr. Hence, theoutage probability for the direct transmission is given byqDT = 1− exp
(−βrWPt d
αSD
). Finally, by comparing the suc-
cess probabilities of the cooperative and direct transmissions,the single-link cooperation gain ρ can be obtained.
VI. NETWORK-WIDE REDUCED SPATIAL FREQUENCYREUSE η
In this section, we characterize the reduction in network-wide spatial frequency reuse by comparing the expectednumbers of concurrent cooperative and direct transmissionsthat can be accommodated within the network coverage area.Taking into account the spatial distributions of both source-destination pairs and relays, we calculate the expected numbersof concurrent direct and cooperative transmissions based on arandomized scheduling scheme [30].
Let P (k, n) denote the probability that k links can bescheduled for concurrent transmissions after checking the firstn links. Denote Q as the interference-free probability betweenany two links. After checking the first n links, there are klinks that can be scheduled concurrently if 1) (k − 1) linksare scheduled after checking the first (n− 1) links, and link ndoes not interfere with the scheduled (k− 1) links; 2) k linksare scheduled after checking the first (n − 1) links, and linkn interferes with at least one of the scheduled k links. Hence,we have
P (k, n) = P (k−1, n−1)Qk−1 +P (k, n−1)(1−Qk). (15)
With the initial values P (1, 1) = 1, P (1, 2) = 1 − Q,and P (2, 2) = Q, we can iteratively calculate P (k,N)for all k ≤ N . Hence, the expected number of concurrent
transmissions can be calculated by NE =N∑k=1
kP (k,N). To
calculate the expected number of concurrent transmissions, theinterference-free probability between any two direct (coopera-tive) links should be derived. The interference-free probabilitybetween two direct links depends on the distance between thesource and destination, while the interference-free probability
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qMRCCT =
[1− exp
(−M
2dαSD
)]exp
(−2λR
∫ dSD2 tanω
0
∫ dSD2 −
ytanω
ytanω−
dSD2
I(x, y)dxdy
)(13)
where
I (x, y) =− exp
[−M
(dαS0R
+ dαRD0
)]+ exp
[−M2
(dαSD + dαRD0
)−MdαS0R
][1− exp
(−M2 d
αSD
)][(dRD0
/dSD)α − 1]
. (14)
Si Di
NIRi
LIRi
2
SDd
2
SDd
L
DTD
S
DTD
I
DTR
Fig. 3: Approximated IR of a direct link.
between two cooperative links is further affected by the sizeof relay selection region and relay density.
As the IR of a direct (cooperative) link is the combinationof the IRs of the source and destination (and relays), it canbe approximated as an elliptical region [1]. We calculate theinterference-free probability between any two direct (cooper-ative) links in the following two subsections.
A. Direct Link
As shown in Fig. 3, we approximate the IR of a direct link,Li, by an elliptical region and refer to it as node interferenceregion (NIRi). Any node of other active links (e.g., Lj , j 6=i) should locate outside NIRi to avoid interrupting link Li.However, to guarantee both Sj and Dj locate outside NIRi isnot trivial, as the locations of Sj and Dj are not independentand they are placed dSD apart. For simplicity, we introduce alink interference region (LIR), which is also an elliptical regionbut larger than the NIR, to capture the interference relationshipamong different links. Specifically, LIRi is an elliptical regioncentered at the center of link Li, and the lengths of the semi-major axis and semi-minor axis are given by
DLDT = RIDT + dSD
DSDT = RIDT +
dSD2.
(16)
As illustrated in Fig. 3, the LIR radii are increased by dSD/2from those of the NIR. Thus, for any two direct links (e.g., Liand Lj), if the center of link Lj is outside LIRi, both Sj andDj are guaranteed to locate outside NIRi, which implies thattwo links are interference-free. Note that the reverse condition,the center of link Li locating outside LIRj , is not required, as
S Dx
ω ω
tan2
SDd ω
2SDd
y
2SDd−
O
V1
V2TR(V2)
Fig. 4: The upper half of the relay selection region.
the interference relationship between the nodes is reciprocal.The elliptical LIR area of a direct link is given by ADT =
πDLDTD
SDT . Because of the uniform distribution of nodes, the
interference-free probability between any two direct links canbe calculated by QDT = 1− ADTAN , where AN is the networkcoverage area.
B. Cooperative Link
As discussed in Section IV, all potential relays should beprotected from interference and the IR of a cooperative link isdetermined by the relative location of the furthermost potentialrelay in each side of the link. Due to the symmetry, we takethe upper half of the relay selection region, as shown in Fig.4, as an example. The furthermost potential relay refers to thepotential relay that has the largest Y-coordinate. Denote V1
as the Y-coordinate of the furthermost potential relay. As thepotential relays are randomly distributed within DR0(ω), V1
is a random variable, which takes value in[0, dSD2 tanω
]. To
calculate the average size of the IR occupied by a cooperativelink, we calculate the expected value of V1 as follows.
Proposition 3. The expected value of the Y-coordinate ofthe furthermost potential relay in the upper half of the relayselection region, E[V1], is given by
E [V1] =dSD
2tanω
−√π tanω
4λRerf
(dSD
2
√λR tanω
).
(17)
From (17), E[V1] increases with λR and ω. This is because,with an increase of λR and ω, the probability of having afaraway potential relay increases. Let RF1 and RF2 denotethe furthermost potential relays at both sides, at (0, E(V1)) and(0,−E(V1)) on average, respectively. The IR of a cooperativelink is calculated according to the average locations of thefurthermost potential relays.
The IR of a cooperative link, Li, is also approximated as anelliptical region, referred to as cooperative node interference
8
Si Di
CNIRi
CLIRi
2
SDd
RF1
RF2
I
CTR
S
CTD
L
CTD
Fig. 5: Approximated IR of a cooperative link.
region (CNIRi). Similarly, as shown in Fig. 5, we define acooperative link interference region (CLIRi), which is centeredat the center of link Li and the lengths of the semi-major axisand semi-minor axis are given by
DLCT = RICT + dSD
DSCT = RICT +
dSD2
+ E(V1).(18)
For any two cooperative links (e.g., Li and Lj), if thecenter of CLIRj locates outside CLIRi, both Li and Lj cantransmit concurrently without interrupting each other. Theelliptical CLIR area can be calculated by ACT = πDL
CTDSCT .
Accordingly, the interference-free probability between any twocooperative links is given by QCT = 1− ACTAN .
The calculation of the interference-free probability betweenany two links is slightly conservative, as some links maybe blocked although they do not actually cause collisions.According to the simulation results in Section VII, the re-sults calculated by this method is rather accurate and thecomputation complexity is quite low. The expected numbersof concurrent direct and cooperative transmissions, NDT andNCT , can be calculated by substituting QDT and QCT into(15). By comparing these two numbers, the network-widereduced spatial frequency reuse η can be obtained. Finally,we can derive the network throughput and network throughputgain by substituting the corresponding values into (4) and (5),respectively.
VII. NUMERICAL RESULTS
This section presents analytical (A) and simulation (S)results for both direct and cooperative transmissions in awireless ad hoc network. In the simulation, a circular networkcoverage area with radius 2000 m is considered. Based on[32] and [33], the interference ranges of direct and cooperativetransmissions, RIDT and RICT , are set to be 60 m and70 m, respectively. The transmission rate (r) and channelbandwidth (B) are normalized to be 1 Mbit/s and 1 MHz,respectively. The reception threshold of direct transmissions(βr) is set to be
√2−1, which is calculated based on Shannon’s
1 2 3 4 5 6 7
x 10−3
0.8
0.85
0.9
0.95
1
Relay Density (λR
)
Suc
cess
Pro
babi
lity
(1 −
q)
SC (A)MRC (A)DT (A)SC (S)MRC (S)DT (S)
Fig. 6: Success probability versus relay density (in nodes/m2) when ω = π/6and dSD = 50 m.
formula and the normalized reception threshold of cooperativetransmissions (i.e., β2r = 1). In addition, we set Pt = 0.06mW, W = −50 dBm, and α = 2. The simulation results areobtained by averaging 105 realizations of the random networktopology.
A. Transmission Success Probability and Single-Link Cooper-ation Gain
In this subsection, we study the impact of relay density λR,angle of relay selection region ω, and link distance dSD on thetransmission success probability and single-link cooperationgain.
Fig. 6 shows the success probabilities of direct transmission(DT) and cooperative transmissions (SC and MRC) versus therelay density with parameters ω = π/6 and dSD = 50 m,where the analytical results are obtained based on (8) and (13).When the relay density is low, the cooperative transmissionwith an SC receiver performs worse than the direct transmis-sion, because the probability of having a reliable relay is lowand the cooperative transmission requires a higher receptionthreshold for the SNR. It is observed that the success probabil-ities of both cooperative transmission schemes increase withthe relay density, while that of the direct transmission does notchange. This is due to the fact that, with an increase of λR,more potential relays are available for relay selection, whichresults in a higher probability of selecting a reliable relay.By combining the signal from the direct link, the cooperativetransmission with an MRC receiver outperforms that withan SC receiver. However, the performance gap between thecooperative transmission schemes reduces with an increase ofλR, because the channel quality of the forward link becomesbetter, which reduces the importance of the direct link forpacket decoding.
Fig. 7 illustrates the single-link cooperation gain achievedby cooperative transmission versus the angle of relay selectionregion for dSD = 50 m and 40 m when λR = 0.003nodes/m2. The cooperation gain increases with the size of
9
0.2 0.4 0.6 0.8 10.9
0.95
1
1.05
1.1
1.15
1.2
1.25
Angle of Relay Selection Region (ω)
Sin
gle−
Link
Coo
pera
tion
Gai
n (ρ
)
SC (A)MRC (A)SC (S)MRC (S)
dSD
=40 m
dSD
=50 m
Fig. 7: Single-link cooperation gain versus angle of relay selection region fordSD = 50 m and 40 m when λR = 0.003 nodes/m2.
50 100 150 200 250 30040
60
80
100
120
140
160
180
200
220
240
Number of Links (N)
Exp
ecte
d N
umbe
r of
Con
curr
ent T
rans
mis
sion
s (N
E)
CT (A)DT (A)CT (S)DT (S)
Fig. 8: Expected numbers of concurrent direct and cooperative transmissionsversus total number of links when ω = π/3, λR = 0.003 nodes/m2, anddSD = 50 m.
relay selection region, as more spatial resources are allocatedto each cooperative link and more potential relays are availablefor spatial diversity. Similarly, the performance gap betweenthe cooperative transmission schemes shrinks with an increaseof ω. The cooperation gain at dSD = 40 m is smaller than thatat dSD = 50 m. This observation shows that the cooperativetransmission is preferable when the link distance is large (i.e.,the channel quality of the direct link is poor).
B. Expected Number of Concurrent Transmissions andNetwork-Wide Reduced Spatial Reuse
Fig. 8 plots the expected numbers of concurrent directtransmissions and cooperative transmissions (CT) versus thetotal number of links with parameters ω = π/3, λR = 0.003nodes/m2, and dSD = 50 m. The expected numbers of con-current transmissions increase with the total number of links.As expected, the expected number of concurrent cooperative
0.2 0.4 0.6 0.8 10.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
Angle of Relay Selection Region (ω)
Net
wor
k−W
ide
Red
uced
Spa
tial R
euse
(η)
N=300 (A)N=200 (A)N=300 (S)N=200 (S)
Fig. 9: Network-wide reduced spatial frequency reuse versus angle of relayselection region for N = 200 and 300 when λR = 0.003 nodes/m2 anddSD = 50 m.
transmissions is always smaller than that of concurrent directtransmissions, as the enlarged IR due to relay transmissionsreduces the network-wide spatial frequency reuse. The gapbetween the expected numbers of concurrent direct and coop-erative transmissions expands with an increase of N . This isbecause the adverse effect of the enlarged IR becomes moresignificant as N increases.
Fig. 9 shows the network-wide reduced spatial frequencyreuse versus the angle of relay selection region for N = 200and 300 with parameters λR = 0.003 nodes/m2 and dSD = 50m. The spatial frequency reuse reduces as the angle of relayselection region increases. As ω increases, each cooperativelink occupies more spatial resources on average for spatialdiversity, which reduces the spatial resources available forspatial reuse. The reduction in network-wide spatial frequencyreuse at N = 300 is larger than that at N = 200. Thisobservation shows that, when the link density is high, thecooperative transmission is more likely to cause link blockage(i.e., the limitation of the cooperative transmission becomesmore significant). Due to the conservative calculation of theinterference-free probability between any two links, thereexists a small deviation between the analytical and simulationresults.
C. Network Throughput
In this subsection, we account for both the transmissionsuccess probability and expected number of concurrent trans-missions, and study the impact of the total number of links,size of relay selection region, relay density, and link distanceon the network throughput.
Fig. 10 illustrates the network throughput of the direct andcooperative transmissions versus the total number of linkswhen dSD = 50 m. The network throughput of all schemesincrease with N . With a small relay selection region and ahigh relay density, as shown in Fig. 10(a), both cooperativetransmission schemes outperform the direct transmission, as
10
50 100 150 200 250 30020
40
60
80
100
120
140
160
180
200
220
Number of Links (N)
Net
wor
k T
hrou
ghpu
t (T
)
SC (A)MRC (A)DT (A)SC (S)MRC (S)DT (S)
(a) ω = π/6 and λR = 0.006 nodes/m2
50 100 150 200 250 30020
40
60
80
100
120
140
160
180
200
Number of Links (N)
Net
wor
k T
hrou
ghpu
t (T
)
SC (A)MRC (A)DT (A)SC (S)MRC (S)DT (S)
(b) ω = π/3 and λR = 0.0002 nodes/m2
Fig. 10: Network throughput versus total number of links when dSD = 50m.
the single-link cooperation gain outweighs the reduction innetwork-wide spatial frequency reuse. On the other hand,with a large relay selection region and a low relay density,as shown in Fig. 10(b), the direct transmission achieveshigher network throughput than both cooperative transmissionschemes. This implies that the cooperative transmission is notalways beneficial and its effectiveness depends on the size ofrelay selection region and relay density.
Fig. 11 plots the network throughput gain versus the angle ofrelay selection region with parameters λR = 0.003 nodes/m2,dSD = 50 m, and N = 300. With the variation of the angle ofrelay selection region, there exists a peak point of the networkthroughput gain. Take the cooperative transmission with MRCas an example. The network throughput gain increases with ωwhen ω < 2π/9, and decreases with ω when ω > 2π/9. Notethat when ω > 2π/9 the increasing rate of the single-linkcooperation gain is smaller than the decreasing rate of the
0.2 0.4 0.6 0.8 10.85
0.9
0.95
1
1.05
1.1
Angle of Relay Selection Region (ω)
Net
wor
k T
hrou
ghpu
t Gai
n (G
)
SC (A)MRC (A)SC (S)MRC (S)
Fig. 11: Network throughput gain versus angle of relay selection region whenλR = 0.003 nodes/m2, dSD = 50 m, and N = 300.
1 2 3 4 5 6 7
x 10−3
0.9
0.95
1
1.05
1.1
1.15
Relay Density (λR
)
Net
wor
k T
hrou
ghpu
t Gai
n (G
)
SC (A)MRC (A)SC (S)MRC (S)
dSD
=50 m
dSD
=40 m
Fig. 12: Network throughput gain versus relay density (in nodes/m2) fordSD = 40 m and 50 m when ω = π/6 and N = 200.
network-wide reduced spatial frequency reuse. This impliesthat the size of relay selection region can be set to balancethe tradeoff between spatial diversity and spatial reuse. Inaddition, we observe that the direct transmission outperformsthe cooperative transmission with SC when ω < π/9, as thenumber of potential relays is not large enough to enhance theoverall network performance. Comparing the results in Figs.7 and 11, due to the reduced spatial frequency reuse, thenetwork throughput gain is always smaller than the single-link cooperation gain, and the cooperative transmission thatis beneficial for a single source-destination pair may not bebeneficial for the whole network. Hence, the effectiveness ofcooperation should be evaluated from a perspective of overallnetwork performance.
Fig. 12 shows the network throughput gain of both coopera-tive transmission schemes versus the relay density for dSD =40 m and 50 m with parameters ω = π/6 and N = 200.The network throughput gain increases with the relay density,
11
as more potential relays are available for each cooperativelink. When dSD = 40 m, the cooperative transmission withSC is beneficial only if λR > 3 × 10−3 nodes/m2, whilethe cooperative transmission with MRC is beneficial only ifλR > 1.4×10−3 nodes/m2. This implies that a smaller numberof potential relays is required for the cooperative transmissionwith MRC, at the cost of requiring higher implementationcomplexity. On the other hand, the network throughput gainat dSD = 40 m is smaller than that at dSD = 50 m. Thisconfirms that the effectiveness of cooperation depends on thelink distance and the cooperative transmission is effectivewhen the link distance is large.
VIII. CONCLUSIONS
In this paper, we investigate the network throughput ofcooperative communication in a wireless ad hoc network withrandomly positioned single-hop source-destination pairs andrelays. The objective is to evaluate the effectiveness of co-operation from a perspective of overall network performance.We construct a diamond-shaped relay selection region to studythe tradeoff between spatial diversity gain of a single link andreduced spatial frequency reuse of the whole network. Wederive the network throughput of the proposed cooperationscheme in terms of the total number of links, relay density, sizeof relay selection region, and link distance. The cooperativetransmission is not always beneficial and its effectivenessdepends on all these influential factors. Due to the reducedspatial frequency reuse, the network throughput gain is alwayssmaller than the single-link cooperation gain. Extensive sim-ulations are conducted to validate the performance analysis.The analytical results can be used to evaluate the networkperformance and provide guidance on the design of large-scale networks. For further work, we aim to take account ofthe protocol overhead incurred by coordination signaling andrelay selection, in order to fully understand the benefits andlimitations of cooperative communication.
APPENDIX A. PROOF OF PROPOSITION 1
The probability that the direct link fails in the first sub-time-slot is given by
P (γS0D0 < β2r) = P(HS0D0 <
β2rW
PtdαSD
)(a)= 1− exp(−MdαSD)
(19)
where (a) follows from the exponential distribution of HS0D0 .Similarly, with a relay located at Zi instead of at a distance
dSD away, the success probabilities of the source-relay andrelay-destination links can be expressed as
P (γS0Ri ≥ β2r) = exp(−MdαS0Ri
)P (γRiD0 ≥ β2r) = exp
(−MdαRiD0
).
(20)
As the potential relays form a homogeneous PPP, theprobability of existing k relays within DR0 (ω) is given by
P (K0 = k) =
(λRADR0(ω)
)kk!
exp(−λRADR0(ω)
) (21)
where ADR0(ω) =d2SD
2 tanω represents the area of DR0 (ω).Outage event E1 means that no potential relays have a
reliable link to the source. Outage event E2 means that noqualified relays have a reliable link to the destination whenthe decoding set is not empty. Hence, outage event (E1 ∪ E2)is equivalent to the event that no potential relays have a reliablelink to both the source and destination. Given that k potentialrelays locate within DR0 (ω), we have
P (E1 ∪ E2 |K0 = k )
= E
[k∏i=1
[1− P (γS0Ri ≥ β2r) · P (γRiD0 ≥ β2r)]
].
(22)
As k potential relays are uniformly distributed withinDR0 (ω), we have
P (E1 ∪ E2 |K0 = k )
(a)=
(∫DR0(ω)
[1− P (γS0Ri ≥ β2r)P (γRiD0 ≥ β2r)] ds
ADR0(ω)
)k≡J k
(23)
where (a) follows from the probability generating functional(PGFL) of the BPP [21].
Combining (21) and (23), we can obtain (24), shown at thetop of the next page.
Consider a rectangular coordinate system with origin at linkcenter O, as shown in Fig. 2. Due to the constrained relayselection region, the coordinate of any qualified relay shouldsatisfy one of the following two constraints{x ∈
[y
tanω −dSD
2 ,− ytanω + dSD
2
], if y ∈
[0, dSD2 tanω
]x ∈
[− y
tanω −dSD
2 , ytanω + dSD
2
], if y ∈
[−dSD2 tanω, 0
].
(25)Due to the symmetry, we focus on the upper half of the
constrained relay selection region, where y ∈[0, dSD2 tanω
].
By substituting (19), (20), and (24) into (6), we get the resultin (8).
APPENDIX B. PROOF OF PROPOSITION 2
As the qualified relay with the best channel to the destina-tion is selected, the outage probability can be expressed as
qMRCCT = P
(γS0D0
+ max{γR0
bD0, γS0D0
}< β2r
)= P
(γS0D0 + max
Zi∈Φ0
{γRiD0} < β2r, 2γS0D0 < β2r
)= P
(γS0D0
+ maxZi∈Φ0
{γRiD0} < β2r
∣∣∣∣γS0D0<β2r
2
)× P
(γS0D0
<β2r
2
).
(26)
With HS0D0following an exponential distribution, we have
P(γS0D0 < β2r/2) = 1− exp(−MdαSD/2).By selecting the best relay, the outage event is equivalent
to that all qualified relays are in outage. The conditionalprobability of transmission failure with MRC can be obtained
12
P (K0 = 0) +
∞∑k=1
P (K0 = k) · P (E1 ∪ E2 |K0 = k )
= exp(−λRADR0(ω)
)+
∞∑k=1
(λRADR0(ω)
)kk!
exp(−λRADR0(ω)
)· J k
= exp[−λRADR0(ω) (1− J )
]= exp
[−λR
∫DR0(ω)
P (γS0Ri ≥ β2r) · P (γRiD0≥ β2r)ds
].
(24)
P(γS0D0
+ maxZi∈Φ0
{γRiD0} < β2r
∣∣∣∣γS0D0<β2r
2
)= exp
[−∫DR0(ω)
(1− P
(γS0D0
+ γRD0< β2r
∣∣∣∣γS0D0<β2r
2
))µ (ds)
].
(28)
P(γS0D0
+ γRD0< β2r, γS0D0
< β2r
2
)= P
(d−αSDHS0D0
+ d−αRD0HRD0
< M,HS0D0< M
2 dαSD
)=∫ M
2 dαSD
0
∫MdαSD−(dSDdRD0
)αx
0 exp (−x) exp (−y) dydx
=
1− exp(−M2 d
αSD
)+
exp(−MdαRD0)−exp(−M2 (dαSD+dαRD0
))(dRD0/dSD)
α−1, if dSD 6= dRD0
1− exp(−M2 d
αSD
) [1 + M
2 dαSD exp
(−M2 d
αSD
)], if dSD = dRD0
.
(31)
as
P(γS0D0
+ maxZi∈Φ0
{γRiD0} < β2r
∣∣∣∣γS0D0<β2r
2
)=E
[ ∏Zi∈Φ0
P(γS0D0
+ γRiD0< β2r
∣∣∣∣γS0D0<β2r
2
)].
(27)
As different source-relay links experience independentchannel fading, the set of qualified relays Φ0 is an independentthinning of ΦR. Statistically, a relay closer to the source hasa higher probability to receive the packet successfully in thefirst sub-time-slot. Hence, the probability of successful packetreception at each relay is location-dependent, which resultsin an inhomogeneous PPP Φ0. According to the PGFL of thePPP [21], we obtain (28), where µ (ds) represents the intensitymeasure.
The intensity measure of Φ0 is equal to the average numberof qualified relays in DR0(ω). It is given by
µ (DR0 (ω))(a)= E
[ ∑Zi∈ΦR
1 (Zi ∈ Φ0 ∩DR0(ω))
](b)=
∫DR0(ω)
λR exp(−MdαS0Ri
)ds
(29)
where 1(·) is the indicator function, (a) follows from thedefinition of intensity measure, and (b) follows from theCampbell’s theorem [21].
According to the definition of conditional probability, we
have
P(γS0D0
+ γRD0< β2r
∣∣∣∣γS0D0<β2r
2
)
=P(γS0D0 + γRD0 < β2r, γS0D0 <
β2r
2
)P(γS0D0 <
β2r
2
) .
(30)
In (30), we should first calculateP(γS0D0 + γRD0 < β2r, γS0D0 <
β2r
2
). As both HS0D0
and HRD0follow an exponential distribution, we obtain (31).
Due to the constrained relay selection region, dSD is notequal to dRD0
. By substituting (29), (30), and (31) into (28),we obtain (13).
APPENDIX C. PROOF OF PROPOSITION 3
Let random variable V2 = dSD2 tanω−V1, as shown in Fig.
4. The expected value of V2 can be expressed as
E [V2] =
∫ dSD2 tanω
0
v · fV2(v)dv
=
∫ dSD2 tanω
0
FV2(v)dv
(32)
where fV2(v), FV2
(v) represent the probability density func-tion and complementary cumulative density function of ran-dom variable V2. Note that FV2(v) is the same as the prob-ability that there are no potential relays within the shaded
13
triangular region. Hence, we have
FV2(v) = P (V2 > v)
= P (No potential relays within TR(v))
(a)= exp
(− λR
tanωv2
) (33)
where (a) follows from the definition of the PPP, andTR(v) represents the shaded triangular region and its areais ATR(v) = v2
/tanω.
By substituting (33) into (32), we have
E [V2] =
√π tanω
4λRerf
(dSD
2
√λR tanω
). (34)
As E[V1] = dSD2 tanω − E[V2], we get the result in (17).
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Yong Zhou (S’13) received the B.Sc. and M.Eng.degrees from Shandong University, Jinan, China,in 2008 and 2011, respectively. He is currentlyworking towards his Ph.D. degree at the Departmentof Electrical and Computer Engineering, Universityof Waterloo, Canada. His research interests includecooperative networking and relay selection.
Weihua Zhuang (M’93-SM’01-F’08) has been withthe Department of Electrical and Computer Engi-neering, University of Waterloo, Canada, since 1993,where she is a Professor and a Tier I Canada Re-search Chair in Wireless Communication Networks.Her current research focuses on resource allocationand QoS provisioning in wireless networks, and onsmart grid. She is a co-recipient of several best paperawards from IEEE conferences. Dr. Zhuang was theEditor-in-Chief of IEEE Transactions on VehicularTechnology (2007-2013), and the Technical Program
Symposia Chair of the IEEE Globecom 2011. She is a Fellow of the IEEE, aFellow of the Canadian Academy of Engineering, a Fellow of the EngineeringInstitute of Canada, and an elected member in the Board of Governors and VPMobile Radio of the IEEE Vehicular Technology Society. She was an IEEECommunications Society Distinguished Lecturer (2008-2011).