Outage Probability of Wireless Ad Hoc Networks with Cooperative Relaying

0.18 MB·English

by  M. Mohammadi

1
Outage Probability of Wireless Ad Hoc Networks
with Cooperative Relaying
Mohommadali Mohammadi†,Himal A. Suraweera‡,andXiangyun Zhou†
† Research School of Engineering, The Australian National University, Australia
‡ Singapore University of Technology and Design, Singapore
Email: [email protected],[email protected],[email protected]
Abstract—In this paper, we analyze the performance relay selection is performed and [4] investigated the out-
of cooperative transmissions in wireless ad hoc networks age performance of opportunistic relay selection for an
with random node locations. According to a contention interference-freerandomnetwork.Theauthorsin[5]pro-
3 probability for message transmission, each source node posed four decentralized relay selection methods based
1 can either transmits its own message signal or acts as a
on the available location information or received signal
0
potential relay for others. Hence, each destination node
2 strength and authors in [6] defined a quality of service
can potentially receive two copies of the message signal,
n one from the direct link and the other from the relay (QoS) region for relay selection to guarantee a target
a QoS at the destination. Common to all these studies is
link. Taking the random node locations and interference
J
into account, we derive closed-form expressions for the the simplifiedassumptionthatthe directlink isneglected
4
outageprobabilitywithdifferentcombiningschemesatthe or at most selection combining (SC) is used, where the
2
destination nodes. In particular, the outage performance destinationonlyselectsonelinkfromtherelayanddirect

] of optimal combining, maximum ratio combining, and links for data detection. To the best of our knowledge,
T
selection combining strategies are studied and quantified.
I the ultimate benefit of cooperative transmission utilizing
.
s both the relay and direct links in such networks has not
c I. INTRODUCTION been studied in the existing literature. Our main goal is
[
Large-scale decentralized wireless systems such as ad to fill this important gap and enhance the fundamental
1
understanding of cooperative relaying in large-scale ad
v hoc networks have attracted significant attention over
7 the last decade due to their wide range of practical hoc networks.
8 Inthispaper,weanalyzetheoutageperformanceofan
applications. The multiuser nature of these networks
6
opportunistic cooperative ad hoc network. The locations
5 motivates the use of cooperative transmission in which
1. additional links via relay nodes are established to en- of all potential transmitting nodes are modeled as a
0 hance the quality of communication between the source PPP. Each potential transmitter is allowed to transmit its
3 message at a given time slot according to a contention
node and destination node [1]. However, there are two
1
probability. Hence, the potential transmitters at a given
: key features, namely indiscriminate node placement and
v time slot are divided into a group of active source nodes
network interference, which make the design and analy-
i
X and a group of idle nodes, where the latter becomes the
sis of cooperative communication a challenging task in
r wireless ad hoc networks. potential relays for the former. The decode-and-forward
a
(DF) protocol is assumed at the relays. We consider two
Studies on relay selection in interference-free and
relay selection schemes,namely, best relay selection and
deterministic networks have shown that opportunistic
random relay selection. For each relay selection scheme,
relaying where a single relay is chosen to aid the
we study the outage performance of different signal
source-destination transmission can guarantee a signifi-
combining methods at the destination, namely, optimal
cantperformance improvementwhilst having low imple-
combining (OC), maximum ratio combining (MRC), as
mentation complexity [2]. In wireless ad hoc networks,
well as SC. Our main contribution is the derivation of
however, interference and random node locations (e.g.,
closed-form expressions for the outage probability with
due to high mobility) need to be taken into account in
various combining schemes in the interference-limited
any meaningful analysis. Relay selection methods based
cooperative ad hoc networks.
onstochasticgeometrymodels,wheretherelaylocations
follows a Poisson point process (PPP), have been inves-
II. SYSTEM MODEL
tigated in a few recent works [3]–[6]. Specifically, [3]
Consider a large-scale wireless ad hoc network with
studied the throughput scaling laws when opportunistic
transmitter-receiver pairs. The locations of all transmit-
ting nodes are modeled as a homogeneous PPP denoted
This work was done while the first author was visiting the
Australian National University. by Φ = xk with density λ on the plane R2, where
{ }
2
x denotes the location of node k. Each transmitter has popular choice of such a selection region is a sector
k
a uniquely-associated receiver at a distance d away in defined by a maximum angle φand a maximum distance
a random direction and is not a part of the PPP. All d [5], [8]. A sectorized selection region is considered in
s
transmitters or receivers are assumed to be identical and this paper.Nevertheless,performanceanalysiswith other
equippedwith one antenna.We consideran interference- different choices of selection regions such as a circular
limited setting [7], i.e., the thermal noise is assumed to area with radius d can be derived in a similar way. In
s
be negligible. the ensuing text, the indicator that (2) holds is denoted
by 1(x x Φt x ) .
S 2 S
→ | \{ }
A. Channel Model 2) Relaying Phase: In the second stage (odd time
Signal propagation is subject to both small-scale mul- slots), each destination node informs only one of the
tipathfadingandlarge-scalepathloss.Theinstantaneous potential relays if any to retransmit the message with
channel from node x to x can hence be modeled as repetition code. Similar to the recent work in [6], two
1 2
relay selection methods are considered, namely, best
g12 =h12ℓ(x1 x2), (1) relay selection and random relay selection. In particular,
−
the best relay selection method chooses the potential
where h captures the small-scale fading and is mod-
12 relay with the best signal strength to the destination as
eled as an exponential random variable (RV) with unit
variance1 and ℓ(x x )= x x α characterizes the R=argmax hkDℓ(xk xD) , (3)
large-scale path lo1s−s fo2llowking1p−ow2erklaw with path loss xk∈Φro { − }
exponent α. where xD presents the destination location and Φro de-
notes the set of potential relays for the S D pair.
−
B. Cooperative Transmission Protocol On the other hand, the random relay selection method
randomly selects one out of all potential relays with
We considera time-slotted Aloha protocol and restrict
equal probability to forward the source message. The
the number of hops between any source-destination (S
− motivation behind the use of best and random relay se-
D) pair to be two. Similarly to [4], [5], we adopt a two-
lectionisto studythetrade-offbetweenperformanceand
stage cooperative transmission protocol as follows:
complexity of relay selection. Best relay selection which
1) Broadcasting Phase: In the first stage (even time
exhibits a superior performance compared to random
slots), for a given contention probability, p, for transmis-
relay selection has a high implementation complexity
sion, the active nodes (source nodes) from Φ transmit
since it requires high signalling overheads and channel
and all other nodes in Φ remain idle. The locations of
state information (CSI) from all potential relays. On
thesourcenodesinthisstagefollowahomogeneousPPP,
the other hand, at the expense of some performance
denoted as Φt, with intensity pλ. On the other hand, the
loss, random relay selection is particularly suitable for
idle nodes form another independent homogeneous PPP
low-complexity relay systems. We denote the set of all
denoted as Φr, with intensity (1 p)λ.
Foreachsourcenode,a selecti−onregion is defined2. transmitting relays, i.e., the relays selected by all source
The idle nodes in Φr located inside the selAection region nodes, as Ψ.
arerequiredto listento the transmissionfromthis source Finally, the signals transmitted by the source and the
node. If any of these idle nodes successfullydecodesthe selected relay are combined at the destination node. We
source message, it becomes a member of the source’s considerthreesignalcombiningtechniques:namely,OC,
potential relay. In other words, a node belonging to Φr MRC, and SC [9], which will be described in detail in
located at x is said to be a potential relay of the source
2 the next section. Note that only the direct S-D link can
nodeatx ,providedthatx isinsidetheselectionregion
S 2 be used when no potential relay is available.
of the source node and
To analyzethe performance of of the consideredwire-
h ℓ(x x )
S2 S − 2 β, (2) less random network, we focus on a typical transmitter
IΦt ≥ locatedatthe origin. Similarly to [4], we will usea polar
where IΦt = xk∈Φt\{xS}hk2ℓ(xk −x2) is the aggregate coordinate system to facilitate the analysis in which the
interference rePceivedat x2 and β is the target (minimum) typicalsource, S,is located atxS =(0,0),its destination,
signal-to-interference ratio (SIR) for data detection3. A D, is at x = (d,0), and an arbitrary relay node is at
D
x=(r,θ). Hence, Φr now denotes the potential relay set
1In what follows, we will use the notation x∼E(µ) to denote x o
of the typical source node at the origin.
that is exponentially distributed with mean µ.
2In practice, selecting a suitable relay from a defined region
with a small number of relays is desirable. As the implementation III. OUTAGE PROBABILITY
complexity increases with the number of relays, a carefully selected
In this section, we derive the outage probability of
regioncanbeusedtotakeintoaccountbothprotocolcomplexityand
the described cooperative transmission protocol. Firstly,
performance gain.
we note that the performance of the typical destination
3Notethatforβ>1,eachidlenodecandecodethemessagefrom
at most one source node, hence can only serve asthepotential relay nodeis subjectto two setsof interferers; allconcurrently
for at most one source node [5]. transmitting source nodes in the broadcasting phase and
3
all concurrently transmitting relays in relaying phase. gSD,β gSR,β (SIR ,β) . (8)
OC
Stinpaetciiofincailnly,brthoeadcinatsetirnfegreanncde preolwayeirngsepenhasaetsthceandbese-, O IΦtD !\ O IΦtR ![O !
WenowinvestigatetheoutageprobabilityofOCreceiver
respectively, written as
for best relay selection and random relay selection.
IΦtD = hkDℓ(xk−xD), (4a) 1) Best Relay Selection: In best relay selection, as-
xk∈ΦXt\{xS} suming non-empty Φr, the relay having the best channel
o
IΨ = hmDℓ(xm xD), (4b) gain of the forward channel is selected, cf. (3). Since
−
xm∈XΨ\{R} computing the cumulative density function (cdf) of the
where R is the selected relay node for the typical S D receivedSIRfromrelayingphasedoesnotyieldaclosed-
pair. Similarly, the interference powerseen by relay R−in form expression,we obtain a lowerbound for the outage
the broadcasting phase can be written as probability in following proposition.
Proposition 1: The outage probability for cooperative
IΦtR = hnRℓ(xn−xR). (5) transmissionprotocolwith OC receiveratthe destination
xn∈ΦXt\{xS} and best relay selection is given by
From here on, we denote the channel gain between S
φ(1 p)λ
gaSnRd, DandasthgeSDc,htahneneclhagnaninelbgeatwineebnetwReeanndSDanads RgRDas. PoOuCt,BS=γ 1,ηβδd2 (cid:18)γ(cid:18)1, β−δη γ 1,ηβδd2s (cid:19)
Furthermore, we denote the outage event for link with (cid:0) (cid:1) 1 PBS +PB(cid:0)S , (cid:1) (9)
× − OC OC
channel gain g and interference power I by (g/I,β).
O where PBS denotes the pro(cid:0)bability(cid:1)that co(cid:1)mbined SIR
Hence (g /I ,β), is the outage eventoverthe direct OC
S-D linOk aSnDd tΦhtDe corresponding outage probability is drops below the target SIR and is lower bounded
given by [10] as (10) at the top of next page wherein dRD =
√d2+r2 2rdcosθ and η = Λ c with Λ being the
Pd=1−E 1(xS→xD|Φt\{xS}) =γ 1,ηβδd2 , (6) intensity o−f the interferers Iin the rIelaying phIase.
where γ(, ) i(cid:8)s the incomplete gam(cid:9)ma f(cid:0)unction (cid:1)[11, Eq. Proof: See Appendix A for the derivation of PBS
· · OC
(8.350.1)], with γ(1,x)=1 exp( x) and η =pλc, with and Appendix B for the derivation of Λ .
I
− −
c=δπΓ(δ)Γ(1 δ) and δ = 2. Note that (10) is a numerically integrable expressionand
− α
can be easily evaluated in MAPLE.
A. Optimum Combining 2) Random Relay Selection: In random relay selec-
tion,assumingnon-emptyΦr,destinationrandomlypicks
With optimum combining, signals received from the o
a single relay from Φr.
direct and relayed links are weighted to maximize the o
Proposition 2: The exact outage probability for co-
SIR at the destination [9, Ch. 11]. This scheme requires
operative transmission protocol with OC receiver at the
the instantaneous CSI of all interferers to be known
destination and random relay selection is given by
at the receiver, and hence, demands significant system
φ(1 p)λ
complexity. As such we consider OC as an important POC,RS=γ 1,ηβδd2 γ 1, − γ 1,ηβδd2
out βδη s
theoreticalbenchmarkto quantifytheperformanceofthe (cid:18) (cid:18) (cid:19)
(cid:0) (cid:1) 1 PRS +PR(cid:0)S , (cid:1)(11)
considered network. × − OC OC
Performance of opportunistic relaying with OC re- where PRS is given in (12)(cid:0)at the to(cid:1)p of ne(cid:1)xt page.
ceiver has been investigated in [12], where transmitted OC
Proof: Following similar steps as in the best relay
signals in broadcasting and relaying phase are impaired
selection scheme and employing
by co-channel interference coming from a deterministic
interferer. The outage performance of optimum combin- FRD(β)=1 E 1(xk xD Ψ xR ) (13)
− { → | \{ }}
ing at a multi-antenna receiver with interferers located =1 Pr(h ℓ(x x )>βI x Φr)
according to a PPP was studied in [13]. Here we extend − kD k− D Ψ| k ∈ o
the result in [13] to the case of cooperative communica- =1− exp −βδ ηId2RD+ηr2 rfr,θ(r,θ)drdθ,
tions. For simplicity, we ignore the correlation between ZA
(cid:0) (cid:2) (cid:3)(cid:1)
the transmitted signal from any source and that from for the cdf of the received SIR from the relay link, we
the corresponding relay in order to obtain analytically obtain POC,RS which concludes the proof.
out
tractable result. Following the derivation in [13], the
resulting SIR at the typical destination after performing
B. Maximal Ratio Combining
OC is given by
gSD gRD Note that MRC is the optimal combining scheme
SIR = + , (7)
OC
IΦt IΨ in the absence of interference [9, Ch. 11]. Although
D
suboptimal in the presence of interference, MRC is an
which is the sum of received SIRs from the direct link
and the relay link. Therefore, the overall outage event attractivereceiversinceitdoesnotrequirethe CSIofthe
for the cooperative transmission scheme with OC is [14] interferers, thus serves as a low complexity alternative
compared to OC.
4
PBS γ(1,ηβδd2) ηδd2
OC
≥ − ×
β
yδ−1exp ηyδd2 (1 p)λ exp( η(β y)δr2) 1 exp( η (β y)δd2 ) rf (r,θ)drdθ dy, (10)
I RD r,θ
− − − − − − − −
Z0 (cid:18) ZA h i (cid:19)
β
PRS = γ(1,ηβδd2) ηδd2 yδ−1exp ηyδd2 (β y)δ η d2 +ηr2 rf (r,θ)drdθdy, (12)
OC I RD r,θ
− − − −
Z0 ZA (cid:16) (cid:17)
(cid:2) (cid:3)
When MRC is employed, the resulting SIR can be 100
expressed as
SIR = (gSD+gRD)2 . (14) 10−1
MRC
gSDIΦt +gRDIΨ
D
Theoveralloutageeventforthecooperativetransmission bility10−2
schemewith MRC canbemathematicallywritten as[14] Proba
gSD,β gSR,β (SIR ,β) . (15) Outage10−3
MRC
O(cid:18)IΦt (cid:19)\ O IΦtR ![O ! 10−4
1) Best Relay Selection: We remark that in this case,
Directtransmission
SCReceiver
with MRC receiver at the destination, derivation of the MRCReceiver
10−5 OCReceiver
outage probability is rather involved and thus we only 10−4 10−3 10−2 10−1
λ
comparethe performanceusingMonte Carlo simulations
Fig.1. Outageprobabilityversusλfordifferentcombiningreceivers
in Section IV. withrandomrelayselection.Simulationresultsareshownwithdashed
2) Random Relay Selection: In random relay selec- lines.
tion, assuming non-empty Φr, the destination randomly
selects one out of all potentioal relays with equal proba- PSBCS=1−E{1(xk →xD|Ψ\{xR})}. (20)
bility. Note that we obtained a lower bound for PBS in (27)
SC
Proposition 3: The outage probability of the MRC
which yields a lower bound for the outage probability
receiver with random relay selection is given by
of the SC receiver at the destination with best relay
PMRC,RS=γ 1,ηβδd2 γ 1,φ(1−p)λγ 1,ηβδd2 selection.
out βδη s 2) Random Relay Selection: The outage probability
(cid:18) (cid:18) (cid:19)
(cid:0) (cid:1) 1 PRS +P(cid:0)RS , ((cid:1)16) of randomrelay selectionfollows from (19) by replacing
× − MRC MRC
PBS with PRS , given in (13).
where PRS is the probabi(cid:0)lity that co(cid:1)mbined (cid:1)SIR at SC SC
MRC
MRC receiver drops below the target SIR and is given
IV. NUMERICAL AND SIMULATION RESULTS
in (17) at the top of the next page.
In this section, we study the accuracy of the derived
Proof: See Appendix C.
analytical results and compare the outage probability
C. Selection Combining of OC, MRC and SC schemes with best and random
Instead of using OC and MRC which require exact relay selection. In all simulations, we have set α = 4,
knowledge of the CSI, a system may use SC which sim- β = 3 dB, d = 10 m, ds = 7 m and φ = π3, unless
plyrequiresSIRmeasurements.Indeed,SCisconsidered stated otherwise. To ensure a fair comparison, the SIR
as the least complicated receiver [9, Ch. 11]. With SC thresholdofcooperativetransmissionissettobetwiceas
receiver at the destination, outage occurs if neither the much as direct transmission threshold. This is because,
direct nor the relayed link can support the target SIR.
in the broadcasting phase, the source uses half of the
Hence, the outage event is [14]
channel uses and in the relaying phase, the relay uses
gSD,β gSR,β gRD,β , (18) the remaining channel uses.
O(cid:18)IΦt (cid:19) O IΦtR ! O(cid:18) IΨ (cid:19)! The performance of OC and sub-optimal combining
\ [
schemes (MRC and SC) can be further ascertained by
1) Best Relay Selection: The outage probability is
referring to Fig. 1, where the probability of outage as a
given by
function of node density, λ is shown for random relay
φ(1 p)λ
PSC,BS =γ 1,ηβδd2 γ 1, − γ 1,ηβδd2 selection. Analytical expressions in (12), (17), and (20)
out βδη s
(cid:18) (cid:18) (cid:19) are confirmed as they are seen to follow the simulations
(cid:0) (cid:1) 1 PBS +PB(cid:0)S , (cid:1)(19)
× − SC SC tightly. The performance improvement of all three com-
where (cid:0) (cid:1) (cid:1) bining schemes compared to the direct transmission is
5
α
PRS =1 1 exp d2 βδ(η+η +ψ) dRD exp d2βδ(η+η +ψ) rf (r,θ)drdθ, (17a)
MRC −ZA1− dRdD α(cid:20) (cid:16)− RD I (cid:17)−(cid:18) d (cid:19) (cid:16)− I (cid:17)(cid:21) r,θ
α α α α
ψ= η d (cid:0) (cid:1)G23 d −δ,−1,0 + ηI dRD G23 dRD −δ,−1,0 (17b)
Γ(δ) dRD 33 dRD (cid:12) 0, 0, 1! Γ(δ) d 33 d (cid:12) 0, 0, 1!
(cid:18) (cid:19) (cid:18) (cid:19) (cid:12) − (cid:18) (cid:19) (cid:18) (cid:19) (cid:12) −
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
100 particular, we have obtained closed-form expressionsfor
the outage probability of OC, MRC, and SC receivers
10−1 at the destination with random relay selection. We have
also derived two useful tight lower bounds for the OC
10−2
bility and SC receivers with best relay selection. The accuracy
ba of the analytical results has been validated using Monte
Pro10−3
Outage10−4 Carlo simulations.
APPENDIX A
10−5 DSSCCireRRceetccTeeiirvvaeenrrs,,mSLiiosmwsiueolrnatBioonund PROOF OF PROPOSITION 1
MRCReceiver,Simulation The proof of Proposition 2 is accomplished in four
OCReceiver,Simulation
OCReceiver,LowerBound
10−6 steps as follows:
10−4 10−3 λ 10−2 10−1 Step 1: The outage probability corresponding to the
Fig.2. Outageprobabilityversusλfordifferentcombiningreceivers
event g /I ,β is given by (6).
with best relay selection . O SD ΦtD
Step 2:(cid:16)The outage(cid:17)event O gSR/IΦtR,β is equivalent
noticeable for all intensities. Note that although increas- to the eventthat the potential r(cid:16)elay set Φro(cid:17)is empty. Note
ingtheintensityofnodesresultsinincreasingthenumber that Φro which is derived by thinning the PPP Φr, is still
a PPP and Marking theorem of Poisson processes [15]
of qualified relay nodes,the interference causedby these
gives its density as
relay nodes in relaying phase is increased. Hence, for
highervaluesofintensitytheperformanceofallschemes Λ (x)=(1 p)λE 1(x x Φt x )
o S k S
− → | \{ }
are converging to the same values. =(1 p)λexp ηβδ r2 . (21)
(cid:8) (cid:9)
− −
The effect of best relay selection on the outage per-
Therefore, the outage proba(cid:0)bility cor(cid:1)responding to de-
formance of combining schemes is investigated in Fig. 2
sired outage event is given by
and the tightness of proposed lower bound for OC and
SC receiver are validated. Comparing Fig. 1 and Fig 2 P =exp( µ ( )), (22)
r o
− A
revealsthat bestrelay selection significantlyoutperforms
where µ ( ) denotes the mean measure of Φr and
the random relay selection as expected. o A o
Our observation of the relation between the outage µ ( )= Λ (dx), (23)
o o
A
performance of combining schemes and selection re- ZA
gion parameters which due to the space limitation are in which is the selection region and dx denotes a
A
not shown in simulations, reveals that: 1) There is an two-dimensional variable of integration over the polar
area. Finally, by substituting (21) into (23) the outage
optimal value of the parameter φ for each combining
probability in (22) is obtained for sectorized selection
scheme that achieves the maximum success probability.
region as
2) Increasing the parameter φ beyond its optimum value
does not degrade the success probability. The reason is P =exp φ(1−p)λγ 1,ηβδd2 . (24)
that although enlarging the selection region increases r − βδη s
(cid:18) (cid:19)
(cid:0) (cid:1)
the possibility of being relay nodes in this area and
Step 3: The outage probability corresponding to the
consequently increases the intensity of interferer set for
outage event (SIR ,β) can be written as4
the second hop, the relay selection strategy only selects O OC
a single relay, and thus the intensity of the interferers for POBCS=Pr(SIRSD+SIRRD <β)
the second hop is upper bounded. Therefore, the success β
probability remains constant. = FRD(β−y)fSD(y)dy, (25)
Z0
4In general, correlation of node locations in wireless ad hoc net-
V. CONCLUSION
work,makestheinterferencetemporallyandspatiallycorrelated[16].
However, our derivation in (25) does not include the impact of cor-
The performance of relay selection schemes along
relation. The justification of this assumption isprimarily to preserve
with different combining schemes for cooperative trans-
analytical tractability and simplicity, however, simulations validate
missions in ad hoc networks have been studied. In this assumption.
6
where F() and f() denotes cdf and probability density is one relay in potentialrelay set which for a typical pair
· ·
function (pdf) of the RV, respectively. f (y) can be is mathematically described by
SD
found simply by taking the first order derivation of (6),
q =1 P =1 exp( µ ( )). (29)
which yields − r − − o A
Therefore,inthecaseofasectorizedselectionregion,the
f (y)=ηd2δyδ−1exp( ηd2yδ). (26)
SD
− intensityofinterferersetin secondstageoftransmission,
A lower bound for FRD(β) is obtained as ΛI, is given by
FRD(β)=1−E{1(xk →xD|Ψ\{xR})} ΛI =pλ 1−exp −(1−p)λ exp −ηβδ r2 dx
(cid:18) (cid:18) ZA (cid:19)(cid:19)
=1−Pr(cid:18)xmk∈aΦxro{hkDℓ(xk−xD)}>βIΨ(cid:19) =pλγ 1,φ(1η−βδp)λγ 1,ηβδd2s (cid:0). (cid:1) (30)
(cid:18) (cid:19)
=1 E exp βIΨ (cid:0) (cid:1)
−  −ℓ(x x ) 
xkY∈Φro (cid:18) k− D (cid:19) APPENDIX C
(=a)1 Eexp 1 exp βIΨ Λ (dx) PROOF OF PROPOSITION 3
o
− − − −ℓ(x x )
(cid:26) (cid:18) ZA(cid:20) (cid:18) k− D (cid:19)(cid:21) (cid:19)(cid:27)
(b) βI Before deriving the outage probability we will need
1 exp 1 E exp Ψ Λ (dx)
≥ − − − −ℓ(x x ) o the following lemma.
(cid:18) ZA(cid:20) (cid:26) (cid:18) k− D (cid:19)(cid:27)(cid:21) (cid:19)
Lemma 1: Given three RVs u (µ ), v (µ ),
(=c)1−exp − 1−exp(−ηIβδd2RD) Λo(dx) , (27) and z (1), the moment generatin∼g fEunc1tion (M∼GEF)2of
(cid:18) ZA (cid:19) W =u∼zE/(u+v), (s)=E exp( sW ) is given by
where (a) follow(cid:2)s from the genera(cid:3)ting functional of 1 MW1 { − 1 }
s
the PPP, Φro with intensity Λo(x)5 and (b) follows by MW1(s)=1−s+1×
using the the Jensen’s inequality. (c) holds by taking the
µ 1 µ 1 1,0
eoxfpcehcatantnioenl ogvaienrsIΨandwhtehree gtheeneerxatpinognenfutinacltidoinstarlibouftioΨn "1−µ21s+1G2222 µ21s+1 (cid:12)−0,0!#, (31)
(cid:12)
have been used. Moreover, η = cΛ where Λ is the (cid:12)
intensity of interferers for sIecond Ihop transmIission. where Gmpqn s | ab11······abqp denotes the Meij(cid:12)(cid:12)er’s G-function
The details of intensity evaluation for the second hops’ defined in [(cid:16)11, Eq. (9(cid:17).301)]. With appropriate changes
interferers are deferred to Appendix B. Plugging (27) ofthemeanindicesin(31),theMGFofW =vz/(u+v),
2
together with (26) into (25), yields the lower bound on can be expressed as
PBS in (25). s
OC (s)=1
MW2 −s+1×
Step 4: Referring to the outage event in (8), the overall
µ 1 µ 1 1,0
outage probability of the cooperative transmission with 1 2 G22 2 − , (32)
OC receiver and best relay selection is given by " −µ1s+1 22 µ1s+1 (cid:12)(cid:12) 0,0!#
(cid:12)
PoOuCt,BS =Pd (1−Pr) 1−POBCS +POBCS (28) Proof: See Appendix D (cid:12)(cid:12)
Following similar steps as in the OC case with random
Plugging (6), (24) a(cid:0)nd (25) i(cid:0)nto (28)(cid:1)and aft(cid:1)er some
relay selection, we get the outage probability as (16).
manipulations gives the desired result in (11).
What remains to calculate is then to determine the
outage probability PRS . Let us define
APPENDIX B MRC
INTENSITY OF THE INTERFERER SET u,gSD, v ,gRD. (33)
Deriving the intensity of interferer set for the second- The RVs, u and v are exponentially distributed with
hop transmission is equivalent to exploit how many parameter µ1 = dα and µ2 = dαRD, respectively. We
selected relays are there for transmission in the second recall that for random relay selection case, the RV gRD
hop. Provided that the potential relay set is not empty, is considered as an arbitrary exponential RV. It can be
since each source only selects a single relay in both best shown that the SIR in (14) is of the form of [17, Eq.
relayselectionandrandomrelay selection,it is clearthat (28)], i.e.,
the intensity of interferer set is at most pλ, which is the c 2
intensity of the sourcenodes.Let usdenote the probabil- SIRMRC = L| s|ν 2, (34)
ity that an arbitrary source node successfully selects one i=1| i|
irnelasyecnoonddesbtyagqe. Tofhecno,otpheeraintitveenstirtaynosmf aiscstiiovnerperloaytoncooldeiss cwhbeerein|gcst|h2e=cuha+nvn,eνlic=oecffi†sccii/eP|ncts|b,eatnwdeLen=th|Φeti|n+te|Ψrfe|rweirthi
i
pqλ, which is also the intensity of the interferers. Note and destination and being the cardinality of a set.
that q is proportional to the probability that at least there Therefore, it can be s|h·o|wn that ν ’s are independent of
i
c [17] and thus PRS can be written as
s MRC
5Let ν(x) : R2 → [0,1] and RR2|1−ν(x)|dx < ∞. When Φ u v
is Poisson of intensity λ, the conditional generating functional is PRS =Pr (u+v)<β I + I . (35)
E{Qx∈Φν(x)}=exp(cid:0)−λRR2[1−ν(x)]dx(cid:1). MRC (cid:18) (cid:20)u+v ΦtD u+v Ψ(cid:21)(cid:19)
7
1 u v
PRS = 1 E µ exp µ β I + I
MRC − µ µ 1 − 2 u+v ΦtD u+v Ψ −
(cid:26) 1− 2 (cid:20) (cid:18) (cid:18) (cid:19)(cid:19)
u v
µ exp µ β I + I
2 1 Φt Ψ
− u+v D u+v
(cid:18) (cid:18) (cid:19)(cid:19)(cid:21)(cid:27)
(=a)1 E 1 µ (µ βℓ(x x )) (µ βℓ(y x ))
− µ µ  1 MW1 2 k − D MW2 2 k − D −
 1− 2 xk∈ΦYt\{xS} yk∈YΨ\{R}


µ (µ βℓ(x x )) (µ βℓ(y x )) (36)
2 MW1 1 k − D MW2 1 k − D 
xk∈ΦYt\{xS} yk∈YΨ\{R} 


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