A performance analysis of multi-hop ad hoc networks with adaptive antenna array systems

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by  Olfa Ben Sik Ali

A performance analysis of multi-hop ad hoc
networks with adaptive antenna array systems
Olfa Ben Sik Ali and Christian Cardinal Franc¸ois Gagnon
Electrical Engineering Department Electrical Engineering Department
E´cole Polytechnique de Montre´al E´cole de Technologie Supe´rieure
P.O. Box 6079, Montre´al, Qc, Canada 1100, rue Notre-Dame Ouest Montre´al, Qc, Canada
Email: olfa.ben-sik-ali,christian.cardinal @polymtl.ca Email:[e-mail]
{ }
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Abstract—Based on a stochastic geometry framework, we the link capacity by providing some form of diversity. In
0
2 establishananalysisofthemulti-hopspatialreusealohaprotocol the presence of concurrent transmissions, the signals from
(MSR-Aloha) in ad hoc networks. We compare MSR-Aloha
multiple antennas can be combined to mitigate interference,
n to a simple routing strategy, where a node selects the next
andconsequently,the link reliabilityis improved[7], [8]. The
a relay of the treated packet as to be its nearest receiver with a
J forwardprogresstowardthefinaldestination(NFP).Inaddition, performanceofmulti-antennasystemsinthecontextofadhoc

0 performancegainsachievedbyemployingadaptiveantennaarray networks has been previously studied in [9], [10], [11], [12]
2 systems are quantified in this paper. We derive a tight upper (and references therein). However, unlike the work presented
bound on the spatial density of progress of MSR-Aloha. Our
inthecurrentpaper,thesepreviousworksonlyconsidersingle-
] analyticalresultsdemonstratethatthespatialdensityofprogress
T hop communication. The main contribution of this paper is
scalesasthesquareroot of thedensityof users,andtheoptimal
I contentiondensity(thatmaximizesthespatialdensityofprogress) the derivation of a closed-form expression for the expected
.
s is independent of the density of users. These two facts are progressofpackets(i.e.,themeandistancecoveredinonehop
c consistent with the observations of Baccelli et al., established by a transmitted packet toward its destination) under MSR-
[ through an analytical lower bound and through simulations in
Aloha and NFP schemes, and with nodes employing adaptive
[1] and [2].
2 antenna array systems. The mean progress allows to compute
v the spatial density of progress of the network, i.e., the mean
8 I. INTRODUCTION
distance traversed by all emitted packets per unit area toward
7
An ad hoc network is a collection of autonomous nodes their intendeddestinations in a single time slot, which in turn
1
3 that communicate in a decentralized fashion without relying providesdirectinsightintothe networktransportcapacity[1],
. on a pre-established infrastructure or on a control unit. The [13]. The rest of this paper is organizedas follows. Section II
1
0 design of communication protocols and the analysis of their describes the system model and the communication protocols
0 performance limits in this class of networks have been the considered. In section III, we briefly discuss some related
1 subject of intense investigation over the last decade. This worksand position the contributionof this paper. SectionsIV
:
v design involves the definition of strategies and procedures andVpresentouranalyticalresultsaswellassomenumerical
i necessary for transferring data between the nodes of the examples. Finally, section VI concludes the paper.
X
network, namely, the development of medium access rules
r II. SYSTEMMODEL
a androutingalgorithms.Theanalysisof thereliabilityof these
protocols in the context of ad hoc networks is more complex A. Network model
than in the context of cellular or controlled networks because We adopt the so-called stochastic geometric representation
of the distributed nature of the former. New lines of research of ad hoc networks, which is described as a planar network
have been introduced using analytical tools from the theory formed by a set of nodes that are randomly located on the
of stochastic geometry. Stochastic geometry represents the points of a homogeneous Poisson point process (PPP) with
nodes of the network as elements of a point process and density λ nodes per unit area. We assume that each node
studies their average behaviour under predefined communi- has an infinite number of packets to transmit, and access the
cationstrategies[3], [4]. Thispaperfollowsthis methodology commonmediumaccordingtotheslottedAlohaprotocol,with
and proposes an analytical evaluation of the performance of a predefinedtransmission probabilityp. Thus, bythe property
two communicationprotocolsundera randomaccessstrategy, of independent thinning of PPPs [3], the sets of transmitters
namely, the multi-hop spatial reuse Aloha protocol (MSR- and receivers form two independent homogeneousPPPs with
Aloha) [2], and the nearest receiver with forward progress density λt =λp and λr =λ(1 p), respectively.
(NFP) routing [5], [6]. In addition, this paper considers the −
B. Channel and capture models
quantification of the performance improvement that could be
reached by employing adaptive antenna array systems. In We assume that every node uses a single transmit antenna
point-to-pointcommunication,multiple antennas can increase and L receive antennas. A transmitted signal undergoes both
large-scale fading with a path-loss exponent greater than 2, specificroute(intermsofrelayslist)isdeterminedinadvance.
and small-scale Rayleigh fading. We assume that the channel At each time slot, a source (or a relay) node authorized to
coefficientsdonotvaryduringthetransmissionofonepacket. transmit selects the next relay, among the set of nodes in
A datapacketissaidtobesuccessfullycapturedbyareceiver place, according to some predefined rules. In NFP routing, a
node if the signal-to-interference-plus-noiseratio (SINR) per- transmitterselectsthenextrelayto beits closestnon-emitting
ceived by this node exceeds a prefixed threshold β. node that lies in the direction of the final destination of the
Formally, let us denote by Φ = X ,i N and Φ = packet processed. Selecting the closest receiver allows the
t i r
Y ,i N the PPPs corresponding{to the t∈ransm}itter and the probability of successful communication to be maximized.
i
{ ∈ }
receiver sets, respectively; where the variables X and Y are However, this scheme implies short paths, which means a
i i
random locations of transmitters and receivers, respectively. smallspatialreusefactor.IntheMSR-Alohascheme,thenext
Consider an emitting node located at X and a receiver node relayisselectedtobethe closestnodetothefinaldestination,
j
locatedatY .ThesignalemittedbyX arrivesatY corrupted among the set of receivers that successfully captured the
k j k
by interference and noise. Thus the received signal vector is data packet. This scheme provides the maximum possible
expressed as: progress toward the destination in each time slot, but needs
the implementation of an elaborate relay selection procedure.
x= X Y α/2h s + X Y α/2h s +n,
| j− k|− jk j | i− k|− ik i Figure1showsanexampleofasnapshotofthenetworkatan
Xi∈ΦXt\{Xj} (1)
where s is the signal emitted by node X ; h is the
i i ik
Y(cid:13)
channel propagationvector between nodes X and Y , that is 3(cid:13)
i k Y(cid:13)
distributed accordingto the multivariate complex Normal law 2(cid:13)
withdimensionL;nisacomplexGaussiannoisevector,with
q(cid:13)
XavnairteiaannnncdaeaYσrk2ra;pyaenrpdrdoiαcmeesinssisnitoghn,e;a|pXwatieh−i-glohYstks|veeisxcpttohoreneidsnista.tapInpnclieesdtbaettitoswtietcheanel Y(cid:13)5(cid:13) X(cid:13) o(cid:13) zr(cid:13)2(cid:13)o(cid:13)(cid:13)2(cid:13) q(cid:13) o1(cid:13) (cid:13)z1(cid:13)(cid:13) Y(cid:13) D(cid:13)X(cid:13)o(cid:13)
1(cid:13) 1(cid:13)
received signal, which is chosen based on the statistics of the
data received and optimized under a given criterion. When Emitter(cid:13)
the optimum vector, in the sense of SINR maximization, is Y(cid:13)
4(cid:13) Receiver(cid:13)
applied, the resulting SINR is [7], [8]:
Feasible(cid:13) receiver(cid:13)
SINR = X Y αhT R 1h , (2)
jk | j − k|− jk − jk
where R is the interference-plus-noise covariance matrix ex- Fig. 1: Simulation Results
pressed as: R = X Y αh hT +σ2I ,
the operator T denPotXesi∈tΦhet\{HXejr}m|itiain−opke|r−ator,ikandikIL is thLe arbitrarytimeslot.ThenodeXo isauthorizedtotransmit,and
identity matrix with dimension L×L. In our paper [12], we has a packet for DXo. According to the NFP scheme, Y2 is
haveestablishedthe followingkeyresultontheprobabilityof designatedtobethenextrelay,sinceitistheclosestreceiverto
successful reception: Xo inthedirectionofDXo.Because,inthecasepresentedby
Proposition 1 ([12]): Let Xi and Yj be a transmitter and a figure 1, Y2 can successfully capture the signal from Xo, the
receivernode.Employingtheoptimumcombiningdetector,the progresstowardD ,i.e.,thedistancetravelledbythepacket
Xo
probability of successful communication in a Poisson field of towardD ,isequalto X D Y D .Thislastquantity
interferers and Rayleigh fading channel is: Xo | o Xo− 2 Xo|
could be approximated by z =r cos(θ ) if X D 1.
2 2 o2 | o Xo|≫
Ps(λt,dij,β,L)= P(SINRij ≥β) Amongthesetofnon-emittingnodes,onlyY1,Y2,Y3,Y4,and
Y can successfully receive the packet. Applying the MSR-
=L−1(λt∆β2/αd2ij+σ2β)k exp(−λt∆β2/αd2 −σ2β), A5loha scheme, Y1 is the next relay, since it is the closest to
X k! ij the destination, and thus provides the best progress.
k=0
(3)
where d = X Y , ∆=2π/αΓ(2/α)Γ(1 2/α), and Γ III. RELATEDWORKS
ij i j
| − | −
denotes the gamma function. The idea of designing routing protocols with the notion of
In order to simplify the mathematical analysis, the noise progress is first introduced in [5], where the authors propose
term will be ignored in the next sections. This simplification the most forward within radius (MFR) routing. In MFR rout-
is reasonable since ad-hoc networks are interference-limited. ing, an emitter selects its next relay, among the nodes within
some given range from it, to be the nearest receiver to the
C. Communication protocol
destination. Baccelli et al. [1] propose the more sophisticated
Wewillnowdescribethetwocommunicationschemescon- selection rule of the MSR-Aloha scheme described above.
sidered.Asourcenodehasapacketthatitwishestodeliverto However,intheiranalyticalframework,theyfindthisselection
a distant destination. Unlike traditional routing protocols, no rule to be difficult to manipulate, and so they apply some
modificationsto it. These modificationswill be discussed and Proof: The PPPs Φ and Φ are stationary (invariance
r t
compared to our framework in the next sections. In [14], the by translation and rotation). Thus, we consider without any
authors propose and analyze the longest edge routing (LER), lossofgenerality,a typicalemitterlocatedatthe centerofthe
which applies a similar selection rule as MSR-Aloha. MSR- network(X =(0,0))withanintendeddestinationlocatedon
o
Alohaand LERdifferin one keyaspectthatis that LERdoes the horizontalaxis. The distributionof the distance separating
not consider the direction of the intended destination, which X fromits closest receiver in the directionof the destination
o
is a challenging analytical aspect. It should be noted that all verifies [3], [1]:
works cited do not consider the use of adaptive antenna array λrπ
systems, which is one of the new aspects proposed in this P(|YN∗FP|≤r)=exp(−2r2). (8)
work. Finally, practical implementation issues and complete
Moreover, the angle between the direction of Y and
simulationpackagesofMSR-Aloharoutingareconsideredand N∗FP
the horizontal axis is uniformly distributed in [ π/2,π/2].
detailed in [1], [2]. −
Consequently, the average progress is:
IV. PERFORMANCE ANALYSIS OF NFPAND MSR-ALOHA
SCHEMES progNFP(λ,p)= EYN∗FP |YN∗FP|cosθYN∗FPPs(λt,|YN∗FP|,β)
(cid:2) (cid:3)
π
A.BPerfoobrelemgoifnogrmthurlaotuigohn mathematical analyses, we must for- =Z 2π Z0∞rcosθP(θ)P(|YN∗FP|=r)Ps(λt,r,β)drdθ.
−2 (9)
mally define some notions described in the previous section.
whereE []denotestheexpectationwithrespecttotherandom
Let X Φ be an arbitrary emitting node. x
i t ·
∈ variable x. Replacing P (λt,r,β) by its expression given by
Definition 1 (Random set of feasible receivers): The random s
relation (9), we get the result of theorem 1.
setoffeasiblereceiversforX Φ isformedbythesubsetof
i ∈ t 2) MSR-Aloha routing: As mentioned in section III, the
nodeinΦ X thatcancapturethepacketofX .Denoting
r∪{ i} i MSR-Aloha scheme was previously analyzed by Baccelli et
this subset by , we have [1]:
Di al. in [1]. However, rather than evaluating the mean progress
= Y Φ :SINR β X (4) accordingtotherule(6),theauthorsconsideredanapproxima-
i j r ij i
D { ∈ ≥ }∪{ } tiononit,whichconsistedinreplacingtheeventofsuccessful
Definition2 (Relay selection rules for NFP and MSR-Aloha): reception, i.e., Y by its probability of occurrence. In
k o
Denoting by YN∗FP and YM∗SR the next relays selected by Xi other words, the fo∈lloDwing modified rule is used:
according to the NFP and MSR-Aloha schemes, respectively,
we have: Y˜M∗SR =arg max PS(λ,dok,β)rkcosθk. (10)
Yk∈Φr
YN∗FP =argYk∈Φrm:|θiink|≤π/2|Yk−Xi|, (5) TPhroepfoosliltoiownin2g:pTrohpeomsietiaonn pisrodgermesosnsotbratateindeidnw[1h]e.n using the
and
modified selection rule (10), denoted as prog (λ,p), is a
MSR
Y =arg max cos(θ )Y X , (6)
M∗SR Yk∈Di ik | k− i| lrouwleer(6b)o,uannddoisnethxpermesesaendparso:gress associagted to the selection
where θ is the angle between the two segments emerging
ik
1
from Xi and pointing to the direction of its intended destina- prog (λ,p)= H˜(p,β), (11)
tion and to the direction of Y , respectively (Figure 1). MSR β1/α λp2∆exp(1)
k
is Nanotaeppthraotx,iminatdieofinnoitniotnhe2,prthoegreqsusa,nwtithyicchosis(θviker)y|Yakc−curXatie| wheregH˜(p,β)=R011−expp(cid:16)−1−pp2βG2(/zα)∆(cid:17)dz, and G(z)=
whenthefinaldestinationisfurtherawayfromthetransmitter. 2 arccos zexp(t) dt.
Definition 3 (Spatial density of progress): The expected Rt:√2exepxp(t()1)t≤1/z (cid:18)√2exp(1)t(cid:19)
The lower bound (11) is difficult to evaluate numerically.
progress values for NFP and MSR-Aloha are computed as:
Moreover, it concerns the case of a receiver with a single
progNFP = E |YN∗FP|cosθYN∗FPPs(λt,|YN∗FP|,β) and antenna,andcannotbeeasilygeneralizedtothecaseofmulti-
progMSR = E[|Y(cid:2)M∗SR|], respectively. The spatial den(cid:3)sity of antenna systems. We propose a direct manipulation of the
progress is simply equal to the transmission density times the
selection rule (6), and we establish the following theorem.
expected progress.
Theorem 2: The mean progress of MSR-Aloha verifies:
B. Spatial average of progress derivation
1
1) NFP routing: progMSR(λ,p)≤ β1/α√λp∆H(p,L,β). (12)
Theorem 1: The expected progress for NFP routing is:
The function H(p,L,β) does not dependent on λ, and is
1 L−1 Γ(k+3/2)(β2/α∆)kpk(1 p) expressed as:
prog (λ,p)= − .
NFP √λ kX=0 k!((1−p)π/2+∆β2/αp)k+3/2 H(p,L,β)= ∞1 exp 1−p F(z) dz, (13)
(7) Z − (cid:18)− p β2/α∆(cid:19)
0
where F(z) = L−1 i Γ(1/2+k)Γinc(1/2+i−k,z2), and V. DISCUSSION AND NUMERICAL RESULTS
i=0 k=0 2k!(i k)!
Γinc(k,z) = z∞Ptk−1exPp(−t)dt is the in−complete gamma Analytical results show that the expected progress depends
function. R on the density λ only through the factor 1/√λ. Thus, the
Proof: Consider again the typical emitter X and its spatial density of progress scales as √λ, and the optimal
o
intended final destination placed at the horizontal axis. The contention density, i.e, the value of p that maximizes the
eventthattheprogress,denotedZ,islessthansomevaluez is spatial density of progress, is independent of λ. This result
equivalenttotheeventthatallthenodesthatcapturethepacket is consistent with the scaling law of Gupta and Kumar [13],
are situated in the half-plane D = (x,y) : x z ((x,y) and with the observations of Baccelli et al., made through
z
denotestheCartesiancoordinatesofa{pointinthe≤plan}e).Then simulations in [2]. In the next, we present numerical and
we have: simulationresultsforNFPandMSR-Aloha.Inthesimulations,
the network area is set at 106m2 with a density of nodes
1 = (1 1 1 ), (14) λ = 10 3. The experimental mean progress is obtained over
(Z<z) YjY∈Φr − (SINRoj>β) (Yj∈/Dz) 1000 in−dependent realizations of the network and results are
normalized to a density equal to 1.
where 1 denotes the indicator function, which is equal to 1
ψ
when the propriety ψ holds, and to 0 otherwise. From (14),
we have: 0.12
MSR-Aloha,ourupperbound
MSR-Aloha,lowerboundof[1]
0.1 NFP,analytic
TaitnoPhgpe(erZofxrubipgn<a(hcbttizilohλi)tnay=anl(dgE1eosnΦfiedrar,efΦa(ttoPiXnfPgP)Y()1jYfd∈6ΨuX)Φn(rccf)(to1i)or[r1−n=e5as]1lp.,Eo(STn[ΦI1hd[N4Qes]R.tpXoorjC∈o>tohΦbβnea)fbs1e(ei(xxlqYip)tujy]r∈e/enDigsstszeliy)noe),eqn(r1utaho5.atef)-l d(p), spatial density of progressprog000...000468 MNFSPR,-Asilmouhlaa,tsioimnulation
− −
progress distRribution verifies: 0.02
P(Z <z)=EΦt(cid:20)exp(cid:18)−λrZR21(SINRY>β)1(Y∈/Dz)(cid:19)dY(cid:21). 00 0.1 0.2 0p.,3 transmiss0io.4n probabil0it.y5 0.6 0.7 0.8
(16)
Fig. 2: Simulation and analytical Results for MSR-Aloha and
Applying the Jensen’s inequality to the last relation, we get:
NFP routing, with α=4, β =1, λ=1 and L=1.
Figure 2 presents the spatial density of progress as a
P(Z <z) exp λr P (λt, Y 2,β)1 dY .
≥ (cid:18)− ZR2 s | | (Y∈/Dz) (cid:19) function of the transmission probability, when the number of
(17) receive antennas is set at 1. In the same figure, the lower
The integral in the right hand side is denoted by InL, and bound on the MSR-Aloha spatial density of progress given
evaluated as follows:
by [1] is plotted. We observe that our analytical results are
L−1 ∞ ∞ (γ(x2+y2))i very accurate. The lower bound of [1] is indeed closer to
InL(z)= Xi=0Zz Z−∞ i! exp(−γ(x2+y2))dyd(1x8.) tshcehecmuerv,ewhoifchNFisParroeustuinltgotfhtahnetaoppthroexicmuravtieonofcoMnsSiRde-Areldohina
where γ = λt∆β2/α. Substituting x and y by u = γx2 and [1] that is explained as follows. The probability of capture
v =γy2, respectively, we get: decays exponentially with the distance, and consequently,the
maximization of the product of the probability of success
L−1 i ∞
In (z) = 1 1u(i−k)−1/2exp(−u)du· and the distance (selection rule (10)) is almost equivalent to
L Xi=0Xk=0γk!(i−k)!Zγz2 2 taking the nearest receiver to the transmitter, i.e., the NFP
∞vk−1/2exp(−v)dv (19) scheme. Figure3showsthatemployingadaptiveantennaarray
Z systems significantly improvesthe spatial density of progress.
0
L−1 i Γ(1/2+k)Γ (1/2+i−k,γz2) Forexample,withonlyoneadditionalantenna,a200%gainis
= inc . (20) observed. This improvementis due to the additional diversity
2γk!(i−k)!
XX
i=0k=0 and interference cancellation capability provided by multi-
The mean progress yields to: antennasystems.ObservethattheMSR-Alohaschemeismore
sensitivetothetransmissionprobabilitypthanisNFProuting,
∞
prog (λ,p)≤ 1−exp(−λrIn (z))dz. (21) and this can be explained as follows: when p increases, the
MSR Z L
0 interference becomes severe, leading to a decrease in the
Applying the substitution w = √γz, we get the result of distance over which packets can be captured, and thus, to a
theorem 2. significant decrease in prog . For the NFP scheme, the
MSR
since it provides a better progress with a smaller number of
0.045
MSR-Aloha,simulation transmission attempts.
0.04 MSR-Aloha,upperbound Finally, several other simulations were done that show the
0.035 NFP,analytic influence of the parameters β and α, and these simulations
d(p), spatial density of progressprog00..0000..12005523 L=2 L=3 mnaraeemTanhneoilsytdeppMnareSspsieRteyrn-AdtoeelfdorihvphaereodrageVnsrddieIm.usNsepCFlotePOofNcratolCwoulLsaoteUicndkScg-Iof.oOomOfNrmmuspruaerncexiescp.uarlettissosnqiounsatsnratfitofeyrgittehhsee,
0.01
L=1 improvement achieved through the use of adaptive antenna
0.005
array systems. Analytical and simulation results show that
00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 MSR-Alohaprotocolishighlyefficientintermsofspatialden-
p, transmission probability
sity of progress and optimal contention probability. However,
Fig. 3: Spatial density of progress as a function of the this scheme calls for implementation of a sophisticated relay
transmission probability and the number of receive antennas, selectionprocedure,whichmayintroduceadditionaloverhead.
with α = 3, β = 10, and λ = 1. For some value of p the A fairer performance comparison of MSR-Aloha and other
experimental curves are above the upper bound. This fact is communication protocols must therefore take this aspect into
due to edge effects. consideration.
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