Analysis of Uncoordinated Opportunistic Two-Hop Wireless Ad Hoc Systems

0.11 MB·English

by  Radha Krishna Ganti

1
Analysis of Uncoordinated Opportunistic Two-Hop
Wireless Ad Hoc Systems
Radha Krishna Ganti and Martin Haenggi
Department of Electrical Engineering
University of Notre Dame
Notre Dame, IN 46556, USA
{rganti,mhaenggi}@nd.edu
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Abstract—Weconsideratime-slottedtwo-hop wirelesssystem relay. In [5] the relay selection is based on the channel state
0
2 in which the sources transmit to the relays in the even time information that is fed back to the source. In [2] the relays
slots (first hop) and the relays forward the packets to the
estimate the channel using channel reciprocity theorem and
n destinationsintheoddtimeslots(secondhop).Eachsourcemay
use timers to select the best relay. In [8] a relay selection
a connect to multiple relays in the first hop. In the presence of
J interferenceandwithouttightcoordinationoftherelays,itisnot method (GeRaF) based on the distance from the destination

5 clear which relays should transmit the packet. We propose four is considered. TDMA based contention is used to resolve the
1 decentralized methodsof relayselection, some based on location relaystransmittingtothesamedestination.Inallthesemethod
information and others based on the received signal strength someformofchannelcontentionandfeedbackisusedtoselect
] (RSS).Weprovideacompleteanalyticalcharacterizationofthese
T methodsusingtoolsfromstochasticgeometry.Weusesimulation the relays. In the methods we analyze relays are selected in a
I resultsto compare these methodsin terms of end-to-endsuccess completely distributed fashion without any channel feedback.
s. probability. Thismakestheserelayselectionschemessuitableinscenarios
c with moderate to high mobility.
[
Themaincontributionofthispaperisthecompleteanalyti-
1 I. INTRODUCTION caldescriptionofatwo-hopwirelessnetworkwithinterference
v We consider a two-hop wireless ad hoc network in which when the nodes and relays are distributed as a Poisson point
0
the sources are distributed randomly on the plane and each process on the plane. The techniques developed in this paper
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sourcehasadestinationatadistanceRinarandomdirection. can be used to analyze other position based relay selection
1
2 In addition there exists a set of relays (different from the methods.
. destinations) which assist the sources. In the first hop each
1
source transmits and will be decoded by any node listening
0 II. SYSTEMMODEL
9 (relaysanddestinations)withsignal-to-interferenceratio(SIR)
0 greater than a fixed threshold T. In the second hop, some of We assume that the sources form a Poisson point process
v: the relays which were able to decode information in the first (PPP) φs of intensity λs on the plane. The relays are also
assumed to form a PPP φ of intensity λ on the plane. Each
i hop,transmit.So in the firsthopeach packetmay bereceived r r
X source x ∈ φ has a destination denoted by r(x) (not a part
bymanyrelays,hencemultiplecopiesofthesamepacketmay s
r exist at different relays. In networks with low or no mobility of φs or φr) at a distance R in some random direction. We
a
assume that the fading between any two nodes is Rayleigh
multiple copies may be avoided by setting the routing tables
distributed so thatthe powersare exponentialwith unitmean.
apriori and relays rejecting packets if the source-destination
A transmitterlocated atx cancommunicatewith a receiverat
pair is not in its routing table. Without routing tables, the
y if SIR(x,y |φ)>T. The SIR is defined as
relay nodes having the same packet may have to coordinate
with each other and then choose one among them to transmit h g(x−y)
xy
the packet. In a mobile wireless network such a coordination SIR(x,y |φ)=
h g(z−y)
incurs significant overhead and also restrict the number of z∈φ\{x} zy
P
relays per source-destination pair to one. More importantly where φ is the transmittingset, g(x) is the path loss function,
such a restriction may in effect reduce the probability of and h is the power fading coefficient between nodes x and
xz
success. It is not clear how to choose a subset of these z.WeassumeT >1,i.e.,anarrowbandsystemwhichimplies
intermediaterelaysin a distributedfashionso astoreducethe at most one transmitter can connect to any receiver. The path
interferenceandincreasethe probabilityofpacketdelivery.In loss function g(x) is assumed to depend only on kxk, to
this paper, we analyze the success probability of such a two- monotonically decrease with kxk, and lim g(a)a2 = 0
a→∞
hopschemetakingtheinterferenceandthespatialstatisticsof to guarantee finite mean interference We restrict the number
the transmitting nodes into account. of hops between any source-destination pair to be two. So a
Related Work: In [3], relay selection is based on the SINR. source can reach its destination either in a single hop or can
All the relay-destination channel states are assumed to be use the relays to reach the destination. We can assume that
known at the destination, and the destination chooses the the sourcestransmitin the eventime slots and a subset of the
2
relays in the odd time slots. 3) In the second hop, the destination r(x) can decode the
Notation: packetfroma relay y ∈φ if the SIR(r(x),y |ψ)>T.
r
• We define The success probabilities of all these methods is analyzed in
the Appendix.
1(x→y |φ)=1(SIR(x,y |φ)>T).
A. Method 1: All relays transmit
1(x →y | φ) represents a random variable that is equal
Thisisthemostbasicschemewhereallrelayswhichreceive
tooneifatransmitteratxisabletoconnecttoareceiver
in the first time slot transmit in the second hop:
y when the transmitting set is φ.
• We define for x∈φs Nx =φˆr(x).
φˆ (x)={y ∈φ : 1(x→y |φ )}. As we shall see, this method has the worst performance
r r s
because of the high interference present in the second hop
φˆ (x) denotes the cluster of relays to which the source
r (specially when the relay density is high).
x is able to connect in the first hop.
Metric: We analyze the success probability for the direct B. Method 2: RSS-based selection
connection between the source-destination and the two-hop
When a relay node is able to connect to a source node, the
connection between them separately. Let P1 denote the prob- relay node has information about the RSS and could use that
ability that a source can connect to its destination directly in
information to make a decision to transmit in the next hop.
the first hop. More precisely we define
In Method 1, we do not utilize any information regardingthe
1 SIR received at the relay. In this method we utilize the RSS
P = lim E 1(x→r(x)|φ ). (1)
1 a→∞λtπa2 x∈φsX∩B(o,a) s Sinfiosrtmheatsiotrnentogtmhaokfeththeeddeesciriesidonsi.gWnaelhaanvdeIRiSsSth=eiSn+terIfewrehnecree
The relays which were able to connect to some source in the observed. Since the relay was able to decode the source, we
firsthoparethepotentialtransmittersinthesecondhop.Inthe have S >T and thus
I
relayselectionmethodsstudiedinthenextsection,asubsetof RSS S
these potential transmitters N ⊆φˆ (x) are selected for each I ≤ ≤ .
x r 1+T T
x∈φ to transmit in the next hop. Let the probability that a
s Hence a small value of RSS implies low interference (and
relay can connect to its intended destination (determined by 1+T
hencemaysee few interferer’sin thesecondhop).Thismight
the source to which it connectsin the first hop) in the second
also mean smaller S which implies that the relay is far from
hop be P , defined as
2 the source. A large value of RSS impliesa largeS and hence
1+T
P = lim 1 E 1−f(x), (2) wouldindicatearelayclosetothesource.Thismaypotentially
2 a→∞λtπa2 x∈φsX∩B(o,a) alsoimplymoreinterferenceattherelay.Sointhesecondhop
we give higherpriority to nodesthat observesmaller value of
where f(x) = y∈Nx(1−1(y →r(x)|ψ)) and ψ = 1R+STS. A relay y ∈ φˆr(x) transmits the packet in the second
∪x∈φsNx. In the Qabove equation 1 − f(x) is equal to one time slot with probability
if and only if at least one node belonging to φˆ (x) is able
r RSS(y)
to connect to the destination r(x). Here we are assuming no exp −δ , (3)
(cid:18) 1+T (cid:19)
cooperative communication between nodes which have the
same information, so relays belonging to the same cluster where RSS(y) is the RSS that the relay y observes and δ
φˆ (x) also interfere with each other in the second hop. Then represents a parameter to be chosen so that δRSS(y) is not
r 1+T
the success probability is given by too large. We have chosen an exponential penalty just for
convenience and the effect of the penalty function should be
P =1−(1−P )(1−P ). investigated further. So we have
s 1 2
RSS(y)
Intheaboveequationweassumedthatthesuccessprobability N = y ∈φˆ (x): U <exp −δ (4)
x r y
(cid:26) (cid:18) 1+T (cid:19)(cid:27)
ofthedirectconnectionisindependentofthetwohopsuccess
probability. We also used the spatial ergodic property of PPP where Uy is a set of i.i.d uniform random variables in [0,1].
in defining P and P . We observe that Methods 1 and 2 are completely decen-
1 2
tralized and require no information about the location of any
node. Hence these methods of relay selection work perfectly
III. LOCATION-UNAWARERELAY SELECTION
even with high mobility and incur zero overhead.
Inallthefourmethodsdescribedbelow,arelayhastomake
a decision whether to transmit in the second hop if it is able IV. LOCATION-AWARESELECTION
to connect to some source in the first hop. More precisely: In the location-aware based methods we assume that each
1) The destination r(x) can directly decode a packet from node has knowledge about its own location and each source
x∈φ in the first hop if the SIR(x,r(x) |φ )>T. knowsthe spatial locationof its destination.Also each packet
s s
2) A subset of the relays N ⊆ φˆ (x) are chosen in a has information about the location of the source from which
x r
distributed manner to transmit in the second hop. it originated and the location of its destination in its header.
3
A. Method 3: Sectorized relay selection λ=1,g(x)=|x|−3,θ=π/6
s
0.04
After a relay receivesa packet in the first hop, it calculates
the angle between the relay-source and the source destination 0.035
and makes a decision based on this information. More pre-
cisely a relay y ∈ φˆ (x) transmits the packet in the second 0.03
r
time slot if ∠yxr(x)<θ. So for this method, we have 0.025
Nx = y ∈φˆr(x): ∠yxr(x)<θ . Ps 0.02
n o
Inthismethodwearereducinginterferencebychoosingrelays 0.015
R=1
in a sector. If the angle θ is properly chosen the sector may
0.01 R=1.2
contain only one relay, i.e., |N | = 1 which would eliminate
x R=1, Center relay
the intra-cluster interference. 0.005 R=1.2, Center relay
0
0 1 2 3 4 5 6 7 8
B. Method 4 : Distance-based selection λ
r
In this method, we select the relays in φˆ (x) dependingon
r Fig. 1. Comparison of Method 3 versus the center-scheme. The success
their distance from the destination r(x). A relay y ∈ φˆr(x) probability versus the relay intensity λr is plotted for various values of the
transmits the packet in the second time slot with probability sourcedestination distance R.
2ky−r(x)k
exp −ǫ (5)
(cid:18) R (cid:19)
R=0.8, g(x)=|x|−4, T=3
We have normalizedthe distance by R/2 so that nodescloser 0.12
to the source are not over penalized. ǫ is a normalizing
parameter which we will choose later. So we have
0.1
N = y ∈φˆ (x): U <exp −ǫ2ky−r(x)kR−1
x r y
n (cid:0) (cid:1)o 0.08
where U is a set of i.i.d uniform random variables in [0,1].
y
In thismethod,we observethatrelaysclose to the destination Ps0.06 λ=1,λ=1
s r
have a higher chance of transmitting than those closer to the
λ=1,λ=1.2
s r
source. One could replace ky −r(x)k in (5) with ky −xk
0.04 λ=1,λ=2
and exp with 1− exp. Then the source need not know its s r
Theory λ=1,λ=1
destination location but one would loose the directionality of s r
0.02 Theory λ=1,λ=1.2
relay selection. s r
Theory λ=1,λ=2
In the location-aware methods proposed in this section, s r
0
each source needs to know where its destination is located 0 0.2 0.4 0.6 0.8 1
θ/(π/4)
and maintaining this information would require a significant
overhead in a mobile network. Fig.2. Comparisonofthesimulationandtheoretical results forMethod3.
V. SIMULATION RESULTS
Inthissectionwecomparethefourmethodsdescribedinthe
above section by simulations. For the purpose of simulation that the intensity of relays required to outperform the center-
weconsiderasquare[−30,30]2inwhichthenodesarelocated method decreases with increasing source destination distance
and use Monte-Carlo method to evaluate the results. We use R. From Figure 2, we observe that the probability of success
T =3 for all the simulations. as computedfrom the theory matches the simulationsclosely.
InFigure1, we compareMethod3with a schemein which In Figure 3, the success probability is plotted for the four
the relays are apriori chosen. In this scheme each source- methods described in previous sections. We first observe that
destination pair has one relay assisting in communication. all the methods have an optimal value of the parameters δ,
The relay is centered halfway between the source and the θ and ǫ that achieve the maximum success probability. We
destination. Intuitively such a scheme is the best single-relay also observe that Method 1 has the worst performance. This
schemeintermsofend-to-endsuccessprobability.Weobserve is because of the high interference caused by all the relays
that using uncoordinated relays yields a better performance (that have received a packet in the first hop) transmitting in
than selecting a relay apriori. This is because of the selection the second hop.We observethat Method 3 i.e., the sectorized
diversity that occurs due to fading and the node locations. selectionmethodperformsthebest.AlsoobservethattheRSS
Also observe that there is an optimal value of relay intensity based selection has twice the P as compared to Method 1 in
s
that achieves the maximum value of P . In practice this can whicheveryrelaytransmits.We alsoobservethattheboththe
s
be achieved by starting with large number of relays and location-aware schemes outperform Method 1 and Method 2
using ALOHA-like thinning in the second hop. We observe for the particular values of T =3,λ =1,λ =1.5.
s r
4
R=1.4, g(x)=|x|−4, T=3,λ=1, λ=1.5 where(a) followsfromthe Campbell-Mecketheorem[7] and
s r
0.035 (b) follows from he fact that p (z) depends only on kξ−zk.
ξ
First hop: A point process is completely characterized by
0.03
Method 1 its PGFL and so we will evaluate the PGFL of the relays
Method 2 which can connect to source ξ ∈ φ , i.e., the cluster φˆ (ξ).
0.025 s r
Method 3 Let 0≤v(x)≤1. The PGFL of φˆ (ξ) is given by
Method 4 r
0.02
Ps Gξ(v(x)) = E 1−(1−v(x))1(ξ →x|φs) (8)
0.015 xY∈φr
(=a) Eexp −λ (cid:2) (1−v(x))1(ξ →x|φ )dx
0.01 r s
(cid:18) R2 (cid:19)
(b) (cid:2)
0.005 ≥ exp −λ (1−v(x))p (x)dx , (9)
r ξ
(cid:18) R2 (cid:19)
0
0 0.2 0.4 0.6 0.8 1 where (a) follows since φ is a PPP, and (b) follows from
r
Jensen’sinequality.From the PGFL we observethat the point
Fig.3. Probabilityofsuccessforthefourmethodsdescribedintheprevious
section. Thex-axis represents δ forMethod 2,4θ/π forMethod 3andǫ/6 processconsistingofrelayswhichconnectto the originis not
forMethod4. a PPP. (b) would have been an equality if 1(ξ →x |φ ) are
s
independent for different x and the resulting process would
be a PPP. Butfor the sake ofanalysis, we makethe following
VI. CONCLUSION
assumptions and justify them by simulations.
Inthispaper,wehavehaveanalyzedthesuccessprobability
1) We assume that the spatial distribution of φˆ (ξ) is an
inatwohopsystemtakinginterferenceandspatialdistribution r
inhomogeneousPPP with intensity λ p (ξ). Since T >
of the nodes into account. We have provided an analytical r ξ
1, φˆ (ξ )∩φˆ (ξ )=∅, ∀ξ ,ξ ∈φ .
solution for all the four methods using some approximations. r 1 r 2 1 2 s
2) We also assume φˆ (ξ ) is independentof φˆ (ξ ) for all
Thismethodofanalysiscanbeeasilyextendedtoanyposition r 1 r 2
ξ ,ξ ∈φ .
based relay selection. We have shown that uncoordinated se- 1 2 s
lectionofrelaysincreasesthesuccessprobabilityascompared Wewillshowtheresultsobtainedbythisassumptionareclose
to selecting a relay for each source-destination pair apriori. to the actual by simulation. From Figure 2 we observe that
the simulation results (for Method 3) are very close to that
predicted by theory making the above assumptions. This is
APPENDIX
intuitivesince manytermsin (8) are independentandthusthe
The probability that a destination located at z can decode
bound in (9) is very tight.
the packet transmitted by a source ξ when the interference is A subset of relays N ⊆ φˆ (ξ) for each ξ ∈ φ transmit
ξ r s
caused by φ is
s in the second hop depending on the relay selection method.
p (z) = P(ξ →z |φ ) (6) Thisisbasicallyathinningofthepointprocessφˆr(ξ).Wewill
ξ s
now derive the intensity of the point process N for different
= P(SIR(ξ,z |φ )>T) (7) ξ
s methods.We will denote the spatial intensity of N by ∆ (z)
ξ ξ
= P(hξz > g(ξT−z)xX∈φshxzg(x−z)) Manedthwoedh1a:vSeinEc[e|NNξξ∩=Aφ|ˆ]r=(ξ)(cid:1),Aw∆eξh(azv)edz∆fξo(rza)n=y Aλr⊂pξ(Rz2)..
Method 2: From (11), we have
(=a)E exp(cid:18)−g(ξT−z)hxzg(x−z)(cid:19) ∆ξ(z) = λrE exp −δRSS(z) 1(ξ →z |φs)
xY∈φs (cid:18) (cid:18) 1+T (cid:19) (cid:19)
(=b)exp(−λs(cid:2) β(ξ−z,y)dy) (a) exp −λs(cid:1)R2 g(ξ−(zδ)g+(ξ(−δgz()ξ+−Tz))g+(Ty))g(y)dy
R2 = λr(1+T) (cid:16) (cid:17),
1+T +δg(ξ−z)
where
1 where (a) follows from a procedure similar to the evaluation
β(x,y)=
1+ g(Tx)g(y)−1 of pξ(z).
Method 3: Given ξ and r(ξ), we have
(a) follows from the exponential distribution of h , and (b)
ξz
follows the probability generating functional (PGFL) of the ∆ (z)=λ p (z)1(∠zξr(ξ)<θ).
ξ r ξ
PPP[1],[7].Alsoobservethatp (z)dependsonlyonkξ−zk.
ξ
Direct transmission: So from (1) we have
Method 4: Given ξ and r(ξ), we have
1
P = lim E 1(x→r(x)|φ )
1 a→∞λtπa2 x∈φsX∩B(o,a) s ∆ξ(z)=λrpξ(z)exp(cid:0)−ǫkz−r(ξ)kR−1(cid:1)
(=a) lim 1 (cid:2) p (r(x))dx (=b) P (R), The average numberof relays in a cluster Nξ that transmit in
a→∞πa2 B(o,a) x o the second hop is (cid:1) ∆ξ(z)dz.
5
Second hop: The transmitting set in second hop is given process [4], [6] with an additional cluster at the origin. The
by ψ = N . Since T > 1, at most one transmitter Laplace transform of the interference in this case is given by
ξ∈φs ξ
belongingSto Nξ can connect to r(ξ). So the probability that Eexp −s y∈ψ′hyRg(y−R) which is equal to
nonodefromN canconnecttor(ξ)denotedbyf(ξ)isgiven (cid:16) (cid:17)
ξ P
by
f(ξ) = 1− 1(z →r(ξ)|ψ). E exp(−shyRg(y−R))(=a)Gψ′(cid:18)1+sg(1y−R)(cid:19)
yY∈ψ′
zX∈Nξ  
where G (.) is the PGFL of the process ψ′ and (a) follows
ψ
So P˜2 (same as P2 without the limit) is given by by Laplace transform of the fading. So we have
P˜2 = E 1(z →r(ξ)|ψ) E[1(z →R|ψ′)]
(=a) λtx(cid:2)∈Bφ(tXo∩,Ba)(oE,az)X∈zXN∈ξN1ξ (z →r(ξ)|Nξ∪ψ)dξ (=a) EξY∈φsyY∈Nξ 1+ Tg1g((zy−−RR))EyY∈No 1+ Tg1g((zy−−RR))
(=b) λt(cid:2) (cid:2) E[∆ξ(z)1(z →r(ξ)|Nξ∪ψ)]dzdξ (=b) E exp −β˜(z−R,ξ) exp −β˜(z−R,R)
B(o,a) ξY∈φs (cid:16) (cid:17) (cid:16) (cid:17)
where (a) and (b) follow from the Campbell-Mecke theorem
andSlivnyak’stheorem.Weincluded∆ξ(x)intheexpectation (=c) exp(cid:18)−β˜(z−R,R)−λs(cid:2) 1−exp(cid:16)−β˜(z−R,ξ)(cid:17)dξ(cid:19),
operator because in Methods 3 and 4, ∆ (z) depends on the
ξ
random variable r(ξ). We now show that the inner integral where (a) follows from assumption 2, (b) follows from a
does not depend on ξ. We also have ∆ξ(z) = ∆o(ξ−z) for technique similar to the derivation of pξ(z), (c) follows from
d the PGFl of PPP, and where
Method 1 and 2. For Methods 3 and 4 we have ∆ (z) =
ξ
∆o(ξ − z) where the equality is in distribution. Using the β˜(z,ξ)=(cid:2) β(z,y)∆˜ (y+ξ)dy.
substitution z′ ← z−ξ, the stationarity of ψ, and the above R2 o
property of ∆ξ(z) we have From the above equation, (12) and (11) we have,
1 (cid:2) 2π(cid:2) (cid:2)
P = E[∆ (z)1(z →R |ψ∪N )]dzdν P = ∆˜ (z+R)
2 o ν o 2 o
2π 0 R2 R2
(10)
(cid:2)
twhheeirseotRroνpi=c n(aRtucreoso(fν)N,Rasnind(νψ))w. eFohravMeethods 1 and 2 by ·exp(cid:18)−β˜(z,R)−λs R21−exp(cid:16)−β˜(z,ξ)(cid:17)dξ(cid:19)dz,
o
where ∆˜ (z) = ∆ (z) for Method 1 and 2. For Method 3,
(cid:2) o 0
P2 = ∆o(z)E1(z →R|ψ∪No)dz (11) ∆˜o(z)= πθλrpξ(z). For Method 4
R2
Due to space constraints we will only describe how to derive ∆˜ (z)= λrp (z)(cid:2) 2πexp −ǫkz−R kR−1 dν.
o ξ ν
P for Method 3. Method 4 can be analyzed in a similar 2π
2 0 (cid:0) (cid:1)
fashion. From (10) and the definition of ∆ (z) for Method
o
3, we have P = REFERENCES
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