The Throughput of Slotted Aloha with Diversity

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by  Andrea Munari

The Throughput of Slotted Aloha with Diversity
Andrea Munari, Michael Heindlmaier, Gianluigi Liva and Matteo Berioli
Abstract—Inthispaper,asimplevariation of classical Slotted a slot are supposed to be available. The different observa-
Aloha is introduced and analyzed. The enhancement relies on tions are associated to K relays, and, for each of them,
adding multiple receivers that gather different observations of
the transmitted packets are subject to independent fading
thepacketstransmittedbyauserpopulationinoneslot.Foreach
coefficients. Collisions are regarded as destructive, and the
observation, the packets transmitted in one slot are assumed to
be subject to independent on-off fading, so that each of them is system is complemented by having relays deliver what they
either completely faded, and then does not bring any power or have decoded to a centralized gateway.
4
interference at the receiver, or it arrives unfaded,and then may SA with space (antenna) diversity was analyzed in [13]
1
or may not, collide with other unfaded transmissions. With this
0 under the assumption of Rayleigh fading and shadowing,
model,anoveltypeofdiversityisintroducedtotheconventional
2 with emphasis on the two-antenna case. With respect to [13],
SA scheme, leading to relevant throughput gains already for
n moderate number of receivers. The analytical framework that we introduce in our analysis a simplified channel model.
a we introduce allows to derive closed-form expression of both In particular, the uplink wireless link connecting user i and
J throughputandpacketlossrateanarbitrarynumberofreceivers, relay j is described by a packet erasure channel with erasure
5 providinginterestinghintsonthekeytrade-offsthatcharacterize probability ε , following the on-off fading model [14]. The
1 the system. We then focus on the problem of having receivers fadingisassuim,jedtobeindependentforeachpairofuser-relay
forward the full set of collected packets to a final gateway
] using the minimum possible amount of resources, i.e., avoiding pair. Despite its simplicity, the model is accurate enough for
I delivery of duplicate packets, without allowing any exchange some cases of interest. As an example, it captures the main
N
of information among them. We derive what is the minimum featuresofaninteractivesatellitenetworkwithsatellitelocated
. amount of resources needed and propose a scheme based on
s on different orbits, and where the line-of-sight link between
random linear network coding that achieves asymptotically this
c users and relays may be blocked whenever an obstacle lies
[ bound without the need for the receivers to coordinate among
them. betweenauserandasatellite (here,thesatellitesplaytherole

1 of relays).
v Under this fading model, elegant exact expressions for the
0 I. INTRODUCTION
system throughput as a function of the number of relays
6
A renewed interest for Aloha-like random access (RA) are derived, yielding deep insights in the gains provided by
6
3 protocols led recently to the development of new high- diversity in SA protocols. We further provide an analysis on
. throughput uncoordinated multiple-access schemes [1]–[10]. how the link between the relays and the centralized gateways
1
0 Theseschemessharethefeatureofcancellingtheinterference (also referredto as downlink)shall be dimensioned,assuming
4 caused by a packet whenever (a portion of) it is successfully the relays to be uncoordinated. A bound on the downlink
1 decoded. Among the aforementioned works, a specific class capacity is derived, which is achieved by a random linear
v: is based on the diversity slotted Aloha (DSA) protocol intro- coding approach based on Slepian-Wolf coding.
i ducedin[11]enhancedbysuccessiveinterferencecancelation The rest of the paper is organized as follows. We start in
X
(SIC). In [4], [5] it was shown that the SIC process can be SectionIIbydefiningthesystemmodelthatisusedtodevelop
ar well modeled by means of a bipartite graph. By exploiting our framework. Section III provides a thorough analysis of
the graph model, a remarkably-high capacity (e.g., up to the system uplink, characterizing it in terms of throughput
0.8[packets/slot]) can be achieved in practical implementa- and delivery reliability, whereas in Section IV we study how
tions,whereasforlargemediumaccesscontrol(MAC)frames to effectively deliver collected packets to a common gateway
itwasdemonstratedthatfullyefficiency(1[packets/slot])can without resorting to coordination and information exchange
be substantiallyattained[8],[10],[12].A furtherkeyingredi- among relays. In Appendix, we also investigate, for the two-
entto attainlargethroughputgainsdealswith theexploitation receivercase,anextensionoftheconsideredschemethattakes
of diversity. As an example, the approaches proposed in [1], advantage of successive interference cancellation techniques.
[4]–[6], [9]–[11] take advantage of time diversity to resolve
collisions.
II. SYSTEMMODELAND PRELIMINARIES
Inthispaper,wedevelopandanalyzeasimpleyetpowerful
relay-aided slotted Aloha (SA) scheme which enjoys space Throughout this paper, we focus on the topology depicted
diversity. More specifically, K independent observations of inFig.1, whereaninfinitepopulationofuserswanttodeliver
informationintheformofdatapacketstoacollectinggateway
Andrea Munari, Gianluigi Liva and Matteo Berioli are with In-
(GW). The transmission process is divided in two phases,
stitute of Communication and Navigation of the Deutsches Zentrum
fu¨r Luft- und Raumfahrt (DLR), D-82234 Wessling, Germany (e-mail: referred to as uplink and downlink, respectively. During the
{andrea.munari,gianluigi.liva,matteo.berioli}@dlr.de). former,dataaresentinanuncoordinatedfashionoverashared
Michael Heindlmaier is with the Lehrstuhl fu¨r Nachrichtentechnik, Tech-
wireless channel to a set of K receivers or relays, which, in
nische Universita¨t Mu¨nchen (TUM), D-80290 Munchen Germany (e-mail:
[email protected]). turn,forwardcollectedinformationtotheGWinthedownlink.
Gateway
be triggered when the relays are considered jointly, since
independent channel realizations may lead them to retrieve
different information units over the same time slot. In order
downlink toquantifythisbeneficialeffect,welabelapacketascollected
when it has been received by at least one of the relays, and
Krelays we introduce the uplink throughput T as the average
up,K
number of collected packets per slot. Despite its simplicity,
...
such a definition offers an effective characterization of the
beneficialeffectsof diversity,by properlyaccountingforboth
ε uplink thepossibilityofretrievinguptomin{u,K}distinctdataunits
or multiple times the same data unit over a slot, as will be
discussedindetailsinSectionIII-A.Ontheotherhand,T
up,K
also quantifies the actual amount of information that can be
...
retrievedbythesetofreceivers,providinganupperboundfor
theoverallachievableend-to-endperformance,andsettingthe
Users
target for the design of any relay-to-GW delivery strategy.
Fig.1. Reference topology forthesystemunderconsideration. For the downlink phase, we focus on a decode and for-
ward (D&F) approach, so that each receiver re-encodes and
transmits only packets it has correctly retrieved during the
As to the uplink, time is divided in successive slots, and
uplinkphase,orpossiblylinear combinationsthereof.A finite
transmissionparametersin termsofpacketlength,codingand
downlink capacity is assumed, and relays have to share a
modulation are fixed such that one packet can be sent within
commonbandwidthtocommunicatetotheGWbymeansofa
onetimeunit.Usersare assumedto beslot-synchronized,and
TDMA scheme. In order to get an insightful characterization
SlottedAloha(SA)[15]isemployedasmediumaccesspolicy.
of the optimum achievable system performance, we assume
Furthermore, the number of users accessing the channel in
relay-to-GWlinks to be error free, and let resourceallocation
a generic slot is modelled as a Poisson-distributed r.v. U of
fortheD&Fphasebeperformedideallyandwithoutadditional
intensity ρ, with:
cost by the central collecting unit.
ρue−ρ
We then complement our study in Appendix I by consid-
Pr{U =u}= . (1)
u! ering, for the simplified K = 2 scenario, an amplify and
The uplinkwireless linkconnectinguser i andreceiverj is forward (A&F) approach. In this case, relays simply deliver
describedbyapacketerasurechannelwitherasureprobability an amplified version of the analog waveform (possibly the
εi,j, where independent realizations for any (i,j) pair as outcome of a collision) they received, whereas the GW per-
well as for a specific user-receiver couple across time slots formsdecodingrelyingonsuccessiveinterferencecancellation
are assumed. For the sake of mathematical tractability, we (SIC)techniques.Thegoalofsuchaninvestigationistoderive
set εi,j = ε, ∀ i, j. Following the on-off fading description a characterization of the gains that are achievable by jointly
[14], we assume that a packet is either completely shadowed, processing signals incoming at different receivers. Along this
not bringing any power or interference contribution at a lineofreasoning,wewillfocusonanidealizeddownlink,such
receiver,or it arrivesunfaded.While, on the one hand,such a thatinformationcanbereliablydeliveredtothecollectingunit
modelisespeciallyusefulto developmathematicallytractable at no cost in terms of bandwidth.
approachesto the aim of highlightingthe key tradeoffsof the
considered scenario, it also effectively captures effects like
A. Notation
fadingandshort-termreceiverunavailabilitydue,forinstance,
to the presence of obstacles. Throughoutour investigation,no Prior to delvinginto the details of ourmathematicalframe-
multi-user detection capabilities are considered at the relays, work,we introducein the followingsome usefulnotation.All
so thatcollisionsamongnon-eraseddataunitsareregardedas the variables will be properly introduced when needed in the
destructive and prevent decoding at a receiver. discussion, and the present section is simply meant to offer a
Within this framework, the number of non-erased packets quick reference point throughoutthe reading.
that arrive at a relay when u concurrent transmissions take K relaysareavailable,and,withintimeslott,thecountably
place follows a binomial distribution of parameters (u,1 − infinite set of possible outcomesat each of them is labeled as
ε) over one slot. Therefore, a successful reception occurs Ω := {ωt,ωt,ωt,...,ωt } for each t = 1,2,...,n. Here,
t 0 1 2 ∞
with probability u(1 − ε)εu−1, and the average throughput ωt denotes the erasure event (given either by a collision or
0
experienced at each of the K receivers, in terms of decoded by an idle slot), while ωt indicates the event that the packet
j
packets per slot, can be computed as: of the j-th user arriving in slot t was received. According
∞ ρue−ρ to this notation, we define as Xkt the random variables with
Tsa = u! u(1−ε)εu−1 =ρ(1−ε)e−ρ(1−ε), (2) alphabetN,whereXkt =jifωjt wastheobservationatrelayk.
uX=0 When needed for mathematical discussion, we let the uplink
corresponding to the performance of a SA system with operate for n time slots. In this case, let An be the set of
k
erasures. On the other hand, a spatial diversity gain can collectedpacketsaftern timeslotsatreceiverk, whereAn (
k
n {Ω \ωt}. That is, we do not add the erasure events to
t=1 t 0 0.8
SAn. The number of received packets at relay k after n time K = 2, ε = 0.1
slokts is thus |An|. 0.7 SA, ε = 0.1
k K = 2, ε = 0.5
In general, the complement of a set A is indicated as A. SA, ε = 0.5
0.6
We write vectors as lowercase underlined variables, e.g., w,
while matrices and their transposes are labeled by uppercase 0.5
letters, e.g., G and GT. up0.4
T
III. A CHARACTERIZATION OFSYSTEMUPLINK 0.3
With reference to the topology of Fig. 1, we first consider 0.2
the uplink phase. In order to gather a comprehensivedescrip-
0.1
tion of the improvements enabled by receiver diversity, we
characterize the system by means of two somewhat comple- 0
0 2 4 6 8 10
mentarymetrics:uplinkthroughput(SectionIII-A)andpacket ρ
loss rate (Section III-B).
Fig. 2. Average uplink throughput vs channel load under different erasure
probabilities. Blackmarkersindicate theperformance inthepresenceoftwo
A. Uplink Throughput
receivers, whereaswhitemarkers reportthebehaviorofpureSA.
Let us focus on the random access channel, and, following
the definition introduced in Section II, let C be the number
of packets collected by the relays over one slot. C is a r.v. Pluggingtheseresultsinto(3)weget,aftersomecalculations,
withoutcomesintheset{0,1,2,...,K},wherethemaximum a closed-formexpressionfor the throughputin the uplink and
valueoccurswhentheK receiversdecodedistinctpacketsdue thus, as discussed, also for the end-to-end D&F case with
to different erasure patterns. The average uplink throughput infinite downlink capacity:
can thus be expressed by conditioning on the number of
T =2ρ(1−ε)e−ρ(1−ε)−ρ(1−ε)2e−ρ(1−ε2). (6)
concurrent transmissions as: up,2
∞ ρue−ρ K The trend of T is reported in Fig. 2 against the channel
T =E [E[C|U]]= cPr{C =c|U =u}. up,2
up,K U u! load ρ for different values of the erasure probability, and
uX=0 Xc=0
(3) compared to the performance in the presence of a single
WhileEq.(3)formulaholdsforanyK,thecomputationofthe receiver, i.e., Tsa. Eq. (6) conveniently expresses Tup,2 as
collectionprobabilitiesintrinsicallydependsonthe numberof twice the throughput of SA in the presence of erasures,
available relays. In this perspective,we articulate our analysis reduced by a loss factor which accounts for the possibility
by first considering the two-receiver case, to then extend the of having both relays decode the same information unit. In
results for an arbitrary topology. this perspective, it is interesting to evaluate the maximum
1) The Two-Receiver Case: Let us first then focus on throughputTu∗p,2(ε)aswellastheoptimalworkingpointρ∗(ε)
the case in which only two relays are available. Such a achieving it for the system uplink. The transcendental nature
scenario allows a compact mathematical derivation of the of (6) does not allow to obtain a closed formulation of these
uplink throughput, as the events leading to packet collection quantities, which, on the other hand, can easily be estimated
at the relays set can easily be expressed. On the other hand, bymeansofnumericaloptimizationtechniques.Theresultsof
it also represents a case of practical relevance, as it can be thisanalysisarereportedinFig.3,wherethepeakthroughput
instantiated by simply adding a receiver to an existing SA- Tu∗p is depicted by the black curve as a function of ε and
basedsystem.WhenK =2,thesituationforC =1caneasily compared to the performance of SA, which clearly collects
beaccountedfor,sinceasinglepacketcanbecollectedassoon on average at most 0.36 pkt/slot regardless of the erasure
as at least one of the relays does not undergo an erasure, i.e., rate. In ideal channel conditions, i.e., ε = 0, no benefits can
withoverallprobability1−ε2.Ontheotherhand,byvirtueof be obtained by resorting to multiple relays, as all of them
thebinomialdistributionofU,theeventofcollectinga single would see the same reception set across slots. Conversely,
information unit over one slot occurs with probability higher values of ε favour a decorrelation of the pattern of
packets that can be correctly retrieved, and consequently
Pr{C =1|U =u}=2u(1−ε)εu−1 1−u(1−ε)εu−1
improve the achievable throughput at the expense of higher
+u(1−ε)2ε2(u(cid:2)−1), (cid:3)(4) loss rates. The result is a monotonically increasing behavior
for T∗ (ε), prior to plummeting with a singularity to a null
where the former addend accounts for the case in which one up,2
throughput for the degenerate case ε = 1. Fig. 3 also reports
relay decodesa packet while the other does not(either due to
(circled-white markers) the average throughput obtained for
erasures or to a collision), whereas the latter tracks the case
ρ = 1/(1 − ε), i.e., when the uplink of the system under
of having the two relays decoding the same information unit.
consideration operates at the optimal working point for a
Conversely,arewardoftwopacketsisobtainedonlywhenthe
single-receiverSA,showingatightmatch.Infact,eventhough
receivers successfully retrieve distinct units, with probability
the abscissa of maximum ρ∗(ε) may differ from this value
Pr{C =2|U =u}=u(u−1)(1−ε)2ε2(u−1). (5) (they coincide only for the ideal case ε=0), the error which
referencetoasingleslott,letΩ :={ωt,ωt,ωt,...,ωt }for
0.8 t 0 1 2 ∞
each t = 1,2,...,n be the countably infinite set of possible
0.75 outcomes at each relay, where ωt denotes the erasure event
ut 0
hp 0.7 while ωt indicates the event that the packet of the j-th user
g j
ou0.65 arriving in slot t was received. Let us furthermore define as
able thr 0.6 XwhktetrheeXratnd=omj vifarωiatblwesaswtihtheaolbpshearbveattiXon=at{r0e,l1a,y2k,.,.s.o,∞th}at,
m achiev0.05.55 SKA=2, numerical Xthek1,uXplk2in,.kk.o.p,Xeraknteisfoajrnni.it.idm.esesqloutesn,caendfoirnedaiccahterealasyAknk.tWheesleett
mu0.45 K=2, approximation of packets collected at receiver k over this time-span, where
Maxi 0.4 KK==22 ww// AA&&FF,, naupmpreorxicimalation Ank ( nt=1{Ωt\ω0t} (i.e., we do not add the erasure events
to An)S. The number of received packets at relay k after n
0.35 k
timeslotsisthus|An| and,withreferencetothisnotation,we
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 k
ε prove the following result:
Proposition 1: For an arbitrary number of K relays, the
Fig. 3. Maximum uplink throughput vs erasure rate. Theblack continuous throughputTup,K is given by
line reports the performance Tu∗p,2 of a two-receiver scheme, while white K
bcierhcalevdiormoafrkpeurrseSinAdi.cGatreayTcuupr,v2e(s1a/n(d1m−arεk)e)r,saanredtothbeedroeftteerdredlintoethsheoawmspltihfye Tup,K = (−1)k−1 K ρ(1−ε)ke−ρ(1−εk) (8)
(cid:18)k(cid:19)
andforward case,thatwillbetreated inAppendixI. Xk=1
Proof: We have |An| = n , where
k t=11{Xkt6=0} 1{E}
denotestheindicatorrandomvariPablethattakesonthevalue1
iscommittedwhenapproximatingT∗ withT (1/(1−ε))
up,2 up,2 iftheeventE istrueand0otherwise.Thethroughputseenby
can easily be shown numerically to never exceed 0.6%, due
a single relay can then be written as T = E[ ] =
to the very small slope of the function in the neighborhood Pr{Xt 6= 0}, and does not depend onupt,h1e spec1ifi{cXktr6=ec0e}iver
of ρ∗(ε). We can thus provide a very precise estimate of the k
being considered. By the weak law of large numbers,
peak uplink performance for a specific erasure rate as:
|An|
2 Tup,1 = lim k (9)
T∗ (ε)≃ −(1−ε)e−1−ε, 0≤ε<1. (7) n→∞ n
up,2 e
or, more formally,
which once again compactly captures the behavior of a two- |An|
receiverscenarioby quantifyingthe losswith respectto twice nl→im∞Pr(cid:26)(cid:12) nk −Tup,1(cid:12)>ǫ(cid:27)=0 for some ǫ>0. (10)
the performanceof SA. In this perspective,two remarksshall (cid:12) (cid:12)
bemade.Firstofall,inordertoapproachtheupperbound,the Similarly, for(cid:12)(cid:12)K relays we(cid:12)(cid:12)have
system has to be operated at very high load, as ρ∗ ≃ 1/(1− | K An|
T = lim k=1 k (11)
ε)). These working points are typically not of interest, since up,K n→∞ S n
very low levels of reliability can be provided by a congested
By the inclusion-exclusionprinciple (see, e.g., [16]), we have
channelwithhigherasurerates.Nevertheless,thepresenceofa
K
second receivertriggersremarkableimprovementsalreadyfor
An = (−1)|S|−1|In| (12)
loss probabilities that are of practical relevance, e.g., under (cid:12)(cid:12)(cid:12)k[=1 k(cid:12)(cid:12)(cid:12) S⊆{1,.X..,K},S6=∅ S
harsh fading conditionsor for satellite networks.Indeed, with (cid:12) (cid:12)
ε = 0.1 a ∼ 15% raise can be spotted, whereas a loss rate (cid:12) (cid:12) with ISn = Ank (13)
of 20% already leads to a 50% throughput gain. Secondly, k\∈S
the proposed framework highlights how no modifications in Here, In denotes the set of packets that all the relay nodes
S
terms of system load are needed with respect to plain SA for specified by S ={k ,k ,...,k } have in common:
1 2 |S|
a two-receiver system to be very efficiently operated. Such
n
a result is particularly interesting, as it suggests that a relay |In|= An = (14)
node can be seamlessly and efficiently added to an already S (cid:12)(cid:12)(cid:12)k\∈S k(cid:12)(cid:12)(cid:12) Xt=11{06=Xkt1=Xkt2=...=Xkt|S|}
operating SA uplink when available, triggering the maximum (cid:12) (cid:12)
Due to symm(cid:12)etryin(cid:12)the setup, the value of |In| only depends
achievable benefit without the need to undergo a re-tuning of S
on the cardinality of S but not the explicit choice, so that
the system which might be particularly expensive in terms of
|In|=an for k =|S|, and,
resources. S k
K K
2) The General Case, K > 2: Let us now focus on the K
An = (−1)k−1 an.
agbelnee.raWlthoipleolcoognycreepptuoartlelydianppFliigc.ab1l,ew, htheereaKpprroelaacyhsparreesaevnateild- (cid:12)(cid:12)(cid:12)k[=1 k(cid:12)(cid:12)(cid:12) Xk=1 (cid:18)k(cid:19) k
(cid:12) (cid:12)
to computethe uplinkthroughputin the two-receivercase be- As Xk1,Xk2,.(cid:12)..,Xkn (cid:12)are i.i.d., by the weak law of large
comescumbersomeasK grows,dueto the rapidlyincreasing numbers we have:
number of events that have to be accounted for. In order to |In|
lim S =Pr[{06=Xt =Xt =...=Xt }]. (15)
characterize Tup,K, then, we follow a different strategy. With n→∞ n k1 k2 k|S|
1.6 SA
K = 2 4.5
K = 4
1.4 K = 10 4
1.2 3.5
1 ∗ρK 3
Tup0.8 ∗,up,K2.5
T 2
0.6
1.5
0.4
1 T∗
0.2 up,K
0.5 ρ∗K
0
0 2 4 6 8 10 5 10 15 20 25 30 35 40 45 50
ρ K
Fig.4. Average uplink throughput vs channel load fordifferent numberof Fig.5. MaximumachievablethroughputT∗ asafunctionofthenumber
up,K
relays K.Theerasureprobability hasbeensettoε=0.5. ofrelays K foranerasurerate ε=0.5.Thegraycurvereports theload on
thechannel ρ∗ neededtoreachT∗ .
K up,K
We can compute the latter probability as
linear and that exhibits a logarithmic-like trend in K.
Pr{06=Xt =...=Xt } ∆Tup =Tup,K −Tup,K−1
k1 k|S| K
K−1
=Xu Pr{06=Xkt1 =...=Xkt|S||U =u}Pr{U =u} =kX=1(−1)k−1(cid:18)k−1(cid:19)ρ(1−ε)keρ(1−εk) (17)
= ∞ e−ρρu u (1−ε)εu−1 |S|
u! (cid:18)1(cid:19)
uX=1 (cid:0) (cid:1)
B. Packet loss probability
=(1−ε)|S|ρe−ρ(1−ε|S|) (16)
The aggregate throughput derived in Section III-A repre-
sents a metric of interest towards understanding the potential
Aslim ank =(1−ε)kρe−ρ(1−εk),thepropositionfollows. of SA with diversity when aiming at reaping the most out
n→∞ n
of uplink bandwidth. On the other hand, operating an Aloha-
The performance achievable by increasing the number of basedsystemattheoptimalloadρ∗ exposeseachtransmitted
K
relays is reported against the channel load in Fig. 4 for a ref- packet to a loss probability that may not be negligible. In the
erenceerasure rate ε=0.2. As expected,T benefitsfrom classical single-receiver case without fading, for instance, the
up,K
a higher degree of spatial diversity, showing how the system probabilityfor a data unit notto be collected evaluatesto 1−
can collect more than one packet per uplink slot as soon as e−1 ≃0.63.Fromthisstandpoint,in fact, severalapplications
morethanfourreceiversareavailable,fortheparametersunder may resort to a lightly loaded random access uplink, aiming
consideration. Such a result stems from two main factors. On at a higher level of delivery reliability rather than at a high
the one hand, increasing K enables larger peak throughput throughput. This is the case, for example, of channels used
overasingleslot,asuptoK differentdataunitscanbesimul- for logon and control signalling in many practical wireless
taneously retrieved. On the other hand, broader receivers sets networks. In order to investigate how diversity can improve
improvetheprobabilityofdecodingpacketsinthepresenceof performance in this direction, we extend our framework by
collisions even when less than K users accessed the channel, computingtheprobabilityζK thatauseraccessingthechannel
by virtue of the different erasure patterns they experience. experiencesa data loss, i.e., that the information unit it sends
The uplink throughput characterization is complemented by is not collected, either due to fading or to collisions, by any
Fig. 5, which reports the peak value for T∗ (solid black of the K relays.
up,K
curve), obtained by properly setting the channel load to ρ∗ To this aim, let O describe the event that the packet of the
K
(whose values are shown by the gray dashed curve), for an observed user sent over time slot t is not received by any
increasing relay population.1 The plot clearly highlights how of the receivers. Conditioning on the number of interferers i,
thebenefitbroughtbyintroducinganadditionalreceivertothe i.e., of data units thatwere concurrentlypresenton the uplink
scheme,quantifiedbyEq.(17),progressivelyreduces,leading channel at t, the sought probability can be written as:
toa growthratefortheachievablethroughputthatislessthan ∞
ζ = Pr[O|I =i]Pr[I =i]. (18)
K
Xi=0
1 As discussed for the K = 2 case, a mathematical derivation of the
optimalworkingpointloadρ∗ isnotstraightforward, andsimplenumerical Here, the conditional probability can easily be determined
K
maximization techniques wereemployed toobtaintheresults ofFig5. recallingthateachoftheK relaysexperiencesanindependent
collected to a central GW. In doing so, we aim at employing
100
the minimum number of resources in terms of transmissions
thathavetobeperformedbytherelays,whilenotallowingany
informationexchange among them. In particular, we consider
10−1
a finite-capacity downlink, where the K receivers share a
common bandwidth to communicate with the GW by means
K10−2 ofatimedivisionmultipleaccessscheme,andweassumethat
ζ
each of them can reliably deliverexactly one packet, possibly
composedofa linearcombinationofwhathasbeencollected,
10−3 KK == 12 over one time unit. We once again focus on a horizon of n
K = 3 slots to operate the uplink, after which the downlink phase
K = 4
K = 5 starts.
10−4 K = 10 We structure our analysis in two parts. First, in Sec-
10−3 10−2 ρ 10−1 100 tion IV-A, we derive lower bounds for the rates (in terms of
downlinkslots allocatedper uplinkslot) that haveto assigned
Fig.6. ProbabilityζK thatapacketsentbyauserisnotreceivedbyanyof to receivers in order to deliver the whole set of data units
therelays. Different curves indicate different values ofK,while theerasure collected in the uplink over the n slots. Then, Section IV-B
probability hasbeensettoε=0.2.
showshowasimpleforwardingstrategybasedonrandomlin-
ear network coding suffices to achieve optimality, completing
erasure pattern, obtainingPr[O|I =i]=(1−(1−ε)εi)K for the downlink phase in Tup,K slots for asymptotically large
an individual packet and K relays with independent erasures values of n.
on all individual links. By resorting to the binomial theorem, Priortodelvingintothedetails,letusintroducesomeuseful
such an expression can be conveniently reformulated as: notation. We denote the L-bit data part of packet of the j-th
user arriving in time slot t as Wt ∈ W, with W = F ∪
K k 2L
Pr[O|I =i]= (−1)k K (1−ε)εi k. (19) e, where e is added as the erasure symbol. We furthermore
(cid:18)k(cid:19) assumethatthereceivercandeterminethecorrespondinguser
kX=0 (cid:0) (cid:1)
through a packet header, i.e., the receiver knows both j and t
On the other hand, the number of interferers seen by a user
after successful reception. As the uplink operates over n time
that accesses the channel at time t still follows a Poisson slots, relay k observes the vector w = [W1,...,Wn]. In
k k k
distribution of intensity ρ, so that, after simple calculations each time slot, the tuple (Wt,Wt,...,Wt ) is drawn from a
1 2 K
we finally get:
joint probability distribution P which is governed by
W1...WK
K theuplink,anddifferentrelaysmightreceivethesamepacket.
ζ = (−1)k K (1−ε)ke−ρ(1−εk). (20)
K
(cid:18)k(cid:19)
kX=0
A. Bounds for Downlink Rates
Fig.6reportsthebehaviorofζK asafunctionofρwhenthe EachrelayktransmitsapacketineachofitsnRk downlink
erasurerateovera singlelinkissetto ε=0.2.Differentlines slots. We are interested in the set of rates (R )K such that
k k=1
indicate the trend when increasing the number of receivers the gateway can recover all packets (with high probability).
from 1 to 10. As expected, when ρ→0, a user accessing the This is essentially the problem of distributed source cod-
channel is not likely to experience any interference, so that ing (SW-Coding [17]), with the following modification: SW-
failurescanonlybeinducedbyerasures,leadingto anoverall coding ensures that the gateway can recover all K observed
loss probability of εK. In this perspective, the availability of strings w , k = 1,2,...,K perfectly. In this setup, the gate-
k
multiple receivers triggers a dramatic improvement, enabling way should be able to recover every packet that was received
levels of reliability that would otherwise not be possible at any relay. However, neither is the gateway interested in
irrespective of the channel configuration. On the other hand, erasures symbols at the relays, i.e. whenever Wt =e for any
k
Eq. (20) turns out to be useful for system design, as it allows k, t, nor in reconstructing each relay sequence perfectly. The
to determine the load that can be supported on the uplink authors in [18] overcame this problem by assuming that the
channel while guaranteeing a target loss rate. Also in this decoderknowsallthe erasurepositionsof thewholenetwork.
casediversitycansignificantlyamelioratetheperformance.As This assumption applies in our case as packet numbers are
shown in Fig. 6, for example, a target loss rate ζ = 5·10−2 supposed to be known via a packet header. Let all erasure
is achievedby a three-andfour-receiverscheme under6- and positions be represented by Γ.
10-foldlargerloadscomparedtotheK =2case,respectively. The rates (R ,...,R ) are achievable [17] if
1 K
IV. DOWNLINKSTRATEGIES Rk ≥H(WS|WS,Γ), ∀ S ⊆[1,2,...,K] (21)
The analysis carried out in Section III has characterized kX∈S
the average number of packets that can be decoded at the where W = (Wt ,Wt ,...,Wt ), denotes the observa-
S k1 k2 k|S|
relay set when SA is used in the uplink. We now instead tions at some time t at the subset of receivers specified by
consider the complementary task of delivering what has been S = {k ,k ,...,k }. Γ has the effect of removing the
1 2 |S|
influence of the erasure symbols on the conditionalentropies. The gateway collects all incoming packets and obtains the
Computingthe entropieshoweverrequiresthe full probability system of linear equations
distribution P which is a difficult task in general. By
W1...WK
different means, we can obtain the equivalent conditions: cT1 G1 0 ... 0 wT1
Proposition 2: The rates (R )K have to satisfy cT2   0 G2 ... 0 wT2 
k k=1  ... = 0 0 ... 0  ...  (25)
    
K−|S| cT   0 0 ... G wT 
R ≥T + (−1)k K−|S| ρ(1−ε)ke−ρ(1−εk),  K   K  K 
k up,K
kX∈S kX=1 (cid:18) k (cid:19) cT G wT
| {z } | {z }| {z }
∀S ⊆{1,...,K}
where G∈FnPkRk×nK. Note that some elements of w can
(22) 2L
beidenticalbecausetheywerereceivedbymorethanonerelay
Proof: Consider a subset of relays S ⊆{1,...,K} and
and thus are elements of some w , w ,.... One can merge
their buffer contents An after n time slots. In order to k1 k2
k∈S k these entries in w that appear more than once. Additionally,
satisfy successful recoSvery at the gateway, at least all packets wedropallerasure-symbolsinwanddeletethecorresponding
that have been collected only by nodes in the set S and not
columns in G to obtain the reduced system of equations
byanyoneelse haveto becommunicatedtothegateway.That
is, cT =G˜w˜T (26)
n·Rk ≥(cid:12)(cid:12) Ank\ Ank(cid:12)(cid:12), (23) nwoheerreaswu˜re∈sFym2|ALbno|lsc.oCntlaeianrslyo,nthlyerdeisatrienct receiAvend pealecmkeetnstsanind
kX∈S (cid:12)(cid:12)(cid:12)k[∈S k[∈S (cid:12)(cid:12)(cid:12) w˜. (cid:12)(cid:12)Sk∈S k(cid:12)(cid:12)
(cid:12) (cid:12) Wepartitiontheentriesinw˜into2K−1vectorsw˜ foreach
with S ={1,...,K}\S. Note that S
nonemptysubsetS ⊆{1,2,...,K}:Eachvectorw˜ contains
S
allpacketsthathavebeenreceivedonlybyallrelaysspecified
An\ An =An\ An,
k k k by S and not by anyone else. That is, w˜ corresponds to the
k[∈S k[∈S k[∈S set Pn = An\ An, its lengthSis |Pn|.
S k∈S k k∈S k S
The colTumns in G˜Sand rows in w˜T can be permuted such
so by the inclusion-exclusionprinciple and due to |S|=K−
that one can write
|S|
cT =G˜ w˜ := G˜ ·w˜ , ∀ k =1,...,K.
K−|S| k k k,S S
(cid:12) An\ An(cid:12)=|An|+ (−1)k K−|S| an S⊆{1X,2,...,K}
(cid:12) k k(cid:12) K (cid:18) k (cid:19) k (27)
(cid:12)(cid:12)k[∈S k[∈S (cid:12)(cid:12) kX=1
(cid:12) (cid:12) (24)
(cid:12) (cid:12) Each of the matrices G˜ ∈ FnRk×|PSn| contains only ele-
k,S 2L
with |ISn| = ank for k = |S| as before. By plugging in the ments from F∗2L if k ∈S and is an all-zero matrix otherwise.
value for limn→∞ annk, the proposition follows. Abotctoommpoafctnreexptrpeasegnet.ation for K = 3 is shown in (28) at the
The variables that are involved only in n R equa-
k∈S k
B. Random Linear Coding tions are those in w˜L, L ⊆ S, for eacPh subset S ⊆
{1,2,...,K}.Fordecoding,thenumberofequationshastobe
By meansofProposition2,wehavederiveda characteriza- largerorequaltothenumberofvariables,soanecessarycon-
tion of the rates that have to be assigned to relays in order to ditionfordecodingis thatn R ≥ |Pn|. This is
k∈S k L⊆S L
deliver the whole set of collected packets to the GW. In this satisfied by (23), since P|Pn|= P An\ An ,
L⊆S L k∈S k k∈S k
section, we complete the discussion by proposing a strategy as we show in AppendiPx II (cid:12)S S (cid:12)
(cid:12) (cid:12)
that is capable of matching such conditions, thus achieving A sufficient condition is that the matrix G˜ , k ∈ S
k
optimality.Thesolutionthatweemployisbasedonastraight- representing n R equations has rank |Pn| for
k∈S k L⊆S L
forward application of the well-known random linear coding each subset S P⊆ {1,2,...,K}. Denote the sPet of indices of
schemein [19],andwillthereforeonlybrieflysketchedin the nonzero columns of matrix G˜ as the support of G˜ . Note
k k
following. that a row of matrix G˜ has a different supportthan a row of
k
Each relay k generates a matrix G ∈FnRk×n and obtains matrixG˜ ,fork 6=l.Theserowsarethuslinearlyindependent.
k 2L l
the data part of its nR transmit packets by cT = G wT. It thus suffices to check that all rows of matrix G˜ are
k k k k k
Whenever an element of w was an erasure symbol, the linearly independent. As all nonzero elements are randomly
k
corresponding column of G is an all-zero column. Erasure drawn from F∗ , the probability of linear dependence goes
k 2L
symbolsthushavenocontributionstothetransmitpacketsc . to zero as L grows large, completing our proof, and showing
k
AllotherelementsofG aredrawnuniformlyatrandomfrom that the presented forwarding scheme achieves the bounds of
k
F∗ , where F∗ denotes the multiplicative group of F . Proposition 2.
2L 2L 2L
V. CONCLUSIONS At the gateway side, successive interference cancellation
(SIC) is applied to the collected signals. This approachoffers
In this paper, a simple and practical extension of Slotted
an improvementwheneverthe waveformforwardedby one of
Aloha in the presence of multiple receivers, or relays, has
the relays allows decodingof a packet, say x, while the other
been presented and thoroughly discussed. By means of an
reports a collision given by the superposition of x and one
analytical framework, closed-form expression for the uplink
otherpacket.In thiscondition,the setof relayswouldbe able
throughput (defined as the average number of packets per
to collect only one information unit, whereas, with A&F, the
slot collected by the set of K relays), as well as for the
gateway can subtract the interference contribution of x from
probability that a data unit is not retrieved by any of the
the collision-corrupted waveform and successfully collect the
receivers, have been derived for an arbitrary value of K
second packet as well. Details on the accuracy of this model
undertheassumptionofon-offfading.Remarkablegainshave
onnoisychannelswithactualsignalprocessingtechniquescan
been shown and discussed already for a moderate number
be found in [1], [4].
of receivers. The study is complemented by considering the
ThegainofferedbySICcanthusbecomputedfortheK =
problemofdeliveringthesetofcollectedpacketstoacommon
2 case by simply adding to the uplink throughput derived in
gateway without allowing any information exchange among
Section III-A one additional collected data unit each time the
receivers.Theoreticalboundsforthe amountof resourcesthat
described collision condition is met. Hence, we can write:
have to be allocated to achieve this task have been derived,
and a simple scheme based on random linear network coding ∞ ρue−ρ
G = 2u(u−1)εu−1(1−ε)·εu−2(1−ε)2
has been shown to match such bounds. SIC u!
uX=2
=2ρ2ε(1−ε)3e−ρ(1−ε2), (29)
APPENDIX I
AMPLIFY AND FORWARD WITH SICATTHEGATEWAY where, within the summation, εu−1(1−ε) accounts for the
correct reception at one relay while εu−2(1−ε)2 enforces a
The framework developed in this paper has focused on
collision of exactly two packets at the other relay, for a total
characterizing the performance achievable by a SA system
of 2u(u−1) configurationsthat can be solved with SIC. The
with receiver diversity when a D&F scheme is implemented
average number of collected packets at the GW per uplink
at intermediate nodes. On the other hand, restricting relays
slot, which we refer to as T , is thus simply expressed as
to simply forward what they have successfully retrieved in A&F
T =T +G :
the uplink prevents the GW from performing joint decoding A&F up,2 SIC
on possibly uncorrelated signals. In order to go beyond this T =2ρ(1−ε)e−ρ(1−ε)−ρ(1−ε)2e−ρ(1−ε2)
A&F
limitation, we consider in this appendix the possibility for
+2ρ2ε(1−ε)3e−ρ(1−ε2). (30)
receiverstosendinthedownlink,onaslot-basis,anamplified
version of the analog waveform they perceive even in the The obtained trend is plotted in Fig. 7 against the channel
presence of a collision, following an amplify and forward load ρ for an erasure probability of ε = 0.2, showing a
(A&F) approach. For the sake of mathematical tractability, 20% and 66% improvement in peak throughput compared to
wefocusontheK =2case,andwemodelrelaystoinstantly the performance achieved under the same conditions in the
and reliably deliver information to the GW. The advantage of uplink (i.e., without SIC) and by a SA scheme with single
such an assumptionis twofold.On the one hand,it will allow receiver,respectively.As discussed in Section III-A, a closed-
us once more to identify elegant closed-form expressions for form evaluation of the maximum throughput T∗ is not
A&F
the throughputof A&F with spatial diversity,highlightingthe straightforward, due to the transcendental nature of the terms
fundamental tradeoffs that arise in the presence of multiple thatdefinethemetric.Nevertheless,inthetwo-receivercase,a
receivers. On the other hand, despite its ideality, the model goodapproximationisonceagainofferedbyevaluatingT
A&F
under consideration is representative for several scenarios of at ρ=1/(1−ε), obtaining, after some calculations:
practical interest, in which the bandwidth available in the
2
downlink is much larger than the one of the uplink. Satellite T∗ (ε)≃ −e−1−ε(1−3ε+2ε2), (31)
networks, as well as topologies where multiple base stations A&F e
or access points are connected to a coordinating unit via a wherealossfactorofe−1−ε(1−ε)(1−2ε)≤e−1−ε(1−ε)is
wideband backbone may be examples in this direction. exhibited with respect to the upper bound provided by twice
w˜T
{1}
w˜T 
{2}
cT G˜ G˜ 0 0 G˜ G˜ 0 G˜ w˜T
1 1 1,{1} 1,{1,2} 1,{1,3} 1,{1,2,3}  {3} 
cT2 =G˜2w˜T = 0 G˜2,{2} 0 G˜2,{1,2} 0 G˜2,{2,3} G˜2,{1,2,3}w˜T{1,2}  (28)
cT3  G˜3  0 0 G˜3,{3} 0 G˜3,{1,3} G˜3,{2,3} G˜3,{1,2,3}ww˜˜T{T1,3} 
 {2,3} 
w˜T 
 {1,2,3}
subset L ⊆ S. Second, we show that this subset is unique.

K = 2, w/ SIC Let again L be the subset of largest cardinality such that b∈
K = 2, w/o SIC
0.6 SA Bl, ∀l ∈ L and choose a different subset V ⊆ S, V =6 L.
Then, either b 6∈ B or b ∈ B . The element b
l∈L l s∈S\L s
0.5 is thus only incluTded in l∈LBl\Ss∈S\LBs.
0.4 By choosing Bk = AnkT\ k∈S¯ASnk, the result follows by
up elementary set operations. S
T
0.3
ACKNOWLEDGEMENT
0.2
The authors would like to thank Prof. Gerhard Kramer for
0.1
the fruitfuland insightfuldiscussions throughoutthe develop-
ment of the present work.
0
0 1 2 3 4 5 6 7
ρ
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