On the stability of asynchronous Random Access Schemes

0.53 MB·English

by  Alessio Meloni

On the Stability of Asynchronous Random Access
Schemes
Alessio Meloni and Maurizio Murroni
DIEE - Department of Electrical and Electronic Engineering
University of Cagliari
Piazza D’Armi, 09123 Cagliari, Italy
Email: {alessio.meloni}{murroni}@diee.unica.it
Abstract—Slotted Aloha-based Random Access (RA) tech- the same packet. It has been demonstrated that the diversity
5 niques have recently regained attention in light of the use of created by multiple transmissions is beneficial in case of
1 Interference Cancellation (IC) as a mean to exploit diversity
smallchannelloadwhileachievesworseresultsfrommoderate
0 created through the transmission of multiple burst copies per
channel loads on. The idea behind the use of IC in DSA is
2 packet content (CRDSA). Subsequently, the same concept has
been extended to pure ALOHA-based techniques in order to to further exploit the advantage of sending multiple copies by
n
boosttheperformancealsoincaseofasynchronousRAschemes. tryingtorestorealsothecontentofcollidingpackets.Thisnew
a
J Inthispaper,throughputaswellaspacketdelayandrelatedsta- access scheme is known as Contention Resolution Diversity
8 bility for asynchronous ALOHA techniques under geometrically SlottedAloha(CRDSA)[4]andworksasfollows.InCRDSA,
distributedretransmissionsareanalyzedbothincaseoffiniteand
2 terminals transmit packets in a given frame (composed of a
infinite population size. Moreover, a comparison between pure
ALOHA, its evolution (known as CRA) and CRDSA techniques certainnumberofslots)byplacingtwopacket’sburstcopiesin
]
T is presented, in order to give a measure of the achievable gain tworandomlychosenslotlocations.Eachburstcopycontainsa
thatcanbereachedinaclosed-loopscenariowithrespecttothe pointeridentifyingtheslotpositionofitstwin.Atthereceiver,
I
. previous state of the art. if at least one burst copy of a given packet can be correctly
s
c decoded, its signal content is removed from all other involved
Index Terms—Random Access, Aloha, Interference Cancella-
[
tion,CRDSA,CRA,Stability,EquilibriumContour,PacketDelay slotsthankstoIC.Byiterativelyrepeatingthisprocedure,itis

1 possibletorestorethecontentofthosepacketsthathadalltheir
v burst copies interfering, if at least one burst copy interfered
2 I. INTRODUCTION with bursts belonging to correctly decoded packets that are
7
thus eligible for IC. As a result, while original SA technique
0 Despite its almost 40 years-long life since the original idea
reachesapeakthroughputT (cid:39)0.37[pkt/slot],CRDSAboosts
7 was published [1], Aloha and its successive evolutions such
0 as Slotted Aloha (SA) [2] and Diversity Slotted Aloha (DSA) the performance up to T (cid:39)0.55[pkt/slot].
. Further studies have regarded the optimization of the num-
1 [3]havebeenalwaysusedinmanyrandomaccessapplication
0 scenarios (such as initial terminal login in satellite commu- ber of copies per packet (namely burst degree) to be sent.
5 In particular [5] and [6] deal with the use of more than two
nications), especially in case of long propagation delay and
1 copies per packet, demonstrating by means of simulation that
directive transmissions that do not allow carrier sensing and
:
v collision avoidance as for example in 802.11 DCF. Basically the results can be beneficial in terms of maximum achievable
i throughput and/or in terms of Packet Loss Ratio depending
X all Aloha-based techniques have in common the capability to
on the chosen burst degree. For example, the use of 3 copies
allow transmissions from a number of terminals in a multi-
r
a access channel without the need of coordination among them, per packet yields to a throughput peak T (cid:39) 0.68[pkt/slot]
while using 5 copies can lower the Packet Loss Ratio down
eventhoughthismeansthatthepossibilityofcollisionbetween
to PLR =1·10−6 for load values up to G=0.6[pkt/slot].
two or more different packets is present. Recently, these
Afterwards, the same idea has been extended to the case of
techniques and in particular synchronous access schemes (i.e.
Irregular Burst Degree, known as Irregular Repetition Slotted
thoseinwhichthechannelisdividedintoslots)havereceived
Aloha (IRSA) [7] and renamed in the DVB-RCS2 Lower
new interest in light of a breakthrough idea that consists in
Layer Satellite Specification [8] as Variable Rate - Contention
introducing Interference Cancellation (IC) in DSA schemes.
ResolutionDiversitySlottedAloha(VR-CRDSA).Inthiscase
Differently from SA in which packets are sent just once (or
thenumberofcopiespereachpacketischosenaccordinglyto
once per communication feedback in case of a system using
a given burst degree probability distribution that is optimized
retransmissions), in DSA multiple burst copies are sent for
via differential evolution, allowing to reach throughput values
A.MelonigratefullyacknowledgesSardiniaRegionalGovernmentforthe uptoT (cid:39)0.8[pkt/slot]inpracticalimplementations.Lastbut
financialsupportofhisPhDscholarship(P.O.R.SardegnaF.S.E.2007-2013 notleast,aswiththebirthofSlottedAlohatechniquesacertain
-AxisIVHumanResources,Objectivel.3,LineofActivityl.3.1.).
interestontherelatedstabilityincaseofretransmissionscame
(cid:13)c2013 IEEE. The IEEE copyright notice applies. DOI:
10.1109/IWCMC.2013.6583667 out [9] [10], also the birth of CRDSA has given place to
someworksthatanalyzeitsstabilityincaseofretransmissions entire loss of the packet’s burst; the second case considers
and compare its performance with the one for SA [11] [12]. the application of a strong FEC able to allow decoding if the
CRDSA has currently been introduced as option for Random amount of interference is limited. In any case, similarly to
Access communication in DVB-RCS2 [8] and its use has what happens in CRDSA, an iterative IC process is started at
beendiscussedinaquasi-real-timesatellitemobilemessaging the receiver in order to remove bursts belonging to correctly
systems [5]. decoded packets thanks to the knowledge of their location
In a recent paper the same concept behind CRDSA has within the frame from the correctly decoded burst.
been applied to Pure Aloha giving birth to a technique called Consider the assumption of ideal channel estimation and
Contention Resolution Aloha (CRA) [13]. For this reason, perfect Interference Cancellation. In the first case (i.e. where
in this paper the analysis in terms of stability that has been no FEC is used), at each iteration packet bursts are attempted
carriedoutforCRDSAisextendedtothecaseofasynchronous to be decoded only if the burst is not overlapping with other
RA schemes. This analysis uses the same tools adopted for bursts.InFigure1,Packet 1hasacopythatdidnotinterfere
synchronousaccessschemes,withthenecessarymodifications during transmission, therefore it can be decoded and the
needed in order to take into account the differences between interferenceoftheotherburstcopycanberemovedinorderto
the two techniques. The paper is organized as follows. In recover the content of Packet 2. In the second case in which
Section II an overview of the considered asynchronous access a strong FEC is applied, not only bursts without interference,
scheme isgiven. SectionIII presentsthe definition ofstability butalsothosesatisfyingacertainthresholdintermsofamount
as well as the model used for the measure of the stability of interference power are decoded. Let us define as in the
whenusingretransmissions.SectionIVdealswithamodelfor original CRA paper [13] the rate R = R · log M where
C 2
the computation of the delay associated to received packets. R isthecodingrateandM themodulationindex.Moreover
C
Finally, in Section V a comparison both in terms of stability the normalized MAC channel load is defined as G = Ntx·τ
TF
and in terms of delay is carried out between CRDSA, CRA with N indicating the number of transmitted packets, while
tx
and pure ALOHA. Section VI concludes the paper. T(G) = G[1−PLR(G)] represents the throughput in terms
of portion of load successfully decoded. Notice that PLR (i.e.
II. SYSTEMOVERVIEW the Packet Loss Ratio) depends on the frame size T , the
F
Thescenarioconsideredinthispaperisamulti-accesschan- burst degree distribution (defined from [7] as the probability
nel for satellite communications, in which a certain number of having a certain number of copies per packet through
ofterminalscommunicatetoagateway(e.g.agroundstation) the following polynomial Λ(x) = (cid:80)dΛdxd, where Λd is
via satellite. Differently from synchronous access schemes, in the probability that a given packet will have burst degree
this case the channel is divided into frames but each frame d), the rate R, the maximum number of iterations for the
is not subdivided into slots. When a terminal has a packet decoding process Imax and the SNIR. In [13] the decoding
to transmit, it waits for the beginning of the next available threshold has been approximated with the Shannon bound.
frame in order to place d copies of that packet. Let us call As claimed by the authors, even though this threshold is
t the beginning of a frame, T the frame duration and τ a quite optimistic, it can be considered valid for moderate to
0 F
genericburstduration.Thedcopiesofapacketareplacedwith high SNIR with properly designed schemes and represent the
startingtimewithintheinterval[t ;t +T −τ],withuniformly ground base for the study of the performance with real codes.
0 0 F
distributedprobabilityandsothatburstcopiesbelongingtothe Setting C =R=log2(1+SNIR), the decoding threshold is
same packet are not overlapping. SNIRdec,dB =10·log10(2R−1). In order for a burst to be
decoded, its SNIR must be at least equal to SNIR . The
dec
SNIR of each burst can be computed as
Pkt 1
Pkt 2 P SNR
Pkt 3 SNIR= = (1)
x·P +N x·SNR+1
Pkt 4 t
where x represents the degree of interference for a certain
t T
0 F burst as a sum over all interference contributions expressed
Figure 1: Example of a generic frame at the receiver for CRA. In with a value between 0 and 1. For example, in case of no
lightcolor(green)portionsoftheburstnotoverlapping.Indarkcolor interference x = 0, in case of 50% overlapping with another
(blue) portions of the burst overlapping with other bursts burst x = 0.5 and in case of 50% interference with n other
bursts,x=0.5·n.InFigure2resultsforanopenloopscenario
At the receiver, each frame contains a certain number of (i.e. without retransmission of lost contents) are illustrated for
bursts as depicted in Figure 1. Any burst will have or not a ALOHA and the representative case of CRA with Λ(x)=x2.
certaindegreeofinterferenceduetotransmissiontimeoverlap Theparametervalueschosenforsimulationsarethesamethat
with other bursts. Notice that differently from CRDSA, in will be used throughout the paper: T =100 ms, τ =1 ms
F
which interference can only occur for the whole burst, in this for every packet, M =4 (QPSK), R =1/2, I =50 and
C max
case also partial interference can occur. In [13] two cases are a number of simulation rounds per channel load point equal
analyzed: the first case assumes that any overlap results in to 104.
istheexpectedchannelloadinframej duetoretransmissions
1
ALOHA, Λ(x)=x, NO FEC and
0.9 CRA, Λ(x)=x2, NO FEC Gj = (NTOT −NB(j−1))·τ ·p0 (3)
ALOHA, Λ(x)=x, SNR=2dB T T
0.8 CRA, Λ(x)=x2, SNR=2dB F
0.7 ALOHA, Λ(x)=x, SNR=10dB is the expected channel load of frame j due to new transmis-
CRA, Λ(x)=x2, SNR=10dB sions. Finally Gj = Gj +Gj is the expected total channel
0.6 T B
ut load of frame j.
p
h
ug0.5 The aim of the definitions above is to find the equilibrium
o
Thr0.4 contour in a plane having as axis the number of backlogged
users and the channel load due to new transmissions. As a
0.3
matter of fact, equilibrium contour is defined as the locus
0.2 of points for which the expected channel load due to new
transmissions is equal to the expected throughput, so that
0.1
the communication can be considered as stable and the total
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 expected channel load Gj is equal frame after frame. The
Normalized channel load
expected number of new transmissions at the equilibrium can
Figure 2: Open loop throughput results be defined as
G =T(G)=G [1−PLR(G)] (4)
T
III. STABILITY Instabilityconditions,alsothenumberofbackloggedusers
AssumethecaseinwhichCRAhasbeenchosenasrandom remains the same frame after frame. Therefore
access scheme and a certain number of users N (either finite G·T
u N =N (1−p )+ FPLR(G) (5)
or infinite) participate in the described scenario. We assume B B r τ
each user to be in one of two possible states: Thinking (T) or
from which
Backlogged (B). N = G·PLR(G)·TF (6)
B τ ·p
r
(1-p0)+p0∙(1-PLR) PLR∙p0 (1-pr)+(PLR∙pr) Equations (4) and (6) describe the equilibrium contour. This
contour, together with the expected channel load due to new
transmissionsinEquation3(knownaschannelloadline)gives
T B
a model for the computation of the stability.
Consider the examples in Figure 4. Each channel load line
can intersect the equilibrium contour in one or more points
(1-PLR)∙p
r
(i.e. for one or more N values). These intersections are
B
Figure 3: Markov Chain for user state referred to as equilibrium points since GOUT = GT holds.
The rest of the points of the channel load line belong to
Users in T state send a packet at the beginning of the next one of two sets: those on the left part of the plane with
frame with probability p . Assuming that users are acknowl- regard to the equilibrium contour represent points for which
0
edged about the outcome of the transmission at the end of GOUT > GT, thus situations that yield to decrease of the
the frame in which they transmitted, if the packet is correctly backlogged population; those on the right part of the plane
decoded the user stays in T state. Therefore the probability with regard to the equilibrium contour represent points for
of staying in Thinking state is equal to the probability that a which GOUT < GT, thus situations that yield to growth of
user does not send any packet plus the probability that a user the backlogged population.
sends a packet that is correctly received at the first attempt. Therefore, an intersection point is defined as a stable
Ontheotherhand,ifauserisunsuccessfulinitsfirstattempt, equilibrium point with coordinates (GST,NBS) if it enters the
it switches to backlogged state. In B state, a user attempts left part of the plane for increasing NB since for NB < NBS
retransmission with probability pr. In case the retransmission the result is that GOUT < GT and for NB > NBS we find
ends up successfully the user comes back to Thinking state at that GOUT >GT so that the equilibrium point acts as a sink.
the end of the frame in which it retransmitted its packet while With the same reasoning, an intersection point is defined as
in case of no retransmission or unsuccessful retransmission, it an unstable equilibrium point with coordinates (GUT,NBU) if it
stays in B state. enterstherightpartoftheplaneforincreasingNB.Inthiscase
Let us define Nj as the number of backlogged packets at it can be proven that as soon as a statistical fluctuation from
B
the end of frame j and N as the total number of users, the equilibrium point occurs, the number of backlogged users
TOT
so that NB divergesfrom(GUT,NBU).Asamatteroffact,asexplained
N(j−1)·τ ·p in [11], the model is based on the expected behavior while in
Gj = B r (2)
B T realitytheobtainedvaluesoscillatearoundtheexpectedvalue.
F
number of users is valid and we can assume that the point of
saturation is found for N →∞. Notice that in this case the
B
formula for the channel load used so far is no longer valid.
Channel
saturation However, if Poisson arrivals with expected value λ in terms
oCphearantninegl point oCphearantninegl of new packets to transmit are considered, the channel load
NB Channel load linepoint NB Channel load line point line can be expressed asGT = λT·Fτ (7)
G GT As a matter of fact, for a finite number of users the number
T
of new transmissions is binomially distributed, while for a
(a)Stablechannel (b)Unstablechannel(finiteM)
number of users that goes to infinity we can consider the
binomial distribution converging towards the poissonian one.
Finally Figure 4d shows another example of globally stable
e
n
d li Channel equilibrium point. However, in this case the intersection point
a saturation
el lo point occurs for throughput close to zero. Therefore the point
NB oCphearantninegl Chann NB Channel load line icsondseidfienreeIddV.oavsPeArclChoKaandEneTedlD. EsaLtAuYratIiNonSTpAoBinLtECanHdANthNeELcShannel is
point
G G Assumingastablechannelsothatonlyagloballystableand
T T
operating point is present, it is of interest to know the delay
(c)Unstablechannel(infiniteM) (d)Overloadedchannel
associated to successfully transmitted packets. For a generic
Figure 4: Examples of stable and unstable channels packet, it is possible to do so using the discrete-time Markov
chain in Figure 3. T is assumed to be our discrete time
F
unit. Therefore, the packet delay D can be calculated as
pkt
the number of frames that elapse from the beginning of the
In Figure 4a an example of stable equilibrium point is frameinwhichthepacketwastransmittedforthefirsttime,till
given. Being this point the only one of intersection it is also a theendoftheoneinwhichthepacketwascorrectlyreceived.
globally stable equilibrium point and we consider the related
channelasstablesincethecommunicationwillkeepoperating

indefinitely around that point. On the other hand, if the point 1−PLR, for f =1
of equilibrium is not the only one as in Figure 4b it takes 
Pr{D =f}=
the name of locally stable equilibrium point. In particular pkt
the illustrated example shows two locally stable equilibrium P·[1L−Rp[pr+(P1L−RPpLR]f)−]·2 , for f >1
points: one for a good throughput value, thus called channel r r
(8)
operating point in the sense that is the point in which we
Based on Equation 8 the expected packet delay can be
want the communication to operate; one for throughput close
written as
to zero called channel saturation point since in that state
too many users are in backlogged state and thus any packet
∞
transmission has hard times in being successful. Consider a Av[D ]=(cid:88)f ·Pr{D =f} (9)
pkt pkt
scenario in which the communication starts from N = 0.
B f=1
Thecommunicationwillkeepbeingaroundtheoperatingpoint
as long as statistical fluctuations are small enough to keep Equation 9 can also be rewritten in a form that is more
N < NU. At a certain point however, the instability point practical for our analysis, by means of Little’s Theorem. As a
B B
will be crossed and in small time the saturation point will be matter of fact, in a stable system the average number of users
reached. Depending on the communication settings, there is inBstateisequaltotheaveragetimespentinbackloggedstate
also a certain probability to exit from the state of saturation multipliedbythearrivalrateofnewpacketsG (thatweknow
T
and come back around the channel operating point. However, to be equal to G at the operational point). Therefore
OUT
this probability is generally small and considered negligible.
Figure 4c represents the same scenario as in Figure 4b (i.e. NO·τ
Av[D ]= B (10)
the case of unstable channel) but for an infinite number of pkt GO ·T
OUT F
users. In this case the channel load due to new transmissions
is independent on the actual number of backlogged users. where the presence of T /τ in the formula has the aim of
F
Nevertheless the same discussion as for the case with finite normalizing the delay to the frame unit.
V. COMPARISONOFRANDOMACCESSTECHNIQUES
400
Beforestartingtheanalysisoftheresults,itisusefultohave ALOHA Λ(x)=x
amoresolidcomprehensionoftheroleofthreekeyparameters 350 CRA Λ(x)=x2
forthecommunication:theprobabilityofnewtransmissionp , CRA Λ(x)=x3
0
the probability of retransmission p and the total population 300 SA Λ(x)=x
r CRDSA Λ(x)=x2
N .Thefirsttwoparametershaveinfluenceonthechannel
TOT 250 CRDSA Λ(x)=x3
load line while the retransmission probability influences the
shape of the equilibrium contour. In particular, defining a
NB200
generic line y = m·x+q with x = G and y = N , p is
T B 0
inversely proportional to m. Therefore, fixing q, a decrement 150
ofp determinesachangefortheslopeofthechannelloadline
0
thatbecomessteeperwhileanincrementofp hastheopposite 100
0
effect on the slope. N has the same graphical meaning
TOT
50
of q. In other words, fixing p (i.e. the line slope) changing
0
N corresponds to changing the point of intersection with
TOT 0
the y-axis since for G =0, N =N . Finally, as shown 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
T B TOT G
in Figure 5c, a decrement of the retransmission probability T
determines a shift upwards of the equilibrium contour. Figure 6: EquilibriumcontourforpureALOHAandCRAwhenno
FEC is used.
p− total number of users must not be bigger than NTOT (cid:39) 60;
r
on the other hand we can see that for Pure Aloha, almost
NB p0+ NB NT+OT NB 400 users can take part in the communication still ensuring
a channel operating point around the throughput peak. If the
p0− N− p+ design constraints require the use of CRA together with a
TOT r bigger number of users, we know from Figure 5c that it is
possible to decrease the retransmission probability for back-
G G G
T T T loggedusersp .Neverthelessthestabilitycomesatthecostof
r
increased packet delay. This can be qualitatively understood
(a)Changingp0 (b)ChangingM (c)Changingpr
considering that decreasing p , the peak throughput remains
r
Figure 5: Graphical representation of the result for increments and thesamewhilethecorrespondingnumberofbackloggedusers
decrements of p , M and p
0 r N increases. Therefore from Little’s theorem an increase
B
in the average packet delay is expected. Finally, it can be
Figures 6-8 show results for the settings outlined in Sec-
seen that without the use of FEC the results of Aloha and
tion II. In the same figures, also the results for Slotted Aloha
CRA are worse than those for SA and CRDSA. As a matter
and CRDSA are reported, assuming the same settings and a
of fact the results for synchronous techniques give place to
comparable frame size of 100 slots, since TF = 100. Notice
τ equilibrium contour with identical shapes but bigger in value
that the aim of this section is to give a qualitative analysis
ofthroughputaswellasinwidthofthecurvebelowthepeak.
rather than precise numerical results. In fact, the obtained
Similar considerations can be done in Figure 7 for the case
results are based on the Shannon Bound while in practical
in which FEC is used and the SNR is quite low (2 dB). In
implementations a real code must be considered. Therefore a
fact, concerning asynchronous techniques the same reasoning
quantitativeanalysiswouldbeofunnoticeableimportance.On
as for the previous case applies. Moreover, concerning the
the other hand a qualitative analysis is still of big value since
comparison with synchronous techniques, we can see that SA
itcanprovethegeneralvalidityofthetechniqueandhighlight
and CRDSA still outperform asynchronous techniques even
pros and cons with regard to the state of the art.
though the performance of the two gets closer.
Figure 6 shows that even when no FEC is used, CRA can
reach higher values of throughput than Pure Aloha, if the FinallyforhighSNR(10dB)asinFigure8, asynchronous
communication is designed properly so that the channel is techniques outperform synchronous ones. In particular, it can
stable.However,throughputresultsarefarfromthoseobtained benoticedthatwhileforCRDSAtheburstdegreedistribution
for CRDSA. In addition, having a stable channel in CRA Λ(x) = x3 is always better than Λ(x) = x2, in CRA when
assumes that the total number of users participating is small the SNR is high enough Λ(x) = x2 appears to be the best
enough so that only one point of intersection is present. For solution.HoweveralsointhiscasePureAlohastillallowsthe
example, in the case of CRA with Λ(x) = x3, if we want participationinastablecommunicationofahighernumberof
an expected throughput close to the peak (i.e. T (cid:39) 0.3) the users N with regard to CRA.
TOT
this increment of packet delay would still allow asynchronous
400 ALOHA Λ(x)=x access schemes to be more efficient than Pure Aloha or not
350 CRA Λ(x)=x2 from a packet delay perspective. We want to remark that
CRA Λ(x)=x3 obtained results for CRA represent an upper bound, since the
300 SA Λ(x)=x Shannon Bound has been considered as decoding threshold
CRDSA Λ(x)=x2 for received bursts. This is the reason why in this paper the
250 CRDSA Λ(x)=x3 analysis has been accomplished in a qualitative and graphical
manner rather than comparing the various techniques and
NB200 burst degree distributions with numerical strictness. A very
recent work proposed in [14] and called ECRA (Enhanced
150
CRA) shows the possibility to outperform CRA in terms of
100 throughput and Packet Error Rate and sets the more realistic
Random Coding Bound as decoding threshold. While those
50 resultsstilldonotconstituteapracticalcaseusingarealcode,
they constitute an interesting step forward towards the case of
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 arealscenario.Thepresentedanalysiscanbeaswellextended
G
T tothisrecentevolutionandfutureworksshouldconsiderthese
latest findings rather than CRA.
Figure7:EquilibriumcontourforALOHAandCRAwithassociated
FEC with R =1/2 and SNR=2 dB
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showing that under the constraint of channel stability, despite
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theobtainedthroughputboostCRAsupportsasmallernumber forpossiblepublication,arXiv:1203.4693
of users than pure ALOHA and does not appear convenient [13] Kissling, C. ; , Performance enhancements for asynchronous random
access protocols over satellite, in Communications (ICC), 2011 IEEE
in low SNR scenarios with respect to synchronous access
InternationalConferenceon,june2011,pp.1-6.
schemes. As a matter of fact, designing CRA to support a [14] Clazzer, F.; Kissling, C.; ”Enhanced Contention Resolution Aloha -
bigger number of users requires a decrement of the retrans- ECRA”,9thInternationalITGConferenceonSystems,Communications
andCoding-SCC2013
mission probability that yields to an increase on the average
packet delay. Therefore further studies could investigate if