Random Access in DVB-RCS2: Design and Dynamic Control for Congestion Avoidance

1.4 MB·English

by  Alessio Meloni

1
Random Access in DVB-RCS2: Design and
Dynamic Control for Congestion Avoidance
Alessio Meloni, Student Member, IEEE, Maurizio Murroni, Senior Member, IEEE
Abstract—In the current DVB generation, satellite terminals around 0.36 [packets/slot] despite the possibility of collision
areexpectedtobeinteractiveandcapableoftransmissioninthe among bursts from different users. Nevertheless, in prac-
return channel with satisfying quality. Considering the bursty
tice SA works with very moderate channel load to ensure
natureoftheirtrafficandthelongpropagationdelay,theuseofa
acceptable delay and loss probability (PLR = 10−3 for
RandomAccesstechniqueisaviablesolutionforsuchaMedium
Access Control (MAC) scenario. In this paper Random Access 10−3[packets/slot]).
5 communication design in DVB-RCS2 is considered with partic- This gap has been recently filled with the introduction of
1 ular regard to the recently introduced Contention Resolution a new technique named CRDSA that is able to boost the
0 Diversity Slotted Aloha (CRDSA) technique. The paper presents
throughput even up to values close to 1[packets/slot]. This
2 a model for design and tackles some issues on performance
evaluation of the system by giving intuitive and effective tools. technique is based on the transmission of a chosen number
n
Moreover,dynamiccontrolproceduresabletoavoidcongestionat of replicas for each burst payload as in Diversity Slotted
a
J thegatewayareintroduced.Resultsshowtheadvantagesbrought Aloha(DSA)[6].However,differentlyfromDSA,inCRDSA
by CRDSA to DVB-RCS2 with regard to the previous state of each burst copy has a pointer to the location of the other
6
the art.
2 replicas of the same burst payload so that when the frame
IndexTerms—Retransmission,SuccessiveInterferenceCancel- arrives at the receiver a Successive Interference Cancellation

] lation, Slotted Aloha, Random Access, Congestion Avoidance, (SIC) process is accomplished. The SIC process consists in
T
Dynamic Control, Satellite Broadcasting, Interactive Terminals
removing the content of already decoded bursts from the slots
I
. in which collision occurred. Doing so, it is possible to restore
s
c the content of bursts that had all their replicas colliding thus
[ I. INTRODUCTION boosting the throughput. Nevertheless the use of interference

1 Current DVB standards such as Digital Video Broadcasting cancellationcanbenefitfrompowerunbalanceamongdifferent
v over Satellite are increasingly going towards the need of sources [7] [8].
1 high-speed bidirectional networks for consumer interactivity Considering these recent findings on Aloha-based RA and
6
[1]. Nevertheless in consumer type of interactive satellite the fact that in [2] contention control mechanisms to avoid
3
6 terminals, users generate a large amount of low duty cycle congestion are claimed to be out of scope for the document,
0 and bursty traffic with frequent periods of inactivity in the inthispaperacompletemodelforthedesignofacongestion-
. return link. Under these operating conditions, the traditionally free systemusing CRDSA is presented. To do so, we rely on
1
0 used Demand Assignment Multiple Access (DAMA) satellite the analysis we carried out in [9], in which the need for such
5 protocol does not perform optimally, since the response time a model was firstly introduced.
1 for the transmission of short bursts can be too long. For this The main contributions of this paper are: the presentation
v: reason, in the recently approved specification for the next ofastabilitymodelandofsomecrucialevaluationparameters
i generation of Interactive Satellite Systems (DVB-RCS2) [2], (packet delay and FET) developed in [10] for SA and here
X
the possibility of sending logon, control and even user traffic adapted to CRDSA; a thorough analysis of the design aspects
ar using Random Access (RA) in timeslots specified by the to take into account in such a system; the adaptation of
NetworkControlCenter(NCC)isprovidedtoReturnChannel the dynamic control limit policies developed for SA and the
Satellite Terminals (RCST). In particular, two methods are analysisoftheirconveniencewhencomparedtostaticCRDSA
considered for RA: the first one is Slotted Aloha (SA) [3] [4], design; the extension of all these results to the case of infinite
the second is called Contention Resolution Diversity Slotted population that is usually not considered even though in some
Aloha (CRDSA) [5]. scenarios is the most appropriate population representation.
SA represents a well established RA technique for satellite The remainder of this paper is organized as follows. In
networks in which users send their bursts within slots in a SectionIIabriefoverviewoftheCRDSAschemeisgivenand
distributedmanner,i.e.withoutanycentralentitycoordinating the congestion problem that arises when introducing retrans-
transmission. This allows to reach an average throughput missions in such a scheme is stated. Section III presents the
stability model used for analysis and design in the subsequent
A.MelonigratefullyacknowledgesSardiniaRegionalGovernmentforthe sectionsanddefinestheconceptofstability.SectionIVregards
financialsupportofhisPhDscholarship(P.O.R.SardegnaF.S.E.2007-2013
thedefinitionofaMarkovchainforpacketdelaycomputation
-AxisIVHumanResources,Objectivel.3,LineofActivityl.3.1.).
(cid:13)c2014 IEEE. The IEEE copyright notice applies. DOI: anditsvalidation.InSectionVthedefinitionofFirstExitTime
10.1109/TBC.2013.2293920 is given and the formulas for its calculation are presented;
The authors are with the Department of Electrical and Electronic Engi-
moreover the adaptation of the model for computation reduc-
neering(DIEE),UniversityofCagliari,PiazzaD’Armi,09123Cagliari,Italy
(e-mail:[email protected];[email protected]. tionandfortheextensiontothecaseofinfinitepopulationare
2
discussed. Section VI exploits the model and the evaluation
parameters introduced in the previous sections by giving a USER 1 1 1
general overview of how they are used in the design phase. USER 2 2 2
Section VII compares the advantages and disadvantages of USER 3 3 3
t
variousAloha-basedRAtechniquesandpacketdegreechoices. USER 4 4 4
FinallySectionVIIIintroducessomedynamiccontrolpolicies
already used in [11] for SA while Section IX and X discuss N
s
their application respectively to the case of finite and infinite
population. Section XI concludes the paper.
Figure 1. Example of frame at the receiver for CRDSA with 2 copies per
packet;plainslots indicate thatatransmissionoccurredfor thatuserinthat
slot.
II. CRDSASYSTEMOVERVIEW
Consider a multi-access channel populated by a total num-
ber of users M. Users are synchronized so that the channel is has been correctly received (see User 4), the contribution of
divided into slots and N consecutive slots are grouped in a the other copies of the same packet can be removed from the
s
frame.Theprobabilitythatatthebeginningofaframeanidle other slots thanks to the knowledge of their location provided
user will send a new packet is p . When a frame starts, users by the pointers contained in the decoded packet. This process
0
willing to transmit place d copies of the same packet over the might allow to restore the content of packets that had all their
N slots of that frame. The number of copies d can be either copiescolliding(seeUser2)anditerativelyotherpacketsmay
s
the same for each packet or not. The first case is known as be correctly decoded up to a point in which no more packets
Constant Replication CRDSA (CR-CRDSA); the second case can be restored (see User 1 and 3) or until the maximum
is known as Variable Replication CRDSA (VR-CRDSA). In number of iterations I for the SIC process is reached, if a
max
this second case the number of copies is randomly generated limit has been fixed.
using a pre-determined probability mass function [12]. If a packet has not been successfully decoded at the end of
Packetcopiesarenothingelsethatredundantreplicasexcept the SIC process and a retransmission request is sent by the
for the fact that each one contains a pointer to the location NCC, a feedback is created between the transmitter and the
of the others. These pointers are used in order to attempt receiverside.Thisfeedbackcouldcausethechanneltogettoa
restoring collided packets at the receiver by means of SIC, stateofcongestioninwhichtheloadistoobigbecauseoftoo
i.e. by removing the interfering content of already decoded many packets to retransmit and causes excessive packet loss
packets thanks to their location’s knowledge. Nevertheless, duetosimultaneousdestructivetransmissions.Forthisreason,
the physical structure of the burst copies used is equal to a careful design of the system based on a stability model is
the one used for dedicated access with respect to the burst of great interest for practical DVB-RCS applications.
construction,thewaveform,thetimeslotstructureandtheburst
reception.
III. STABILITYMODEL
Throughout the paper perfect SIC and ideal channel are
Consider the RA communication system presented in Sec-
assumed, which means that the receiver is always able to
tion II. Each user can be in one of two states: Thinking (T)
perfectly apply SIC without errors and no external sources of
orBacklogged(B).UsersinTstategenerateanewpacketfor
disturbance such as noise that might require SNR estimation
transmission with probability p over a frame interval; if so,
[13] are present. Therefore, the only cause of unsuccessful 0
decoding of packets is interference among them1. Given these no other packets are generated until successful transmission
for that packet has been acknowledged. Users in B state have
conditions,considerFigure1representingapracticalexample
unsuccessfullytransmittedtheirpacketandkeepattemptingto
of SIC. Each slot can be in one of three states:
retransmit it with probability p over a frame interval. In the
• no packet’s copies have been placed in a given slot, thus r
followings, we assume that users are acknowledged about the
the slot is empty;
success of their transmission at the end of the frame in which
• only1packet’scopyhasbeenplacedinagivenslot,thus
the packet has been transmitted (i.e. immediate feedback).
the packet is correctly decoded;
Neverthelessthisconstraintwillberelaxedinthelastsections.
• more than 1 packet’s copy has been placed in a given
slot, thus resulting in collision.
InDSA,ifallcopiesofagivenpacketcollidedthepacketis A. Equilibrium Contour
surelylost.InCRDSA,ifatleastonecopyofacertainpacket Defining
• Nf : backlogged users at the end of frame f
1For the case in which non-ideal channel estimation is considered, the B
raepapdreerciacbalneirmefpearcttoon[5th]ewChReDreSAitphearsforbmeeanncdeefmoromnsotrdaetreadtetShNatRt.hMeroereiosvnero, • GfB = NB(fN−s1)pr : expected channel load2 of frame f due
as suggested by one of the reviewers that we kindly thank, we specify here to users in B state
thatalsosituationssuchasoutagethataffectthenumberofbackloggedusers
are not be a problem as long as the channel is nominally stable. In fact, if
there is only one point of stability, the channel will rapidly set back to the 2Inthispaper,withchannelloadwerefertothelogicalchannelloadthat
operationalpointaftertheoutageperiod.Thissentenceisanticipatedherebut is the number of packets sent normalized over the frame size, regardless of
willbeclearerinthesubsequentsections. thenumberofcopiessentperpacket.
3
• GfT : expected channel load of frame f due to users in 500 500
T state
450
450
• GfIN =GfT +GfB : expected total channel load of frame 400 400 Channel
f 350 saturation
350 point
•• PlfGoraaLfOmdUReTsfuf(cG=cefIsNGsf,fIuNNlls(y1,dtr−,aInmPsmaLxiRt)te(:GdexfIinNpef,crNatemsd,edpf,aIc,mki.eaetx.l)to)hsrs:ourpagathirotpuootff. NB122350500000 Channel load lineoCphpearoanitnnitnelg NB122350500000 Channel load line oCphpearoanitnnitnelg
100 100
The equilibrium contour can be written as 50 50
0 0
0 0.2 0.4 0.6 0 0.2 0.4 0.6
G G
T T
G =G =G (1−PLR(G ,N ,d,I )) (1)
T OUT IN IN s max (a) Stablechannel (b) Unstablechannel(finiteM)
that represents the locus of points for which at any time the p0 =0.143 p0 =0.143
expected channel load due to users in T state is equal to the p =0.5 p =1
r r
expected throughput G . Notice that the frame number f
OUT M =350 M =350
has been omitted, since in equilibrium state this condition is
expected to hold for any frame. (GG,NG)=(0.49,8.2) (GS1,NS1)=(0.495,3.9)
T B T B
From the definition above we can also gather that the (GU,NU)=(0.44,42.7)
expectednumberofbackloggedusersremainsthesameframe T B
after frame. Therefore (GST2,NBS2)=(6·10−3,346)
G PLR(G ,N ,d,I )N
N = IN IN s max s (2)
B
p
r 500 500
Equations (1) and (2) completely describe the equilibrium 450 450
contour on the (GT,NB) plane for different values of GIN3. 400 400 Channel
350 d line 350 satpuoraintiton
Boplof0.aNedsCqoohultfiiainaclnireeb.nreTtaihlunhamlditosatptphdwroeooilvnienietqdsesu.esieIlnitnbhtfreiaaiulckmtpn,aowrcwaoemlnehtedoatgvueeerrsinotcofsoetntlchfsoetiinstuasiatcdentuetrahienledfiscnMthaitabenialnsinteeydlt NB112230505000000 oCphearantninelg Channel loa NB112230505000000 Channel load line
point
points for a certain scenario. 50 50
Consider M and p0 to be constant (i.e. stationary input). 00 0.2 0.4 0.6 00 0.2 0.4 0.6
G G
Given a certain number of backlogged packets, the channel T T
load line expresses the expected value of channel load due to (c) Unstablechannel(infiniteM) (d) Overloadedchannel
users in T state. For the finite population case, the channel λ=40 p =0.18
0
load line can be defined as p =0.5 p =1
r r
G = M −NBp (3) M →∞ M =350
T 0
N
s (GS1,NS1)=(0.4,1.7) (GS,NS)=(6·10−3,346)
T B T B
while for M → ∞ the channel input can be described as a
(GU,NU)=(0.4,107.32)
Poisson process with expected value λ [thinking users] [4] so T B
that (GS2,NS2)=(0.4,∞)
λ T B
G = (4)
T
Ns Figure2. ExamplesofstableandunstablechannelsforCRDSA(2replicas)
for any N (i.e. the expected channel input is constant and withNs=100andImax=20.Stableequilibriumpointsaremarkedwith
B ablackdot.
independent of the number of backlogged packets).
3Concerning the values used for the Packet Loss Ratio, it is known from
C. Definition of stability
theliterature[5]thattherelationbetweenPLR(GIN)andGIN cannotbe
tightly described in an analytical manner. For this reason PLR values used
Figure 2 shows the equilibrium contour and the channel
in this work are taken from simulations. However, once simulations for the
openloopcaseareaccomplished,theycanbeusedinthestabilitymodelto load line for various cases. The equilibrium contour divides
see how the performance changes when modifying some crucial parameters the (N ,G ) plane in two parts and each channel load line
B T
such as pr, p0 and M without the need of running brand new simulations. can have one or more intersections with the equilibrium
Moreoversimulatedvaluescanbesubstitutedwithtightanalyticalequations
incaseanywillbefoundinthefuture. contour. These intersections are referred to as equilibrium
4
points since GOUT = GT. The rest of the points of the IV. PACKETDELAYMODEL
channel load line belong to one of two sets: those on the
Once we are sure that the channel is working at its channel
left of the equilibrium contour represent points for which
operating point, we would like to know what is the expected
G > G , thus situations that yield to decrease of the
OUT T
distribution and average delay associated to packets that are
backlogged population; those on the right represent points for
successfully received at the gateway. This can be described
which G < G , thus situations that yield to growth of
OUT T
using a discrete-time Markov chain with the two states B and
the backlogged population.
T (Figure 3).
Therefore we can gather that an intersection point where
Theedgesemanatingfromthestatesrepresentthestatetran-
the channel load line enters the left part for increasing
sitionsoccurringtouserswhichdependonp andtheexpected
backlogged population corresponds to a stable equilibrium r
PLR at the channel operating point. If a packet transmitting
point. In particular, if the intersection is the only one, the
for the first time receives a positive acknowledgment the user
point is a globally stable equilibrium point (indicated as
stays in T state while in case of negative acknowledgment
GG,NG) while if more than one intersection is present, it
T B the user switches to B state until successful retransmission
is a locally stable equilibrium point (indicated as GS,NS).
T B has been accomplished. We assume a frame duration to be
If an intersection point enters the right part for increasing
the discrete time unit of this Markov chain. Therefore the
backloggedpopulation,itissaidtobeanunstableequilibrium
packet delay D is calculated as the number of frames that
point (indicated as GU,NU). pkt
T B elapse from the beginning of the frame in which the packet
was transmitted for the first time, till the end of the one in
which the packet was correctly received.
D. Channel state definition
Figure 2(a) shows a stable channel. The globally stable 1-PLR PLR (1-p)+(PLR p)
r r
equilibrium point can be referred as channel operating point
in the sense that we expect the channel to operate around that
T B
point. With the word around we mean that due to statistical
fluctuations, the actual G and G (and thus also G and
T B IN
G )maydifferfromtheexpectedvalues.Infact,theactual
OUT
(1-PLR)p
values have binomially distributed probability for GB and r
G with finite M and Poisson distributed probability for G
T T
with infinite M. Nevertheless, the assumption of taking the Figure3. MarkovChainforthePacketDelayanalysis.
expected value has been already validated in [9].
Figures 2(b) and 2(c) show two unstable channels respec- According to the definition given above, the delay distribu-
tively for finite and infinite number of users. Analyzing this tion is entirely described by
two figures for increasing number of backlogged packets, the
first equilibrium point is a stable equilibrium point. Therefore 
1−PLR, for f =1
the communication will tend to keep around it as for the 
stable equilibrium point in Figure 2(a) and we can refer
Pr{D =f}=
pkt
ttoo tihteoanbcoevaegmaeinntiaosnecdhasntanteisltiocpalerflauticntguaptiooinnst,. tHheowneuvmebr,erduoef P·[1L−Rp[pr+(P1L−RPpLR]f)−]·2 , for f >1
r r
backlogged users could pass the second intersection and start (5)
to monotonically increase. which results in the average delay
In the case of finite M, this increment goes on till a new
intersection point is reached, while in the case of infinite M ∞
(cid:88)
the expected number of backlogged users increases without Av[Dpkt]= f ·Pr{Dpkt =f} (6)
any bound. In the former case, this third intersection point is f=1
another stable equilibrium point known as channel saturation
Equation 6 can alternatively be rewritten in a simpler and
point, so called because it is a condition in which almost any
more intuitive form by means of Little’s Theorem [10] [15]
user is in B state and G approaches zero. In the latter
OUT considering that the average number of backlogged users in a
case, we can say that a channel saturation point is present for
stablesystemisequaltotheaveragetimespentinbacklogged
N →∞.
B state,multipliedbythearrivalrateofnewpacketsG (thatwe
T
FinallyFigure2(d)showsthecaseofanoverloadedchannel.
knowtobeequaltoG attheoperationalpoint).Therefore
OUT
In this case there is only one equilibrium point corresponding
to the channel saturation point. Therefore, even though the
N
channel is nominally stable, the point of stability occurs in Av[Dpkt]= G B·N (7)
OUT s
a non-desired region. For this reason this case is separated
anddistinguishedfromwhatisintendedinthisworkasstable where the presence of N in the formula has the aim of
s
channel. normalizing the delay to the frame unit.
5
pr Av[Dpkt]ana Av[Dpkt]sim PLR 500
p=1
r
1 543.48 541.89 0.99816 450 p=0.4
0.4 3.17 3.16 0.465 r
p=0.2
0.2 2.01 2.12 0.168 r
400
0.05 2.53 2.66 0.0713 pr=0.05
0.005 5.08 5.22 0.02 p=0.005
350 r
TableI
ANALYTICANDSIMULATEDAVERAGEPACKETDELAYATTHECHANNEL 300
OPERATINGPOINTFORCRDSAWITHNs=100slots,Imax=20,
M =350ANDp0=0.18. NB250
200
150
A. Model validation and retransmission probability’s effect
100
Figures4and5showsomeresultsbasedontheexampleof
overloaded channel given in Section III, in order to validate
50
themodelanddemonstratetheeffectofp onthepacketdelay.
r
Inparticularpr isprogressivelydecreasedinordertoevaluate 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7
how distribution and average packet delay change. Figure 4 GT
represents the channel load line and equilibrium curves for
several pr values. Figure4. EquilibriumcurvesforCRDSAwithNs =100slots,Imax =
When pr is decreased, the equilibrium contour moves up- 20,M =350andp0=0.18.
wards and the point of equilibrium is found for bigger values
of throughput while N decreases. Therefore, from Little’s
B 1
theorem, it is intuitive to expect that users will spend less p=0.005
pr=0.05
time in B state. If we keep decreasing p after the throughput r p=1
r p=0.2 r
peak is reached, the throughput starts to decrease while the 0.8 r
number of backlogged users increases again thus yielding to
p=0.4
an increase of the average packet delay. This is confirmed by 0.6 r
the results given in Table I.
Figure5showsthatwhilehavingasmallp isbeneficialfor
r 0.4
thePLR,italsomeansthatusersinBstatewillgenerallywait
longer before retransmitting a packet. As a consequence, the
smaller the p value the less steep the cumulative distribution 0.2
r
curve is, so that at a certain point curves with bigger p
r
overcome it in terms of probability to receive a packet before 0
100 101 102 103 104
a certain time deadline. Finally the simulated average packet D
pkt
delay associated to the examples in Figure 5 is reported in
Table I together with the expected results, demonstrating the
Figure 5. Packet delay cumulative distribution for CRDSA with Ns =
validity of the model. 100slots,Imax=20,M =350andp0=0.18.
V. FIRSTEXITTIME
In Section III the concept of unstable channel has been backlogged users NB = i. To calculate each pij value we
introduced. Beside knowing that a certain channel is unstable, need to know the probability that having a certain number of
it is also of big interest to quantify how much the channel is backlogged users j and a certain number of thinking users
unstable. The First Exit Time (FET) calculates the time spent M −j, respectively b backlogged users and t thinking users
around the operational point by an unstable channel before have transmitted and s=t−(i−j) out of t+b packets have
leaving the stability region. To calculate this parameter we been correctly received. Therefore
consideradiscrete-timeMarkovchainwhichusesthenumber
ofbackloggedusersN asdescribingstate,sinceequilibrium
B
points and more generally the state of the communication can M−j j (cid:18) (cid:19)
(cid:88) (cid:88) M −j
be uniquely identified with it. p = pt(1−p )M−j−t·
ij t 0 0
t=0 b=0 (8)
(cid:18) (cid:19)
j
A. Transition matrix definition · pb(1−p )j−b·q(s|t+b)
b r r
Let us define P as the transition matrix for this Markov
chain, with each element p denoting the probability to pass
ij
from a number of backlogged users N = j to a number of whereq(s|t+b)istheprobabilitythatspacketsarecorrectly
B
6
Figure 6. P matrix for CRDSA with Ns =100, Imax =20, M =350,
p0=0.143,pr =1.
Figure7. EvolutionofBf forthePmatrixinfigure6.
decoded if t+b packets are transmitted4. Let us then call Q
eventhoughforthetypicalsettingsusedinthesescenariosthe
the matrix with elements q corresponding to probability
s,t+b
last case is really rare. Nevertheless this highlights one of the
q(s|t+b).
advantages of calculating the P matrix: it tells us theoretically
In Figure 6 an example of the resulting P matrix is shown.
if there is any probability of coming back to the region of
Accordingly to the bidimensional representation used in Sec-
stability and in the case the answer is no, tells us what is the
tionIII,wecanseeherethattherearetwopeakscorresponding
no-return value of N .
tothetwoexpectedlocallystableequilibriumpoints.However, B
it is more fruitful for our considerations to look at this graph
B. State probability matrix and FET calculation
from an (i,j) plane perspective plotting a line for i = j.
Points below this line represent decrements of the number of Based on the P matrix, it is now possible to calculate the
backlogged packets while points above represent increments. probabilityBf+1tobeinacertainstateN =iatframef+1,
i B
Non-zero probability points look like a wake close to the line givenacertainprobabilitydistributionBf =[Bf Bf ··· Bf ]
0 1 M
i=j, highlighting that possible transitions are found only for at frame f:
smallchangesinthenumberofbackloggedpacketscompared
to the total population M. Taking single values of j, we can Bf+1 p0,0 ··· p0,M  Bf
0 0
rou•ghfnlooyrnsi-dmzeeanrlotlifvpyarlo3uberasebgoiifloijtnies(scoloafsrteehtefoowtuhanekdceh:baontnhelfoorpeirnactrienagspeoainntd) B1f...+1=p1...,0 ... ...... ·B...1f (9)
decrease of NB; BMf+1 pM,0 ··· pM,M BMf
• for values of j close to M (i.e. close to the channel
GeneralizingEquation9,thestatedistributionateachframe
saturation point) non-zero probabilities are found both
(f +1) can always be calculated from the initial probability
for increase and decrease of N ;
B distribution B0 = [B0 B1 ··· BM] and the matrix Pf+1.
• for the rest of the j values non-zero probabilities are 0 0 0
Given the state probability matrix, it is now possible to
almost exclusively found for i > j so that the number
calculate the average and cumulative distribution of the FET
of backlogged users monotonically increases.
fromagiveninitialstate.Inthisexampleweconsiderasinitial
Therefore in this particular case the system has probability
state B0 = [ 1 0 0 ··· 0 ] that is the case in which the
zero of returning to the stability region after N crosses 120.
B communication starts with zero backlogged users.
We want to highlight here that the presence of a no-return
Figure7reportstheevolutionframeafterframeofthestate
point is scenario-dependent in the sense that for a different
probability for the P illustrated in Figure 6. As we can see,
set of values M, p , p , N it is not necessarily true that
0 r s when f increases the probability of being around the locally
the probability of coming back from a certain value N is
B stable point in the region of high throughput decreases while
zero: sometimes could be negligible, sometimes could not,
the probability of being in saturation increases.
4 Forthesamereasonclaimedinfootnote3forthePLR,alsoq(s|t+b)
valuesarecalculatedfromsimulationsoverabigamountofruns(over106per C. Reduced transition matrix
(t+b)value)sinceanalyticalformulasabletotightlydescribetheprobability
The entire description of any state of this Markov chain is
for a packet to be correctly decoded have not been found yet. Nevertheless
inthefollowingsamethodtoreducethiscomputationaleffortisintroduced. of great interest but most of the times impractical, mainly
7
1 190
0.9 180
on0.8 170
ncti0.7
u
Distribution F00..56 Av[FET]115600 SAinmaulylatitcion
e 0.4 140
v
ati
ul0.3
m With absorbing state (Av[FET]=114) 130
u
C0.2 Without absorbing state (Av[FET]=160)
120
0.1
0 110 50 100 150 200 250 300
0 200 400 600 800 1000 1200 1400
f Nabs
B
Figure8. FETcumulativedistributionforCRDSAwithNs=100,Imax= Figure 9. Simulated and analytic average FET when increasing NBabs for
20,M =350,p0=0.18,pr =1. CRDSAwithNs=100,Imax=20,M =350,p0=0.18,pr =1.
because the dimension of the P matrix is equal to M2. validityofthisapproximation.Figure8showsthedistribution
Thereforeitcouldtakeahugecomputationalefforttocalculate for the results obtained with the reduced matrix and the
it. Nevertheless it is of interest to know the probability that simulated results. Regarding simulations, if NB crosses the
the stability region around the operational point has been left value (cid:100)NBU(cid:101) but it comes back to the stability region, the
in a certain moment, regardless of the actual state once we FET is reset and the next exit is waited to set it again. For
enter the region that yields to saturation. the analytic results the FET is calculated accordingly to the
In fact, it has been already pointed out in the description reduced matrix in Equation 10.
of Figure 6 that the further the communication gets from the As already pointed out in the description of Figure 6,
first point of local stability, the smaller is the probability that when the number of backlogged packets NB is still close to
the communication will come back to the stability region. For the unstable equilibrium point, a certain probability that the
this reason, as a first approximation, it is possible to consider communication will come back from the instability region is
an absorbing state Nabs = (cid:100)NU(cid:101) + 1 that groups all the present. This is the reason why in Figure 8 the two curves
B B
probabilities for states from (cid:100)NU(cid:101)+1 to M. Doing so, our show a noticeable difference. However, this problem can be
P matrix can be rewritten as B solved by considering NBabs = (cid:100)NBU(cid:101)+1+∆ as absorbing
state, where ∆ is a positive integer big enough to ensure that
  the probability of false exits from the stability region (i.e. the
p ··· p 0
0,0 0,NBU probabilitythatNB crossesthestabilityregionbutcomesback
 ... ... ... ... tovalues<NBabs)issufficientlylow.Doingso,simulationand
  analytical results with the approximation of absorbing state
P = 
 pNBU,0 ··· p1,NBU 0 perfectly match. Figure 9 shows how the difference among
  the two average FETs decreases while increasing the chosen
1−((cid:80)NBU p ) ··· 1−((cid:80)NBU p ) 1 value for Nabs.
α=0 α,0 α=0 α,NU B
B In conclusion, we can state that analytic results with the
(10)
approximation of absorbing state represent a lower bound
The P matrix in Equation 10 takes much less effort than
for the actual FET. Therefore during this kind of analysis
calculating the entire P matrix since only ((cid:100)NU(cid:101) + 1)2
B a tradeoff between tightness of the results and simplicity of
elements need to be computed thus speeding up calculations.
Consideringthat(cid:80)NBU p mustbeequalto1,theprobability computation is present.
α=0 α,j
togettotheabsorbingstateNabs fromacertainstatej canbe
B E. Reduced transition matrix for infinite population
computed as p = 1−((cid:80)NBU p ). Moreover, since Nabs
i,j α=0 α,j B Themodelpresentedisalsoadaptabletothecaseofinfinite
isanabsorbingstate,anyprobabilitytoleavethisstateiszero
population. In fact the probability transition matrix P can be
while the probability of staying in the absorbing state is equal
adapted by considering
to 1.
D. Absorbing state validity (cid:88)∞ (cid:88)j λte−λ (cid:18)j(cid:19)
p = · pb(1−p )NB−b·q(s|t+b)
Sincereducingthetransitionmatrixbymeansofanabsorb- ij t! b r r
t=0b=0
ing state is an approximation, we now aim at verifying the (11)
8
Although in this case P has infinite size, it has been just
shown that is possible and practical to reduce the P matrix
by means of an absorbing state. With this expedient, also this
case can be treaten since P is transformed to an equivalent
matrix with finite size. NB p0+
VI. DESIGNSETTINGS
p−
In this section we will resume the role of some crucial 0
parameters such as M, p , p and explain how the commu-
0 r
nication state changes when changing them and what are the
G
T
relations to be considered when designing DVB-RCS2 com-
munications in RA mode. In fact, since optimization depends
(a) Changingp0
on constraints and degrees of freedom of the particular study
case, a deep investigation of the relation among these crucial
parametersisneededinordertounderstandtradeoffsthathave
to be faced. Moreover, the same discussion is introductory
to understand how control policies can help to obtain better
performance and a stable channel even in case of statistical
B M+
fluctuations that could yield to instability. N
A. Degrees of freedom
M−
Consider Figure 10. The first thing we can notice is that
M and p0 only influence the channel load line while chang- G
T
ing p corresponds to modifying the equilibrium contour. In
r
Figure 10(a) we can see that p0 influences the slope of the (b) ChangingM
channel load line that becomes steeper when increasing p .
0
This is intuitive to understand since for fixed M, if pA <pB
0 0
thenGA <GB.Figure10(b)showsthatM shiftsthechannel
T T
load line up and down if the population size is respectively
increasing or decreasing. From Equation 3 we can notice that pr−
the value M is nothing else that what is known in literature as
q, i.e. the intersection of the line with the y-axis. Finally pr NB
determines a shift upward of the channel load line for smaller
values of p (Figure 10(c)). As already shown, if the value
r
pr is sufficiently small, it is possible to stabilize a channel p+
initially unstable. This represents the ground base for one of r
the control policies shown in the next section. G
T
B. Constraints (c) Changingpr
Concerning possible constraints, we can identify the fol-
lowings
1) stability
2) maximize p0 or guarantee a value p0 ≥pm0 in Fwihgeunrein1c0r.easiGnrgapohricdaelcrreeapsriensgenotnateioonfothfethpearcahmaentgeerss’invatlhuees(.GT,NB)plane
3) maximize M or guarantee a value M ≥Mmin
4) minimize the delay or guarantee that a certain value
Av[D ]max is not exceeded 3)
pkt
G ·N
5) maximize the throughput or guarantee a value T s +N ≥Mmin (13)
B
GOUT ≥GmOUinT p0
For constraint 1) it has been previously shown that the 4)
stability corresponds to having a single intersection between NB ≤Av[D ]max (14)
the channel load line and the equilibrium curve (i.e. a single GT ·Ns pkt
point of equilibrium) before the throughput peak. Constraints 5)
from 2) to 5) can be graphically represented on the (GT,NB) GT ≥GmOUinT (15)
plane by the following formulas:
In other words, each formula simply represents an admis-
2)
G ·N sible area where the operational point has to be, as illustrated
T s ≥pmin (12)
M −N 0 in Figure 11.
B
9
the intersection of the respective admissible areas. Therefore
we can conclude that the designed model not only allows to
forecast stability and throughput of the communication, but
also to easiliy represent its constraints. This enables a design
B that does not need any use of complicated formulas in order
N
to understand feasible settings.
Notice that the choice of N has not been considered as
s
a design parameter. This choice have basically two reasons:
first of all in DVB-RCS2, the number of slots allocated for
GT RA is dynamically decided by the NCC; secondly, the best
choiceforN canalwaysbeconsideredbetween100and200
(a) p0 constraint since a lowesr value would degrade too much the throughput
performancewhileabiggervaluewoulddegradetoomuchthe
delay performance.
VII. PACKETDEGREEANDRATECHNIQUE
B In the previous sections we have concentrated on the best
N
selection of p , p and M once the number of replicas per
0 r
packet (packet degree) is chosen. However also the number
of packet copies is a design parameter5. For this reason, this
sectionanalyzesthechoiceonthenumberofcopiesperpacket
GT to be sent. Moreover, a comparison on the advantages and
disadvantages of CRDSA with regard to SA are outlined.
(b) M constraint
100
10−1
B
N
SA
CRDSA with 2 copies
LR10−2 CRDSA with 3 copies
P
G
T
10−3
(c) Delayconstraint
10−4
0 0.5 1 1.5 2
Channel load [G]
NB Figure12. PacketLossRatiofordifferentpacketdegreeswhenNS =100
slots
Figure 12 shows that up to moderate channel loads, a
greater packet degree is always convenient not only in terms
of PLR but also in terms of throughput and packet delay. But
G
T
considering stability to be one of our constraints, we can see
(d) Throughputconstraint from Figure 13 that for pr = 1 SA offers the most reliable
performance since after the throughput peak any CRDSA
curverapidlydegradeswithN whileSAdegradesinaslow-
B
Figure 11. Graphical representation of the admissible areas for various paced manner. As a result, for moderate channel load lines
constraints. with G ≤ 0.4 for N = 0, SA is able to support up to
T B
5In this paper we compare various packet degrees under the assumption
of equal received power per replica. For discussion on the power efficiency
Anystudycasewillbecharacterizedbyoneormoredegrees whenconsideringtransponder’saveragepowerlimitsreferto[16].Moreover
theadditionaloverheadduetotheneedofmorepointerswhenmorecopies
offreedomandoneormoredesignconstraints.Incaseofmore
aresenthasbeenomitted,sinceitcanbeconsideredneglibilewithregardto
than one design constraint, the resulting constraint will be theburstsizeespeciallywhenfewcopies(nomorethan4or5)areconsidered.
10
dynamicmanner.Inthissectionweanalyzecontrolprocedures
500
of the limit type [11] able to ensure stability in the case of
450 stationary parameters. However the same reasoning can also
400 beappliedinthecaseofdynamicones(suchasnon-stationary
SA
M) with the necessary modifications.
350 CRDSA with 2 copies
CRDSA with 3 copies Two simple yet effective dynamic control procedures are
300 considered: the input control procedure (ICP) and the re-
NB250 transmission control procedure (RCP). These two control
procedures are based upon a subclass of policies known as
200
control limit policies, in which the space of the policies is
150 generally composed of two actions plus a critical state that
100 determines the switch between them, known as control limit.
Inthiscasethecontrollimitisacriticalthresholdforacertain
50
number of backlogged users Nˆ .
B
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
GT A. Input Control Procedure
This control procedure deals with new packets to transmit.
Figure13. EquilibriumcontourfordifferentpacketdegreeswhenNS =100 In particular, two possible actions are possible: accept (action
slots
a) or deny (action d) and the switch between them is deter-
mined by the threshold Nˆ as previously mentioned.
B
approximately 450 users ensuring stability, while for CRDSA
with 2 replicas the admissible population is approximately B. Retransmission Control Procedure
50% less and for CRDSA with 3 replicas the admissible M
As the name says, the retransmission control procedure
is less than 150.
deals with packets to retransmit and in particular with their
However, as previously said, the operational point for
retransmission probability. In particular two different
CRDSA always occurs in a point with higher throughput,
retransmission probabilities p and p are defined
smaller backlogged size and smaller packet delay. Moreover r c
that represent respectively the action taken in normal
in Section V it has been shown that depending on the channel
retransmission state (action r) and in critical state (action c).
loadlinetheactualFET couldbesobigthatinstabilitycould
From the definition above it is straightforward that it must be
be accepted. For example, for M = 450 and p = 0.089 the
0 p >p . The switch between these two modes is determined
computedFETispracticallyzero,inthesensethateventhough onrce agcain by the threshold Nˆ .
we know that a certain probability exists, it is so small that B
the computation resulted to be zero due to its littleness.
Considering M =250 and p =0.2 SA presents a globally
0
stable behaviour around its throughput peak (G = 0.36)
OUT
for NG =65 while CRDSA with 3 replicas presents a locally
B
stable equilibrium point with throughput equal to 0.5 , NS
B
close to 0 and unstable equilibrium point for NU ≈43.
B
is Uapspinrogxifmoramteullyas10in4 fSraemctieosn. IVn tthhies coabstea,iniefdwaevaerreagweilFliEnTg NB NB pc
to maintain the advantages of CRDSA with 3 replicas (e.g. p
r
its lower PLR) some countermeasures to ensure stability
are required. In the next Section we will show simple yet
effectivecontrolpoliciesabletocounteracttheeventofdriftto G G
T T
saturation.Afterthat,theconvenienceofusingsuchadynamic
policy in the case of finite and infinite users’ population will
(a) ICP (b) RCP
be discussed.
Figure14. Controllimitpolicyexamples.
VIII. CONTROLLIMITPOLICIES
If we are in the case in which the considered channel is Figure 14 graphically represents these 2 cases. As we can
not stable, some kind of solution to counteract the possibility seebothpoliciesaccomplishthesametaskofensuringchannel
ofdrifttosaturationisrequired.Twostraightforwardsolutions stability. However, while ICP controls the access of thinking
couldbe:useasmallervaluefortheretransmissionprobability users, RCP controls the access of backlogged users.
givingthenrisetoalargerbackloggedpopulationforthesame
throughput value; allowing a smaller user population size M IX. ONTHEUSEOFDYNAMICCONTROLPOLICIESFOR
thus resulting in a waste of capacity. FINITEPOPULATION
Athirdsolutionistheuseofcontrollimitpoliciestocontrol In this section we discuss the result of the application of
unstablechannelsbyapplyingthecountermeasuresaboveina dynamic control policies for CRDSA with 2 and 3 replicas