Table Of ContentExploiting the Path Propagation Time Differences
in Multipath Transmission with FEC
Maciej Kurant
LCA - School of Communications and Computer Science
EPFL, CH-1015 Lausanne, Switzerland
Email: maciej.kurant@epfl.ch
relaynode
kdatapackets
9 datAabssttrraecatm—Wfroemcoanssiidnegrleasoturracnesmtoissaiosningolfeadedsteilnaayt-isoenn.siTtihvee n−kredundancy packets P2,t
0 2
reliability of this transmission may suffer from bursty packet n 2
0 losses - the predominant type of failures in today’s Internet. FEC(n,k) n1 P1,t1
2
n Aprnoteecfftectthievedaatnadbwyelal FstourdwieadrdsoElurrtoiornCtoorrtehcitsiopnro(FblEeCm) icsodtoe nFECpackets n3 P 3,t 3
a and send the FEC packets over multiple paths. In this paper
source s destination d
J we show that the performance of such a multipath FEC scheme
1 can often be further improved. Our key observation is that the
1 propagationtimesontheavailablepathsoftensignificantlydiffer, Fig. 1. Illustration of a multipath system with R = 3 paths P1,P2,P3
typicallyby10-100ms.Weproposetoexploitthesedifferencesby between source s and destination d. t1,t2,t3 are the corresponding path
] appropriatepacketschedulingthatwecall‘Spread’.Weevaluate propagation times.k datapackets arecomplimented withn−k redundancy
I our solution with a precise, analytical formulation and trace- packets,andtheresultingnFECpackets aresplitontothethreepathsusing
N driven simulations. Our studies show that Spread substantially therates n1,n2 andn3,respectively.
. outperforms the state-of-the-art solutions. It typically achieves
s
c two-tofive-foldimprovement(reduction)intheeffectivelossrate.
[ Orconversely,keepingthesamelevelofeffectivelossrate,Spread and (3) schedule the packets. The previous studies proposed
significantlydecreasestheobserved delaysandhelpsfightingthe techniques to solve (1-3) as a function of the statistical loss
1 delay jitter. properties of the paths [4,5,10].
v
However,thereareotherimportantparametersaffectingthe
9 I. INTRODUCTION
7 performance of the multipath FEC system. In particular, in
4 We considera transmissionofa delay-sensitivedata stream thispaperweshowthatthepropagationtimesontheavailable
1 from a single source to a single destination. How to improve multiple paths often significantly differ. These differences, in
1. the reliability of this transmission? Traditional ARQ (Au- turn,canbeexploitedtoimprovethesystemreliability.Below,
0 tomatic Repeat-reQuest) mechanisms often cannot be used, we explain and motivate our approach on concrete examples
9 as they impose additional and usually unacceptable delays and measurements.
0 of at least one RTT (Round Trip Time). A more applicable
v: technique is to introduce some type of redundancy, e.g., A. Propagation times on direct and indirect paths may differ
i Forward Error Correction (FEC). Clearly, due to the delay significantly
X
constraints, a FEC block must be of limited length [1]. This, In Fig. 2 we study the path propagationtime differencesin
r
a in turn, makes it inefficient against bursty packet losses [1] thereal-lifeInternet.Themeasurementswerecollectedbyrun-
- the predominant type of losses in today’s Internet [2]. A ning all-to-all traceroutesbetween 326 nodes in DIMES [12].
good solution to this problem is to assign the FEC packets These nodesare usually privatehostslocatedat differentsites
to multiple pathsspanningthe sourceand the destination[3]– around the world. (We obtained similar results for measure-
[10].AnillustrationofamultipathFECsystemispresentedin ments on PlanetLab [13].)
Fig. 1. Theoretically, the multiple paths could be constructed For each source-destination pair we construct a set of R
with the help of source routing, but this technique is not yet paths. We always include the direct path P with propagation
1
fully available in the Internet. A more practical alternative is time t . Each of the remaining R−1 paths is indirect, i.e.,
1
the usage of overlay relay nodes that forward the traffic (as it uses some overlay relay node to forward the traffic. We
in Fig. 1). If the resulting paths are statistically independent, choose uniformly at random a number C of candidate relay
whichisespeciallylikelyformulti-homedhosts,thentheloss nodes among the remaining 324 DIMES nodes. This results
burstsgetaveragedoutandFECregainseffectiveness.Similar in C candidateindirectpaths. From them we select the R−1
performance benefits due to multipath were also observed in indirectpathsfollowingtheintuitiveselectionproceduregiven
the context of Multiple Description Coding [11]. in [5]. For R=2 paths we choose the indirect candidate path
When designing a system that splits a FEC block across thatisthemostIPlinkdisjointwiththedirectpathP .Clearly,
1
multiple paths, we have to (1) select some paths out of all thisminimizesthelosscorrelationbetweenP andP .Ifthere
1 2
candidates, (2) assign the transmission rates to these paths, are more paths that achieve the minimal IP overlap, then the
2
0.015 R=2 paths 0.012 R=3 paths A) delays P1
0.0125 0.01 P2
∆t=50ms
0.01 0.008
pdf0.0075 pdf0.006 B) no FEC πB∗ =1.00%
0.005 0.004
0.0025 0.002
C) Simm π∗ =0.553%
0 0 (6,0) B
0 50t2−10t01 150 0 ma50x(t2,1t003)−t1150
C=1 C=5 C=50 D) S(i3m,3m) πB∗ =0.148%
R=2 t2−t1 [ms] 43 37 18 ∆t=50ms (stateoftheart)
R=3 max(t2,t3)−t1 N/A 69 39
E) Sspr π∗ =0.113%
Fig. 2. The difference between propagation times on the direct path P1 (3,3) B
and the best indirect paths P2 and P3. We present the results for R = 2
(twopaths:onedirect andoneindirect) andR=3(thedirect pathandtwo F) Sspr π∗ =0.016%
indirectpaths).Thehistograms(top)showthedistributionofpropagationtime (4,2) B
(proposedsolution)
differences forC =5available candidate indirect paths. Thetable (bottom)
showsthemedians ofthesedistributions forC=1,5and50.Theaverages
(notshown)aresystematically higherthanthemedians. G LG
START LG1
Fig. 3. Illustration of various packet schedules and their performance
pornoecweeidthsitmheilsamrlya,lleexstcepprtopthaagtantioowntwimeecoisnskiedpetr.tFhoeraRgg>reg2atwede amnedasPur2edwiitnhthideenetfifceacltivfaeilulorsesdriastteribπuB∗tio.nWs.eTuhseedtwatoa ipnadcekpeetsndaernetgpeanthesratPed1
values of IP overlap and propagation time, i.e., summed over at the source every T = 5ms and coded with FEC(6,4). (A) The path
all R(R−1)/2 possible path pairs. propagation time t2 on path P2 is ∆t=t2−t1 =50ms larger than the
pathpropagationtimet1onpathP1. (B)NoFEC,singlepath,thepackets
According to Fig. 2, for R = 2, the best indirect path P2 are sent at times 0,5,10,15ms. (C) FEC on P1 only, packets are sent
has propagation times larger by typically 0...75ms than the assoonastheyaregenerated, i.e.,weusethe‘Immediate’ scheduleSimm.
directpathP (seet −t intop-lefthistogram).Thisdifference (D)PacketsalternatebetweenP1 andP2withequalratesratesn1=n2=3,
1 2 1 asin[5,10].ThetotalFECblockdelayresultingfromthisschemeservesasa
gets larger for a smaller number of candidates C (table at the maximalFECblockdelayinthefollowingscenarios. (E)Packetsalternate
bottom). between P1 and P2 with equal rates, but the three packets sent on P1 are
Moreover, the path propagation time differences grow sig- maximally spread. (F)Packets are split between P1 and P2 with optimal
ratesn1=4,n2=2,maximally spread.
nificantly with the number of paths R used in the system.
As shown in Fig. 2, already for R = 3 the medians of the
distributions are roughly doubled compared to R = 2, and on P with inter-packet times T, then the effective loss rate
1
typicallyP1 is faster than the slower of the two indirectpaths after FEC decoding is πB∗ =0.553% (C). Following [5,10],
by max(t2,t3)−t1 ≃0...150ms. we now split the packets equally between P1 and P2, which
We conclude that in the real-life Internet the propaga- decreasesπ∗ to0.148%(D).Thissolutionrepresentsthestate
B
tion time differences on multiple paths between a source- of the art. Note that now the last FEC packet on path P
2
destination pair are significant, typically reaching severaltens reaches the destination t = t +4·T =170 milliseconds
FEC 2
of milliseconds. after the generationof the first FEC packetatsource.In other
words, in this case the application using multipath FEC must
B. The differencesin propagationtimes canbe exploitedby a
accept the (maximal) delay equal to t . However, we can
FEC
multipath FEC system
achieve far better results still respecting this delay constraint.
We propose to exploit these path propagation time differ- Forinstance,wecanappropriatelyincreasethepacket-spacing
ences when designing a multipath FEC system. Our solution onP andachieveπ∗ =0.113%(E).Finally,wegetevenmore
1 B
is easy to implement and can bring significant performance significant improvement by sending 4 packets on P and 2
1
gains. Let us take a concrete example described in Fig. 3. packets P , i.e., by applying unequal rates on the paths (F).
2
Assume that there exist two paths between the source and This results in π∗ =0.016%, which is almost one order of
B
the destination, the direct path P , and an indirect path P magnitude smaller than (D).
1 2
created by employing another peer that works as a relay. Let In other words, we exploit the differences in path prop-
t =100msandt =150msbethepropagationdelaysonP agation times by spreading the packets in time, such that
1 2 1
and P , respectively. So the path propagation time difference the maximal allowed delay is respected. The gain over the
2
is ∆t = 50ms (Fig. 3A). Let the two paths be lossy with state of the art measured in the effective loss rate π∗ may
B
the same loss rate 1% and the same average loss burst length be very significant (here 0.016% vs 0.148%, i.e., almost ten-
of 10ms, but independent. The data packets are generated at fold). Moreover, some results may seem counterintuitive. For
the source every T =5ms. If no form of packet protection is instance, it may be better to use only one path than to use
used,thenthedatapacketlossrateobservedatthedestination, two (un-spaced) paths. It also turns out that even if the loss
or the effective loss rate, is π∗ =1% (B). Assume now that distributions on the paths are the same, the optimal rates
B
we useFEC(6,4)toprotectthepackets.Ifwesendallpackets assigned to these paths are not necessarily equal.
P, E probability, expected value 3
C. Organization of this paper
s sourcenode
The remainder of this paper is organized as follows. In d destination node
T (constant) interval between two consecutive data packets
Section II we fully specify our model, which allows us to
atsources
preciselystatetheproblemwearesolving.Next,inSectionIII R numberofindependentpathsbetweensourcesanddesti-
we derive exact analytical expressions for the effective loss nationd
rate π∗ under multipath FEC and an arbitrary schedule. In Pr rthpath
B tr propagation delayonPr
SectionIVwedescribethe‘Immediate’schedulerepresenting πB(r),1/µ(Br) theaveragelossrateandlossburstlength onpathPr
the state of the art, and propose a ‘Spread’ schedule that n,k,(n−k) the number of FEC, data, and redundancy packets in a
exploitsthedifferencesinpathpropagationtimes.InSectionV FECblock,respectively
we evaluate our solution analytically, by simulations and by nr numberofFECpackets assignedtoPr (rateofpathPr)
kr numberofdatapackets assignedtopathPr
trace-driven simulations fed with real-life Internet traces. In Tr (constant) spacingofthenr packets onpathPr
Section VI we discuss the related work. Finally we conclude F,D numberoflostFECanddatapacketsbeforeFECrecovery
π∗ effective loss rate, i.e., the expected fraction of lost data
the paper and propose future directions. The details of some B
packets atthedestination aftertheFECrecovery
calculations are put in Appendix. tFEC FEC block transmission time, i.e., the time between the
generation of the first FEC packet at source s and the
II. MODELAND PROBLEM STATEMENT scheduleddeliveryofthelatestFECpacketatdestinationd
S=(T,R) packet scheduling: Theithpacket inaFECblockissent
The packets, called data packets, are generatedat sources, attimeT(i)overpathR(i)
with constant inter-arrival time T. There exist R paths be-
tween sender s and destinationd, with the propagationdelays TABLEI
t1,...,tR, respectively. BASICNOTATIONUSEDINTHISPAPER.
A. Path losses
Thepathsareassumedto beindependent.We modelbursty T
losses on each path by the popular two-state Gilbert model. 4 3 2 1 Data packets generated
at the source
Its basic version is a Discrete Time Markov Chain (DTMC), 6 5 3 1
andcapturesthelosscorrelationsduetoqueuingonbottleneck P1 4 2 Schedule S =(T,R),
links, when the path is sampled at some constant rate (e.g., P R=(2,1,1,2,1,1)
2
1/T). However, as we vary the sampling rates, DTMC is Time FEC(6,4)
nhoigthseurffilocsisenbt.urIsntdineeesds,uonndtehrethsaempeacpkaetthiwnteerevxapleorfieTnce=m5umchs TT((66)) T(5) T(4) T(3) T(2)T=(T1)=0 nn21==24,,kk21==22
thanofT =100ms[1].Forthisreasonweusethecontinuous-
Fig. 4. An illustration of a schedule S = (T,R) on R=2 paths with
timeversionofthe Gilbertmodel[4,14]thatnaturallyaccom-
FEC(6,4).Fourdatapacketsnumbered1-4aregeneratedatthesourceatequal
modates different sampling rates. It is a two-state stationary intervals T; the first one specifies time t = 0. The n−k = 2 redundancy
Continuous Time Markov Chain (CTMC) {X (t)}. The state packets are numbered 5and 6.According tothe schedule S =(T,R), the
r
ithFECpacket issentattimeT(i)≥0overpathR(i).
X (t) at time t assumes one of the two values: G (‘good’)or
r
B (‘bad’). If a packet is sent at time t and X (t) = G then
r
thepacketistransmitted;ifX (t)=B thenthepacketislost.
r we consider a systematic1 FEC, i.e., a scheme where the first
We denoteby π(r) and π(r) the stationaryprobabilitiesthat
G B k packets are the k data packets (unchanged). The remaining
the rth path is good or bad, respectively. Similarly, let µG(r) n−kpackets,calledredundancypackets,carrytheredundancy
and µ(r) be the transition rates from G to B and from B to information. The destination uses the redundancy packets to
B
G, respectively.Inthis paperwe use two meaningful,system- recoversome ofthelostdata packetsasfollows.LetF bethe
dependentparameterstospecifytheCTMCpacketlossmodel: number of lost FEC packets and let D be the number of lost
• the average loss rate πB(r), and data packets of a FEC block, both before the FEC recovery
• the average loss burst length (in seconds) 1/µB(r). (note that D contributes to F). If F ≤ n−k then all the n
FEC packets and hence all the k data packets are recovered.
All other parameters can be easily derived from these two,
In contrast, if F > n−k, then no FEC recovery is possible
because
and D data packets are lost.
µ(r) µ(r)
πG(r) = µ(r)+Bµ(r) and πB(r) = µ(r)+Gµ(r). (1) C. Packet scheduling
G B G B
Finally,thepacketsaresentaccordingtosomeschedulethat
B. Multipath FEC defineswhenandonwhichpatheachFECpacketissent.More
precisely, we denote by S = (T,R) the schedule of packets
We use a FEC(n,k) scheme to protect the data packets
in a FEC block, where T and R are vectors of length n. The
againstlosses(seeFig.1).Thismeansthatk datapackets(not
necessarily consecutive) are encoded as one FEC block of n
1The non-systematic FEC is easier to handle, but also less efficient. We
packets,calledFECpackets.Inparticular,asin[4,6,10,15,16], showitsanalysis inAppendix.
4
ith FEC packetis sent at time T(i) overpath R(i), as shown A. The effective loss rate π∗ for an arbitrary schedule
B
in Fig. 4. The time is counted from the generation (at the
Let c be a n-tuple representing a particular failure config-
source) of the first data packet of the FEC block. Denote by
uration; c , 1≤i≤n, takes the value G (resp., B) if ith FEC
i
t the FEC block transmission time, i.e., the time between
FEC packet is transmitted (resp., lost). By considering all possible
the generation of the first FEC packet at source s and the
failureconfigurationsc we can computethe effectivelossrate
scheduled delivery of the latest FEC packet at destination d. π∗ for a given schedule S as follows:
B
Given a schedule S, t can be easily computed as
FEC
1
π∗ = D(c)·P(c), (3)
tFEC =1m≤ia≤xn(cid:16)T(i)+tR(i)(cid:17). (2) B k Xallc
For a given schedule, t can be interpreted as the total where D(c) is the number of lost data packets (after the FEC
FEC
delay imposed by the multipath FEC system on the delay- recovery)for a given failure configurationc. For a systematic
sensitive application using it. Indeed, if the first packet of a FEC(n,k) we have
FEC blockis lost andneedsto be reconstructedby FEC, then 0 if n 1 ≤n−k
we have to wait up to t until the destination is reached by D(c)= i=1 {ci=B}
FEC (cid:26) k 1 othPerwise.
the other FEC packets necessary for the reconstruction of the i=1 {ci=B}
lostpacket.Inpractice,however,aconstraintislikelytocome In order to coPmpute the probability P(c) of a failure con-
from the delay-constrained application itself, as the maximal figuration c, we consider the R paths separately, as follows.
acceptabledelaytFEC.Inthiscaseourgoalistodesignagood Denote by T(r) the vector of length nr with departure times
schedulerespectingthisconstraint,whichistheapproachused of packets scheduled by S on path P . Similarly, let c(r) be
r
in this paper. ann -elementvectorwiththefailureconfigurationonpathP
r r
The schedulealso implicitlydefinestherate nr of pathPr, defined by c. As the R paths are independent, the probability
i.e., the number of FEC packets sent on Pr. Similarly, let kr P(c) is
be the number of data packets among the n packets sent on R
r
P . Clearly, n =n and k =k. P(c)= P(c(r)), (4)
r r r r r
P P rY=1
D. Effective loss rate π∗ and problem statement
B where P(c(r)) is the probabilityof a failure configurationc(r)
Our ultimate goal is to send a stream of data packets over onpathP .ThederivationofP(c(r))fortheContinuousTime
r
(possiblymultiple)lossychannelsinawaythatminimizesthe Markov Chain loss model is straightforward. Indeed, denote
losses observed at the destination, given a maximal value for by p(r)(τ) the probability of transition from state i to state j
i,j
tFEC. Therefore,we adopta naturalperformancemetric called on path P in time τ, i.e.,
r
effective loss rate π∗. It is defined as the expectedfractionof
B
lostdatapacketsobservedatthedestinationdafteranattempt pi(,rj)(τ)=P[Xr(τ)=j|Xr(0)=i].
of FEC decoding. Now the problem can be stated as follows:
Given the path loss properties (π(r), 1/µ(r) and t for From classical Markov Chain analysis we have:
B B r
every path Pr), the FEC parameters (n and k) and maximal p(r) (τ) = π(r)+π(r)·α
G,G G B
FEC block transmission time tFEC, find the schedule S that p(r) (τ) = π(r)−π(r)·α
minimizes the effective loss rate π∗. G,B B B (5)
B p(r) (τ) = π(r)−π(r)·α
We approachthisproblemin two steps. First, in SectionIII B,G G G
wederiveanexactanalyticalformulafortheeffectivelossrate p(r) (τ) = π(r)+π(r)·α
B,B B G
π∗ foragivenscheduleS.Second,inSectionIVweintroduce
B whereα=exp −(µ(r)+µ(r))τ .NowP(c(r))canbeeasily
a schedule that exploits the differences in path propagation G B
times and outperforms the schedules proposed to date. computed. For e(cid:0)xample, for c(r)=(cid:1)GBB we have
P(c(r)=GBB)=π(r)·p(r) (τ )·p(r) (τ ),
III. EXACT ANALYTICALDERIVATION OF THEEFFECTIVE G G,B 1 B,B 2
LOSSRATEπB∗ whereτi isthetime intervalbetweentheithand(i+1)thFEC
In order to design a good schedule we must be able to packetscheduledbyS onpathP ,i.e,τ =T(r)−T(r).More
r i i+1 i
evaluate it. In this section we derive the exact analytical generally,
expressionfortheeffectivelossrateπ∗ foragivenscheduleS.
B nr−1
We consider two cases. First, we derive πB∗ for an arbitrary P(c(r)) = π(r) p(r) (T(r)−T(r)). (6)
schedule S. The resulting formula is simple but computa- c(1r) iY=1 c(ir),c(ir+)1 i+1 i
tionally expensive and untractable for larger sizes n of the
Finally, we plug (6) and (4) to (3), to obtain
FEC block. Next, we derive π∗ assuming that on each path
B
separately the packets are evenly spaced. This constraint is 1 R nr−1
compatible with the schedule we propose later and results in π∗ = D(c) π(r) p(r) (T(r)−T(r)). (7)
a computationally lighter formula for π∗. B k Xallc rY=1 c(1r) iY=1 c(ir),c(ir+)1 i+1 i
B
concatenation of events, e.g., G[a] (resp., [a]B) means tha5t
b b
anyb outof ablockofa consecutivepacketsarelostandthat
103
o this block is preceded by a good packet (resp., followed by a
rati102 badpacket).WecannowcomputeP(Fr =jr)byconditioning
e on the state of the first packet that conforms the packet loss
m
ti101 stationary distribution:
100 P(Fr =jr)=P(G [nrj−r1])+ P(B [njrr−−11]) =
8 10 12 14 16 18 20
=π(r)·P([nr−1] | G)+ π(r)·P([nr−1] | B), (9)
n - size of a FEC block G jr B jr−1
where P([a] |q), q ∈ {G,B}, is the probability that any b
b
Fig.5. Thetimecomplexityoftheeffectivelossrateπ∗ underanarbitrary out of a consecutive packets are lost given that this block
B
schedule(III-A)vs.theeven-spacedschedule(III-B):timeratioistheruntime is preceded by a packet in state q. Although no general
ofEq.(7)dividedbytheruntimeofEq.(11).HereweuseFEC(n,0.7n)on
twoidentical paths. closed form of P([ab] |q) is known, it can be calculated by
the recursive approach first proposed in [17] and extended
e.g. in [4,15]. We show in Appendix the details of this
B. The effective loss rate π∗ for even spacing on paths computation. It takes π(r), 1/µ(r) and T as parameters, and
B B B r
Equation (7) allows us to compute the effective loss rate directly uses the relations (5) above.
π∗ for any schedule S. However, evaluating (7) is computa- In order to find E[Dr|Fr = jr], we first derive P(Dr =
B
tionallyexpensivebecause the main sum is overall 2n failure i,Fr=jr). Letusconsiderthekr data packetsandthe nr−kr
configurations. Thus it can be applied to relatively small n redundancy packets separately, and additionally condition on
only.Fortunately,wecansignificantlyreducethecomputation the state of the last data packet as follows.
complexity by assuming that on each path Pr separately the P(Dr =i, Fr =jr)=
packetsareevenlyspaced,i.e.,forall1≤i≤nr−1theintervals =P([kr−1]G)·P([nr−kr]|G) + P([kr−1] B)·P([nr−kr]|B) =
T(r)−T(r)arethesameandequaltoaconstantthatwedenote i jr−i i−1 jr−i
byi+T1r. Iindeed, this constraintleads us to a formulationof πB∗ =P(G [kri−1])·P([njrr−−kir]|G) + P(B [ki−r−11])·P([njrr−−kir]|B) =
(below) that may take orders of magnitude less time to solve =π(r)P([kr−1]|G)P([nr−kr]|G) + π(r)P([kr−1]|B)P([nr−kr]|B).
G i jr−i B i−1 jr−i
than (7), as shown in Fig. 5.
Thefirstequalityusesthe Markovpropertyofthelossmodel:
In order to compute π∗ under the even-spacing case, we
lookcloseratthepacketslBostoneverypathseparately.Denote P(Dr=i,Fr =jr |last datapacketis q)=
by Fr and Dr the number of FEC and data packets lost on =P(Dr=i |lastdata packetisq)·P(Fr=jr |lastdatapacketis q),
pathPr,respectively(bothbeforeFECrecovery).Nowwecan whereq ∈{G,B}.NowitiseasytocalculateE[Dr|Fr =jr],
rewrite the total number of lost FEC packets as F = rFr because
adnedcotmhepotostiatilonnnulmeabdesroufsltoostthdeatfaopllaocwkientgsadserDiva=tioPnroDf πPrB.∗T:his E[Dr|Fr =jr] = kr i· P(DPr(=Fi,=Frj=) jr). (10)
π∗ = 1 P(F =j)·E[D|F =j]= Xi=0 r r
B k
j=n−k+1 We plug (9) and (10) into (8) and obtain a complete formula
X
1 n for the effective loss rate π∗:
= kj=nX−k+1 0j1≤+Xj1..,+..,PjjR(RF=≤1=jjj1,..,FR=jR)·E[D|F1=j1,..,FR=jR]= πB∗ = k1j=nX−nk+1 0≤Xj1,..,jR≤j rY=RB1(cid:16)πG(r)·P([njr−r1]|G)+ πB(r)·P([njrr−−11]|B)(cid:17)!·
1 n R R j1+..+jR=j
= kj=nX−k+1 0j1≤+Xj1..,+..,jjRR=≤jj rY=1P(Fr=jr)·Xr=1E[Dr|Fr =jr(]!8) · Xr=R1Xik=r0i·πG(r)·P([kri−1]|GπG()r)·P·P([(n[jrnr−−jrk−rir1]]||GG))++ππB(B(rr))··PP(([[njkirr−r−−−1111]]||BB))·P([njrr−−kir]|(B1)1!),
AccordingtoEquation(8),inordertoevaluateπB∗,forevery whereeverytermoftypeP([a]|G)orP([a]|B)iscalculated
path P separately we need to calculate two components: (i) b b
r through the set of recursive equations given in Appendix.
theprobabilityP(F =j )thatj FECpacketsarelost,and(ii)
r r r
the expectednumber E[D |F =j ] of lost data packets given Tothebestofourknowledge,Equation(11)isthefirstexact
r r r
solution of this model. Indeed, all previous works used some
that j FEC packets were lost. We achieve this by applying
r approximationsof E[D |F =j ]. In [4] the authors approxi-
an approach similar to the one used in [15] in the context of r r r
mate E[D |F =j ] by assuming that any configuration of j
a single path FEC, as follows. r r r
lossesamongthenFECpacketsisequallylikely;we callthis
We considera pathP anda setof alln FECpacketssent
r r
on P with equal packet interval T . Denote by [a] the event approach ‘Golubchik’. In [6,16] the authors use an intuitive
thatarnyboutofaconsecutivepackretsarelost.2 Wbe allowfor linear formula, i.e., E(Dr|Fr = jr) = nkrrjr. Although not
mentioned in the papers this is only an approximation that is
2Theformof[ab]isinspiredbythesimilaritywiththebinomialcoefficient. exactonlywhenkr,nr →∞;we referto itas ‘Proportional’.
6
) in[4,6,7,9].Incontrast,whentheratesdiffer,amoreelaborate
prox.xact approachshouldbeused.Forthispurposeweadoptthecredit-
ape based technique proposed in [10], as follows. Each path is
(
]r]r associated with a credit initially equal to 0. Before each FEC
jj
== packet transmission the credit of every path P is increased
r r r
FF
|r|r by nr/n. Next, the path with the largest credit is selected to
DD
transmit this packet; the credit of this path is decreased by 1.
eE[E[
This scheme is iterated until all n FEC packets are sent.
j TheImmediateschedulecanbeinterpretedasthefollowing
function:
Fig.6. Approximations ofE[Dr|Fr =jr]normalizedbythecorrectvalue Simm =Immediate(n1...nR, T)
givenby(10).Herenr =10,kr =8,πB(r)=0.01and1/µB(r)=2. Two examples of Immediate schedules Simm are given in
Fig. 3: (C) is a single-path schedule, i.e., with n = 6 and
1
Weillustratethedifferencesbetweentheseapproximationsand n2 =0, whereas in (D) we use two paths and n1 =n2 =3.
the real values in Fig. 6.
B. ‘Spread’ packet scheduling Sspr - our proposal
IV. THEDESIGN OF THESCHEDULES Under Immediate, all packets are sent as soon as they
In the previous section we derived an exact analytical are generated, i.e., according to (12). Instead, we propose to
formulafortheeffectivelossrateπ∗ underagivenscheduleS. spread the packets evenly in all the available time on each
B
Here we focus on the design of a good schedule that results path. We call this schedule Spread Sspr = (Tspr,Rspr).
in small π∗. Comparedwith Immediate,Spreadadditionallytakes the path
B
Not all schedules are applicable in practice. Indeed, both propagationtimes t1...tR and the maximalFEC block delay
(i) the maximal allowed FEC block transmission time t tspr as parameters, i.e.,
FEC FEC
and (ii) the packet interval T at the source impose important Sspr =Spread(n ...n , T, t ...t , tspr).
scheduling constraints. We say that a schedule is feasible if 1 R 1 R FEC
all the three following conditions are satisfied: The design of Spread is not straightforward. Indeed, as
thek datapacketsare generatedatthesourcewith spacingT,
C1 T(i) ≥ (i−1)·T for 1≤i≤k, i.e., no data packet is
the paths are inter-dependent, which may easily lead to the
sent before it is generated at the source.
violation of the constraint C1. For example, if we schedule
C2T(i)≥(k−1)·T fork<i≤n, i.e.,noredundancypacket
packet 1 on P at time T(1)= 0 (and k >1), then no other
is sent before all data packets have been generated (we need 1
packet on any path can be scheduled before time t=T.
to collect all data packets in order to create the redundancy
We guaranteethe feasibility of Spread as follows. First, we
packets).
orderthe paths accordingto their rates, starting from the path
C3 T(i)+tR(i) ≤ tFEC for 1≤i≤n, i.e., all FEC packets with the highest rate. (When two paths have the same rate,
should arrive at the destination before the deadline. we take the one with a higher path propagation time first.)
We assume that the path rates n ,...,n are fixed. There We consider the paths one by one following this order. For
1 R
are usually a variety of feasible schedules. Below we discuss each path P we spread the packets evenly on time interval
r
two classes of schedules we use in this paper. The first one, [t(r),tspr−t ],wheret(r) takesthesmallestpossiblevaluethat
FEC r
called Immediate, reflects the state of the art, whereas the satisfies the feasibility condition. (The value of t(r) usually
second one, Spread, is our proposal. grows with the number of paths processed.) We iterate this
algorithm until all paths have been scheduled.
A. ‘Immediate’ packet scheduling Simm - state of the art
We present two examples of Spread schedules Sspr in
We denote by Immediate the schedule Simm = Fig. 3. We use tspr =170ms and two different sets of rates:
FEC
(Timm,Rimm) that represents the approach used in [4]– n =n =3 in (E) and n =4, n =2 in (F).
1 2 1 2
[7,9,10]. As the name suggests, Immediate sends the data Spread is very effective. Indeed, we can prove that
packets as soon as they are generated, i.e., every time inter- Theorem 1: The Spread schedule is optimal for the repeti-
valT.Theredundancypacketsuse the same spacingT. So in tion code FEC(n,1).
general Proof of Theorem 1 Under FEC(n,1) every data packet is
replicated and sent in n copies; the reception of at least one
Timm(i)=(i−1)·T for 1≤i≤n. (12)
suchcopyleadstoasuccess.Asthereisonlyonedatapacket,
ThisspecifieswhentheFECpacketsaresent,butnotonwhich all the redundancy packets (i.e., the duplicates of the data
path. A good and commonly used guideline for Rimm is to packet)canbegeneratedalreadyattimet=0.Thiseliminates
spreadthepacketsoneachpathseparatelywith(roughly)even all the time dependenciesbetween the paths. Therefore,every
spacing [10]. When the rates are equal, i.e., n =n =...=n , path P separately must maximize the probability of at least
1 2 R r
then thisboilsdownto a simple round-robinscheduleapplied one successful transmission. It is achieved by even spreading
on the maximalallowed time interval[0,tspr−t ]. (Theproof D. Capacity constraints 7
FEC r
for the under repetition code on a single path can be found
SofarwehaveconsideredthecasewhereeverypathP can
r
in [14].) This, in turn, is exactly what Spread returns under
beassignedwithanyrate0≤n ≤n.Inpractice,however,P
r r
FEC(n,1). (cid:4) may have a relatively limited capacity, which would impose
a direct constraint on n . Fortunately, integrating these con-
r
Spread builds on even packet spreading - a simple and
straints in our model is straightforward. Indeed, it is enough
widely accepted guideline that is often thought of as leading to respect them when computing the rates n ...n in Sspr
1 R opt
to the optimal solution. Indeed, its optimality was proven for and Sspr in IV-C.
some particular cases [14]. But, surprisingly, this is not a imm
general result. Consider for example FEC(4,3) on a single V. PERFORMANCE EVALUATION
path (i.e., R = (1,1,1,1)) with loss rate π(1) = 1%
B Inthissectionweevaluateourapproachfirstinsimulations
and average loss burst length 1/µ(1) = 5ms, and available and next on real-life traces.
B
time interval equal to 15ms. The even spreading schedule
S = ((0,5,10,15),R) yields π∗ = 0.53%. But the optimal A. Simulation results
1 B
schedule (foundwith optimizationtoolsof Mathematica[18]) The goal of simulations is twofold. First, we verify the
is S1 =((0,7.16,12.51,15),R)and yields πB∗ =0.50%. correctness of our analytical results. Second, we can test our
ThismeansthatSpreaddoesnotguaranteeoptimalityinthe idea in a fully controlled environmentand study the effect of
generalFEC(n,k)case.However,weshowlaterinsimulations various parameters on the results.
thatitusuallyleadsto almost-optimalsolutionsandisthusan
1) Default values of parameters: If not stated otherwise,
effective and practical rule of thumb.
in our simulations we use the following default values. The
data packets are generated at the source with interval T =
C. Comparison of Simmand Sspr: Optimal schedules Simm
opt 5ms. Next they are encoded by systematic FEC(10,8) and
and Sspr, and loss rate improvement γ.
opt sent over R independent paths. For the sake of simplicity we
It was shown in previous studies that Immediate multipath speak mainly of systems with R = 2 paths: P and P . It
1 2
is better than a single path communication. The main point allowsusto describethe pathpropagationtimedifferencesby
we make here is that once we allow for multipath, the Spread a single parameter ∆t = t −t that takes the default value
2 1
schedule Sspr that we propose in this paper is significantly ∆t = 100ms. Finally, the paths P1 and P2 have the same
better than the Immediate schedule Simm representing the averagefailurerateπ1 =0.01andtheaveragelossburstlength
state of the art. equal to 1/µ =10ms.
1
In order to demonstrate this, we compare the performance 2) The effectivelossrateπ∗ asafunctionof∆t: InFig.7
B
of Simm and Sspr in terms of their effective loss rates. What we plotthe effectiveloss rate π∗ as a functionof ∆t for four
B
rates n1...nR and what FEC block transmission time tFEC different schedules. Our first observation is that the results
should we use to make this comparison meaningful and fair? obtained in a simulation of the model (circles) fit precisely
WeshouldallowImmediateandSpreadtooptimizetheirrates the analytical curves (plain lines).
n1...nR independently,giventhattheyimposeidenticalFEC Next,wecomparetheperformanceofvariousschedules.As
block transmission times timm = tspr. More precisely, we the loss propertiesof the two paths are identical, the previous
FEC FEC
assume that the FEC parameters n and k are fixed, and we techniquesdescribedin[4,5,10]splitthe FECpacketsequally
proceedintwo steps. First, we optimizetheratesn1...nR of between P1 and P2. This results in the optimal Immediate
IImt rmeseudlitastei,nstuhcehothpatitmthaleIemffmecetdiviaetelosscshreadteulπeB∗Soiimpstmm.inTimhiisz,eidn. sscchheedduullee uSsoiempstmmu=ltipSa(it5mh,5mt)ra,nis.em.,iswsiiotnh,nit1is=nont2su=rp5r.isiAngs tthhaist
tsrueerstnut,lsFtpiEsnrpCge=ciinfitetiFhmsEeCmtiFomEpaCmtnimdaaoslpsSthipmorwiezaneditnshceh(2er)ad.tueIlsnenSth1osep.ptr.s..e3ncoRndofstSeppr,eawde, sSacfoifhmpeetcmdtutlhseiegSnp(iie1mfir0fmc,0oa)rn.mtNlayontcoeeuatoplfseorafntohyramot,sfbtythheecmons.isntrgulectipoant,h∆Itmdmoeesdinaotet
Finally, we define the relative effective loss rate improve- In contrast, in Spread Sspr we use the same rates as
(5,5)
ment γ as the relative gain in πB∗ due to the usage of optimal in Soimptm, but we spread the packets uniformly within the
Spread instead of optimal Immediate, i.e., time budget timm set by Simm (similar schedule is shown
FEC (5,5)
inFig.3E).Itresultsinafurtherdecreaseoftheeffectiveloss
π∗(Simm)
γ = B opt . (13) rate π∗. Thisdifferencemoderatelygrowswith∆t. However,
π∗(Sspr) B
B opt forlarger∆ttherates(5,5)becomesuboptimalunderSpread.
For instance, in the inset in Fig. 7 we show the performance
The metric γ can be precisely evaluated by formulas (7) and
of Spread under various rate configurations (n ,n−n ); the
(11). The values of γ can be easily interpreted; for example, 1 1
minimum is reached for (7,3). As descried in IV-C, allowing
γ >1 means that Spread performs better than Immediate.
forthisrateoptimizationleadstotheoptimalSpreadschedule
3Note that Soimptm and Sospptr are optimal subject to their construction Sospptr. Its advantage over S(i5m,5m) grows roughly exponentially
constraints presented inIV-AandIV-B,respectively. with ∆t.
8
70
10-2 SinglepathImmediateS(i1m0m,0) s] 60
m
[ 50
∗B ImmediateoptimalSoimptm=S(i5m,5m) (stateoftheart) prEC40
π sF
t
e − 30
at
ossr 10-3 SpreadS(s5p,r5) immtFEC1200
ectivel 0.002 ∆t=50ms SpreadoptimalSospptr P0r0opag20atio4n0 tim60e d8i0ffe1r0e0nc1e20∆t14[0ms]
ff ∗B
E π simulation
10-4 0.001 analytical Fig.9. ThegaininFECblocktransmissiontimetFECbytheusageofSpread
n51 -6rat7e on8 P19 optimal 1in/sµte(a2d)=of1Im0mmsed,Tiate=.P5amrasm.eFteorrst:hFesEeCp(a1r0a,m8)e,tπerB(s1,)th=eoπpB(t2im)=alI0m.0m1e,d1i/atµe(Br1a)te=s
0 25 50 75 100 125 150 arenB1=n2=5,whichresultsintheeffectivelossrateπB∗(S(i5m,5m))=0.24%.
For Spread we choose the minimal FEC block transmission time tspr such
Propagation time difference ∆t thatπ∗(Sspr)≤π∗(Simm). FEC
B opt B (5,5)
Fig. 7. The effective loss rate π∗ as a function of path propagation time
B
wdiiftfherednactaep∆act.keWtespuasceinFgETC=(105,8a)tothnetwsoouricned.epTehnedelnotsspeasthosn, PP11aannddPP22, optimal Immediate multipath schedule Soimptm uses mainly (or
aremodeledbycontinuoustimeGilbertmodelwiththesameaveragefailure only) the better path, which substantially limits the gain of
rate πB = 0.01 and the average burst length equal 1/µB = 10ms. Four multipath [5,7]. This is illustrated in plot (D) by the dashed
schedules are used: • S(i1m0m,0) - all packets are sent on a single path P1 curve;theratioπ∗(Simm )/π∗(Simm)isthelargestwhenthe
n•w1iSt,hn(s5pi2,nr5t)ecrhv-oasSlepnTre,oadpti•wmiaSthl(liy5mn,5m1b)a=s-edInm2omn=edt5hiae,tevwa•liutehSosoopppftrti∆m-atSl.prraetaeAdsdwdni1ittiho=ntahnlle2y,r=atthe5es, wpaitthhsghroawveinigdedniftBficeareln(l1co0es,0sb)eptrwoBepeenrtioπepBs(t1,)aannddqπuB(i2c)k.ly diminishes
dashed curve shows the effective loss rate of the optimal schedule, where We couldexpecta similar diminishingeffectforthe advan-
packets are not restricted to even spacing on each path, as described in tage γ = π∗(Simm)/π∗(Sspr) of Spread over Immediate.
SectionIV.Theoptimalschedulewasfoundwithnumericaloptimizationtools B opt B opt
ofMathematica[18]. Inset:πB∗ asafunctionofraten1 onpathP1 for Surprisingly, this is not the case; γ remains relatively stable
∆t=50msunderSpread. Inbothfigurestheplainlinesarethetheoretical (3<γ<6)forawiderangeofvaluesofπ(2).Forπ(2) ≈0.25
valuesaccordingtoformula(11),whereasthecirclesaretheresultsobtained B B
inasimulationofthemodel.Thesizeofconfidenceintervals (notshown)is thepathP2becomestoolossy,andbothImmediateandSpread
comparable withthesizeofthecircles. send all packets on P only and thus become equivalent.
1
4) Minimizing t - decreasing delays and fighting jitter:
FEC
Finally, we observe that the performance of the optimal So far we used Spread to minimize the effective loss rate πB∗
Spread schedule Sospptr is very close to the global optimum whilekeepingtheFECblocktransmissiontimetsFpErC notlarger
(dashed curve) where packets are not necessarily evenly- than that of Immediate schedule tiFmECm. Let us now reverse
spaced, as described in Section IV. This confirms the use- the problem: Let us minimize the FEC block transmission
fulnessof the even-spreadguidelinethatwe followin Spread. time tsFpErC of Spread, while keeping its effective loss rate not
larger than that of Immediate, i.e., subject to π∗(Sspr) ≤
3) Loss rate improvement γ as a function of various pa- B opt
π∗(Simm).
rameters: Clearly, there are many parameters that affect the B opt
We plot the results in Fig. 9. The gain timm− tspr in
performanceof the schedules. We study the effect of some of FEC FEC
FEC blocktransmission time is significantand growsroughly
them on the relative loss rate improvementγ in Fig. 8.
linearlywith∆t,astimm−tspr ≃∆t/2.Thereductionoft
First, plot (A) confirms that the advantage of Spread FEC FEC FEC
brings obvious advantages to delay-constrained applications
over Immediate grows with the path propagation time differ-
using the multipath FEC system. First, the effective end-to-
ence ∆t.
end delays get smaller which allows us to reduce the playout
Second, with growing packet interval T at the source, the
timeatthe destination,keepingthe samelevelofthe effective
fixed ∆t becomes a smaller fraction of the entire FEC block
loss rate.
transmission time t . As a consequence, there is relatively
FEC
Another important interpretation is related to the delay
less to exploit and γ drops with T, see plot (B). A similar
jitter, i.e., variations of path propagation times. Indeed, in
phenomenon can be observed in plot (C), where t grows
FEC
this work, as in most previous works on multipath FEC, we
due to an increase of the number n of FEC packets.
Finally, in Fig. 8D we vary the loss rate π(2) of path P . consider the path propagation times constant and focus on
B 2 (correlated) packet losses only. However, as Spread results in
The difference between path loss rates is a crucial parameter
asmallerdelayt ,italsoleavesmorespacetoaccommodate
affecting the performance gain of Immediate multipath over FEC
potential jitter, naturally making Spread more robust to jitter
the single path transmission. Indeed, if out of two paths
than Immediate.
one is very lossy and the other one is very good, then the
9
(A) (B) (C) (D)
γ γ γ γ
nt nt nt
e e e
m m m
e e e
v v v
mpro mpro mpro πB∗(S(i1m0,m0))/πB∗(Soipmtm)
I I I
∆t T n π(1)=0.01 π(2)
B B
Fig.8. RelativelossrateimprovementγduetousageofSpreadinsteadofImmediateasafunctionoffourparameters:(A)pathpropagationtimedifference
∆t, (B)packetgenerationintervalT atthesource, (C)thesizenofaFECblock, (D)lossrateπB(2) ofpathP2.WeconsiderasystemwithR=2paths
andthefollowingdefaultparameters:FEC(10,8),∆t=100ms,T =5ms,π(r)=1%,1/µ(r) =10ms,k=n−2.Allresultsshownhereareanalytical.
B B
Theirregularshapesofthecurvesinthisandotherfiguresareexpected,becausethecomputationofγinvolvestheratesoptimization(seeIV-C).Forinstance,
infigure(D),goingfromlefttoright,theoptimalImmediateandSpreadrates(n1,n2)changegradually(andseparately)from(5,5)to(10,0);everysuch
ratetransition mayintroduce irregularities intheshapeofthecurves.
5) Other FEC parameters n,k: So far we assumed that 1) Datasets: ThetracescomefromtwodifferentPlanetLab
Immediate and Spread use the same general FEC parameters (PL)[13]experiments.Oneverypaththepacketsaresentwith
nandk;onlytheratesonparticularpathscouldbeoptimized. time-interval T, i.e., with the generation rate at the source.
However, in some cases the optimal choice of n and k under Every trace is a sequence composed of symbols G (packet
Spread may differ from that of Immediate, given the same not lost) and B (packet lost).
redundancy k/n. Every time-constrainedexperimenton PlanetLab should be
For example, according to our additional simulations (not designed and interpreted carefully. This is because at any
shown here), for the setting in Fig. 7 and ∆t > 220ms, the point in time most of PlanetLab nodes are overloaded. Not
Spread schedule using FEC(15,12) would outperform Spread only their CPU utilization is at 100%, but more importantly
with FEC(10,8). Similarly, FEC(12,6) would be better for the queueing delays experienced by the running processes
Spread than FEC(10,5) for ∆t>140ms. Note, however, that can be very significant - even up to several seconds between
this phenomenon can be observed only for relatively large two consecutive accesses to CPU. This results in incorrect
values of ∆t that rarely occur in reality. propagation time measurements and packet dropping due to
incomingbufferoverflowatthedestination[19,20].Moreover,
To conclude,the lossrate improvementγ can bemanyfold,
the situation changes dynamically. We minimize the effects
but its exact value strongly depends on various parameters.
of these problems by introducing periodic pauses in packet
First, the advantage of Spread over Immediate grows with
generation and avoiding the highly loaded PlanetLab nodes.
path propagationtime differences∆t, but drops with the data
We use the following two data sets.
packetgenerationintervalT andFECblocksizen.Second,the
optimalImmediateratesarenotalwaysoptimalunderSpread; a) ‘Relays’ - PlanetLab with relays: In this experiment
usually optimal Spread sends more packets on faster paths. every trace is collected on a two-hop overlay path between
Third, although the advantage of Immediate over a single three PlanetLab nodes: source, relay and destination. The
pathtransmissionquicklydiminisheswithgrowingdifferences UDP packets at the source are generated every T = 5ms
between the loss rates of the paths, the advantage of Spread and sent immediately to the relay that forwards them to the
over Immediate is relatively stable. Finally, Spread can also destination. After every one-second-long packet generation
achieve FEC block transmission times much smaller than periodwe introduceone second of idle time in order to avoid
Immediate,stillguaranteingthesameorbetterperformancein droppingpacketsatPlanetLabhostswhentheprobingtrafficis
termsoftheeffectivelossrate.Thisresultsnotonlyinsmaller toobursty.Wecollectedmorethan5’000traces,eachcovering
effectivedelays,butalso in a higherrobustnessto delayjitter. 100 seconds of packet generation time.
InordertofurtherreducetheeffectofoverloadedPLnodes
on the results, for every experiment separately we select the
B. Trace-driven PlanetLab evaluation
source,relayanddestinationrandomlyfrom50currentlyleast
In the previous section we presented analytical and sim- loaded PL nodes. As the load estimate we use the number
ulation results where the packet losses were modeled by of processes queueing for the CPU and I/O devices; it can
the Continuous Time Gilbert Model. As any model, it is be obtained by parsing the file /proc/stat that stores the
only an approximation of reality. In this section we feed information about kernel activity.
our simulations with real-life packet loss traces collected in b) ‘Web sites’ - PlanetLab to popular web sites: This
Internet experiments. data set consists of 2’839 traces used in [10]. They were col-
1 10
lected by sending 16-byte ICMP echo packets from 57 Plan-
‘Relays’
etLab hosts to 55 popular web sites selected from [21]. Next, 0.8 ∆t = 10ms
the ICMP echo-reply packets were captured by Tcpdump,
F0.6
resultingintraceswherepacketstravelfromaPlanetLabnode D
to a web site and back to the original PlanetLab node. The C0.4 ∆t = 50ms
packetsweresenteveryT =2ms.Asabove,everyone-second 0.2 Oracle
Prediction
packetgenerationtime was followedbyone-secondidle time. 0
0 1 2 3 4 5
Each measurementlasted at least 800seconds.As in [10],we
Loss rate improvement γ
split it into 40-second long intervals that we call chunks.
1
Despite the measures we took, in both data sets we find
‘Web sites’
traces with numerous long (100ms and more) blocks of 0.8 ∆t = 10ms
consecutivelosses.Asthisisnotcausedbynetworklosses,but
F0.6
rather by buffer overflow at the nodes due to CPU queueing, CD0.4 ∆t = 50ms
we exclude these traces from our simulations.
2) Simulationsetting: Ina simulationof a R-pathscenario 0.2 Oracle
Prediction
weuseRtraces(oneperpath)randomlychosenfromthepool 0
0 1 2 3 4 5
of all available traces. Thus, by construction,the R traces are
Loss rate improvement γ
independent(typically generated at differenttimes and places
in the Internet). For the sake of simplicity, we restrict the
Fig. 10. The effective loss rate improvement γ (by using Spread instead
presentation to the case R=2.
ofImmediate) intrace-driven simulations underFEC(10,8).WeuseR=2
Our basic metric is loss rate improvement γ. As described independent pathswithreal-life losstraces;theirpropagationtimesdifferby
in Section IV-C, it optimizes the rates of Immediate and ∆t.Weconsidertwodatasets:‘Relays’(top)and‘Websites’(bottom).
Spread. This optimization is based on the observed traces.
One approach to do this is to infer for every path its loss rate
π(r) and the average loss burst length 1/µ(r), feed them into Finally, we find our simple prediction method satisfactory,
B B as the Prediction curve is always close to Oracle.
the model and optimize the rates as in section V-A. However,
this technique has two drawbacks: it introduces errors when
VI. RELATEDWORK
measuringthepathproperties,andassumesaparticularpacket
loss model. We avoid these problems by working directly on The performance of FEC on a single path with correlated
the traces - the optimalrates in Simm andSspr are those that loss failures was studied e.g., in [1,14,15]. One common
opt opt
perform best on a given chunk. conclusion is that the FEC efficiency drops with growing
We present two types of results. In Oracle we choose the burstiness of packet losses.
optimalratesforthe currentlyevaluatedchunk.In contrast,in Multipathtransmissionasawayofde-correlatingthepacket
Prediction we use the optimal rates of the preceding chunk losses and increasing the performance of FEC was first pro-
to evaluate the current chunk. Thus Oracle shows the best posed in [3]. It got more attention recently, e.g., in [4]–
achievableresultsforImmediateandSpreadwithnoprediction [7,9,10].MultipathwasalsostudiedinthecontextofMultiple
errors, whereas Prediction is a practical implementation. Description Coding [11].
3) Results: InFig.10wepresenttheresultsforFEC(10,8). In [5] the authors study a multipath FEC system by simu-
The figure presents the cumulative distribution of the relative lations only,on artificially generatedgraphs. They also give a
loss rate improvement γ for ∆t=10ms and ∆t=50ms. We heuristic to select from a number of candidate paths a set of
consider the cases where optimal Immediate uses both paths highly disjoint paths with relatively small propagationdelays.
(i.e.,n 6=0andn 6=10)andthereisaspaceforimprovement There are a number of approaches to evaluate analytically
1 1
(i.e., π∗(Simm) > 0). In about 90% of cases we observe an theperformanceofmultipathFECwithindependentpathsand
B opt
advantage of Spread over Immediate. For instance, for both bursty path losses. For instance, [4,6,7] and [10] give four
data sets under Oracle with ∆t=50ms, in 50% of cases the different derivations of the effective loss rate π∗ (or related
B
loss rate drops by a factor of 3 or more when we use Spread metrics) in such a setting. However, in all four cases the
instead of Immediate. For smaller ∆t the advantage is less resulting formula is only an approximation of the complete
pronounced,which is in agreement with the results presented solution due to (sometimes very significant) model simpli-
in the previous section. fications. First, [6] [7] use a discrete Gilbert model. Thus
Surprisingly, in roughly 10% of cases Spread performs two consecutive packets on one path are equally correlated
slightly worse than Immediate. A possible explanation is that irrespectivelyofthetimeintervalsbetweenthem,whichmakes
in some traces we can find loss patterns that are periodic, themodelsinherentlyunabletocaptureanyaspectsofvarying
presumably due to other applications running on PlanetLab packetspacing.[10]alsousestheadiscreteGilbertmodel,but
nodes. If such an unnatural loss pattern gets aligned with the adapts the transition matrix appropriately.Second approxima-
packets scheduled by Spread on one or more paths, then the tion comes when computing the number of lost data packets
performance of Spread may drop below Immediate. given that a FEC block cannot be entirely recovered: [4] and
