Global fairness of additive-increase and multiplicative-decrease with heterogeneous round-trip times

TLDR: It is shown that the source rates tend to be distributed in order to maximize an objective function called F/sub A//sup h/ ("F/ sub A//Sup h/ fairness"), which provides some insight into the distribution of rates, and hence of packet loss ratios, which can be expected in a given network with a number of competing TCP or TCP-friendly sources.

Abstract: Consider a network with an arbitrary topology and arbitrary communication delays, in which congestion control is based on additive-increase and multiplicative-decrease. We show that the source rates tend to be distributed in order to maximize an objective function called F/sub A//sup h/ ("F/sub A//sup h/ fairness"). We derive this result under the assumption of rate proportional negative feedback and for the regime of rare negative feedback. This applies to TCP in moderately loaded networks, and to those TCP implementations that are designed to interpret multiple packet losses within one RTT as a single congestion indication and do not rely on re-transmission timeout. This result provides some insight into the distribution of rates, and hence of packet loss ratios, which can be expected in a given network with a number of competing TCP or TCP-friendly sources. We validate our findings by analyzing a multiple-bottleneck scenario, and comparing with previous results (Floyd, 1991, Mathis et al, 1997) and an extensive numerical simulation with realistic parameter settings. We apply F/sub A//sup h/ fairness to gain a more accurate understanding of the bias of TCP against long round-trip times.

Figures

Fig. 3. Parking–lot topology.

Fig. 12. Residual biasagainst long RTT connections, HETRTT, I = 0=I , I = 4, v = 3, w = 2, Kc = K=c.

Fig. 2. Feedback modeling for the HOMRTT case.

TABLE II LSE CORRECTION OF THE TCP THROUGHPUT-LOSS FORMULA (ov er q i 2 [10 4; 10 1]for i 2 [10ms600ms] ).

Fig. 13. Correction of the TCP throughput-los formula (not delayed ACKs).

TABLE I FRACTION OF CAPACITY c GIVEN TO A CLASS0 FLOWS, FOR THE PARKING–LOT, WITH xi >>r i= i FOR ALL i 2 S .

Full text

Globalfairnessofadditive–increaseand
multiplicative–decreasewithheterogeneous
round–triptimes
MilanVojnovi´c,Jean-YvesLeBoudec,andCatherineBoutremans
InstituteforComputerCommunicationsandApplications
SwissFederalInstituteofTechnologyatLausanne(EPFL)
CH-1015Lausanne,Switzerland
mvojnovi@epfl.ch SSC Technical Report SSC/1999/024
Abstract—Consideranetworkwithanarbitrarytopology
andarbitrarycommunicationdelays,inwhichcongestion
controlisbasedonadditive–increaseandmultiplicative–
decrease.Weshowthatthesourceratestendtobedis-
tributedinordertomaximizeanobjectivefunctioncalled
F
h
A
(“
F
h
A
fairness”).Wederivethisresultundertheas-
sumptionofrateproportionalnegativefeedbackandforthe
regimeofrarenegativefeedback.ThisappliestoTCPin
moderatelyloadednetworks,andtothoseTCPimplemen-
tationsthataredesignedtointerpretmultiplepacketlosses
withinoneRTTasasinglecongestionindicationanddonot
relyonre-transmissiontimeout.Thisresultprovidessome
insightintothedistributionofrates,andhenceofpacket
lossratios,whichcanbeexpectedinagivennetworkwith
anumberofcompetingTCPorTCP-friendlysources.We
validateourfindingsbyanalyzingtheparkinglotscenario,
andcomparingwithpreviousresults[1],[2],andanexten-
sivenumericalsimulationwithrealisticparametersettings.
Weapply
F
h
A
fairnesstogainamoreaccurateunderstanding
ofthebiasofTCPagainstlongroundtriptimes.
Keywords—Additive–Increase,Multiplicative–Decrease,
Fairness,Best–Effort,TCP,TCP–Friendly,TCPthroughput-
lossformula,RTT,parking–lot,StochasticApproximation,
ODE,Lyapunov.
I.INTRODUCTION
Thereisacontinuinginterestonthroughputandfair-
nessissuesofTCP[3]congestionavoidance.Thisinterest
isparticularlynourishedbytheproliferationofreal–time
“stream”applicationsovertheInternet(e.g.voice,video)
forwhichitisrequiredtobeTCP–Friendly,i.e.tofairly
coexistwithalreadyexistingTCPapplications.
Inoneofthepioneeringworks,ChiuandJain[4]formu-
latedasetofbasicprinciplesoftheadditive–increaseand
multiplicative–decreasecongestionavoidancetoachieve
efficiencyandfairness,byanalyzingthesimplemodelof
asinglebottleneck.
In[5]Kelly,Maulloo,andTanshowedthatalarge-
scalenetworkdeployingsomespecificformofadditive–
increaseandmultiplicative–decreasecongestionavoid-
ancetendstodistributeratesaccordingtoproportionalfair-
ness.Thisresultiscommonlymisinterpretedasbeingap-
plicabletocongestionavoidanceintheInternetwithTCP.
Recently,Hurley,LeBoudec,andThiran[6]showed
thatinanetworkemployingadditive–increaseandmulti-
plicative–decrease,thesourceratestendtobedistributed
inordertomaximizeanobjectivefunctioncalled
F
A
.The
authorscallthis“
F
A
fairness”.Thisresultisobtained
bythelimitmeanordinarydifferentialequation(ODE)
method,foranetworkoperatingintheregimeofrareneg-
ativefeedback.Thepivotalassumptionofthatworkis
therateproportionalnegativefeedback,whichtheauthors
claimtobemorerealisticthanonewhichdependsexclu-
sivelyontheoverallload[5].However,theresultisre-
strictedtothehomogeneousround–triptime(RTT)case
wheretheratesareupdatedsynchronously.
Inthispaper,weextendthemodelingof[6]tothehet-
erogeneousRTTcase.Ourresultisageneralizationof
F
A
fairness,whichwecall
F
h
A
fairness.Itgivesthedistribu-
tionofratesinaarbitrarynetworkemployingtheadditive–
increaseandmultiplicative–decreasemethodforconges-
tioncontrol,withtheassumptionthatnegativefeedback
israre.Weallowtheround–triptimestodifferfromone
sourcetoanother.Theratestendtomaximizeanobjec-
tivefunctioncalled
F
h
A
,whoseparametersreflecttherate
adaptationalgorithm.Tothebestofourknowledge,this
isthefirstgeneralresultencompassingmanyoftherele-
vantsystemparameters,applicabletoanarbitrarynetwork
topologywithmultiplebottlenecks.Ourresultsallowsto
findafirstorderapproximationofratedistributions;com-
binedwithaloss-throughputformulasuchas[2],[7],this
givesapredictionofthelossrates.Extensivesimulation
resultsconfirmthesepredictions.
Thenoveltyofourapproachisanapplicationofthe
recentweakconvergenceresultsofdecentralizedasyn-
chronousstochasticapproximationalgorithms[8].Our

1
modelessentiallydiffersfrom[6]inthatwedonotas-
sumethatrate–adaptationisperformedsynchronouslyby
allsources;incontrast,weuseanasynchronousmodel
whereeverysourceupdatesitsratebasedonitsownround
triptimeinterval.Unlikethesynchronousmodelin[4],
[5]or[6],thisallowsustoaddressthecasewithdifferent
roundtriptimes.Buteveninthecasewhereallroundtrip
timesareequal,thisgivesamoreaccuratemodel.Indeed,
withthesynchronousmodel,rateadjustmentisbasedon
themostrecentpreviousrates.Inreality,thefeedbackre-
ceivedbyonesourceattheendofoneroundtriptimein-
tervaldependsontheratesduringthepreviousinterval,
shiftedintimebythedelayrequiredforfeedbacktoreach
thesources.Thesynchronousmodelassumesimplicitly
thatfeedbackreachessourcesinstantaneously.Wecallthis
assumption“stolenlag”.Weshowwithourmodelingthat
thestolenlagassumptiondoesnotaffectthedistribution
ofaveragerates;bysimulation,weseehoweverthatitaf-
fectstheamplitudeofoscillations.Notethatourmodel
explicitlyconsidersallcommunicationdelays.
Weassumeinthispaperthatthenegativefeedbackre-
ceivedbysourcesisrare,andisproportionaltothesource
rate.Therarenegativefeedbackassumptionisvalidina
reasonablyloadednetwork;theproportionalassumption
shouldbetruewithactivequeuemanagement[9](e.g.
RED[10])appliedtootherwiseFIFOqueues.Inaddition,
ourmodelassumesasinglerateupdatingperRTT;thisfits
withTCPimplementationsdesignedtocopewithmulti-
plepacketlosseswithinsingleRTT,i.e.thattreatmultiple
packetlosseswithinoneRTTasasinglecongestionsignal,
andavoidre-transmissiontimeouts.
Ourmodeldoesnotincorporatetheeffectofthevari-
ationofRTTforonegivensourcefromonefeedbackin-
tervaltotheother.Itisknownthat,foranetworkwith
fixedwindows[11],thevariationofroundtriptimesdueto
queuesbuildinguphasinitselfacongestionavoidanceef-
fect,whichisnotcapturedbyourmodeling.Anotherlimi-
tationisthatweassumetheratestobepiecewiseconstant,
i.e.tobeadjustedonlyonceperroundtriptime.Thus,the
effectofburstinessatthetimescaleoftheroundtriptimeis
nottakenintoaccount.Incontrast,ourstudycapturesthe
effectofthewindoworrateadaptationmechanismfound
forexamplewithTCPorABR.Ourresultsmaybeusedas
areferencefairnessmeasureinperformanceevaluationsof
TCP–friendlyrateadjustmentalgorithms.
Inthenextsubsectionweoutlineourmainresults.
A.SummaryoftheMainResults
Weconsideranetworkwithmultiplebottlenecksand
heterogeneousround–triptimes.Then,underthecondi-
tionthatthereisnosubstantialqueuingdelayvariation,
andthenetworkisoperatingintheregimeoftherareneg-
ativefeedback,thecollectionofrates
x
=(
x
1
;:::;x
i
;:::
)
isdistributedsuchthat
x
maximizestheobjectivefunction
F
h
A
(
x
)=
X
i
2S
1

i
log
x
i
r
i
+

i
x
i
;
subjecttotheconstraints
P
j
2S
A
l
;
j
x
j

c
l
,
8
l
2L
.Inthe
formula,
S
isthesetofsources,
L
thesetoflinks,
A
l;i
the
routingmatrix(
A
l;i
is
0
or
1
),
c
l
thecapacityoflink
l
,and

i
istheRTTforfloworsource
i
.Thereisoneflowper
source.Therateadaptationparametersare
r
i
(additive–
increaseelement)and

i
(multiplicative–decreasefactor);
theymaydependonsource
i
.
Theaboveresultisappliedtotheparking–lotnetwork
topology;weobtainaclosed-formforthedistributionof
rates.Thisallowsustoverifytheconsistencyofourre-
sultwithexistingworkandwithconductedsimulations.
Wefindthattheresultsin[1]areanasymptoticcase
of
F
h
A
fairnessforsmalladditive–increase/multiplicative–
decreaseratiorelativetoconnectionthroughput.
Wealsogainamoreaccurateunderstandingofthebias
ofTCPagainstlongroundtriptimes.Wepointoutthatit
isimportanttomakethedifferencebetweenabiasagainst
longRTTs(perhapsanundesirablefeature)andabias
againstflowswithmanyhops(perhapsadesiredfeature).
Weseethatthebiasagainstflowswithmanyhopsisinthe
natureofanyrateadaptationalgorithmbasedonadditive–
increaseandmultiplicative–decrease.Incontrast,abias
againstlongRTTscanbeattenuatedwithcorrectionssuch
asmentionedin[1]and[12].Finally,wealsoconfirm
throughputlossformulas,withinthelimitationsofour
modeling.
B.OutlineofthePaper
Thepaperisorganizedasfollows.InSectionII,the
mainresultsarederived.Followingthebasicmodeldef-
initions,feedbackmodelingisdescribedinmoredetail.
Then,asymptoticconvergenceresultsofthedecentralized
asynchronousstochasticapproximationalgorithms[8]are
sketched.Intherestofthesection,objectivefunction
F
h
A
ofthealgorithmofconcernisderivedandanalyzed.In
SectionIII,
F
h
A
resultisappliedtotheparking–lotnetwork
topologyforwhichaclosed-formratedistributioniscom-
puted,andresultsareverifiedthroughnumericalsimula-
tion.InSectionIV,theresultsarediscussedandcompared
totherelatedpreviouswork.Implicationsoftheresultto
theInternetareaddressedinSectionV.InSectionVI,
concludingremarksaregiven.InAppendixAwegivethe
maintheoremoftheunderlyingtheory[8].

2

l;i
l
:
c
l
;
l

i;l
S
i
D
i
Fig.1.Anillustrationofthedefineddelays.
II.DERIVATIONOFTHEMAINRESULTS
A.ModelSetup
Thenotationisdevelopedasfollows.Letset
L
contain
networklinks.Thenlet
c
l
and

l
bethecapacityanddelay
oflink
l
2L
,respectively.Letset
S
comprisesources
(flows)thatareactiveonthegivennetwork.Therout-
ingsettingwedescribebyroutingmatrix
A
=(
A
l;i
;l
2
L
;
i
2S
)
,suchthat
A
l;i
=1
,ifflow
i
traverseslink
l
,and
A
l;i
=0
,otherwise.
1
Further,wedefinecommunicationdelays.Let

i;l
de-
notedelayfromsource
i
tolink
l
,andlet

l;i
bethedelay
fromlink
l
tosource
i
.Then,set

i;j
=
j;l
+
l;i
.Defin-
ing

i
astheRTTofsource
i
,clearly,

i
=
i;l
+
l;i
,for
all
l
suchthat
A
l;i
>
0
.InFig.1,asamplenetworkillus-
tratesdefineddelays.
Let
f

i;n
g
n

0
beanon-decreasing
[0
;
1
)
-valuedse-
quenceofrateupdatingtimesofsource
i
.Then,anum-
berofrateupdatesofsource
i
ontheinterval
[0
;t
)
is
N
i
(
t
)=
P
1
n
=1
1
f

i;n
<t
g
.
Let
f
x
i;n
g
n

0
bea
[0
;
1
)
-valuedstochasticprocess,
where
x
i;n
isarateofsource
i
atthe
n
-thupdate.Then,
defineacontinuoustimeinterpolationonrealtimeas
x
i
(
t
)=
x
i;n
,for
t
2
[

i;n
;
i;n
+1
)
.
For
b

a

0
definea

-algebraoftheform
F
i
;
l
[
a
;
b
)
=

(
x
j
;
k
:
A
l
;
j
;
A
l
;
i
>
0
;
N
j
(
a
,

i
;
j
)

k
<
N
j
(
b
,

i
;
j
))
.
Finally,let
F
i
[
a
;
b
)
=
[
l
:
A
l
;
i
>
0
F
i
;
l
[
a
;
b
)
.
Anadditive–increaseandmultiplicative–decreasealgo-
rithmhasthefollowingform
x
i;n
+1
=
x
i;n
+
r
i
(1
,
I
i;n
)
,

i
I
i;n
x
i;n
;
(1)
where
r
i
and

i
aretheadditive–increaseelementandthe
multiplicative–decreasefactor,respectively.Therandom
sequence
f
I
i;n
g
n

0
isanegativefeedbackindicationwith
valuesin
f
0
;
1
g
.Weassumethatthenegativefeedback
indication
I
i;n
isbasedonthefeedbackreceivedbetween
n
-thand
n
+1
-thrateupdating.Consequently,itturnsout
that
I
i;n
ismeasurableon

-algebra
F
i
[

i
;
n
;
i
;
n
+1
)
.
1
Wedefine
A
l;i
on
f
0
;
1
g
whichcanbeextendedto
[0
;
1]
toaccom-
modateforinstanceloadsharing,etc.[6]
(
n
+1 )
n
(
n
,
1)
s
t

n

s<
(
n
+1 )

x
i;n
+1
x
i;n
,
1
s
,

x
j
(
s
,

)
;j
:
A
l;i
;A
l;j
>
0
x
i;n
x
i
(
t
)
I
i;n
Fig.2.FeedbackmodelingfortheHOMRTTcase.
Nowletusbrieflycommentonthespecialcasead-
dressedin[6],whereitisassumedthatallround–triptimes
areequal,andtheratesareupdatedsynchronously.Fol-
lowingthedefinitionof
I
i;n
inthefullextent,itisrather
easytoseethat
I
i;n
isafunctionof
x
j;n
,
x
j;n
,
1
,and
x
j;n
,
2
,forall
j
2S
suchthat
A
l
;
i
;
A
l
;
j
>
0
,depending
onvaluesof

j;l
.Intherelatedwork[4]–[6],itiscom-
monlyassumedthat
I
i;n
iscomputedbasedon
x
j;n
,forall
j
2S
suchthat
A
l;i
;A
l;j
>
0
,whichisindeedanunreal-
isticassumption.
Letusassumethatallflowstraversinglink
l
haveequal
accessdelaytothatlink.Formally,

i;l
=
j;l
,forall
i;j
2S
,suchthat
A
l;i
;A
l;j
>
0
.Then,itfollowsthat
I
i;n
dependson
x
j;n
,
1
,forall
j
2S
suchthat
A
l;i
;A
l;j
>
0
.
WerefertothisassumptionasaHOMRTTassumption.In
addition,whenever
x
i;n
isusedwhereitshouldbe
x
i;n
,
1
werefertothisasastolenlag.Feedbackmodelingisil-
lustratedinFig.2.
B.FeedbackModeling
First,weintroduceanotionofthelinkcostfunction
g
l
(

): [0
;
1
)
!
[0
;
1]
,where
l
2L
.Atagiventime
t
,thelinkcostisafunctionofthelinkload
f
l
(
t
)=
X
i
2S
A
l
;
i
x
i
(
t
,

i
;
l
)
:
(2)
Onecaninterpret
g
l
(
f
l
(
t
))
asaprobabilityofmarking
asinglepacketattime
t
.Weareconcernedwithneg-
ativefeedbackindication,
I
i;n
,basedonfeedbackre-
ceivedwithin
[

i;n
;
i;n
+1
)
.Letuspartitiontheinterval
[

i;n
;
i;n
+1
)
intonon-overlappingintervals
[
a
k
;b
k
)
such
that
x
j
(
s
)
,isconstantfor
s
2
[
a
k
,

i;j
;b
k
,

i;j
)
,for
all
j
2S
suchthat
A
l;i
;A
l;j
>
0
.
Wedefine
M
i;l
a;b
asanamountofnegativefeedbackre-
ceivedbysource
i
fromlink
l
withininterval
[
a;b
)
,which
isequaltoanumberofmarkedpacketsoftheflow
i
bythe
link
l
withininterval
[
a
,

i
;b
,

i
)
.
Let
P
i;l
a;b
(

)
and
P
i
a;b
(

)
beconditionalprobabilitiesgiven
F
i
;
l
[
a
;
b
)
and
F
i
[
a
;
b
)
,respectively.Analogously,let
E
i;l
a;b
[

]
and
E
i
a;b
[

]
berespectiveconditionalexpectations.

3
Admittingtheinterpretationof
g
l
(

)
asaprobabilityof
markingasinglepacket,itiseasytoseethatwedohavea
binomialconditionalprobability
P
i;l
a
k
;b
k
(
M
i;l
a
k
;b
k
=
m
)=
d
x
i;N
i
(
a
k
,

i
)
(
b
k
,
a
k
)
e
m
!


g
l
(
f
l
(
a

i
k
))
m
[1
,
g
l
(
f
l
(
a

i
k
))]
d
x
i;N
i
(
a
k
,

i
)
(
b
k
,
a
k
)
e,
m
;
(3)
where
a

i
k
iswritteninlieuof
a
k
,

l;i
.Notethat
d
x
i;N
i
(
a
k
,

i
)
(
b
k
,
a
k
)
e
correspondstoanumberofpackets
offlow
i
thatarepresentonlink
l
within
[
a
k
,

i
;b
k
,

i
)
.
Clearly,theexpectedamountofnegativefeedbackis
E
i;l
a
k
;b
k
[
M
i;l
a
k
;b
k
]=
d
x
i;N
i
(
a
k
,

i
)
(
b
k
,
a
k
)
e
g
l
(
f
l
(
a

i
k
))
:
Similarly,let
M
i
a;b
beanamountofnegativefeedbackre-
ceivedwithin
[
a;b
)
bysource
i
fromalllinks
l
suchthat
A
l;i
>
0
.Itfollows
P
i
a
k
;b
k
(
M
i
a
k
;b
k
=0 )=
Y
l
:
A
l;i
>
0
P
i;l
a
k
;b
k
(
M
i;l
a
k
;b
k
=0 )
;
andfrom(3)follows
P
i
a
k
;b
k
(
M
i
a
k
;b
k
=0 )=
=
Q
l
:
A
l;i
>
0
[1
,
g
l
(
f
l
(
a

i
k
))]
d
x
i;N
i
(
a
k
,

i
)
(
b
k
,
a
k
)
e
:
(4)
Bydefinitionof
[
a
k
;b
k
)
wehave
P
i

i;n
;
i;n
+1
(
M
i

i;n
;
i;n
+1
=0 )=
Y
k
P
i
a
k
;b
k
(
M
i
a
k
;b
k
=0 )
:
(5)
Finally,
I
i;n
=1
,ifsource
i
hasreceivedanindica-
tion,within
[

i;n
;
i;n
+1
)
,thatatleastonepackethasbeen
marked,thus
P
i

i;n
;
i;n
+1
(
I
i;n
=1 )=
P
i

i;n
;
i;n
+1
(
M
i

i;n
;
i;n
+1

1)=
=1
,
P
i

i;n
;
i;n
+1
(
M
i

i;n
;
i;n
+1
=0 )
:
(6)
Letusexamine(6)fortheHOMRTTcase.Herewehave
asinglepartitionof
[

i;n
;
i;n
+1
)
,hence,from(4)–(6)fol-
lows
P
i

i;n
;
i;n
+1
(
I
i;n
=1 )=1
,
Y
l
:
A
l;i
>
0
[1
,
g
l
(
f
l
)]
d
x
i;n
,
1

e
;
(7)
where

i;n
+1
,

i;n
=

,forall
i
2S
,
n

0
,and
f
l
standsfor
f
l
(

i;n
,

l;i
)=
P
j
2S
A
l
;
j
x
j
;
n
,
1
.In
thelimitcase
g
l
(

)
!
0
,limiteddevelopmentyields
[1
,
g
l
(
f
l
)]
d
x
i;n
,
1

e
'
1
,d
x
i;n
,
1

e
g
l
(
f
l
)
,thenreplacing
thisin(7),andneglectingthehigherorderproducts,yield
P
i

i;n
;
i;n
+1
(
I
i;n
=1 )
'
X
l
2L
A
l
;
i
g
l
(
f
l
)
d
x
i
;
n
,
1

e
:
(8)
Therefore,itisshownthat,undertherarenegativefeed-
backassumption,(7)degeneratestotherateproportional
feedbackasisimplicitlyassumedin[6].However,note
that(8)dependson
x
i;n
,
1
andnoton
x
i;n
.
C.AsymptoticConvergence
Traditionaltheoryofthestochasticapproximational-
gorithms[13]–[14]isconcernedwithanalgorithmofthe
generalform
x
i;n
+1
=
x
i;n
+

n
H
i;n
(
x
i;n
;
i;n
)
;i
2S
;
where
x
i;n
isdefinedon
R
,
H
i;n
(

):
R

R
!
R
,

i;n
:
R
!
R
isarandomnoise,and

n
astepsize.It
isassumedthatcomponents
x
i;

,
i
2S
,areupdatedsyn-
chronously,facilitatingtheassociationofcontinuousinter-
polation
x
i
(
t
)
todiscreteprocess
f
x
i;n
g
n

0
onthe“natu-
ral”commoniteratetime
f
n
g
n

0
.However,itfollows
thatforanasynchronousupdatingonehastoworkinreal
time,oratleastanappropriatelyscaledrealtime[8].In
general,fordecreasing

n
,inrespectto
n
,convergence
withprobabilityonecanbeobtained,whileforconstant
small

n


,onlyconvergenceinprobabilitycanbe
proven(theweakconvergence).Intherestofthissec-
tionwebrieflysketchresultsof[8]thatareappliedinour
work.Foracompletetreatmentoftheunderlyingtheory
thereaderisreferredto[8].
Let
f


i;k
g
k

0
bearandomsequenceofupdatingin-
tervalsof
f
x
i;k
g
k

0
,
i
2S
.Thendenote(scaled)real
updatingtimeof
x
i;n
as


i;n
=

n
,
1
X
k
=0


i;k
;
(9)
andacontinuousinterpolationontheiteratetime


i
(
t
)=


i;n
,for
t
2
[
n;
(
n
+1 )

)
.Further,let
N

i
(
t
)
isanumber
ofupdatesof
f
x
i;k
g
k

0
before
t=
.Formally,
N

i
(
t
)=

1
X
n
=1
1
f


i;n
<t=
g
:
(10)
Fromthedefinitions,itturnsoutthat
N

i
(


i
(
t
))=
n
,
t
2
[
n;
(
n
+1 )

)
,i.e.
N

i
(

)
isinverseof


i
(

)
.
Then,let
^
x

i
(
t
)=
x

i;n
,
t
2
[


i;n
;

i;n
+1
)
,isacontinuous
interpolationonscaledrealtime,and
^
x

(

)=( ^
x

i
(

)
;i
=
1
;
2
;:::;S
)
,
S
=
jSj
.Fromdefinitionsof“time”pro-
cesses(9)and(10)itfollows
x

i
(
t
)=^
x

i
(


i
(
t
))
,and
^
x

i
=
x

i
(
N

i
(
t
))
.
Furthermore,let


i;j;n
beanon-negativerandomvari-
ablerepresentingascaled(multipliedby

)communica-
tiondelaybetweensource
i
andsource
j
atthe
n
-thrate
updatingofsource
i
.

4
Finally,thedecentralizedasynchronousalgorithmcan
bewrittenintheform
x

i;n
+1
=
[
a
i
;b
i
]

x

i;n
+
H

i;n
(^
x
j
(


i;n
+1
,


i;j;n
))

=
=
x

i;n
+
H

i;n
(^
x
j
(


i;n
+1
,


i;j;n
))+
Z

i;n
;i
2S
;
(11)
where

[
a
i
;b
i
]
(

)
denotesprojectionoftheargumenton
[
a
i
;b
i
]
,fortheconstrained
x
on
C
=[
a
1
;b
1
]

[
a
2
;b
2
]

:::

[
a
S
;b
S
]
,and
Z

i;n
isareflectionterm.
Letforall
i
,
F

i
;
n
and
F
;
+
i
;
n
benon-decreasing

-
algebrasmeasuringthepastdata(including
x

i;
0
,
H

j;k
,and


j;k
,
j
2S
)availableon
[0
;

i;n
+1
)
,and
[0
;

i;n
+1
]
,re-
spectively.Then,with
P

i;n
and
P
;
+
i;n
denoterespective
conditionalprobabilities,andanalogously
E

i;n
and
E
;
+
i;n
conditionalexpectations.
Subsequently,wehavethefollowingconditions.Itis
assumedthat
f
H

i;n
; 

i;n
;
;i;n
g
;
(12)
and


i;j;n
,

;
+
i;j;n
areuniformlyintegrable.Weconsider
theMartingaledifferencenoise[8],forwhichwehave
E

i;n
H

i;n
=
h

i;n
(^
x

j
(


i;n
+1
,


i;j;n
)
;j
2S
)+


i
;
n
;
where
h

i;n
(

)
arereal-valuedfunctionscontinuousin
n
and

,


i;n
isasymptoticallynegligiblenoise,and
sup
n

T= 


i;j;n
!
0
.Therearereal-valuedfunctions
u

i;n
(

)
thatarestrictlypositive(
inf
n;;x;
u

i;n
(
x;
)
>
0
)
andarecontinuousuniformlyin
n
and

,andnon-negative
randomvariables

;
+
i;j;n
suchthat
E
;
+
i;n


i;n
+1
=
u

i;n
+1
(^
x

j
(


i;n
+1
,

;
+
i;j;n
+1
)
;j
2S
)
;
(13)
where
sup
n

T= 

;
+
i;j;n
!
0
;
inprobabilit ya s

!
0
.
Therearecontinuousreal-valuedfunctions

h
i
(

)
suchthat
foreach
x
2
C
,
2
lim
m;n;
1
m
n
+
m
,
1
X
k
=
n

[
h

i;k
(
x
)
,

h
i
(
x
)]=0
:
(14)
Therearecontinuousreal-valuedfunctions

u
i
(

)
suchthat
forall
x
2
C
,
lim
m;n;
1
m
n
+
m
,
1
X
k
=
n
E
;
+
i;n
[
u

i;k
(
x
)
,

u
i
(
x
)]=0
:
(15)
Suppose
lim
m;n;
n
+
m
,
1
X
k
=
n
E

i;n


i;k
=0
;
inmean
:
(16)
2
In(14),(15),and(16),
lim
m;n;

lim
m
!1
;n
!1
;
!
0
,simulta-
neouslyinanyway.
Finally,fromtheTheorem[8](AppendixA)particularly
followsthat,fortheunconstrainedalgorithm,theweak
convergencesubsequence
^
x
isthelimitsetofODE
_
^
x
i
=

h
i
(^
x
)

u
i
(^
x
)
;i
2S
:
(17)
Thus,thelimitmeanODEisthesameasinthesyn-
chronouscase,exceptforanadditionalweightfactorthat
takesintoaccountfrequencyoftheupdating.
D.
F
h
A
Fairness
Weidentify
H

i;n
of(11)inthealgorithm(1)as
H

i;n
=
r
i
,
(
r
i
+

i
x
i;n
)1
f

i;n
<P
i
i;n
+1
(
I

i;n
=1)
g
;
(18)
where
f

i;n
g
n

0
isasequenceofindependentrandom
variablesuniformlydistributedon
[0
;
1]
.
3
Then,
E

i;n
+1
H

i;n
=
r
i
,
(
r
i
+

i
x
i;n
)
P

i;n
+1
(
I

i;n
=1 )
;
where
P

i;n
+1
(
I

i;n
=1 )
isgivenby(6).
Inthelimitcase,as

!
0
,and
n
!1
,weneglect
scaleddelays

i;j
,thentheprobabilityofnegativefeed-
backis
P

i;n
+1
(
I

i;n
=1 )=^
x
i
(

;
,
i;n
+1
)

i;n
X
l
2L
A
l;i
g
l
(
^
f
l
(

;
,
i;n
+1
))
:
For

i;n
=

i
,forall
n>
0
,themeanvectorfieldis

h
i
(^
x
(
t
))=
r
i
,
^
x
i
(
t
)(
r
i
+

i
^
x
i
(
t
))

i
X
l
2L
A
l
;
i
g
l
(
^
f
l
(
t
))
(19)
where
^
f
l
(

)=
P
j
2S
A
l
;
j
^x
j
(

)
.
Seemingly,

u
i
(^
x
)=

i
,thenwith(19)thelimitmean
ODE(17)becomes
_
^
x
i
=
r
i

i
,
^
x
i
(
r
i
+

i
^
x
i
)
X
l
2L
A
l
;
i
g
l
(
^
f
l
)
:
(20)
Followingthesamestepsasin[6]weexpressODE(20)as
_
^
x
i
=^
x
i
(
r
i
+

i
^
x
i
)
@J
h
A
(^
x
)
@
^
x
i
;
(21)
where
J
h
A
(^
x
)=
X
i
2S
1

i
log
^x
i
r
i
+

i
^x
i
,
G
(
^x
)
;
(22)
andbydefinition
G
(^
x
)=
P
l
2L
G
l
(
^
f
l
)
,where
G
l
(

)
isa
primitiveof
g
l
(

)
.
3
Notethatinsteadof
r
i
and

i
itshouldbewritten
r

i
and


i
,where
r
i
=
r

i
and

i
=


i
,butweabusethisfornotationsimplicity.

Frequently asked questions

Q1. What have the authors contributed in "Global fairness of additive–increase and multiplicative–decrease with heterogeneous round–trip times" ?

The authors show that the source rates tend to be distributed in order to maximize an objective function called F h A ( “ F h A fairness ” ). This applies to TCP in moderately loaded networks, and to those TCP implementations that are designed to interpret multiple packet losses within one RTT as a single congestion indication and do not rely on re-transmission timeout. This result provides some insight into the distribution of rates, and hence of packet loss ratios, which can be expected in a given network with a number of competing TCP or TCP-friendly sources. 

Q2. What have the authors stated for future works in "Global fairness of additive–increase and multiplicative–decrease with heterogeneous round–trip times" ?

Further research should concentrate on analyzing the impact of RTT variations due to queueing. 

Q3. What is the nature of the bias against flows?

The authors see that the bias against flows with many hops is in the nature of any rate adaptation algorithm based on additive– increase and multiplicative–decrease. 

Q4. What is the effect of the inverse on the network?

under the condition that there is no substantial queuing delay variation,and the network is operating in the regime of the rare negative feedback, the collection of rates x =( x1;:::;x i;::: ) is distributed such that x maximizes the objective functionF hA(x) = X i2S 1 i logxiri + ixi ;subject to the constraints Pj2S Al ;j xj cl , 8l 2 L.