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USC Computer Science Technical Reports, no. 758 (2002)

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USC Computer Science Technical Reports, no. 758 (2002)

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Content Router Mechanisms and Nash Equilibria
Debojyoti Dutta
¤
Department of Computer Science
University of Southern California
941 W. 37th Place Los Angeles, CA 90089
ddutta@isi.edu
Ashish Goel
y
Department of Computer Science
University of Southern California
941 W. 37th Place Los Angeles, CA 90089
agoel@pollux.usc.edu
ABSTRACT
Game theory and mechanism design are well known tools
to model various scenarios in economics and the social sci-
ences. Recently, several research groups have applied game
theoreticideastocomputernetworksinordertounderstand
diverse phenomena from QoS routing to multicast pricing.
In this paper, we analyze commonly used queueing policies
and indicate that Drop-tail, CHOKe and RED do not pos-
sessNashequilibriawhentheusersaregreedyandandhave
M/M/1/K sources. Then, we present a router queue man-
agement policy, PseudoRED, that imposes a Nash equilib-
rium. Finallywepresentadistributedmechanismwherethe
routermakesaadmissioncontroldecisionbasedonthelocal
information. We show that this scheme, MAX-MIN-AQM,
imposes a Nash equilibrium and the bandwidth allocation
atthisequilibriumismax-minfair.
1. INTRODUCTION
The growth of the Internet has been exponential. Such a
huge system needs to ensure stability along with good per-
formance and easy scalability. Also, a desirable property
would be ensure that everyone is satis¯ed with the service
providedtothem. Thedesignofprotocolsandroutermech-
anisms in the Internet that are very simple and yet very
stable, non dictatorial and e®ective is a challenging task.
Simple schemes lead to easy deployment. Non dictatorial
schemes imply more apparent freedom and choice for end-
users.
¤
Debojyoti is a graduate student of Computer Science at
USC.HeissupportedbyDARPAandtheSpaceandNaval
WarfareSystemsCenterSanDiego(SPAWAR)underCon-
tractNo. N66001-00-C-8066.
y
Ashish is an Assistant Professor of Computer Science at
USC. He is supported, in part, by DARPA and the Space
and Naval Warfare Systems Center San Diego (SPAWAR)
underContractNo. N66001-00-C-8066.
USC-CS-TR April 2002
Thispaperlooksatthedesignofrouterqueuemanagement
techniques from a game theoretic perspective. Speci¯cally,
we study the existence of the Nash equilibria imposed on
usersbydi®erentroutermechanisms. Gametheoryisnota
new tool. It has proved to be useful in areas such as eco-
nomics,transportationengineeringandthemanagementsci-
ences. Recently,severalresearchgroupshaveappliedGame
Theory to computer networks. See [11] and its references
foragood,concisesurvey.
The current Internet is dominated by TCP tra±c. TCP
lacks sel¯shness as the protocol leaves little choice for the
greedy user. However, there has been a recent surge in the
amountofnoncongestionreactivetra±cduetomultimedia
streaming and network games for example. It is not clear
that future applications like Video on Demand will also be
TCPfriendly. Also,distributeddenialofserviceattackslead
totra±cthatisnoncongestionreactiveandoblivioustoso-
cialgood. Thisisanindicationthatnoncongestionreactive
tra±c may be on the rise. Thus, it seems reasonable to
modelInternetusersasbeingsel¯sh,thatis,thosewhoare
not concerned about social good. They are only concerned
abouttheirownqualityofservice. Atthesametime,these
sel¯sh users desire freedom and they prefer non-dictatorial
schemes.
The¯rstroutermechanismtobeimplementedwasdrop-tail
queueing. Unfortunately drop-tail queueing cannot guar-
antee protection of well behaved, congestion reactive °ows
from misbehaving °ows. Several active queue management
techniques have been proposed such as RED [4], FRED[7],
FairQueueing[2],Di®serv[8],CSFQ[13]andCHOKe[10].
Someoftheseschemestrytosolvetheproblemglobalsyn-
chronization by randomizing packet losses. Others try to
solvetheproblemoffairresourceallocation. Othersensure
that misbehaving °ows are punished. Some of them even
try to provide the best services in the presence of sel¯sh,
misbehavingagents. Thequestionthatfollowsimmediately
is whether these schemes impose Nash equilibria on sel¯sh
users.
Hence,thereisaneedtostudyAQMstrategiesthatimpose
Nashequilibriaonsel¯shagents. Awellknowndirectionis
toanalyzetheseschemeswithtoolsfromgametheory. The
goal of this paper is to look at very simple mechanisms for
router design that can be proved to have Nash equilibria
under sel¯sh agent behavior. We stress on the fact that
routermechanismsneedtobesimpletobeeasilydeployable.
We do not consider complex per-°ow, fair AQM strategies
that impose Nash equilibria, but, which may be very hard
toimplementanddeploy.
The format of this paper is as follows. In Section 2 of this
paper we consider some current mechanisms present in the
Internetduetotheexistingprotocolseg. drop-tailandRED
gateways. Wedeterminenetworkconditionswhentheabove
mechanismswillimposeaNashequilibrium. Afterthat,we
propose a new variant of RED that achieves Nash equilib-
rium in Section 3. This is followed by study of per-°ow
mechanisms and we show that per-°ow queueing imposes
Nashequilibrium. Finally,wepresentasimpleroutermech-
anism that imposes a Nash equilibrium and exsures that
this bandwidth allocation at this equilibrium is max-min
fair. Finally we discuss our future directions and conclude
inSection5.
1.1 Related Work
Game theory is an very mature topic. See [9] for an intro-
duction. The applications and challenges of game theory
appliedtocomputernetworksissummarizedin[11].
In [12], the authors solve a very similar problem. They as-
sumeusersaresel¯shandthestabilityarisesfromqueueing
disciplines. Unfortunately,thispaperdoesnotlookatdi®er-
entAQMstrategiesthathavebeenproposedsincethen. In
somesenseourpaperisanextensionofthispaper. Besides,
ourcriterionforNashequilibriumisbasedonthemarginal
change in the goodput of a °ow as it increases its through-
put. The authors base their analysis on the average queue
occupancyofa°ow.
WearenotawareofanyotherworkthatdiscussestheNash
equilibriumpropertiesofREDandapproximatelyfairmech-
anismssuchasCHOKe.
2. EQUILIBRIUM IN THE INTERNET
In this section, we consider current router queue manage-
ment schemes and analyze the existence of Nash equilibria
imposed by these mechanisms on sel¯sh agents. We know
that the existence of Nash equilibrium is an indication of
stability under sel¯sh behavior. First, we consider some
standard router queue management schemes and then we
propose some simple ones of our own. We shall use simple
router models with the help of M/M/1/K queueing theory.
For a good introduction to the background material, refer
to [6].
TCPtra±cisthedominanttra±cintheInternet. However
wedonotconsiderTCPtoexhibitsel¯shbehavior. Itiswell
known([5])thatthereexistsauniqueequilibriumpointin
aTCPnetworkusingM/M/1/Kqueueingtheory.
In this paper, we will focus on non-TCP sel¯sh tra±c. In
thefollowingsubsections,wewillconsidertheinteractionof
routers and CBR tra±c over UDP as sel¯sh behavior can
beachievedbyvariableratetra±coverUDP.
2.1 Drop-tail Queueing
In this section, we consider simple drop-tail routers preva-
lentintheInternet.
From M/M/1/K queueing, we know that the drop proba-
bility of a drop tail router with a bu®er size B and link
utilization ½=
¸
¹
isgivenby
P
drop
=
½
B
(1¡½)
1¡½
B+1
(1)
Thenwecanprovethefollowing.
Theorem 1. For sel¯sh agents pushing CBR tra±c and
routers implementing drop-tail queuing, there is no Nash
equilibrium.
Proof. Consider a single link with a maximum bu®er
size of B and several CBR °ows on this link. Let ¸i, Ti be
thethroughput,goodputofeach°owrespectively. Then Ti
isgivenby
Ti = ¸i(1¡P
drop
) = ¸i:
1¡½
B
1¡½
B+1
(2)
Since, the drop probability at a router is same for all °ows
traversingit,wehave
Ti =
¸i
¸
£T (3)
where
¸ =
X
i
¸i,T =
X
i
Ti and ½ =
¸
¹
(4)
Assume that only one source i changes increases its band-
width. Then
@Ti
@¸i
=
@
@¸i
(
¸i
¸
)+
¸i
¸
dT
d¸
(5)
Now the ¯rst term in the above equation is positive. The
proof of the second term being positive is shown in the ap-
pendix. Thus
@T
i
@¸
i
is positive and there can be no Nash
equilibrium.
2.2 RED
Inthissection,weborrowanapproximatesteadystatemodel
ofREDfrom[3]andweanalyzewhetherthereisanyincen-
tivefor°owstopushmoretra±c.
The model of RED we use is repeated from [3] and is as
follows. We assume that for RED router, ¸ = ½. Let
the average queue length at the router be lq and the drop
probabilityattherouterbe p. TheREDcharacteristicscan
beexpressedby
p =
8
<
:
0 if lq < min
th
(lq¡min
th
)£
pmax
max
th
¡min
th
if min
th
· lq· max
th
1 otherwise
(6)
Aslongasthequeuelengthisbelow min
th
,thedropprob-
abilityiszero. Ifqueuelengthisbetween min
th
and max
th
,
dropprobabilityincreaseslinearlybetween0and pmax. Af-
terthislimitiscrossed,dropprobabilitybecomesunity.
Lemma 1. The steady state average value of the queue-
lengthwithutilization ½anddropprobability pisgivenby
q
l
=
½(1¡p)
1¡½(1¡p)
(7)
Lemma 2. In the steady state, the average queue length
ofaREDrouterisneverlargerthan max
th
Proof. After this point, drop probability will be 1. So
thepacketswillbedroppedforsure.
Also,ifthedropprobabilityseenatthisrouteris pandthe
steadystatequeuelengthisbetweentheinterval min
th
and
max
th
,thedropprobability patthisrouterisgivenby
p =
(
½(1¡p)
1¡½(1¡p)
¡min
th
)£pmax
max
th
¡min
th
(8)
With the above model, the following theorem is easy to
prove.
Theorem 2. For the given model of RED routers and
sel¯shagentswithuncontrolledCBR°ows,thereisnoequi-
libriumpointasweapproachthemaximumthresholdofthe
REDrouter. Thatis,thereisincentiveforeach°owtopush
moretra±cthroughtherouter.
Proof. Weshallanalyzethemodelatthepointwhenthe
linkisoperatingatacriticalload. Thatis,whentheaverage
queue length is max
th
. From Lemma 1 and Lemma 2, we
canwritedown max
th
intermsofthedropprobability, p=
P
drop
,as
max
th
=
½(1¡p)
1¡½(1¡p)
(9)
Wecansolvethisfor pandwritedown
1¡p = (
max
th
1+max
th
)(
1
½
) (10)
Nowthegoodput, Ti,intermsoftheo®eredload ¸i as
Ti = ¸i(1¡p) = (
¸i
½
)(
max
th
1+max
th
) (11)
Now, without the loss of generality, let us assume that the
servicerateofpackets, ¹,is1. Thus, ½= ¸and
Ti = (
¸i
¸
)(
max
th
1+max
th
) = (
¸i
P
i
¸i
)(
max
th
1+max
th
) (12)
Now,takingpartialderivativewithrespectto ¸i,wehave
@Ti
¸i
= (
maxth
1+max
th
)(
¸¡¸i
¸
2
) (13)
Since ¸ > ¸i,
@T
i
¸
i
>0. Notethat ¸= ¸i isatrivialcase,so
we can say that there is incentive for sel¯sh agents to push
moretra±cinpresenceofothersel¯shagents.
Theintuitionbehindtheproofisasfollows. REDpunishes
all°owswiththesamedropprobability. Thus,misbehaving
°ows can push more tra±c and get less hurt (marginally).
Hence there is no incentive for any source to stop pushing
packets.
2.3 CHOKe
In[10]theauthorsdeviseasimpleaggregatestatelesspacket
dropping scheme to provide approximate fairness. Unlike
RED, CHOKe tries to impose harsher penalties on misbe-
having °ows. The basic idea is as follows. A simple FIFO
bu®eriskeptanditsoccupancyistrackedbyusinganexpo-
nentially weighted moving average (EWMA). Like RED, it
hastwothresholds, max
th
and min
th
. Iftheaveragequeue
sizeislessthan min
th
,thepacketisenqueued. Ifthequeue
sizeisbetween max
th
and min
th
,arandompacketfromthe
setofalreadyenqueuedpacketsistakenandcomparedwith
thisincomingpacket. Iftheybothbelongtothesame°ow,
they are both dropped. Else, only the incoming packet is
droppedwiththesametransferfunctionasinRED.
A simple variant model of CHOKe is Front CHOKe where
the front of the queue is used to select the packet for com-
parison. Forthismodel,thegoodput Ti isgivenby
Ti =
¹¸i
¹+2¸i
(14)
when
P
i
¸
i
¹+2¸
i
<1. Beyondthis,accuratemodelsforgood-
put do not exist. In the above regime, it is easy to see the
following.
Theorem 3. When
P
i
¸
i
¹+2¸
i
<1,CHOKedoesnothave
aNashequilibrium.
Proof. Di®erentiating Equation 14 with respect to ¸i,
wehave
@Ti
@¸i
= (
¹
¹+2¸i
)
2
(15)
Thus
@T
i
@¸
i
>> 0. Hence there is incentive for sel¯sh agents
to push more tra±c through a CHOKe router. Hence the
theorem.
2.4 Is the Internet game really stable
One of the fundamental questions that we address in this
paper is whether there exist Nash equilibria in routers cur-
rentlydeployedintheInternet. Itisonlynaturaltoviewthe
wholeInternetasagamewherealmostallusershavesel¯sh
motivesandalmostalltherouterstriestoachievethisobliv-
ioussocialgood. Itisahardquestiontoaskwhetherequilib-
riaexistgiventheheterogeneousnatureofthecurrentInter-
net and the variations in Internet tra±c. As shown above,
the current dominant router queue management strategies
donotimposeNashequilibriainthepresenceofsel¯shUDP
tra±c. Insuchsituations,itisdi±cultfortheInternetgame
tohaveaNashequilibrium.
3. PSEUDO RED
In the last section, we saw that current routers cannot im-
poseNashequilibriaonsel¯shusers. Inthissection,weshow
thatNashequilibriumcanbeachievedbyslightlymodifying
theREDtransferfunction. Wepresentamechanism,Pseu-
doRED,wherethetransferfunctionforthedropprobability
intermsofthetotalo®eredloadcanbewrittendownas
P
drop
=
8
>
<
>
:
0 ½ < ½
th
½
1¡½
¡B
th
B¡B
th
½
th
< ½ < ½
critical
1 otherwise
(16)
Note that ½
th
is the o®ered load that pushes the average
bu®er size to B
th
and ½
critical
is the load that ¯lls up the
bu®er. Also, instead of using the measured queue length,
as in RED, we have used virtual queue lengths using the
M=M=1 model. Now, we ask whether this AQM strategy
hasaNashequilibrium.
Theorem 4. PseudoREDimposesaNashequilibriumon
sel¯shagentsinjectingCBRtra±coverUDP.
Proof. Supposetherearemany°owswitheachsource i
o®eringaloadof ¸i. Thegoodput Ti isrelatedto ¸i by
Ti = ¸i(1¡P
drop
) (17)
Takingpartialderivatives,weget
@Ti
@¸i
= 1¡P
drop
¡¸i£
@P
drop
@¸i
(18)
Now,wecanwrite
¸i:
@P
drop
@¸i
= ¸i:
dP
drop
d¸
=
1
B¡B
th
(
¸i
¸
)
½
(1¡½)
2
(19)
Now the term in Equation 19 is the dominant term that
determines whether
@T
i
@¸
i
<0. Considerscenarioswhere ½is
closeto½
critical
. Weknowthat
½
1¡½
isthenclosetoB¡B
th
.
Also the drop probability P
drop
is close to 1, say it is 1¡²
forsomesmallpositiverealnumber ². Then,
@Ti
@¸i
¼ ²¡
¸i
¸(1¡½)
(20)
Thus, for scenarios with loads such that
¸
i
¸(1¡½)
> ²,
@T
i
@¸
i
is
negative. Thuswecanseethatthereexistconditionswhere
there is no incentive for CBR °ows to arbitrarily ramp up
their bandwidth. Thus Nash equilibrium exists in each of
those scenarios. Compared to RED, our scheme imposes a
Nash equilibrium becuase our PseudoRED punishes more
severelyasthebu®ersget¯lledclosetocapacity.
Corollary 1. If all °ows in a router have equal share
ofthebottlenecklink,PseudoREDensuresequilibriumwhen
the bu®er length B is larger than the number of °ows, say
K.
Proof. Supposewewanttomake thefraction
¸
i
¸(1¡½)
>
1. Then we can guarantee equilibrium for arbitrary values
of ² in Equation 20. Now,
¸
i
¸
=
1
K
Also
1
1¡½
<
B
½
. Thus,
K <
B
½
forall ½
th
< ½ < ½
critical
.
0 5 10 15 20
Number of flows
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Total Offered Load
Goodput
Figure 1: Variance of the throughput and the good-
put at a PseudoRED router as the number of °ows
changes
3.1 Discussion
We now need to analyze the properties of this Nash equi-
librium. We know that there is an equilibrium point when
@T
i
@¸
i
is 0. Also, to ensure whether our solution is valid, we
need to see whether
½
1¡½
< B. We took some sample pa-
rameters, Bt = 10; B = 50; k =
¸
i
¸
= 5;::;10 and tried
to solve the equation,
@T
i
@¸
i
= 0. We found out that as k
increased,thegoodputofthesystemstarteddecreasingdue
to a sudden increase in the drop probability as the bu®er
lengthapproached B. ThisisshowninFigure1. Inthe¯g-
ure,they-axisplotsboththethroughputandgoodputaswe
increasethenumberof°owsinthex-axis. Togeneratethis
grpah,wesolvedtheequation,
@T
i
@¸
i
=0for ¸assumingthat
thevalueof ¸i is
1
k
thofthetotalo®eredload(throughput).
Thenwecalculatedthegoodputforthatloadandthecom-
puterdropprobability. Thus,eventhoughoursolutionhas
a Nash equilibrium, it is clear that the equilibrium points
notleadtoveryhighutilization.
4. FAIRNESS AND EQUILIBRIUM
In[12],theauthorsshowedthattheirfairschedulingscheme
imposes a Nash equilibrium for a single link. In this paper
we show that we can design a mechanism that imposes a
Nash equilibrium for a general network. In this section, we
¯rst we discuss fair queueing strategies. Then, we analyze
a simple distributed router mechanism, MAX-MIN-AQM,
thatensuresaNashequilibriumwithamax-minfairalloca-
tion(foranintroductiontofairnesssee[1]).
4.1 Fair Queuing
Mechanisms such as FRED, CHOKe and CSFQ try to ap-
proximate per-°ow fair queueing. Per-°ow fair queueing at
therouterimposeNashequilibriainnetworkswithasingle
bottleneck link carrying non-congestion reactive tra±c. If
we do per-°ow queueing, we can implement some measure
of fairness F. F is a function that assigns weights wi to
eachvirtualqueueforasingle°ow i. Now,ifanyusersends
more than their Fair Share, the router can decide to drop
the °ow or drop the excess packets. Thus, there can be no
incentiveforanyroutertopushexcesspackets. Thusthere
wouldbeNashequilibria.
Also Per-°ow Fair Queueing results in Nash equilibria in
general networks with multiple bottleneck links. If every
router implements fair queueing, a misbehaving °ow will
getpunishedateveryrouteranditstotalgoodputwouldbe
theminimumhewouldhavegotoveralltheroutershadthat
router been the bottleneck link. Since the above discussion
ensures that there is no incentive for the misbehaving °ow
topushtra±catanyrouter,thereisnoincentiveforanyone
tobegreedyinthegeneralnetworks'case.
Hence we see that per-°ow strategies are dictatorial and
strict. Trivially, they impose Nash equilibrium. But these
schemesaredi±culttodeployandtheyrestrictthesel¯sh-
ness of the end user. We need to design mechanisms that
droppacketsinanobliviousfashionandresultinfairband-
width allocation amongst di®erent °ows. By oblivious we
meanthatarouterdropsapacketwithoutlookingatwhich
°owitbelongsto.
4.2 MAX-MIN-AQM
ThePseudoREDresultsonlyshowthatitispossibletoim-
pose a Nash equilibrium using an oblivious drop strategy.
But it does not predict the properties of the Nash equilib-
rium. Now we present a simple strategy that will ensure
that a Nash equilibrium can be imposed and the that the
bandwidth allocation at this Nash equilibrium is max-min
fair.
Problem: SupposewehaveanetworkG(V;E)withnsource
destination pairs each with a desired tra±c rate. Then we
candescribethedemandsoftheusersasalistoftuplesofthe
form < si;ti;xi >where si, ti, xi standsforthe ithsource,
destination, demands respectively. Each router has knowl-
edge of the °ows that traverse through it. The problem is
to¯ndaroutermechanismthatusesthislocalinformation
that imposes a Nash equilibrium. Also the equilibrium (if
any) bandwidth allocation is max-min fair. Hence we have
thefollowingtheorems.
Theorem 5. MAX-MIN-AQMensuresaNashequilibrium.
Theorem 6. The Nash equilibrium obtained by MAX-
MIN-AQMismax-minfair
Corollary 2. MAX-MIN-AQMisstrategyproof
5. CONCLUSION AND FUTURE WORK
In this paper we have investigated the existence of Nash
equilibrium, in the presence of sel¯sh agents, of existing
routerqueuemanagementschemessuchasdrop-tail,CHOKe
and RED. We have shown that none of these policies can
imposeaNashequilibrium. Wehavealsoshownthatthere
exist simple mechanisms that do impose Nash equilibria.
WeproposetwomechanismsnamelyPseudoREDandMAX-
MIN-AQM that can impose equilibria. Also, we note that
MAX-MIN-AQMensuresthatthebandwidthsallocationat
Nashequilibriumismax-minfair.
We need to explore each of the above proposed policies in
detail. For PseudoRED, we need to explore why the Nash
equilibriumpointdoesnotresultingoodnetworkutlization.
OurconjectureisthatPseudoRED'sdropfunctionbecomes
very harsh as we reach equilibrium. Thus, there is a need
tostudygentlerversionsofPseudoRED.
FinallyweneedtoinvestigatewhetherwecanemulateMAX-
MIN-AQM like strategies in an oblivious fashion without
explicitly requiring the users to declare the bandwidth de-
mandsinabeste®ortInternet.
6. REFERENCES
[1] RishiBhargava,AshishGoel,andAdamMeyerson.
Usingapproximatemajorizationtocharacterize
protocolfairness.InSIGMETRICS/Performance,
pages330{331,2001.
[2] A.Demers,S.Keshav,andS.J.Shenker.Analysisand
simulationofafairqueueingalgorithm.Sigcomm,
1989.
[3] DebojyotiDutta,AshishGoel,andJohnHeidemann.
Fasternetworksimulationwithscenariopre-¯ltering.
Technicalreport,USC/ISITechReport550,
November2001.
[4] S.FloydandV.Jacobson.Randomearlydetection
gatewaysforcongestionavoidance.IEEE/ACM
TransactionsonNetworking,V.1N.4,1993.
[5] R.J.Gibbens,S.K.Sargood,C.VanEijl,F.P.Kelly,
H.Azmoodeh,R.N.Macfadyen,andN.W.Macfadyen.
Fixed-pointmodelsfortheend-to-endperformance
analysisofipnetworks.InProceedingsofthe13thITC
SpecialistSeminar: IPTra±cMeasurement,Modeling
andManagement,Sept2000,Monterey,California,
2000.
[6] L.Kleinrock.QueueingTheoryVolume1.Wiley,1967.
[7] DongLinandRobertMorris.Dynamicsofrandom
earlydetection.SIGCOMM'97,pages127{137,
september1997.
[8] K.Nichols,V.Jacobson,andL.Zhang.Atwobit
di®erentiatedservicesarchitecturefortheinternet.
RFC2638,1999.
[9] MartinJ.OsborneandArielRubenstein.ACoursein
GameTheory.TheMITPress,Cambridge,
Massachusetts,1994.
[10] RongPan,BalajiPrabhakar,andKonstantios
Psounis."CHOKe-Astatelessqueuemanagement
schemeforapproximatingfairbandwidthallocation".
InINFOCOM,2000.
[11] ChristosH.Papadimitrou.Algorithms,games,and
theinternet.InSTOC,2001.
[12] Shenker.Makinggreedworkinnetworks: A
game-theoreticanalysisofswitchservicedisciplines.
IEEETNWKG:IEEE/ACMTransactionson
NetworkingIEEECommunicationsSociety,IEEE
ComputerSocietyandtheACMwithitsSpecial
InterestGrouponDataCommunication(SIGCOMM),
ACMPress,3,1995.
[13] IonStoica,ScottShenker,andHuiZhang.
Core-statelessfairqueueing: Achievingapproximately
fairbandwidthallocationsinhighspeednetworks.
SIGCOMM,1998.
APPENDIX
Lemma 3. In a M/M/1/K queue, the goodput increases
astheo®eredloadincreases.
Proof. Assume the service rate ¹ = 1. Hence o®ered
load ¸= ½,where ½=
¸
¹
. Thegoodput T isgivenby
T = ¸(1¡P
drop
) =
½(1¡½
B
)
1¡½
B+1
(21)
Thus,
T = 1¡
1
1+½+½
2
+:::+½
B
(22)
Thus,weseethat T hasalwaysapositivederivative. Thus,
goodputincreaseswitho®eredload

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Description

Debojyoti Dutta, Ashish Goel. "Router mechanisms and nash equilibria." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 758 (2002).

Asset Metadata

Creator Dutta, Debojyoti (author), Goel, Ashish (author)

Core Title USC Computer Science Technical Reports, no. 758 (2002)

Alternative Title Router mechanisms and nash equilibria (title)

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)

Tag OAI-PMH Harvest

Format 6 pages (extent), technical reports (aat)

Language English

Unique identifier UC16269506

Identifier 02-758 Router Mechanisms and Nash Equilibria (filename)

Legacy Identifier usc-cstr-02-758

Format 6 pages (extent),technical reports (aat)

Rights Department of Computer Science (University of Southern California) and the author(s).

Internet Media Type application/pdf

Source 20180426-rozan-cstechreports-shoaf (batch), Computer Science Technical Report Archive (collection), University of Southern California. Department of Computer Science. Technical Reports (series)

Access Conditions The author(s) retain rights to their work according to U.S. copyright law. Electronic access is being provided by the USC Libraries, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Repository Email csdept@usc.edu

Inherited Values

Title Computer Science Technical Report Archive

Description Archive of computer science technical reports published by the USC Department of Computer Science from 1991 - 2017.

Coverage Temporal 1991/2017

Repository Email csdept@usc.edu

Repository Name USC Viterbi School of Engineering Department of Computer Science

Repository Location Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089

Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)