From Altruism to Non-Cooperation in Routing Games

34 Pages·2008·0.86 MB·English

by  Amar P. Azad

From Altruism to Non-Cooperation in Routing
Games
AmarP.Azad
JointworkwithEitanAltman,INRIA,MAESTROGroup,SophiaAntipolis
RachidElAzouzi,UniversityofAvignon,LIA/CERI.
July 11,2008
AmarP.Azad (INRIA) RoutingGame July11,2008 1/31
Outline
Outline
1 ModelandProblemFormulation
RoutingGame
CooperationParadigm
ProblemFormulation
NetworkTopologywith Cooperation
2 NumericalInvestigation
Experiments
ObservationsSummary
3 Existence andUniquenessofNEP
Assumptions
4 Summary
AmarP.Azad (INRIA) RoutingGame July11,2008 2/31
Outline
Outline
1 ModelandProblemFormulation
RoutingGame
CooperationParadigm
ProblemFormulation
NetworkTopologywith Cooperation
2 NumericalInvestigation
Experiments
ObservationsSummary
3 Existence andUniquenessofNEP
Assumptions
4 Summary
AmarP.Azad (INRIA) RoutingGame July11,2008 2/31
Outline
Outline
1 ModelandProblemFormulation
RoutingGame
CooperationParadigm
ProblemFormulation
NetworkTopologywith Cooperation
2 NumericalInvestigation
Experiments
ObservationsSummary
3 Existence andUniquenessofNEP
Assumptions
4 Summary
AmarP.Azad (INRIA) RoutingGame July11,2008 2/31
Outline
Outline
1 ModelandProblemFormulation
RoutingGame
CooperationParadigm
ProblemFormulation
NetworkTopologywith Cooperation
2 NumericalInvestigation
Experiments
ObservationsSummary
3 Existence andUniquenessofNEP
Assumptions
4 Summary
AmarP.Azad (INRIA) RoutingGame July11,2008 2/31
ModelandProblemFormulation RoutingGame
Outline
1 ModelandProblemFormulation
RoutingGame
CooperationParadigm
ProblemFormulation
NetworkTopologywith Cooperation
2 NumericalInvestigation
Experiments
ObservationsSummary
3 Existence andUniquenessofNEP
Assumptions
4 Summary
AmarP.Azad (INRIA) RoutingGame July11,2008 3/31
ModelandProblemFormulation RoutingGame
System Model
Network: a graphG =(V,L)
V isasetofnodes
L⊆V ×V issetofdirectedlinks.
I = {1,2,...,I} is a set ofuserswhich share thenetworkG.
fi= flowofuseri in linkl .
l
Eachuseri hasa throughputdemandrate ri (which can besplit
amongvariouspath).
Strategy: fi = (fi) isthe routingstrategy ofuseri.
l l∈L
Assumptions:
Atleastonelink exist betweeneach pairofnodes(ineach
direction).
Flowispreservedatall nodes.
AmarP.Azad (INRIA) RoutingGame July11,2008 4/31
ModelandProblemFormulation RoutingGame
Nash Equilibrium
Cost/Utility function Ji(f) = fiT(f).
l l l l
Eachuserseeks to minimize thePcost function Ji, which dependsupon
routingstrategy ofuseri as wellas onthe routingstrategy ofother
users.
NashEquilibrium
Avector˜fi,i = 1,2,...,I iscalledaNashequilibriumifforeachuseri,˜fi
minimizesthecost function giventhatotherusers’routing decisions
are˜fj, j 6= i. In otherwords,
J˜i(˜f1,˜f2,...,˜fI)= minJˆi(˜f1,˜f2,...,fi,...,˜fI),
fi∈Fi
i = 1,2,...,I , (1)
whereFi is the routingstrategy spaceofuser i.
AmarP.Azad (INRIA) RoutingGame July11,2008 5/31
ModelandProblemFormulation RoutingGame
Network Topology
Considerthe followingnetworktopology
LoadBalancingNetwork ParallelLinkNetwork
3
2
l1 l2
l3 l l2
r11 l4 r22 1
1
r1 r2
Jˆi = fiT(f)
l l l
l∈{X1,...4} Jˆi = fliTl(fl)
l∈X{1,2}
AmarP.Azad (INRIA) RoutingGame July11,2008 6/31
ModelandProblemFormulation RoutingGame
Cost Function
Considerthe followingCost function.
LinearCostFunction M/M/1 DelayCostFunction
Used in Transportation Used in Queueing
Networks Networks
Ti =l(fl1i),2=,awihflie+regaisfo,r link Tl(fli) = Cli1−fli ,wherethe
C andf denotethetotal
l l
T(f ) = cf +d forlink i i
l l l capacity andtotalflow of
j j
j = 3,4.
the link l.
i
Forparallellink topologyonlylinkl, i = 1,2 exist whilefor load
i
balancingtopologylink l,i = 3,4 also exist.
i
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