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Competitive Routing in Multi-User Communication Networks
Presentation By: Yuval Lifshitz
In Seminar: Computational Issues in Game Theory (2002/3)
By: Prof. Yishay Mansour
Original Paper: A. Orda, R. Rom and N. Shimkin, “Competitive Routing in Multi-User Communication Networks”, pp. 964-971 in
Proceedings of IEEE INFOCOM'93
Introduction
• Single Entity – Single Control Objective– Either centralized or distributed control– Optimization of average network delay– Passive Users
• Resource shared by a group of active users– Different measures of satisfaction– Optimizing subjective demands– Dynamic system
Introduction
• Questions:– Does an equilibrium point exists?– Is it unique?– Does the dynamic system converge to it?
Introduction
• What was done so far (1993):– Economic tools for flow control and resource
allocation– Routing – two nodes connected with parallel
identical links (M/M/c queues)– Rosen (1965) conditions for existence,
uniqueness and stability
Introduction
• Goals of This Paper– The uniqueness problem of a convex game
(convex but not common objective functions)– Use specificities of the problem (results cannot
be derived directly from Rosen)– Two nodes connected by a set of parallel links,
not necessarily queues– General networks
• Set of m users: • Set of n parallel communication links:• User’s throughput demand – stochastic process
with average:• Fractional assignment• Expected flow of user on link:
Users flows fulfill the demand constraint: • Total flow on link:
Model and Formulation
i Il L
ir
ilf
i il
l
f ri
l li
f f
Model and Formulation
• Link flow vector:• User flow configuration:• System flow configuration:• Feasible user flow – obey the demand constraint• Set of all feasible user flows:• Feasible system flow – all users flows are feasible• Set of feasible system flows:
1( ,..., )ml l lf f f
1( ,..., )i i inf f f
1( ,..., )mf f f
iF
F
• User cost as a function of the system’s flow configuration:
• Nash Equilibrium Point (NEP)– System flow configuration such that no user
finds it beneficial to change its flow on any link– A configuration:
that for each i holds:
Model and Formulation
)( fJ i
Ffff m )~
,...,~
(~ 1
)}~
,...,,...,~
({)( 1min mii
Ff
i fffJfJii
Model and Formulation
• Assumptions of the cost function:– G1 It is a sum of user-link cost function:
– G2 might be infinite
– G3 is convex
– G4 Whenever finite is continuously differentiable
– G5 At least one user with infinite flow (if exists) can change its flow configuration to make it finite
n
ll
il
i fJfJ1
)()(
ilJilJ
ilJ
Model and Formulation
• Convex Game – Rosen guarantees the existence of NEP
• Kuhn-Tucker conditions for a feasible configuration to be a NEP
• We will investigate uniqueness and convergence of a system
Model and Formulation
• Type-A cost functions– is a function of the users
flow on the link and the total flow on the link– The functions in increasing in both its
arguments– The function’s partial derivatives are increasing
in both arguments
),( li
lil ffJ
Model and Formulation
• Type-B cost functions– Performance function of a link measures its
cost per unit: – Multiplicative form: – cannot be zero, but might be infinite– is strictly increasing and convex– is continuously differentiable
lT)(),( ll
ill
il
il fTfffJ
lT
lT
lT
Model and Formulation
• Type-C cost functions– Based on M/M/1 model of a link– They are Type-B functions– If then:
else:– is the capacity of the link
lll fC
T
1
lT
ll Cf
lC
Uniqueness
• Theorem: In a network of parallel links where the cost function of each user is of type-A the NEP is unique.
• Kuhn-Tucker conditions: for each user i there exists (Lagrange multiplier), such that for every link l, if :
then: else: when:
i
0ilf
il
il fK )(
il
il
lil f
JfK
)(
il
il fK )(
Monotonicity
• Theorem: In a network of parallel links with identical type-A cost functions. For any pair of users i and j, if then
for each link l.
• Lemma: Suppose that holds for some link l’ and users i and j. Then, for each link l:
ji rr j
li
l ff
jl
il ff ''
jl
il ff
Monotonicity
• If all users has the same demand then:
• If then
• Monotonic partition among users:
User with higher demands uses more links, and more of each link
mff li
l ji rr 00 j
li
l ff
Monotonicity
• Theorem: In a network of parallel links with type-C cost functions. For any pair of links l and l’, if then for each user i.
• Lemma: Assume that for links l and l’ the following holds:
Then: for each user j.
il
il ff '
'll CC
)()()()( '''' llllllll fTfTfTfT j
lj
li
li
l ffff ''''
Convergence
• Two users sharing two links
• ESS – Elementary Stepwise System– Start at non-equilibrium point– Exact minimization is achieved at each stage– All operations are done instantly
• User’s i flow on link l at the end of step n :
)(nf il
Convergence
• Odd stage 2n-1: User 1 find its optimum when the other user’s 2n-2 step is known.
• Even stage 2n: User 2 find its optimum when the other’s user 2n-1 step is known.
Steps
User 1
User 2
Convergence
• Theorem: Let an ESS be initialized with a feasible configuration, Then the system configuration converges over time to the NEP, meaning:
• Lemma: Let be two feasible flows for user 1. And optimal flows for user 2 against the above. If: then:
1f2f
1~f
2~f
*)(lim fnfn
11 ~ll ff
22 ~ll ff
Diagonal Strict Convexity
• Weighted sum of a configuration:
• Pseudo-Gradient:
•
m
i
ii fJ
1
)(
0)),(),~
()(~
( fgfgff
)(
.
.
)(
),(
1
ff
J
ff
J
fg
m
m
m
i
i
Diagonal Strict Convexity
• Theorem (Rosen): If there exists a vector
for which the system is DSC. Then the NEP is unique
• Pseudo-Jacobian
• Corollary: If the Pseudo-Jacobian matrix is positive definite then the NEP is unique
Symmetrical Users
• All users has the same demand (same source and destination)
• Lemma:
• Theorem: A network with symmetrical users has a unique NEP
m
ff li
l
All-Positive Flows
• All users must have the same source and destination
• Type-B cost functions
• For a subclass of links, on which the flows are strictly positive, the NEP is unique.
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