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

Part I – Parallel links

Users

Links

irlf

mf

f

.

.

1

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

Part II – General Topology

Users

Network

Non-uniqueness NEP1

User 1

User 2

1 2

4

322 ,18

10 ,12

8 ,108 ,16

24 ,14

14 ,240

40

Non-uniqueness NEP2

User 1

User 2

1 2

4

320 ,23

18 ,5

2 ,128 ,16

22 ,18

4 ,1340

40

Non-monotonous

User 1

User 2

1 2

4

3

T(4 ,3)=5

7

4

T(3 ,1)=20

T(1 ,2)=1T(4 ,3)=4

T(3 ,1)=21

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.

Further Research

• General network uniqueness for type-B functions

• Stability (convergence)

• Restrictions on users (non non-cooperative games)

• Delay in measurements – “real” dynamic system