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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

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competitive routing in multi user communication networks

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
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
introduction1
Introduction
  • Questions:
    • Does an equilibrium point exists?
    • Is it unique?
    • Does the dynamic system converge to it?
introduction2
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
introduction3
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
model and formulation
Model and Formulation
  • 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 formulation1
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:
model and formulation2
Model and Formulation
  • 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 formulation3
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
model and formulation4
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 formulation5
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
model and formulation6
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
model and formulation7
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
uniqueness
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:

monotonicity
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:
monotonicity1
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

monotonicity2
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.

convergence
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 :
convergence1
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.

User 2

Steps

User 1

convergence2
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:
non uniqueness nep1
Non-uniqueness NEP1

10 ,12

User 1

40

22 ,18

14 ,2

1

2

3

40

8 ,16

User 2

8 ,10

24 ,14

4

non uniqueness nep2
Non-uniqueness NEP2

18 ,5

User 1

40

20 ,23

4 ,13

1

2

3

40

8 ,16

User 2

2 ,12

22 ,18

4

non monotonous
Non-monotonous

T(3 ,1)=20

User 1

7

T(4 ,3)=4

T(1 ,2)=1

1

2

3

4

User 2

T(3 ,1)=21

T(4 ,3)=5

4

diagonal strict convexity
Diagonal Strict Convexity
  • Weighted sum of a configuration:
  • Pseudo-Gradient:
diagonal strict convexity1
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
Symmetrical Users
  • All users has the same demand (same source and destination)
  • Lemma:
  • Theorem: A network with symmetrical users has a unique NEP
all positive flows
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
Further Research
  • General network uniqueness for type-B functions
  • Stability (convergence)
  • Restrictions on users (non non-cooperative games)
  • Delay in measurements – “real” dynamic system