Competitive Routing in Multi-User Communication Networks

1
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

2
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

3
Introduction

• Questions
• Does an equilibrium point exists?
• Is it unique?
• Does the dynamic system converge to it?

4
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

5
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

6
Model and Formulation

• Set of m users
• Set of n parallel communication links
• Users throughput demand stochastic process
  with average
• Fractional assignment
• Expected flow of user on link
• Users flows fulfill the demand constraint
• Total flow on link

7
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

8
Model and Formulation

• User cost as a function of the systems 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

9
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

10
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

11
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 functions partial derivatives are increasing
  in both arguments

12
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

13
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

14
Part I Parallel links
Links
Users
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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

16
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

17
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

18
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.

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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
• Users i flow on link l at the end of step n

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Convergence

• Odd stage 2n-1 User 1 find its optimum when the
  other users 2n-2 step is known.
• Even stage 2n User 2 find its optimum when the
  others user 2n-1 step is known.

User 2
Steps
User 1
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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

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Part II General Topology
Network
Users
23
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
24
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
25
Non-monotonous
T(3 ,1)20
User 1
7
T(1 ,2)1
T(4 ,3)4
1
2
3
4
User 2
T(3 ,1)21
T(4 ,3)5
4
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Diagonal Strict Convexity

• Weighted sum of a configuration
• Pseudo-Gradient

27
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

28
Symmetrical Users

• All users has the same demand (same source and
  destination)
• Lemma
• Theorem A network with symmetrical users has a
  unique NEP

29
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.

30
Further Research

• General network uniqueness for type-B functions
• Stability (convergence)
• Restrictions on users (non non-cooperative games)
• Delay in measurements real dynamic system