Decentralized Resource Allocation Mechanisms in Networks Tudor - - PowerPoint PPT Presentation

Decentralized Resource Allocation Mechanisms in Networks

Tudor Stoenescu

Information Science and Technology

Caltech

Organization of the Talk

Major issues of resource allocation in

networks

Overview of fundamental issues in

decentralized resource allocation

Development of two network pricing

mechanisms

Implementation in networks Conclusions

Motivation

Integrated services networks support the

delivery of a variety of services to their users

Diversity of information imposes different

requirements on the delivery methods

– (audio, video, file transfer)

Challenge

Design of resource allocation strategies which

guarantee the delivery of different services, each with its own Quality of Service (QoS) requirement, maximize some performance criterion (e.g. network's utility to its users) and satisfy the network’s informational constraints

– Issue: Compatibility with individual objectives

Key Network Features

Informationally decentralized system formed by two types of agents:

Users Network

Users’ Informational Constraints

Preferences over the set of services offered by the

network are private information.

– Preferences are expressed by a utility function

Users are unaware as well as uninterested in the

delivery method used for the requested services

Users are unaware of the other users requesting

services from the network

Network Informational Constraints

Network manager knows the network

topology and the network's resources

– link capacities, buffer size

Network manager is unaware of the number

• f users that may request services, as well

as the users' utilities

Decentralization of information

Major issue

If information were centralized one could use

Math Programming methods

Major issue

If information were centralized one could use

Math Programming methods

But it is not…

Major issue

If information were centralized one could use

Math Programming methods

But it is not… Can we find ways of implementing the

centralized design and still satisfy the informational constraints?

Major issue

If information were centralized one could use

Math Programming methods

But it is not… Can we find ways of implementing the

centralized design and still satisfy the informational constraints?

If we find a method of implementing the

centralized design, can we guarantee that the agents will follow this method?

Organization of the Talk

Major issues of resource allocation in

networks

Overview of fundamental issues in

decentralized resource allocation

Development of two network pricing

mechanisms

Implementation in networks Conclusions

Decentralized Resource Allocation Background

Early 1800’s (beginning of the socialist debate) Late 1800’s – Walrasian school (Pareto, Barone,…) World War I – German economy von Mises – economic calculation (1920’s) Socialist economists of the 1930’s

(Taylor, Dickinson, Lange, Lerner,…)

von Hayek – rebuttal to the socialist arguments

von Hayek’s arguments regarding the weakness of socialist economies

Amount of information exchange and

calculation needed by a central-control system to determine an optimal resource allocation may be too great.

Incentives provided by the market economy

could not be reproduced by any socialist system.

Mechanism Design

Realization Theory

– Informational efficiency – Complexity of information processing

Implementation Theory

Mechanism Design (Realization Theory) E

Mechanism Design (Realization Theory) E A

Mechanism Design (Realization Theory) E A

π

Mechanism Design (Realization Theory) E M A

π

µ

h

Mechanism components

E – Environment A – Action Space M – Message Space π – Goal correspondence µ – Equilibrium message correspondence h – Outcome function

Requirements

1.

For each element of the environment there exist a non-empty set of feasible actions.

Requirements

1.

For each element of the environment there exist a non-empty set of feasible actions.

2.

For each element of the environment the set of feasible actions satisfying the goal correspondence π is non-empty.

Requirements

1.

For each element of the environment there exist a non-empty set of feasible actions.

2.

For each element of the environment the set of feasible actions satisfying the goal correspondence π is non-empty.

3.

The actions generated by π also satisfy some sort

• f optimality criteria.

Requirements

1.

For each element of the environment there exist a non-empty set of feasible actions.

2.

For each element of the environment the set of feasible actions satisfying the goal correspondence π is non-empty.

3.

The actions generated by π also satisfy some sort

• f optimality criteria.

Requirements 1-3 are constraints on the problem type considered.

Requirements

4.

For all

∅ ≠ ∈ ) ( , e E e µ

Requirements

4.

For all

5.

(non-wastefulness)

∅ ≠ ∈ ) ( , e E e µ

E e e e h ∈ ∀ ⊆ ), ( )) ( ( π µ

Requirements

4.

For all

5.

(non-wastefulness) A mechanism satisfying requirements 1 - 5 is called goal realizing.

∅ ≠ ∈ ) ( , e E e µ

E e e e h ∈ ∀ ⊆ ), ( )) ( ( π µ

Requirements

6.

Unbiasedness - mechanism should not favor one group of agents over another.

Requirements

6.

Unbiasedness - mechanism should not favor one group of agents over another.

• 7. Essential single-valued - for any environment, the

rules of the process leads the system to a uniquely determined allocation

Requirements

6.

Unbiasedness - mechanism should not favor one group of agents over another.

• 7. Essential single-valued - for any environment, the

rules of the process leads the system to a uniquely determined allocation A mechanism satisfying requirements 4 through 7 is called satisfactory.

Requirements

• 8. Privacy preserving - all the agents generate their

equilibrium messages based only on their own information about the environment.

Requirements

• 8. Privacy preserving - all the agents generate their

equilibrium messages based only on their own information about the environment. 9. Spot threadedness of µ - the correspondence has a continuous selection around every point in the domain

Requirements

• 8. Privacy preserving - all the agents generate their

equilibrium messages based only on their own information about the environment. 9. Spot threadedness of µ - the correspondence has a continuous selection around every point in the domain 0.a11a12a13… 0.a21a22a23… 0.a31a32a33… 0.a11a21a31a12a22a32…

Requirements

• 8. Privacy preserving - all the agents generate their

equilibrium messages based only on their own information about the environment. 9. Spot threadedness of µ - the correspondence has a continuous selection around every point in the domain A mechanism satisfying requirements 8 and 9 is called regular.

Desired property

A mechanism is said to be informationally efficient

if it is goal realizing and regular and it has a message space of a dimensionality which is minimal among all the other goal realizing and regular mechanisms

Implementation Theory

Studies the constrains on the design of

mechanisms imposed by the divergence of individual incentives from the performance

• bjective

Question:

– Can we design a noncooperative game that

implements the social choice rule in some sort of equilibrium messages (Nash, Bayesian, subgame perfect, undominated strategies, etc.) ?

Implementation Issues

N

E E E E × × × = ...

2 1

E A

π

Implementation Issues

N

M M M M × × × = ...

2 1 N

E E E E × × × = ...

2 1

E A M

π

h

Implementation Issues

N

M M M M × × × = ...

2 1 N

E E E E × × × = ...

2 1

E A M

π

R h

( )

( )

E e N i m e m h e R e m h

i i i

∈ ∀ ∈ ∀

−

] ,..., 2 , 1 [ ), ( ) ( ) (

* *

( )

) ( ),..., ( ), ( : ) (

* * 2 * 1 *

e m e m e m e m

N

=

( )

) ( ),..., ( ), ( ), ( ),..., ( : ) (

* * 1 * 1 * 1 *

e m e m e m e m e m e m

N i i i i + − −

=

i i

M m ∈

Implementation Issues

N

M M M M × × × = ...

2 1 N

E E E E × × × = ...

2 1

E e e e R h ∈ ∀ ⊆ ), ( )) ( ( π

E A M

π

R h

( )

( )

E e N i m e m h e R e m h

i i i

∈ ∀ ∈ ∀

−

] ,..., 2 , 1 [ ), ( ) ( ) (

* *

( )

) ( ),..., ( ), ( : ) (

* * 2 * 1 *

e m e m e m e m

N

=

( )

) ( ),..., ( ), ( ), ( ),..., ( : ) (

* * 1 * 1 * 1 *

e m e m e m e m e m e m

N i i i i + − −

=

i i

M m ∈

Organization of the Talk

Major issues of resource allocation in

networks

Overview of fundamental issues in

decentralized resource allocation

Development of two network pricing

mechanisms

Implementation in networks Conclusions

Mechanism design in the context of networks

Studied two problems

– Unicast with routing and QoS requirements – Multi-rate multicast

Problem Description (Unicast)

Problem Components

1)

Network:

• A collection of nodes with general topology
• Nodes connected by unidirectional links
• Each link has a finite amount of resources
• Services can be delivered over multiple routes
• Services have different Quality of Service (QoS) requirements
• We assume that a relation between resource allocation and

QoS requirement is given. 2)

Users

• Request multiple services from the network
• Unaware of the form of service delivery

Picture of the Unicast Problem

Problem Description Objective

The network must determine the resource

allocation strategies that maximize the network’s utility to its users and satisfy the informational constraints imposed by the network problem

Market mechanism

Network Auctioneer

Checks excess demand Sets link prices

Service Provider

Determines optimal service prices

Users

Demand service based on prices Demand Price per unit

• f service

Reference

• T. Stoenescu and D. Teneketzis. A pricing

methodology for resource allocation and routing in integrated-services networks with quality of service requirements, Mathematical Methods of Operation Research 56 (2002) 2, 151-167

Informational Efficiency

The unicast network pricing mechanism

presented is goal realizing (satisfies requirements 1-5)

Pricing mechanisms are (Pareto) satisfactory

(satisfies requirements 4-7)

Is the network pricing mechanism regular?

(Does it satisfy requirements 8 and 9?)

Informational Efficiency

Proved that the network pricing mechanism

developed has a message space of dimensionality which is minimal among all the regular and goal realizing mechanisms.

Proved that the network pricing mechanism

developed is informationally efficient for the rate allocation problem (where each user requests a single service).

Contributions of Work on Routing

Addresses simultaneous resource allocation and

routing in integrated service networks with end-to- end QoS requirements

Proposes a (goal realizing) market based

mechanism that takes into account the informationally decentralized nature of the problem and leads to a utility maximizing resource allocation and routing

Develops a class of environments for which the

market mechanism is informationally efficient

Motivation of the Multicast Problem

• Efficient transmission of data in real time

applications from one source to many users

─

The source sends one copy of a message to its users and this copy is replicated only at the branching points of a multicast tree

Multi-rate multicast

Motivation of the Multicast Problem

• Efficient transmission of data in real time

applications from one source to many users

─

The source sends one copy of a message to its users and this copy is replicated only at the branching points of a multicast tree

• Types of multicast

–

Single-rate

–

Multi-rate (hierarchical encoding)

Multi-rate multicast

Multi-rate multicast

6 4 3 7 7

Multi-rate multicast

6 4 3 7 7 7 6

Multi-rate multicast

6 4 3 7 7 7 7 6

Objective

Design of resource allocation strategies

which guarantee the delivery of different services, maximize some performance criterion (e.g. network's utility to its users) and satisfy the network’s informational constraints

Challenge

Informationally decentralized nature of the

network problem

– Users – Network

Who is going to pay for the services?

– How is this going to be determined in the absence

• f centralized information?

Picture of a network

Problem Description

Problem Components

1)

Network:

• A collection of nodes with general topology
• Nodes connected by unidirectional links
• The groups of links form multicast trees used for the

delivery of service

• Each link has a finite capacity

2)

Users

• Request services from the network
• Unaware and uninterested of the form of service

delivery

Picture of a network

Problem Description Users

There are N users requesting service from

the network

Each user i’s preferences on the set of the

available services are described by a utility function Ui

Ui, i=1,...,N, are strictly concave and

continuously differentiable utility functions

Problem Description Network

Network receives requests for services from

the users

Each network service is delivered on a

predetermined multicast tree

Problem Description Users (Informational Constraints)

Users know their own preferences over the

set of services offered by the network

– Preferences are characterized by a utility

function

Users are unaware as well as uninterested in

the delivery method used for the requested services

Each user is unaware of the other users’

requests

Problem Description Network (Informational Constraints)

Network manager knows the network

topology and the network's resources

– link capacities, buffer size

Network manager is unaware of the number

• f users that may request services, as well

as the users' utilities

Problem Description Objective

The network must determine the resource

allocation strategies that maximize the network’s utility to its users and satisfy the informational constraints imposed by the network problem

How to achieve the objective

Formulate a centralized resource allocation

problem

Describe a market mechanism (Tâtonnement

process) that achieves the solution of the centralized resource allocation problem and satisfies the informational constraints

Mechanism Design

Derivation of a triple (M,µ, h) for which

commutes, by satisfying:

) ( )) ( ( e e h π µ ⊆

E e ∈ ∀

E M A

π

µ

h

Centralized Optimization Problem (P)

) ( max

, i R r i R r x

x U

∑

∈ ∈

subject to the constraints:

L l c x

l r M m R r

m l

∈ ∀ ≤

∑

∈ ∈

, max

,

R r x r ∈ ∀ ≥ ,

• M = set multicast trees
• R = set of receivers
• Rl,m = set of receivers on multicast tree m downstream link l
• x = vector of users demand
• cl = capacity of link l

Proposed Market Structure

Network

– Auctioneer – Service provider

Users

How the Market Structure Works

The Auctioneer sets prices λl per unit rate on

each link

Based on these prices the service provider

computes the price per unit of service for each user based on an iterative algorithm

The service provider announces the price per

unit of service to the users

How the Market Structure Works

The users determine the amount of

requested service by solving a utility maximization problem

How the Market Structure Works

The auctioneer computes the excess

demand at each link and announces new prices per unit of resource at each link

The process repeats In our work the auctioneer’s update of prices

is done using Scarfs’ algorithm

Market Mechanism

Auctioneer/Resource Provider (excess demand) Service Provider Users

Network Sign of excess demand ? + _

Resource price Demand Service price

Main result

The above described market mechanism (Tâtonnement process) achieves an optimal solution of Problem (P).

) ( )) ( ( e e h π µ =

E e ∈ ∀

Numerical results (inner loop)

Numerical results (inner loop)

Inner loop

5 10 15 20 25 0.5 1 1.5 2 2.5 Iteration Service Price user 1 user 2 user 3 user 4 user 5 user 6 5 10 15 20 25 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Iteration Service Price user 1 user 2 user 3 user 4 user 5 user 6

Numerical results (outer loop)

Outer loop (Scarf’s Algorithm)

20 40 60 80 100 120 0.2 0.4 0.6 0.8 1 1.2 1.4 Iteration Link price link 1 link 2 link 3 link 4 link 5 link 6 link 7 link 8 link 9 link 10 link11 100 200 300 400 500 600 700 0.2 0.4 0.6 0.8 1 1.2 1.4 Iteration Link price link 1 link 2 link 3 link 4 link 5 link 6 link 7 link 8 link 9 link 10 link11

Outer Loop (Eves K1 Algorithm)

100 200 300 400 500 600 700 800 900 1000 0.5 1 1.5 2 2.5 3 Iteration link price link 1 link 2 link 3 link 4 link 5 link 6 link 7 link 8 link 9 link 10 link11

Informational Efficiency

The multicast network pricing mechanism

presented is goal realizing (satisfies requirements 1-5)

Pricing mechanisms is not (Pareto)

satisfactory due to the externalities generated by the common links

Is the network pricing mechanism regular?

(Does it satisfy requirements 8 and 9?)

Informational Efficiency

Proved that the multirate multicast network

pricing mechanism developed is informationally efficient.

Contribution of this work

Developed properties of the optimal service price

given fixed price per unit of rate on each link

Developed an iterative algorithm which satisfies the

properties developed and computes the optimal service price for each user

Presented a (goal realizing) pricing mechanism

which achieves a solution of the decentralized rate allocation multicast network problem

Proved that the multicast pricing mechanism is

informationally efficient

Organization of the Talk

Major issues of resource allocation in

networks

Overview of fundamental issues in

decentralized resource allocation

Development of two network pricing

mechanisms

Implementation in networks Conclusions

Implementation in networks

The above mechanisms do not implement a solution

to the centralized problem

Tsitsiklis et. all (2004) and Hajek et. all (2004) show

that there is an efficiency loss if agents are greedy

One can implement the solution to the centralized

problem by using a mechanism with a message space of higher dimension

– The extra dimensions force the users to act as price takers

Conclusion

Presented some mechanism design background Investigated two different network resource

allocation problems

Developed goal realizing pricing mechanisms for

both problems

Investigated the informational efficiency of the both

pricing mechanisms

Discussed mechanism implementation in networks