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Decentralized Resource Allocation Mechanisms in Networks Tudor Stoenescu Information Science and Technology Caltech

Decentralized Resource Allocation Mechanisms in Networks · Decentralized Resource Allocation Mechanisms in Networks Tudor Stoenescu Information Science and Technology Caltech. Organization

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Decentralized Resource Allocation Mechanisms in

Networks

Tudor Stoenescu

Information Science and TechnologyCaltech

Organization of the Talk

Major issues of resource allocation in networksOverview of fundamental issues in decentralized resource allocation Development of two network pricing mechanismsImplementation in networksConclusions

Motivation

Integrated services networks support the delivery of a variety of services to their usersDiversity 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:

UsersNetwork

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 servicesUsers 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 of 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 methodsBut it is not…

Major issue

If information were centralized one could use Math Programming methodsBut 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 methodsBut 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 networksOverview of fundamental issues in decentralized resource allocation Development of two network pricing mechanismsImplementation in networksConclusions

Decentralized Resource Allocation Background

Early 1800’s (beginning of the socialist debate)Late 1800’s – Walrasian school (Pareto, Barone,…)World War I – German economyvon 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 Aπ

Mechanism Design (Realization Theory)

E

M

µ h

Mechanism components

E – EnvironmentA – Action SpaceM – Message Spaceπ – Goal correspondenceµ – Equilibrium message correspondenceh – 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 of 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 of optimality criteria.

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

Requirements

4. For all ∅≠∈ )(, eEe µ

Requirements

4. For all

5. (non-wastefulness)

∅≠∈ )(, eEe µ

Eeeeh ∈∀⊆ ),())(( πµ

Requirements

4. For all

5. (non-wastefulness)

A mechanism satisfying requirements 1 - 5 is called goal realizing.

∅≠∈ )(, eEe µ

Eeeeh ∈∀⊆ ),())(( πµ

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 efficientif 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 objectiveQuestion:– Can we design a noncooperative game that

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

Implementation Issues

NEEEE ×××= ...21

E Aπ

Implementation Issues

NMMMM ×××= ...21

NEEEE ×××= ...21

E A

M

π

h

Implementation Issues

NMMMM ×××= ...21

NEEEE ×××= ...21

E A

M

π

R h

( ) ( ) EeNimemheRemh iii ∈∀∈∀− ],...,2,1[),()()( **

( ))(),...,(),(:)( **2

*1

* emememem N=

( ))(),...,(),(),(),...,(:)( **1

*1

*1

* emememememem Niiii +−− =

ii Mm ∈

Implementation Issues

NMMMM ×××= ...21

NEEEE ×××= ...21

EeeeRh ∈∀⊆ ),())(( π

E A

M

π

R h

( ) ( ) EeNimemheRemh iii ∈∀∈∀− ],...,2,1[),()()( **

( ))(),...,(),(:)( **2

*1

* emememem N=

( ))(),...,(),(),(),...,(:)( **1

*1

*1

* emememememem Niiii +−− =

ii Mm ∈

Organization of the Talk

Major issues of resource allocation in networksOverview of fundamental issues in decentralized resource allocation Development of two network pricing mechanismsImplementation in networksConclusions

Mechanism design in the context of networks

Studied two problems

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

Problem Description (Unicast)

Problem Components1) Network:

A collection of nodes with general topologyNodes connected by unidirectional linksEach link has a finite amount of resourcesServices can be delivered over multiple routesServices have different Quality of Service (QoS) requirementsWe assume that a relation between resource allocation and QoS requirement is given.

2) UsersRequest multiple services from the networkUnaware of the form of service delivery

Picture of the Unicast Problem

Problem DescriptionObjective

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

AuctioneerChecks excess demand

Sets link prices

Service ProviderDetermines optimal

service prices

UsersDemand servicebased on pricesDemand

Price per unit of 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 requirementsProposes 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 routingDevelops 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

77

Multi-rate multicast

6

4

3

77

7

6

Multi-rate multicast

6

4

3

77

7

76

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

of centralized information?

Picture of a network

Problem Description

Problem Components1) Network:

A collection of nodes with general topologyNodes connected by unidirectional linksThe groups of links form multicast trees used for the delivery of service Each link has a finite capacity

2) UsersRequest services from the networkUnaware and uninterested of the form of service delivery

Picture of a network

Problem DescriptionUsers

There are N users requesting service from the networkEach 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 DescriptionNetwork

Network receives requests for services from the usersEach network service is delivered on a predetermined multicast tree

Problem DescriptionUsers (Informational Constraints)

Users know their own preferences over the set of services offered by the network – Preferences are characterized by a utility

functionUsers are unaware as well as uninterested in the delivery method used for the requested servicesEach user is unaware of the other users’requests

Problem DescriptionNetwork (Informational Constraints)

Network manager knows the network topology and the network's resources – link capacities, buffer size

Network manager is unaware of the number of users that may request services, as well as the users' utilities

Problem DescriptionObjective

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 problemDescribe a market mechanism (Tâtonnementprocess) 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:

)())(( eeh πµ ⊆ Ee ∈∀

E

M

µ h

Centralized Optimization Problem (P)

)(max,

i

RriRrx

xU∑∈∈

subject to the constraints:

Llcx lrMm Rr ml

∈∀≤∑∈ ∈

,max,

Rrx r ∈∀≥ ,0•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 linkBased on these prices the service provider computes the price per unit of service for each user based on an iterative algorithmThe 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 repeatsIn 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).

)())(( eeh πµ = Ee ∈∀

Numerical results (inner loop)

Numerical results (inner loop)

Inner loop

0 5 10 15 20 250

0.5

1

1.5

2

2.5

Iteration

Ser

vice

Pric

e

user 1user 2user 3user 4user 5user 6

0 5 10 15 20 251

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Iteration

Ser

vice

Pric

e

user 1user 2user 3user 4user 5user 6

Numerical results (outer loop)

Outer loop (Scarf’s Algorithm)

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

1.4

Iteration

Link

pric

e

link 1link 2link 3link 4link 5link 6link 7link 8link 9link 10link11

0 100 200 300 400 500 600 7000

0.2

0.4

0.6

0.8

1

1.2

1.4

Iteration

Link

pric

e

link 1link 2link 3link 4link 5link 6link 7link 8link 9link 10link11

Outer Loop (Eves K1 Algorithm)

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

Iteration

link

pric

e

link 1link 2link 3link 4link 5link 6link 7link 8link 9link 10link11

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 linksIs 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 linkDeveloped an iterative algorithm which satisfies the properties developed and computes the optimal service price for each userPresented a (goal realizing) pricing mechanism which achieves a solution of the decentralized rate allocation multicast network problemProved that the multicast pricing mechanism is informationally efficient

Organization of the Talk

Major issues of resource allocation in networksOverview of fundamental issues in decentralized resource allocation Development of two network pricing mechanismsImplementation in networksConclusions

Implementation in networks

The above mechanisms do not implement a solution to the centralized problemTsitsiklis et. all (2004) and Hajek et. all (2004) show that there is an efficiency loss if agents are greedyOne 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 backgroundInvestigated two different network resource allocation problemsDeveloped goal realizing pricing mechanisms for both problemsInvestigated the informational efficiency of the both pricing mechanismsDiscussed mechanism implementation in networks