decentralized resource allocation mechanisms in networks · decentralized resource allocation...
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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
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 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
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.
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
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
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
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)
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?
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
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
Aπ
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
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 ∈∀
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
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