A Scalable Network Resource Allocation Mechanism With Bounded Efficiency Loss

IEEE Journal on Selected Areas in Communications, 2006

Johari, R., Tsitsiklis, J.N.

Presented byMa Man Lok and Wan Wing San

Agenda

• Introduction• Single Link• General Networks• Simulation• Conclusion• Q & A

Introduction

• Network Resource Allocation– Network Traffic has grown exponentially

• User base increases• Applications require increasing resource• Applications require stricter Quality of Service

– Introduce Usage-based Charges• Resolve the allocation of resources to users• Traffic management and congestion control

Introduction

• Congestion Pricing Mechanisms– Objective

• Users should pay for the additional congestion they create

• Encouraging the redistribution of the demand in space or in time

Introduction

• Congestion Pricing Mechanisms– Design

• Simple and Scalable end-to-end implementation• Efficiency of resulting equilibria

Introduction

• Motivation– Recently proposed mechanisms

• Assume users are price taker– They do not anticipate the effect of their strategic

decisions on the prices

– Derive a alternative mechanisms by studying Cournot game

Introduction

• Cournot game– There is more than one firm– All firms produce a homogeneous product– Firms do not cooperate– Firms compete in quantities, and choose

quantities simultaneously

Single Link

• Game– Multiple users compete for a single link– Strategies of the users represent their desired

rates

Single Link

• Model– N users compete for a single link– Each user n has a utility function Un

– Total data rate through the link incur a cost characterized by a cost function C

Single Link

Single Link

• It can be characterized as a optimization problem

Single Link

• Pricing Scheme– Assume users are price takers– Given a price μ > 0, user n choose xn to

maximize

– There exists a vector x and a scalar μ such that

Single Link

• Pricing Scheme– If users are not price takers– Alternative model

• Play a Cournot game to acquire a share of the link• Notation x-n denote the vector of all rates chosen

by users other than n• Given x-n, user n choose xn to maximize

Single Link

• Pricing Scheme– Qn is similar to Pn

• Except the user can anticipate– Nash Equilibrium (NE) exists for this game

Single Link

• Pricing Scheme– Assume

• p(q) = aq + b• Un(0) ≥ 0 for all n

– xs is any optimum solution of the problem– x is any NE of the game

– The worst case efficiency loss is bounded by 1/3

General Networks

• Game• Model• Optimization Problem• Payoff to User• Bound of Efficiency Loss

Game

• Multiple users compete for network resources provided by multiple links

• Strategies of users represent their desired rate on paths which are combination of links

Model

• Assumption 1 & 2 still hold • Network contains J, P and N as set of

links, paths and users respectively• Each path is a combination of some links

– jJ, qP and jq• Each user can own several paths

– nN and qn• Each path is owned by single user only

– qn, q'n', n n' and q q'

Model (cont.)

• Rate allocated to path q: yq 0• Rate allocated to user n: dn = qn yq 0• Total rate on link j: fj = q:jq yq

• Utility of user n: Un(dn)• Cost of link j (overall users): Cj(fj)• Price of link j of user n: j(y) = pj(qn:jqyq)• Total payment of user n:

qnyqjqj(y)

Model (cont.)

• Path-resource incidence matrix A– Ajq = 1 if jq– Ajq = 0 if otherwise

• Path-user incidence matrix H– Hnq = 1 if qn– Hnq = 0 if otherwise

• d = (dn, nN), y = (yq, qP)• Ay = f, Hy = d

Optimization Problem

Payoff to User

• Price taker n

• Price anticipating user n

where y-n = (y1, …, yn-1, yn+1, …, yN)

Bound of Efficiency Loss

• Suppose pj(qj)=ajqj+bj for some aj>0, bj0• Let yS be any solution to the optimization

problem

Bound of Efficiency LossProof Sketch

• Establish relationship of N.E. of choosing rates on paths and N.E. of choosing rates on links

• Reduce analysis to individual games at each link, extend the bound for Single Link

Bound of Efficiency LossRelationship

• Consider another game that each user n has to choose rate djn at each link j

• User n can achieve max. rate by solving max-flow optimization problem

Bound of Efficiency LossRelationship (cont.)

• Denote optimal objective value by zn(dn)• Price at each link j: pj(ndjn)• Total payment of user n: jdjnpj(ndjn)• Payoff to price anticipating user n

• Suppose y is N.E. of game of (Q1, …, QN)Define djn=qn:jqyq follows that

Bound of Efficiency LossRelationship (cont.)

• Un(qnyq) = Un(zn(dn))– yn is feasible for the max-flow problem,

qnyq zn(dn) Un(qnyq) Un(zn(dn))– For case that Un(qnyq) < Un(zn(dn))

qnyq < zn(dn) yn is not optimal and hence contradict with the assumption of N.E and so result follows

• Hence, the following hold at N.E.

Bound of Efficiency LossReducing analysis to individual link

• Let dn* be N.E. of the second game

• Replace Un(zn(dn)) by linear utility function n

Tdn while keeping dn* as N.E. of the new game

• The second game can be decoupled into j Single Link game and hence the bound can be extended from the previous bound

Simulation• Since the General Networks part is simply an extension of

Single Link, only Single Link case is considered• Objective: Test if the bound would be reached easily while

assuming users are homogenous for simplicity• Configuration

– Both functions are non-linear• Utility function Un(x) = 1 – e-kx

• Price function p(x) = epx

– Both functions are linear• Utility function Un(x) = kx• Price function p(x) = px

• Result: achieved aggregate surplus is very close to the optimal value (within 3% loss)

Conclusion

• The scheme proposed by this paper is to– users choose the rate to send on paths– set the link price according to marginal cost of

total rate allocated• By using this scheme, the Efficiency Loss

is bounded above by 1/3

Q & A