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