Resource Allocation and Pricing

R. SrikantUniversity of Illinois

References

• The Mathematics of Internet Congestion Control, Birkhauser, 2004.

• Pricing: Kelly• Distributed Resource Allocation: Kelly,

Mauloo and Tan, Low and Lapsley, Kunniyur and S., Wen and Arcak, Liu, Basar and S.

Resource Allocation

• How much bandwidth should each user get?

• Constraints: x0+x1· cA; x0+x2· cB

User 2

cA cB

User 1

User 0

Utility Functions

• Associate a utility function with each user• Strictly concave, increasing functions• Maximize system utility, i.e., the sum of

the utilities of all the users

User 0

User 1

User 2

)( 11 xU

)( 00 xU)( 22 xU

cA cB

Kelly’s System Problem

subject to

Issues

• Will users truthfully reveal their utility functions?

• If not, can we design a pricing scheme (mechanism) to induce truth-telling?

• Is there a distributed algorithm to compute the prices?

Computing Source Rates

Source Algorithm

• Source needs only its path price:

Computing Lagrange Multipliers

• Dual problem:

Link Algorithm

• Gradient algorithm• Link needs to only know its arrival rate

Network Solution

• Network doesn’t know the utility function• Choose Ur(xr)=wr log xr

• Allow users to choose wr

Proportional Fairness

• If the utility function is of the form Ur(xr)=wr log xr,

then the optimal allocation satisfies

Pricing

• Can the network choose a pricing scheme to achieve fair resource allocation?

• Suppose that the network charges a price qr ($/bit) where qr=l2 rpl

• User’s strategy: spend wr ($/sec.) to maximize

Optimal User Strategy

• Equivalently,

Distributed Computation

• With the optimal choice of wr, the controller becomes

• We have already seen that this solves

Price Takers vs. Strategic Users

• Kelly Mechanism: Users are price takers, i.e., user does not know the impact of its action on the price

• Strategic users:

Efficiency and Competition

• Price takers: selfish users can maximize social welfare

• Strategic users: Competition leads to loss of efficiency, i.e., social welfare is not maximized

• Question: by how much? • Answer: Workshop

Recap

• Goal: Maximize social welfare• Hard code programs into computers to

achieve proportional fairness based on user bids {wr}

• Selfish, price-taking users naturally bid to maximize social welfare

• Reasonable for the Internet• Small number of resources: strategic

users

Convergence

• Approximate computation of Lagrange multipliers

• Associate a price function with each link: fl(yl), where yl is the arrival rate into the link

Solution

• User r’s depends only on its path price• Link price depends only on the total arrival

rate into the link

Congestion Control

Stability

• Note that

• V(x) is the resource allocation objective• V(x) is a Lyapunov function:

Recall Primal-Dual Algorithm

• Is this algorithm also stable?

Lyapunov function

Vickrey-Clarke-Groves (VCG) Mechanism

• Seller asks for {Ur(xr)}

• Computes x*=arg maxx r Ur(xr)• The presence of user r reduces the utility

to other users. Charge this reduction as the price to user i:

Truth-Telling is optimal

• Net utility for user i:

• If truth-telling is not optimal, we have a contradiction:

Comparing Kelly and VCG

• VCG requires each user to give the entire utility function

• Kelly requires each user to submit a bid• VCG: computation is not decentralized • Kelly: computation of prices is distributed

among users and resources

Other Pricing

• Maximize revenue, with little or no regard for social welfare

• Peering, transit and access charge arrangements across multiple ISPs

• Resource allocation and pricing in wireless networks (both cellular and ad hoc networks)

Modeling Delays

• Delay in receiving congestion feedback

• Tr: RTT (round-trip time)

Window Flow Control

• W: Window size of a source• W is the number of unacknowledged

packets that can be in the network• x: Transmission rate (packets/sec.)• T: Round-trip time. Amount of time it

takes to receive an ack for a packet

Differential Equation - I

• q(t): Probability of packet loss at time t

Differential Equation - II• Additive Increase-Multiplicative Decrease

(AIMD)

Delays

Delay from source r to link l

Delay from link l to source r

Network Stability?

Sources Links

x y

pq

yl=r xr(t-df(r,l))

qr=l pl(t-db(r,l))

Arrivals and Departures

• The number of sources has been assumed to be a constant

• On a slower time-scale, files (sources) arrive and depart

• If the fast time-scale algorithms are designed well, congestion control can be viewed as an instantaneous resource allocation process

Connection-Level Model

• Files arrive according to some point process• Each file brings a random amount of work

(bits)• File departs when the work is finished• Between arrivals and departures, resources

allocated to each flow according to the system problem described earlier

• Is the connection-level model (stochastically) stable?

Part II

• Resource allocation and control

• Stability conditions for a network with delays

• Connection-level stability