resource allocation and pricing
DESCRIPTION
Resource Allocation and Pricing. R. Srikant University 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. - PowerPoint PPT PresentationTRANSCRIPT
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