Resource Allocation and Pricing. R. Srikant University of Illinois

Transcription

1 Resource Allocation and Pricing R. Srikant University of Illinois

2 References The Mathematics of Internet Congestion Control, Birkhauser, Pricing: Kelly Distributed Resource Allocation: Kelly, Mauloo and Tan, Low and Lapsley, Kunniyur and S., Wen and Arcak, Liu, Basar and S.

3 Resource Allocation User 1 User 0 c A c B User 2 How much bandwidth should each user get? Constraints: x 0 +x 1 c A ; x 0 +x 2 c B

4 Utility Functions User 1 U 1 x ( 1 ) User 0 U 0 ( x 0 ) c A User 2 U 2 ( x 2 ) 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 c B

5 subject to Kelly s System Problem

6 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?

7 Computing Source Rates

8 Source Algorithm Source needs only its path price:

9 Computing Lagrange Multipliers Dual problem:

10 Link Algorithm Gradient algorithm Link needs to only know its arrival rate

11 Network Solution Network doesn t know the utility function Choose U r (x r )=w r log x r Allow users to choose w r

12 Proportional Fairness If the utility function is of the form U r (x r )=w r log x r, then the optimal allocation satisfies

13 Pricing Can the network choose a pricing scheme to achieve fair resource allocation? Suppose that the network charges a price q r ($/bit) where q r = l2 r p l User s strategy: spend w r ($/sec.) to maximize

14 Optimal User Strategy Equivalently,

15 Distributed Computation With the optimal choice of w r, the controller becomes We have already seen that this solves

16 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:

17 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

18 Recap Goal: Maximize social welfare Hard code programs into computers to achieve proportional fairness based on user bids {w r } Selfish, price-taking users naturally bid to maximize social welfare Reasonable for the Internet Small number of resources: strategic users

19 Convergence Approximate computation of Lagrange multipliers Associate a price function with each link: f l (y l ), where y l is the arrival rate into the link

20 Solution User r s depends only on its path price Link price depends only on the total arrival rate into the link

21 Congestion Control

22 Stability Note that V(x) is the resource allocation objective V(x) is a Lyapunov function:

23 Recall Primal-Dual Algorithm Is this algorithm also stable?

24 Lyapunov function

25 Vickrey-Clarke-Groves (VCG) Mechanism Seller asks for {U r (x r )} Computes x * =arg max x r U r (x r ) The presence of user r reduces the utility to other users. Charge this reduction as the price to user i:

26 Truth-Telling is optimal Net utility for user i: If truth-telling is not optimal, we have a contradiction:

27 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

28 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)

29 Modeling Delays Delay in receiving congestion feedback T r : RTT (round-trip time)

30 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

31 Differential Equation - I q(t): Probability of packet loss at time t

32 Differential Equation - II Additive Increase-Multiplicative Decrease (AIMD)

33 Delays Delay from source r to link l Delay from link l to source r

34 Network Stability? x y l = r x r (t-d f (r,l)) y Sources Links q q r = l p l (t-d b (r,l)) p

35 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

36 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?

37 Part II Resource allocation and control Stability conditions for a network with delays Connection-level stability