1. A General Framework for Wireless Spectrum Auctions Sorabh Gandhi, Lili Cao, Haitao Zheng, Subhash Suri ( Department of Computer Science University of California, Santa Barbara ) Chiranjeeb Buragohain ( Amazon.com, Seattle, USA ) IEEE DySPAN(2007)

2. Outline Introduction Preliminaries and related work Spectrum auction framework ◦ PLPD ◦ Auction-clearing problems ◦ Optimal clearing algorithm Fast auction clearing algorithm Experimental results Practical consideration Conclusion 2

3. Introduction (1/4) Long-term spectrum leases result in significant over-allocation and under-utilization Auction is a promising way to provide efficient allocation of scarce resources [3] ◦ Sellers can improve revenue by pricing based on buyer demand ◦ Buyers benefit since the resources are assigned to whom value them most Auction-based allocation is widely-used ◦ Energy markets [3], treasury bonds [2] 3 [2] BINMORE, K., AND SWIERZBINSKI, J. Treasury auctions: Uniform or discriminatory? Review of Economic Design 5, 4 (2000), 387–410. [3] BORENSTEIN, S. The trouble with electricity markets: Understanding californias restructuring disaster. Journal of Economic Perspectives 16, 1 (2002).

4. Introduction (2/4) In this paper, we consider how to efficiently auction spectrum to satisfy user demands while maximizing system revenue 4

5. Introduction (3/4) Because of the requirement to minimize radio interference, there are some new challenges: ◦ Radio interference constraints ◦ Supporting diverse demands ◦ Online multi-unit allocations  Compact bidding language and efficient allocation are needed Assumptions in this paper ◦ Fixed power requirement and focus solely on channel allocation  spectrum is divided in to number of homogeneous channel ◦ Centralized auctions 5

6. Introduction (4/4) We consider the problem of real-time dynamic spectrum auction to distribute spectrum ◦ Focus on computational-efficient channel allocation ◦ By restricting bids and radio interference constraints 6

7. Preliminaries and Related Work (1/3) Auctions have been widely used to provide efficient allocation of scare resources ◦ Multi-unit auctions Auction system produces financial efficiency and provides efficient bidding process and fast execution [17] Pricing models: ◦ Uniform pricing  Simple; Fairness [20] ; Collusion among bidders [4] ◦ Discriminatory pricing  More revenue 7 [17] KRISHNA, V. Auction Theory. Academic Press, 2002. [20] P. MALVEY, C. ARCHIBALD, S. F. Uniform price auctions : Evaluation of the treasury experience. http://www.treasury.gov/offices/domestic-finance/debtmanagement/auctions-study/upas2.pdf.

8. Preliminaries and Related Work (2/3) Spectrum auctions: ◦ Allocate transmit power to minimize interference [13], and users use the same spectrum band ◦ Use demand responsive pricing framework [15] ◦ Propose a hybrid pricing model to reduce the frequency of auctions [21] Interference constraints: ◦ Spectrum auction differs from conventional auctions ◦ Interference-constrained resource allocation ◦ Use different spectrum frequency to avoid interference 8 [13] HUANG, J., BERRY, R., AND HONIG, M. Auction mechanisms for distributed spectrum sharing. In Proc. of 42nd Allerton Conference (September 2004). [15] ILERI, O., SAMARDZIJA, D., SIZER, T., AND MANDAYAM, N. B. Demand responsive pricing and competitive spectrum allocation via a spectrum server. In Proc. of DySpan’ 05 (November 2005). [21] RYAN, K., ARAVANTINOS, E., AND BUDDHIKOT, M. M. A new pricing model for next generation spectrum access. In Proc. of TAPAS (August 2006).

9. Preliminaries and Related Work (3/3) Conflict graph ◦ Vertices: access point ◦ Edge: interference Consider A and B: ◦ Assume spectrum consists of M channels ◦ represents spectrum assigned to A ◦ if the kth channel is assigned to A, and otherwise 0 ◦ Interference constraints: F A ∩F B = ∅ ◦ In this case, f A + f B ≤ 1, where f A = |F A |/M, f B = |F B |/M ◦ Auction clearing problem becomes: 9

10. Spectrum Auction Framework － PLPD (1/3) Piecewise linear price-demand(PLPD) bids ◦ Expressive and concise bids, and lead to low- complexity clearing algorithms ◦ Bidder i uses continuous linear demand curves to describe the desired quantity of spectrum f i at each per-unit price p i ◦ Any PLPD curve can be expressed as a conglomeration of a set of individual linear pieces 10

11. Spectrum Auction Framework － PLPD (2/3) A simple example of linear demand curve: ◦ Demand curve: ◦ Quantity f i (p i ) and revenue generated R i (p i ): 11

12. Spectrum Auction Framework － PLPD (3/3) PLPD has advantages ◦ Simple and highly expressive ◦ Single bid covers different pricing options ◦ Quadratic revenue function 12

13. Spectrum Auction Framework － Auction-Clearing Problems (1/2) Uniform pricing ◦ The auctioneer sets a clearing price p ◦ Each bidder obtains a fraction of spectrum f i (p)=(b i - p)/a i and produces a revenue of R i (p)=(b i p - )/a i ◦ Assume bidders 1 to n are in increasing order of b i, i.e., and b 0 =0 ◦ The auction clearing problem becomes 13

14. Spectrum Auction Framework － Auction-Clearing Problems (2/2) Discriminatory pricing ◦ The clearing prices vary across i ◦ The optimization problem becomes 14 (-a i f i + b i ) * f i

15. Spectrum Auction Framework － Optimal Clearing Algorithm If we allocate a specific channel to one bidder, none of its neighbor in the conflict graph can use the channel [16] proposed an optimal algorithm to resolve interference conflicts ◦ Result in a linear programming problem with an exponentially large number of constraints ◦ Not feasible for large number of bidders 15 [16] JAIN, K., PADHYE, J., PADMANABHAN, V., AND QIU, L. Impact of interference on multi-hop wireless network performance. In Proc. of Mobicom’03 (2003).

16. Fast Auction-Clearing Algorithm Linearize the interference constraints ◦ Node-ALL interference constraints(NI) ◦ Node-L interference constraints(NLI) Clearing algorithm for different pricing models ◦ Clearing algorithm for uniform pricing(CAUP) ◦ Clearing algorithm for discriminatory pricing(CADP) Schedule spectrum usage 16

17. Fast Auction-Clearing Algorithm － Linearize Interference Constraints (1/4) Assume the spectrum is finely partitioned into a large number of channels Each buyer i obtains a normalized allocation of { f i : i = 1, 2,..., n} where f i ≤ 1.0 Example: ◦ A 1MHz spectrum band is divided into 100 channels of 10kHz ◦ A buyer i with f i = 0.143 ◦ Obtains channels 17

18. Fast Auction-Clearing Algorithm － Linearize Interference Constraints (2/4) Node-ALL interference constraints(NI) ◦ Constraint: restrict i and every neighbor of i to use different spectrum channels ◦ N(i) : the set of neighbors of i ◦ n : the total number of nodes It is more restrictive than necessary 18

19. Fast Auction-Clearing Algorithm － Linearize Interference Constraints (3/4) Node-L interference constraints(NLI) ◦ Define the notion of “left of” ◦ Nodes i and j locate at (x i,y i ) and (x j,y j )  If x i < x j, node i is to the left of node j  If x i = x j, node with smaller index is to the left to another node ◦ Constraint: every neighbor of i to the left of i, and i itself should be assigned with different channels 19 the set of neighbors of i lying to its left

20. Fast Auction-Clearing Algorithm － Linearize Interference Constraints (4/4) To illustrate our algorithm, we start from a simple model where each buyer pays a fixed per-unit price: p i (f i ) = b i, a i = 0 Problem: ◦ Can be solved by linear programming (LP) ◦ The quality of the solution produced by this LP is bounded by the following worst case error guarantee, proved by [6] : 20 Use NLI constraints [6] BURAGOHAIN, C., SURI, S., TOTH, C., AND ZHOU, Y. Improved throughput bounds for interference-aware routing in wireless networks. In UCSB Technical Report 2006-13 (2006).

21. Fast Auction-Clearing Algorithm － for Different pricing models (1/3) Clearing algorithm for uniform pricing(CAUP) ◦ Under NLI, the optimization problem becomes: ◦ Step 1: find the feasible region of p subject to interference constraints  Lemma 2: There exists a unique price p T where for any p, p ≥ p T, the channel allocation according to (17) will satisfy the constraints defined by (16), and for any p, p < p T results in allocations that violate the constraints. ◦ The feasible region of p is [p T, b n ]. Let b j −1 ≤ p T < b j 21 Use NLI constraints

22. Fast Auction-Clearing Algorithm － for Different pricing models (2/3) Clearing algorithm for uniform pricing(CAUP) ◦ Under NLI, the optimization problem becomes: ◦ Step 2: search for the revenue-maximizing p  Divide the region of p into intervals (p T, b j ], (b j, b j+1 ],..., (b n−1, b n ] => in each interval, revenue R(p) is a quadratic function 22 Use NLI constraints The proof can be found in [11] [11] GANDHI, S., BURAGOHAIN, C., CAO, L., ZHENG, H., AND SURI, S. A general framework for wireless spectrum auctions. UCSB Technical Report, 2007.

23. Fast Auction-Clearing Algorithm － for Different pricing models (3/3) Clearing algorithm for discriminatory pricing(CADP) ◦ Under NLI, the optimization problem becomes: ◦ Use separable programming [12] to approximately solve a special class of non-linear programs using linear programming 23 The proof can be found in [11] Use NLI constraints

24. Fast Auction-Clearing Algorithm － Schedule Spectrum Usage Given spectrum allocations {f i }, we need to schedule the actual usage patterns, that is, assign index of channel to each buyer ◦ Follow the “left of” order ◦ Start from the leftmost node, assign to it the initial portion of the spectrum ◦ For every next node i, find the rightmost node which are left to the i, refer to R i ◦ Assign to i the portion of its allocated spectrum starting from where the assignment of R i finishes 24

25. Experimental Result (1/2) Experiment environment ◦ In our discussion, wireless service providers randomly deploy their access points(buyer) to serve users ◦ Assume every buyer wants to support users within a fixed radius(0.05) ◦ Conflict exists if two access points are within 0.1 ◦ Spectrum available is normalized to 1 Consider three types of bidding curves 25

26. Experimental Result (2/2) Use the following performance metrics: Here examines: ◦ Performance of two pricing models ◦ Performance of the proposed algorithm ◦ Impact of bidding behavior ◦ Impact of node density ◦ Algorithm execution time 26

27. Experimental Result － Uniform vs. Discriminatory Pricing 27 Increase network size: 0 -> 1300 Increase average conflict degree: 0 -> 10 At small network sizes, the difference between uniform pricing revenue and discriminatory pricing revenue is small => The uniform price depends on the maximum level of conflict At small network sizes, the difference between uniform pricing revenue and discriminatory pricing revenue is small => The uniform price depends on the maximum level of conflict

28. Experimental Result － Optimal vs. Approximation Algorithms 28 Use the discriminatory pricing model Optimal solution: Use the randomized algorithm [16] for 200000 iterations to get the optimal revenue Optimal solution: Use the randomized algorithm [16] for 200000 iterations to get the optimal revenue The approximation is always within 10% of the optimal solution The computation time of optimal solution is 2000 times slower than the proposed algorithm(100 nodes) The approximation is always within 10% of the optimal solution The computation time of optimal solution is 2000 times slower than the proposed algorithm(100 nodes) [16] JAIN, K., PADHYE, J., PADMANABHAN, V., AND QIU, L. Impact of interference on multi-hop wireless network performance. In Proc. of Mobicom’03 (2003).

29. Experimental Result － Impact of Bidding Behaviors (1/2) 29 Buyers randomly choose their bidding curve (conservative, normal, aggressive) Buyers randomly choose their bidding curve (conservative, normal, aggressive) Uniform pricing: Aggressive bidders take over all the spectrum Uniform pricing: Aggressive bidders take over all the spectrum Discriminatory pricing: Aggressive bidders get a large portion of the spectrum and their allocation increases with network size Discriminatory pricing: Aggressive bidders get a large portion of the spectrum and their allocation increases with network size

30. Experimental Result － Impact of Bidding Behaviors (2/2) 30 Compare the total revenue generated by different bidders under both pricing models

31. Experimental Result － Impact of Node Clustering (1/4) In practice, wireless service provider might deploy access points with dense user populations, known as hotspots In this experiment: ◦ Randomly deploy 200 nodes ◦ Then deploy the next k(0 ≦ k ≦ 150) nodes in a clustered region 31

32. Experimental Result － Impact of Node Clustering (2/4) 32 For the size of 200 of less, random and clustered deployments produce the same topology Buyers’ bidding curves are normal For the size of 200 of less, random and clustered deployments produce the same topology Buyers’ bidding curves are normal Over 200 nodes － Uniform pricing: Revenue drops with the clustering Over 200 nodes － Uniform pricing: Revenue drops with the clustering Over 200 nodes － Discriminatory pricing: Converge very fast to a constant value, corresponding to a full utilization inside the cluster Over 200 nodes － Discriminatory pricing: Converge very fast to a constant value, corresponding to a full utilization inside the cluster

33. Experimental Result － Impact of Node Clustering (3/4) 33 Under discriminatory pricing model k=100 (total 300 nodes) Under discriminatory pricing model k=100 (total 300 nodes) To maximize revenue and utilization, pricing should depend on the conflict condition (price should be high at places with high demand and scarce resources) To maximize revenue and utilization, pricing should depend on the conflict condition (price should be high at places with high demand and scarce resources)

34. Experimental Result － Impact of Node Clustering (4/4) 34 How can a node in a clustered area obtain more spectrum? (Investigate the impact of bidding behavior in the clustered area) How can a node in a clustered area obtain more spectrum? (Investigate the impact of bidding behavior in the clustered area) Same clustering scenario, pick a buyer i when k=0 Then add k nodes to the cluster (increase the competition around i) Model i’s bidding behavior using p i (f i ) = ci (- f i + 1), where c i is aggressiveness Same clustering scenario, pick a buyer i when k=0 Then add k nodes to the cluster (increase the competition around i) Model i’s bidding behavior using p i (f i ) = ci (- f i + 1), where c i is aggressiveness

35. Experimental Result － Algorithm Complexity 35

36. Practical Considerations Identify interference constraints ◦ The auctioneer measures the network interference ◦ Individual point scan radio signals and report ◦ Clients sense radio signals [19] Decentralized auction systems [7] Iterative bidding and heterogeneous channels ◦ Adjust the bids according to the auction feedback ◦ In the case of heterogeneous channels, defining a standard price-quantity relationship is important ◦ Both issues can be addressed by combining computational and non-computational approaches 36 [7] CAO, L., AND ZHENG, H. Spectrum allocation in ad hoc networks via local bargaining. In Proc. of SECON (September 2005). [19] MISHRA, A., BRIK, V., BANERJEE, S., SRINIVASAN, A., AND ARBAUGH, W. A client-driven approahc for channel management in wireless LANs. In Proc. of IEEE Infocom (2006).

37. Conclusion Propose a spectrum auction framework ◦ Fast and efficient allocation ◦ PLPD ◦ Two pricing model ◦ Low-complexity market-clearing algorithm ◦ Experiments to verify the performance Conclude that to maximize revenue and utilization, pricing must be determined based on local demand and availability of resources 37