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)
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OutlineIntroductionPreliminaries and related workSpectrum auction framework
◦ PLPD◦ Auction-clearing problems◦ Optimal clearing algorithm
Fast auction clearing algorithmExperimental resultsPractical considerationConclusion
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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]
[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).
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Introduction (2/4)
In this paper, we consider how to efficiently auction spectrum to satisfy user demands while maximizing system revenue
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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
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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
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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
[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.
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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[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).
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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: FA∩FB = ∅◦ In this case, fA + fB ≤ 1, where fA = |FA|/M, fB = |FB|/M◦ Auction clearing problem
becomes:
},...,,{ 21AM
AAA SSSF 1A
kS
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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 fi at each per-unit price pi
◦ Any PLPD curve can be expressed as a conglomeration of a set of individual linear pieces
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Spectrum Auction Framework- PLPD (2/3)
A simple example of linear demand curve:◦ Demand curve:◦ Quantity fi(pi) and revenue generated Ri(pi):
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Spectrum Auction Framework- PLPD (3/3)
PLPD has advantages◦ Simple and highly
expressive◦ Single bid covers different
pricing options◦ Quadratic revenue
function
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Spectrum Auction Framework-Auction-Clearing Problems (1/2)
Uniform pricing◦ The auctioneer sets a clearing price p◦ Each bidder obtains a fraction of spectrum
fi(p)=(bi - p)/ai and produces a revenue of Ri(p)=(bip - )/ai
◦ Assume bidders 1 to n are in increasing order of bi, i.e. , and b0=0
◦ The auction clearing problem becomes
2p
nbbbb ...321
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Spectrum Auction Framework-Auction-Clearing Problems (2/2)
Discriminatory pricing◦ The clearing prices vary across i◦ The optimization problem becomes
(-aifi + bi) * fi
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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
[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).
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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
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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 { fi : i = 1, 2, . . . , n} where fi ≤ 1.0
Example: ◦ A 1MHz spectrum band is divided into 100 channels
of 10kHz◦ A buyer i with fi = 0.143◦ Obtains channels 14100143.0
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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
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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 (xi,yi) and (xj,yj)
If xi < xj, node i is to the left of node j If xi = xj, 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
the set of neighbors of i lying to its left
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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: pi(fi) = bi, ai = 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] :
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).
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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 pT where for any p, p ≥
pT , the channel allocation according to (17) will satisfy the constraints defined by (16), and for any p, p < pT results in allocations that violate the constraints.
◦ The feasible region of p is [pT , bn]. Let bj−1 ≤ pT < bj
Use NLI constraints
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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 (pT, bj], (bj, bj+1], . . . , (bn−1,
bn] => in each interval, revenue R(p) is a quadratic function
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.
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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
The proof can be found in [11]
Use NLI constraints
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Fast Auction-Clearing Algorithm- Schedule Spectrum Usage
Given spectrum allocations {fi}, 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 Ri
◦ Assign to i the portion of its allocated spectrum starting from where the assignment of Ri finishes
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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
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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
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Experimental Result-Uniform vs. Discriminatory Pricing
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
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Experimental Result-Optimal vs. Approximation Algorithms
Use the discriminatory pricing model
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)
[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).
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Experimental Result- Impact of Bidding Behaviors (1/2)
Buyers randomly choose their bidding curve(conservative, normal, aggressive)
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
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Experimental Result- Impact of Bidding Behaviors (2/2)
Compare the total revenue generated by different bidders under both pricing models
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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
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Experimental Result- Impact of Node Clustering (2/4)
For the size of 200 of less, random and clustered deployments produce the same topologyBuyers’ bidding curves are normal
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
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Experimental Result- Impact of Node Clustering (3/4)
• 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)
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Experimental Result- Impact of Node Clustering (4/4)
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 pi(fi) = ci (- fi + 1), where ci is aggressiveness
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Practical ConsiderationsIdentify 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[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).
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ConclusionPropose 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