A General Framework for Wireless Spectrum Auctions Sorabh Gandhi, Lili Cao, Haitao Zheng, Subhash...

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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A simple example of linear demand curve:◦ Demand curve:◦ Quantity fi(pi) and revenue generated Ri(pi):

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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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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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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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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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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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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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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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• 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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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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Experimental Result－Algorithm Complexity

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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

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