Combinatorial Auction with Time-Frequency Flexibility in Cognitive Radio Networks

1. Combinatorial Auction with Time-FrequencyFlexibility in Cognitive Radio Networks Mo Dong*, Gaofei Sun*, Xinbing Wang*, Qian Zhang** *Department of Electronic Engineering Shanghai Jiao Tong University, China **Department of Comp. Sci. and Engin HK Univ. of Sci. and Tech., HongKong

2. Outline Introduction System Model and Problem Formulation Solution of CAP Under General Flexibility Model Solution of CAP Under Modified Model Simulation Results and Future Works 2

3. Outline Introduction Background Objectives System Model and Problem Formulation Solution of CAP Under General Flexibility Model Solution of CAP Under Modified Model Simulation Results and Future Works 3

4. Background • Dynamic Spectrum Access Auctions • Motivation • Under-utilized wireless spectrum • Provide incentive for primary users • General Framework Spectrum Opportunity Allocation Mechanism Payment Mechanism SUs’ bids

5. Background • Dynamic Spectrum Access Auctions • General Framework • Auction mechanisms • Periodic Auction [G. Kasbekar TON’10] , [A. Gopinathan INFOCOM’11] , [L. Chen] • Online Auction [H. Zheng INFOCOM’11] , [X. Li DySPAN’10] , [S. Sodagari JSAC’11] Spectrum Opportunity Allocation Mechanism Payment Mechanism SUs’ bids

6. Background • Dynamic Spectrum Access Auctions • General Framework • Auction mechanisms • Single Auction [S. Kasbeka TON’10] • Double Auction [W.Saad, INFCOM’10] Spectrum Opportunity Allocation Mechanism Payment Mechanism SUs’ bids

7. Background • Dynamic Spectrum Access Auctions • General Framework • Auction mechanisms • Periodic Auction • Online Auction • Single Auction • Double Auctino Spectrum Opportunity Spectrum Opportunity Allocation Mechanism Payment Mechanism This assumption doesn’t fit the real scenario Fixed and Homogenous Fixed time span SUs’ bids Fixed bandwidth

8. Background • Dynamic Spectrum Access Auctions • An example of SUs’ flexible requirements Frequency Spectrum Pool of Primary Operator SU1: Rigid and inflexible spectrum offered by PUs Cannot cater to the flexible requirements Spectrum Opportunity Allocation Mechanism Time-frequency flexible requirements SU2: SUs: Payment Mechanism SU3: SUs’ bids Time Auction Period

9. Objective • DSA mechanism based on combinatorial auction • Consider the SUs’ requirements varying over time and frequency • Achieve efficiency, truthfulness & low computational complexity

10. Outline Introduction System Model and Problem Formulation System Model Problem Formulation Solution of CAP Under General Flexibility Model Solution of CAP Under Modified Model Simulation Results and Future Works 10

11. System Model • Four-layer cognitive radio network model • Primary Operator based system

12. System Model • Four-layer cognitive radio network model • Periodic Auction

13. System Model • Four-layer cognitive radio network model • Heterogeneous and flexible requirements

14. System Model • Model of PO (seller)’s spectrum opportunity • Time-frequency divide • We now denote the time interval as and the frequency interval as and every spectrum that starts from as . • Because of the stable assumption, we change the notation of to ,where . 1 2 3

15. System Model • Model of SUs(buyers)’ time-frequency flexible requirements • There are SUs denoted as • The bid of SU is denoted as , where and is the valuation of • The winning SU is denoted as • The payments of SUs are • The net utility of SU is:

16. Problem Formulation • Cognitive Radio Winning SUs Determination Problem(CRWDP): • For every subset , let denote the value of buyer for . We use to denote that the buyer wins the bundle and to denote the buyer loses or does not bid on . The winner determination problem is defined as follows: • Remarks: • Optimize the social welfare • One spectrum slot is assigned to one SU • One buyer can only have at most one bundle of goods at last • One buyer can only require one bundle of goods

17. Problem Formulation • The Truthful Mechanism Design Problem(TMDP): • For any buyer , • Truthful bid: Truthful utility: • Declared bid: Untruthful utility: • Assume No constraint on • The TMDP problem is to design a payment mechanism such that Spectrum Opportunity Allocation Mechanism Combinatorial Auction Problem (CAP) Payment Mechanism Winner Determination Algorithm SUs’ bids Truthful Payment Mechanism

18. Outline Introduction System Model and Definitions Solution of CAP Under General Flexibility Model Approximation Analysis of the NP-hard CRWDP Approximation Algorithm to Solve CRWDP Truthful Payment Mechanism under the Approximation Algorithm Solution of CAP Under Modified Model Simulation Results and Future Works 18

19. Approximation Analysis of CRWDP • The CRWDP is NP hard to solve unless NP =ZPP • SUs: Vectors; Spectrum Slots: Edges; Values: all set to one • CRWDP is reduced from the maximum independent set problem(MISP) • The upper bound of the approximation ratio of CRWDP is , for any polynomial time algorithm. • The approximation ratio of CRWDP will not exceed that of MISP • The approximation ratio of MISP is • There is an implicated assumption in • the reduction:

20. Approximation Algorithm for CRWDP • Sorting Based Greedy Algorithm for CRWDP • Step 1: Reorder the bids according to a newly defined Norm • Step 2: Allocate spectrum opportunity greedily following the reordered list of norm

21. Approximation Algorithm for CRWDP • Sorting Based Greedy Algorithm for CRWDP • The choice of ordering norm is the critical • The computational complexity is • The approximation ratio of this algorithm reaches the upper bound

22. Approximation Algorithm for CRWDP • Proof of the approximation ratio of the proposed algorithm Omitted

23. Truthful Payment Mechanism • The Truthful Payment Mechanism • Recall is the list in which all buyers are reordered by the norm in the first step of algorithm 1. And for buyer , we denote a buyer as the first buyer following in that has been denied but would have been granted were it not for the presence of . We have the following payment: • pays zero if his bid is denied or does not exist. • If there exists an and ’s bid is granted, he pays , where is the norm of .

24. Truthful Payment Mechanism • The Proof of Truthfulness • If an auction mechanism fits the following conditions, it is truthful • Ex-post Budget Balance: That means the buyers are all rational so that they will not pay more than their value of the goods. • Monotonicity: A buyer who wins with can still win with any and any , when others’ bids are fixed. • Critical Payment: There exists a critical value for a winner , so that he only needs to pay this critical value to win. That is to say, if others’ bids are fixed, the payment of a certain winner does not depend on how he reports his bid

25. Outline Introduction System Model and Problem Formulation Solution of CAP Under General Flexibility Model Solution of CAP Under Modified Model A Rational Modification of the System Model Optimal Solution for CRWDP-C Truthful Payment Mechanism under the Modified Model Simulation Results and Future Works 25

26. A Rational Modification of the System Model • Additional Constraints on SUs’ Requirements • Full Time Usage • Consecutive Requirement • Change of Expressions in System Model • Goods Set: where is the starting frequency point of the thinterval: • Redefine the as a close interval , which means the SU wants the frequency that starts from and ends at • to denote the highest valuation submitted to auctioneer on . • G = {T|T ⊆ S and T fits the Consecutive • Requirement condition}. 26

27. A Rational Modification of the System Model • The problem of CRWDP-C can be formulated as: • there is one and only , • for all other , . 27

28. Optimal Solution for CRWDP-C • Algorithm achieves optimal solution for CRWDP-C and the computational complexity is . 28

29. Payment Mechanism for the Modified Model • VCG-Based Truthful Payment Mechanism. • We denote the payment for buyer as . We denote as the goods buyer receives in the final allocation. We can see that when buyer wins and when loses. Further, we denote as valuation obtained by buyer in the final allocation. Note that when wins and when loses. We denote as the optimal social welfare obtained in the auction where is absent. Note that despite , all other buyers stay the same when calculating and . We have that the payment is: • Remarks: • VCG-Payment Mechanism is truthful • VCG-Payment Mechanism’s limitation 29

30. Outline Introduction System Model and Problem Fromulation Optimal Solution for CRWDP-C Truthful Payment Mechanism under the Modified Model Simulation Results and Future Works 30

31. Simulation Results and Future Works • Simulation Results under General Model

32. Simulation Results and Future Works • Simulation Results under Modified Model

33. Simulation Results and Future Works • Future Works • Multiple bids can be submitted by SUs • Make the combinatorial auction mechanism an online one

34. Q&A

35. Thank you for listening