WITH dramatically growing demand of spectrum for new

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1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 2, FEBRUARY Multi-Item Spectrum Auction for Recall-Based Cognitive Radio Networks With Multiple Heterogeneous Secondary Users Changyan Yi and Jun Cai, Senior Member, IEEE Abstract In this paper, we consider a spectrum auction system among heterogeneous secondary users (SUs) with various qualityof-service (QoS) requirements and a recall-based primary base station (PBS) that could recall channels after auction to deal with a sudden increase in its own demand. Beginning with proposing a recall-based single-winner spectrum auction (RSSA) algorithm, we further extend our work to allow multiple winners in order to improve the spectrum utilization and propose a recall-based multiple-winner spectrum auction (RMSA) algorithm. A combinatorial auction model is then formulated, and Vickrey Clarke Groves (VCG) mechanism is applied in the payment function. Moreover, the proposed RMSA algorithm focuses on a fair spectrum allocation among heterogeneous SUs and the increase in the PBS s auction revenue. Both theoretical and simulation results show that the proposed spectrum auction algorithm can improve the spectrum utilization with guarantees on SUs heterogeneous QoS requirements. Index Terms Cognitive radio (CR), dynamic spectrum access (DSA), quality-of-service (QoS), spectrum auction, Vickrey Clarke Groves (VCG) mechanism. I. INTRODUCTION WITH dramatically growing demand of spectrum for new wireless devices and applications, current fixed spectrum assignment policy has imposed significant restrictions on spectrum utilization efficiency, which leads to an issue called spectrum scarcity. Therefore, it is imperative to exploit underutilized spectrum in a more intelligent and flexible way [1]. To achieve this goal, dynamic spectrum access based on cognitive radio (CR) has been proposed as a prospective solution that allows secondary users (SUs) to opportunistically access the licensed spectrum on the premise that the services of primary users (PUs) are not degraded because of the interference [2] [4]. Unlike CR based on sensing, where PUs are assumed to be unconscious of SUs activities [5], PUs could take initiative in spectrum marketing by deciding the quantity of spectrum to be leased to maximize their utilities within their interference tolerance. In addition, spectrum marketing is also considered as Manuscript received August 9, 2013; revised February 11, 2014 and March 30, 2014; accepted April 25, Date of publication May 6, 2014; date of current version February 9, This work was supported by the Natural Sciences and Engineering Research Council of Canada under Discovery Grant. The review of this paper was coordinated by Dr. S. Zhong. The authors are with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB R3T 5V6, Canada ( Changyan.Yi@umanitoba.ca; Jun.Cai@umanitoba.ca). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TVT an effective way to realize spectrum sharing since PUs could charge SUs to dynamically use licensed spectrum [6]. In spectrum marketing, auction-based methods are widely used since they can better depict the behaviors of self-serving users and can take into consideration both the individual utility and social welfare of the system [7] [9]. The authors in [10] proposed two auction mechanisms to receive power allocation to achieve social optimality and fairness in underlay spectrum sharing. A real-time spectrum auction framework was formulated in [11] where interference constraints were modeled by linear programming, and the maximum revenue was generated by optimally selecting market clearing price. In [12], the authors considered to accommodate multiple SUs in one band and presented a novel multiwinner spectrum auction, which was proved to be strategy-proof. All these works only considered the scenario that the seller, e.g., the primary base station (PBS), has one channel to auction. Spectrum double auction, which aims to assign multiple licensed channels to multiple SUs, was studied in [13] [16]. The authors in [13] proposed a general framework for truthful double spectrum auctions, where multiple parties can trade spectrum based on their individual needs. The authors in [14] presented a set of new spectrum double auctions that are specifically designed for local spectrum markets. In [15], a truthful double auction mechanism is studied for heterogeneous spectrums where the distinctive characteristics in both spatial and frequency domains are considered. In [16], the authors investigated a discriminating pricing double spectrum auction where bidders are charged different prices for the same item they purchase. However, most of these works presume singleitem (single-channel) or homogeneous demands from all SUs, whereas in reality, it is common for SUs to request multiple heterogeneous number of items (or channels), particularly in multimedia communication scenarios [17] [19]. To meet these practical requirements, the authors in [17] proposed a sealed-bid reserve auction mechanism for multiradio spectrum allocation. The authors in [18] discussed a strategy-proof combinatorial auction for heterogeneous channel allocation with channel spatial reusability. The authors in [19] studied a cooperationbased dynamic leasing mechanism via multiwinner auction over multiple available spectrum bands. Recently, spectrum auction has been also studied by considering potential temporal reusability. For example, [20] discussed the situation where each bidder requests a number of channels for a certain time interval, whereas in [21], the authors modeled the arrivals of SUs requests as Poisson processes and proposed a general framework for truthful online double spectrum auction IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 782 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 2, FEBRUARY 2015 Most works in the literature assumed that the auctioned channels are exclusively occupied by the winning SU(s). Such an assumption poses a dilemma for the PBS: either auction idle channels and get auction revenue at the risk of a sudden increase in demand from PUs or reserve spectrum uneconomically. To address this issue, the authors in [22] first introduced the idea of recall-based dynamic spectrum auction in which PUs are granted higher channel access priority so that auctioned channels could be recalled if necessary. However, the work ignored the potential heterogeneity in demands of SUs and considered single winner in the auction only. In this paper, multi-item recall-based spectrum auction is addressed in a CR network consisting of one PBS and multiple SUs. Each SU has heterogeneous quality-of-service (QoS) requirements in terms of spectrum demands and spectrum stability requirements. We begin our discussion with single-winner auction and then extend it to the case with multiple winners. In both scenarios, SUs determine their bids based on both the auction information from the PBS and their own spectrum demands and stability requirements. For the single-winner auction, the second-price sealed-bid (SPSB) model [7] is adopted, whereas in the multiwinner auction, Vickrey Clarke Groves (VCG) mechanism is applied as the payment function to match the requirements of combinatorial auction [23], [24]. In both cases, we redefine the private valuation of spectrum for each SU and redesign the optimal strategies for both SUs and the PBS. For the multiwinner auction, a new channel recall scheme is also proposed to achieve fairness among multiple SUs. Both analytical and simulation results show that the proposed spectrum auction algorithm can improve channel utilization while guaranteeing SUs heterogeneous QoS requirements. The main contributions of this paper are summarized as follows. We develop recall-based spectrum auction methods by considering multiple SUs with heterogeneous channel requirements. New private value functions are proposed by introducing new parameters, called risk factors in singlewinner auctions and spectrum stability factors in multiwinner cases, in order to characterize the effects of channel recall on SUs. A recall-based single-winner spectrum auction (RSSA) algorithm is proposed and is then extended to a recall-based multiple-winner spectrum auction (RMSA) algorithm. A new channel recall scheme is proposed in multiwinner spectrum auction to achieve better fairness among winning SUs. Theoretical analysis is provided to prove that both RSSA and RMSA algorithms are economically incentive for the SUs with different spectrum demands and stability requirements. The rest of this paper is organized as follows. Section II describes the system model and summarizes all important notations used in this paper. Section III defines the singlewinner spectrum auction and analyzes the optimal strategies in RSSA algorithm. The extension to a multiwinner auction, called RMSA algorithm, is introduced in Section IV. Section V provides the performance analyses on the auction revenue of Fig. 1. System model of recall-based spectrum auction. the PBS and SUs utilities. Simulation results are shown in Section VI. Finally, we give a brief conclusion in Section VII. II. SYSTEM MODEL Consider a CR network with N SUs who opportunistically access the unused channels of a PBS. The PBS owns total C units of homogenous and undivided channels. Assume that each PU only requires one channel and the PUs with channel demands would generate a queue at the PBS. We further assume that all PUs obey the first-come first-serve rule. If all available channels have been fully occupied, newly arrived PUs have to wait in the queue. Without loss of generality, the PUs arrive at the PBS following a Poisson process with arrival rate λ so that the interarrival times are independent and identically distributed (i.i.d.) random variables with an exponential distribution. Furthermore, assume that the PUs channel occupancy times are also i.i.d. exponential random variables with service rate μ. Thus, the channel service of PUs could be considered as an M/M/m queuing system, as shown in Fig. 1, where M refers to Markov process and m denotes the number of channels for PUs. Channel is considered as idle if it was not occupied by any PUs; otherwise, it is busy. Note that SUs have no information about PUs random activities. The PBS leases certain number of channels to SUs and, at the same time, provides its PUs with a QoS guarantee. In this paper, we define the mean waiting time in the queue as the measurement of the QoS for PUs [25]. Specifically, we suppose that the mean waiting time of PUs M w cannot be greater than a certain threshold γ. Due to the randomness of PUs arrivals, if the PBS decides to auction some unused channels for economic revenue from SUs, it may suffer a risk that there are no enough channels to deal with a sudden increase in PUs spectrum demands. By considering the higher priority of PUs in this paper, we allow spectrum recall for the PBS, i.e., the PBS could recall some channels from the winning SU(s) in order to satisfy its own PUs demands when necessary. In this way, the newly arrived PUs need to wait if and only if there are no idle channels in the PBS and no more channels can be recalled. Recalled channels will not be returned to SUs until next round of auction. Of course, the auction winner(s) will get corresponding compensation if their channels were recalled

3 YI AND CAI: MULTI-ITEM SPECTRUM AUCTION FOR CR NETWORKS WITH MULTIPLE HETEROGENEOUS SUs 783 TABLE I IMPORTANT NOTATIONS IN THIS PAPER by the PBS. For explanation purpose, we further assume that the PBS would always be truthful in the auction (otherwise, a belief index η(η <1) can be introduced in bidders bidding price to enforce the PBS broadcasting auction information truthfully [22]). Different from traditional works, in this paper, SUs are heterogeneous in spectrum demands and stability requirements. Furthermore, we assume that each SU works on an integral number of licensed channels. Such assumption is commonly employed in the literature, such as [26], which considered the application of Microsoft KNOWS prototype [27]. Let SU i have a spectrum demand and a value V i for channels. Each SU submits a sealed bid b i according to its demand to maximize its expected utility. The auction is carried out frame by frame, and each frame has a length of T. We limit our discussion to small region networks [28], i.e., all SUs are located within the interference range of each other; hence, no spectrum reuse among SUs within a frame is considered. We further let SUs be risk neutral. At the beginning of each T, there is a small period ΔT T used for channel auction. Unlike [22], we focus on the utilities of SUs and consider the spectrum allocation among multiple heterogeneous SUs. For convenience, Table I lists some important notations used in this paper. III. RECALL-BASED SINGLE-WINNER SPECTRUM AUCTION Here, an RSSA algorithm is proposed. The valuation function of SUs is first defined, and the SPSB auction model is applied as the payment rule. After this, optimal strategies for both SUs and the PBS are analyzed. A. Private Values of SUs Without the use of recall-based PBS, the private value of SU i, i.e., v i ( ), should increase with the number of demanded channels.ifsu i wins the auction, then it exclusively occupies the channels, and it could transmit at any available power level without interfering with others. Thus, similar to [22] and [29], in this paper, we define the private value of SU i to be equal to the Shannon capacity it could achieve by obtaining channels, i.e., ( v i ( )= B log P ) t, 0 (1) n 0 B where B is the bandwidth per channel, P t denotes the unified transmit power of all SUs, and n 0 indicates the spectral density of noise. Now, let us consider the situation with a recall-based PBS. In this case, the PBS first divides C channels into two categories (,C ) at the beginning of each auction. channels are auctioned while the remaining C channels are reserved for its PUs. For the purpose of protecting its own PUs, the PBS also determines C r, which is the maximum number of channels that can be recalled. In other words, the PBS has at most C + C r channels for PUs in the following frame in order to guarantee that the average waiting time of the system would not be greater than the threshold γ. Obviously, C r should be less than or equal to. Apparently, for any SU i, its utility would not decrease with the channel recall if C r. It means that even under the maximum channel recalls, there is no effect on SU i if it wins the auction. However, if C r <, the channel recall by the PBS introduces a reduction on the winning SU i s utility. It is not difficult to find that under the worst case, the maximum number of recalled channels from SU i is equal to ( C r ). In order to reflect the effects on SUs due to channel recalls, a particular parameter ρ (0 ρ 1), called the risk factor, is introduced. Hence, if C r <, the benefit of SU i to obtain channels in the recall-based system can be defined as [ v i(c i )= 1 C ] i ( C r ) (1 ρ i ) v i ( ). (2) In (2), [ ( C r )]/ represents the maximum channel recall ratio on the only winner SU i. Obviously, v i () is a decreasing function of C r, which matches the intuition that the more channels the PBS declares to recall, the lower private values SUs may have. Parameter ρ i is used to reflect different attitudes from SUs toward the potential channel recall. SU with larger ρ is more willing to take risk in this recall-based system and has less concern about the channel recall. Note that ρ is a system factor and cannot be changed by SUs arbitrarily. In fact, such factor heavily depends on the SU s traffic type and the QoS requirements. Based on the previous discussions, we can define singleminded SUs in the single-winner spectrum auction with different private value functions. Definition 3.1: For C homogeneous channels and SU i with valuation v( ),SU i is single minded if there is a number of auctioned channels and a number of maximum recalled channels C r such that { vi (C V i = i ), if 0 C r (3a) v i (), if C r <. (3b) Note that it is meaningless to report a demand > since the request of SU i can never be satisfied. Hence, we assume that all demands from SUs are bounded by. Then, from SU s perspective, there are two distinct outcomes: 1) It gets all channels it demands, i.e.,, and does not need to worry about the channel recall; 2) it gets all channels it demands but evaluates with consideration of channel recalls.

4 784 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 2, FEBRUARY 2015 B. Optimal Strategies in RSSA Since all bids are sealed, SU i does not know the bids from others. However, it is natural for all SUs and the PBS to know that all bids follow the same valuation function defined in (3) except their private information. We further assume that no SU would misreport their channel requests. Such assumption is widely used in the auction with heterogeneous bidding requests [17], [30]. If the PBS only allows one winner, no matter how much channels are demanded from each SU, all channels are auctioned to the single winner. Thus, the PBS could simply consider the auction, including same-demand SUs, although each SU may have its specific demand and valuation. 1) Optimal Strategy of SUs: According to the SPSB auction that that highest bidder wins the auction but pays the second highest bid, the optimal strategy for each bidder is to bid the true valuation of its demanded objects [7], i.e., b single i = V i. (4) Similar to the private valuation function in (3), SU i has two outcomes of bids according to the values of,, and C r. In the first condition, b single i = B log 2 (1 + P t /n 0 B), and the channel recall has no impact on SU i. In this case, the bid is monotonically increased with its spectrum demand. In the second condition, b single i = {1 (1 ρ i )[ ( C r )]/ } B log 2 (1 + P t /n 0 B), and the channel recall would affect SU i s service. In this case, the bid varies with SU i s spectrum demand, the maximum recall ratio on SU i, and its risk factor ρ i. The bid would increase as ρ i increases. This is to say that SU, which is not much concerned on the impact of channel recall so as having a larger ρ, would bid higher in the auction. 2) Auction Information Broadcasting by the PBS: In recallbased spectrum auction, SU i with highest bidding price wins the auction and pays the second highest bid, i.e., b 2nd.During [ΔT,T], the PBS recalls C r,a channels after the auction and the actual number of channels recalled from SU i is C r,i. Note that for winner SU i, its demand should be less than the total number of auctioned channels, i.e., 0 <. Thus { 0, if Ci C r,a (5a) C r,i = ( C r,a ), if > C r,a. (5b) Hence, the PBS compensates C r,i b 2nd / back to the winning SU. Then, the final auction revenue of the PBS can be expressed as R single a = C r,i b 2nd. (6) Since the PUs QoS would be always protected because of the spectrum recall, we have max U PBS =maxra single (7) where U PBS denotes the utility of the PBS. Therefore, the optimal strategy for the PBS is to choose the highest bid and maximize its auction revenue. Since is bounded by and relaxing always produces a nondecreasing Ra single, should be as large as possible, or in other words, the PBS should auction all its idle channels at the beginning of each frame. In fact, the maximum number of recalled channels C r can be also determined given M w γ. According to M/M/m queue [31] with the arrival rate λ and the service rate μ, the minimum number of channels needed by the PUs m can be obtained by M w = Q(m, G) μ(m G) γ (8) where G = λ/μ, and Q(m, G) is the queuing probability that is equal to G m /m! Q(m, G) = [(m G)/m] m 1 r=0 (Gr /r!) + G m /m!. (9) Suppose C m. Thus, in order to guarantee the QoS of PUs, the minimum number of total reserved channels at the PBS should satisfy the condition that m (C )+C r or C r m (C ). In addition, C r should not be greater than the number of auctioned channels. Therefore, C r should be ranged as C r m (C ). Since the derivation of m has guaranteed the QoS of PUs, the PBS has no intension to reserve more channels, i.e., C r = m (C ). C. RSSA Algorithm The detailed timeline of the proposed RSSA algorithm is listed as follows. At the beginning of each frame, the PBS broadcasts the auction information, including the number of auctioned channels and the maximal recalls C r based on its current service state. The settings of and C r can be found in Section III-B2. Each SU i receives the auction information and sets up a value V i based on its own spectrum demand, risk factor ρ i, and the maximum channel recall ratio on itself, i.e., [ ( C r )]/. Then, SUs submit sealed bids and their specific demands to the PBS. The PBS determines the only winner by selecting the SU with the highest bidding price and charges it with the second highest bid b 2nd. After the auction, the PBS can recall channels from the winning SU i if necessary to satisfy its own sudden increase in spectrum demand. At the end of T, the PBS refunds SU i with C r,i b 2nd /. IV. RECALL-BASED MULTIPLE-WINNER SPECTRUM AUCTION Since the demand of each SU i, i.e.,, is independent of the number of auctioned channels, it is very likely that the auctioned channels cannot be fully utilized by one winning SU. Thus, the auction revenue could be improved if the PBS picks more than one winner. However, the allowance of multiple winners makes the spectrum auction become a more complicated combinatorial auction problem. In the payment design of the proposed RMSA algorithm, VCG mechanism is adopted. Although VCG mechanism cannot

5 YI AND CAI: MULTI-ITEM SPECTRUM AUCTION FOR CR NETWORKS WITH MULTIPLE HETEROGENEOUS SUs 785 guarantee the maximum auction revenue for the PBS [32], it is the basic payment mechanism in combinatorial auction that can ensure efficiency, incentive compatibility, and individual rationality. Note that revenue-maximizing combinatorial auction mechanism or any approximating auction mechanism, such as virtual valuation combinatorial auctions [33] or LOS mechanism [34], can be also applied in our proposed scheme. A. Strategy of SUs in RMSA Without channel recall, the private value on channel demand of SU i in multiple-winner auction is the same as that in (1), i.e., v i ( )= B log 2 (1 + P t /n 0 B) for 0. With channel recall, the PBS needs to announce and C r at the beginning of each frame. Different from the single-winner case where each SU could figure out the maximum number of channels recalled from itself if it won the auction, such information is not available in multiwinner auction because the number of channels recalled from a winning SU is not only determined by its own demand but also by the demands of other winners. Let W {1, 2,...,N} be the set of winners. Different from the single-winner case, since the auctioned channels may not be fully utilized by winners in W, the maximum channel recall ratio on W is equal to χ multiple = C r ( i W ). (10) Same as the RSSA algorithm, each SU needs to evaluate its private value toward its spectrum demand based on χ multiple. However, term ( i W ) is unpredictable since W cannot be determined before the auction. We approximate χ multiple as C r /. Similar to the risk factor in the RSSA algorithm, we define θ i [0, 1] as SU i s spectrum stability factor in its private value function definition. Although θ i also reflects the attitude of SU i toward channel recall, the physical meaning of θ in RMSA is different from ρ in RSSA. In the single-winner case, since the maximum channel recall ratio on the single winner can be determined before auction, the spectrum stability is only determined by the activity of PUs. However, in the multiwinner case, since the maximum channel recall ratio can only be determined at the system level, i.e., C r /, rather than each winner, the spectrum stability factor may affect both the winner determination and the channel recall ratio on each winner. The definition of single-minded SUs in the RMSA algorithm is given in the following. Definition 4.1: For C homogeneous channels and SU i with valuation V i,su i is single minded if there exist a number of auctioned channels and a number of maximum recalls C r such that [ V i = v i ( )= 1 C ] r (1 θ i ) v i ( ), if 0. (11) Thus, from SU i s perspective, it gets channels it demands but multiplies a channel stability ratio to its valuation. Similar as RSSA algorithm, the larger C r the PBS declares, the lower private values SUs may have. Moreover, SUs with different spectrum demands and stability factors would also lead to different private values. Larger θ i indicates that SU i could be provided a more stable service to gain higher utility. Note again that θ is also predetermined by the system depending on SUs traffic types and transmission requirements and cannot be arbitrarily changed by SUs. In the design of payment function, VCG mechanism requires that all bidders only know their own private values for their demands and each of them has a quasi-linear utility function. Since SU i could determine its valuation of the bundle of channels when they receive the announcement of and C r, we prove the quasi-linearity of our utility function in the following proposition. Proposition 4.1: For SU i {1, 2,...,N} with particular spectrum demand and spectrum stability requirement factor θ i, u i = V i t i is a quasi-linear utility function, where t i denotes the payment of SU i in the auction. Proof: In order to prove that u i = V i t i is a quasilinear utility function, we only need to prove that V i is a concave function of channel demand [7]. Recall that V i =[1 (C r / )(1 θ i )] B log 2 (1 + P t /(n 0 B)) if 0. Since the ratio caused by channel recall, i.e., 1 (C r / )(1 θ i ), is not varied with, the concavity and convexity of V i only depend on the formula of Shannon capacity. In fact, it can be directly proved that the capacity B log 2 (1 + P t /(n 0 B)) is an increasing concave function of bandwidth B [35]. Thus, V i is a concave function of, and u i is a quasi-linear utility. According to VCG mechanism, truthful bidding maximizes any player s utility regardless of other players choices [7]. Hence, all the SUs would truthfully bid in the spectrum auction by honestly telling the PBS their private values and their demands, i.e., b multiple i = V i. (12) B. Actions of the PBS in RMSA Similar to the analysis in Section III-B2 in determining and C r in the RSSA algorithm, the PBS would auction all its idle channels and announce a maximum recall quantity C r based on PUs QoS requirement. Here, we focus on the winner determination of combinatorial auction and the payment charged for each winner. Furthermore, a new channel recall scheme is proposed to achieve some level of fairness in spectrum sharing among heterogeneous SUs. 1) Winner Determination and Payment Design: In our system, each SU tells the PBS its sealed bid and specific spectrum demand. The PBS determines the winner by solving the following optimization problem. Given the bid B = {b 1,b 2,...,b N } and spectrum demand {C 1,C 2,...,C N }, the PBS determines the winners such that max P B C = {x i }, i N s.t. N b i x i i=1 N x i (13) i=1

6 786 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 2, FEBRUARY 2015 where { 1, if SUi is the winner in the auction x i = 0, otherwise. The optimization problem (13) aims to find the set of winners W = {i x i = 1, i N} such that the sum of their bids received by the PBS could be maximized under the constraint that their total spectrum demand is less than or equal to the number of auctioned channels. Furthermore, since we assume that SUs are single minded so that SU i can get either all spectrum it demands or nothing, the maximization problem (13) is actually a 0 1 single knapsack problem that can be solved to optimality in pseudopolynomial time by using dynamic programming or branch-and-bound algorithm [36], [37]. Note that the availability of optimal solution to (13) guarantees the feasibility of VCG payment rule. After deciding the set of winners, the PBS charges the winning SUs according to the VCG mechanism. The payment of SU i is t i = P C B\{b i } P C\{} B\{b i } (14) where PB\{b } denotes the maximum welfare if SU i does not participate in the auction and P C\{} B\{b i } denotes the maximum welfare if SU i does not participate and it takes out its winning channels from the total C channels in the auction. The details of payment rule in VCG mechanism can be found in [7]. 2) Channel Recall Scheme: The VCG payment mechanism is actually designed for buyers with fixed private values. However, in our system model, the utilities of winning SUs may decrease after the auction because of channel recalls. It provides us an incentive to design a new spectrum recall allocation in our RMSA algorithm. For explanation purpose, we first introduce a simple definition of fairness index. Definition 4.2 (Min Max Fairness): For each winner i W, if the actual number of channels recalled on SU i is less than its spectrum demand, i.e., C r,i <, we define a resourceallocation index as f i = C r,i t i (15) where C r,i indicates the actual number of channels SU i obtained, and t i is the payment. Given f i, i W, a min max fairness index can be defined as I min max = min{f i} max{f i }, i {i i W, C r,i < }. (16) Obviously, according to the definition previously shown, the spectrum allocation is more fair when I min max tends to 1. Proposition 4.2: The VCG mechanism is unfair under the situation that multiple homogeneous channels are auctioned among SUs with different stability requirements and the same recall ratio C r,a / on multiple winners is applied. Proof: Consider the system with only two winners, i.e., SUs i and j, both of which have the same spectrum demands, i.e., = C j. According to (11), the difference of their private values only depends on the spectrum stability factor θ. Assume that θ i >θ j. Then, SU i has a larger private value than SU j, which leads to a larger bid, i.e., b i >b j. With the VCG mechanism of item allocation and payment design, it is easy to find that SU i and SU j will get the same number of channels but with t i >t j. Since the recall ratio is the same on both SUs i and j, i.e., C r,a /, the fairness index can be calculated as I = f i = (1 C r,a / ) t j f j t i C j (1 C r,a / ) = t j. (17) t i Thus, this scheme is not fair, particularly for the case that t i t j when θ i θ j. Proposition 4.2 indicates that applying same recall ratio on multiple winners is not reasonable for SUs with different spectrum stability factors. In addition, the channel recall may also affect the auction revenue of the PBS. According to the VCG mechanism, channel recalls will be evenly distributed among winning SUs. The recall compensation is equal to the product of actual spectrum recall ratio and the sum of payments gained from winners, i.e., C r,a / i W t i. Thus, the revenue of PBS can be written as ( R multiple a1 = 1 C ) r,a t i. (18) i W Obviously, the PBS could get more profit and decrease the compensation by recalling more channels from the winners with low payments. Based on this analysis, we propose a simple but effective channel recall scheme as follows. Assume during t [ΔT,T], the winning SU i W uses channels, and totally i W channels are used by SUs. The PBS could recall channels one by one when necessary. Since the PBS knows the payment of each SU and the details of auction mechanism, it can figure out the unit price of each channel. Thus, the channel with lower price has the higher priority to be recalled and the unused channels would be recalled in the first place. At the end of T, the PBS refunds winner SU i with t i C r,i /, where C r,i denotes the number of channels that are actually recalled from SU i. Note that C r,i is heterogeneous for each winner, and it is likely that C r,i = 0 for the winner with high unit price for each channel, whereas the channels may be completely recalled for the winner with low unit price. C. RMSA Algorithm We summarize the timeline of the proposed RMSA algorithm as follows. The PBS broadcasts the auction information, including and C r, at the beginning of each frame. Each SU i receives the auction information and sets up a value V i based on its own spectrum demand, stability factor θ i, and channel recall ratio C r /. Then, SUs submit sealed bids and their specific spectrum demands to the PBS.

7 YI AND CAI: MULTI-ITEM SPECTRUM AUCTION FOR CR NETWORKS WITH MULTIPLE HETEROGENEOUS SUs 787 The PBS determines the winner by solving the optimization problem in (13) and charges the winner SU i W based on the VCG payment rule in (14). After the auction, the PBS can recall channels one by one to meet its own sudden increase in spectrum demand. The channel recall follows the scheme proposed in Section IV-B2. At the end of T, the PBS refunds each winner SU i with t i C r,i /. Note that, although the proposed RMSA algorithm follows the basic idea of recall-based dynamic spectrum auction [22], the system model under our consideration is more general by considering heterogeneous SUs requirements, multi-item auction, and multiple winners. V. P ERFORMANCE ANALYSES Here, economic properties of proposed auction algorithms are analyzed in terms of PBS s auction revenue and SUs utilities. A. Auction Revenue of the PBS Since the PBS has the ability of channel recall, the PUs service would be completely protected. In addition, since PUs always have higher priority to access the channels, the PBS takes no risk on its own QoS degradation but only benefits from dynamic spectrum auction. Hence, we focus on analyzing the auction revenue of the PBS only. In our system model, the arrival of PUs follows Poisson process and spectrum auction is carried out by the PBS frame by frame. We use u(r) to represent the number of PUs who are in service at the end of rth frame. Obviously, u(r) also indicates the number of busy channels at the beginning of frame r + 1. With our proposed auction algorithm, the number of auctioned channels at the beginning of rth frame is = C u(r 1). In addition, the actual number of channels recalled during the rth frame is C r,a = u(r) u(r 1)+d(r), where d(r) denotes the number of all departures during that period. Note that only u(r 1) is known by the PBS at the beginning of the rth frame, whereas u(r) and d(r) are unknown. For single-winner auction with channel recall, the winner determination would be optimal only if the winner i satisfies where b i = v i(,c r,i )= i =argmax i C r,i b i (19) [ 1 C ] i ( C r,i ) (1 ρ i ) v i ( ). (20) Note that (20) is formulated based on the assumption that the accurate amount of channel recall C r,i is known at the beginning of the auction. Obviously, the bidding pattern and winner determination in RSSA might be suboptimal compared with the previous case with complete information since C r,i is actually unknown at the beginning of auction with unknown C r,a. Such deficit of Vickrey auction payment rule on the auction revenue in recall-based systems will be numerically presented by the simulation in Section VI. This shortage also exists in multiple-winner auction under VCG mechanism. In fact, the winner determination and spectrum allocation should satisfy the following conditions. }, where b i = v i( ) and spectrum demand {C 1,C 2,...,C N }, the winner in set W = {i x i = 1, i N} should satisfy Given the bid B = {b 1,...,b i,...,b N max {x i }, i N P C B = s.t. N i=1 b i x i N x i C r,a i=1 x i = 0/1, i {1, 2,...,N}. (21) For the same reason that C r,a is unknown at the beginning of auction, it is impossible for the PBS to find the optimal decision and SUs will not bid nonrecall valuations. We now analyze the effects of the proposed channel recall scheme on the utility of the PBS. After receiving the payments from the winners in W, the PBS rearranges the payments according to an increasing order of the unit price per channel. Let the payment set as T = {t 1,t 2,...,t m }, where m is the number of elements in W, and C t i be the demand of SU who paid t i.ifsu j who paid t j is the last one in W whose channel will be completely recalled, j can be found as ( j j =argmin t k + C r,a ) j k=1 C t k t j+1. (22) j C t j+1 k=1 Therefore, the auction revenue of the PBS can be expressed as R multiple a2 = ( j t i t k + C r,a ) j k=1 C t k t +1 j. C i W k=1 t j +1 (23) We can evaluate the difference with respect to (18) as Δ=R multiple a2 R multiple a1 ( j = t i t k + C r,a j C i W k=1 t j +1 ( 1 C ) r,a = = ( j i W t i t k + C r,a j k=1 ( j C t j +1 t k + C r,a j k=1 C t j +1 k=1 C t k k=1 C t k k=1 C t k ) t j +1 ) t j +1 + C r,a ) t j +1 + C r,a i W t i m t k. k=1 (24) Let δ 1 =(C r,a / ) m k=1 tk, which indicates the compensation in the auction with evenly distributed channel recall ratio, whereas δ 2 = j k=1 tk +((C r,a j k=1 C t k)/c t +1)t j j +1, which represents the compensation in the auction with the

8 788 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 2, FEBRUARY 2015 proposed channel recall scheme. Since j could minimize the compensation in T as presented in (22), thus Δ=δ 1 δ 2 0. Therefore, VCG mechanism with the proposed channel recall scheme could improve the auction revenue of the PBS. B. Analysis on the SUs Utilities Each SU s utility is equal to the difference between its gain and payment. At the beginning of the auction, SU i evaluates channels based on the auction information. However, the winning SU i may not obtain channels due to the potential channel recall. Therefore, we need to investigate the relation between an SU s utility and its private information. Moreover, in order to satisfy the heterogeneous requirements of SUs and provide them a fair spectrum allocation in multiwinner auction, we need to prove that any SU with higher spectrum stability factor, which results in a higher bid for a unit of spectrum, can be guaranteed with a more stable service by the PBS. 1) SUs Utilities in the RSSA Algorithm: Consider the case without channel recall first. With the SPSB rule, the highest bidder wins, but the price paid is the second highest bid [7]. Thus, the expected utility for SU i is { } U i =(V i b 2nd )Pr. b i > max b j (25) j i where V i b 2nd is its net utility, and Pr.{b i > max j i b j } is its winning probability. For the system with channel recall, two cases need to be considered. Case1:0 C r,a. The actual gain of SU i is the same as (1), i.e., G i = v i ( ). Case 2: C r,a <. The actual gain G i can be obtained by (2), except that the amount of obtained channels is replaced by C r,i, i.e., [ G i = 1 C ] i ( C r ) (1 ρ i ) v i ( C r,i ). (26) Then, we can derive the expected utility for SU i in the recallbased system as { } (G i b 2nd )Pr. b i >max b U single j, (27a) j i i = ( ) { } (G i 1 C r,i b 2nd )Pr. b i >max b j (27b) j i with (27a) and (27b) corresponding to case 1 and case 2, respectively. Apparently, the SUs utilities are different because of the SUs heterogeneous requirements. Lemma 5.1: In the RSSA algorithm, the utility of SU i is not monotonically increased with its spectrum demand. is monotonically increased with in case 1. However, this property would not be maintained when continues to increase. Since SU i in case 1 can fully utilize channels but SU i in case 2 is affected by Proof: Apparently, U single i channel recall, U single i has a sudden decrease when reaches the threshold C r,a. Therefore, the utility of SU i cannot consecutively increase with from 0 to. The following lemma shows the impact on SUs utilities caused by their different risk factors. Lemma 5.2: In the RSSA algorithm, the SU with a larger value of risk factor ρ has better utility than the SU with a smaller value. Proof: Obviously, the gain or the bid of SU i in both case 1 and case 2 would be increased with ρ i, i.e., G i / ρ i > 0 and G i / ρ i > 0, and ρ i has nothing to do with the compensation. Thus, we have U single i ρ i 0. (28) Moreover, Pr.{b i > max j i b j } would be also enhanced when ρ i is larger. Therefore, the SU with larger risk factor ρ could be provided a higher chance to win the auction. With the above lemmas, we conclude the advantage of the RSSA algorithm in the following theorem. Theorem 5.1: The RSSA algorithm can provide economic incentive for all the SUs to participate in the auction since their utilities are always nonnegative and their heterogeneous requirements could be satisfied when they win the competition. Proof: Since second price sealed-bid auction is a mechanism that could ensure incentive compatibility and individual rationality to all the players [7], we have U single i > 0 when SU i wins the auction and U single i = 0, otherwise. Moreover, all SUs are assumed to truthfully report their channel demands. With the help of Lemmas 5.1 and 5.2, we can prove that the utility of SU is strictly related to how much it concerns for channel recall but not its channel demand. Hence, all the SUs would follow the rules in the auction. 2) SUs Utilities in the RMSA Algorithm: In multiplewinner auction, the private value model is symmetric since the valuation function of each winner is the same as (11), except the private information. Without the consideration of channel recall, we have U i =(V i t i )Pr.{i W } (29) where V i t i is its net utility, and Pr.{i W } is its winning probability in the knapsack problem of (13). Similarly, with channel recall, according to the valuation function in (11), the gain of winning SU i is G i = [ 1 C r (1 θ i ) ] v i ( C r,i ). (30) Therefore, the expected utility in RMSA can be formulated as [ ( U multiple i = G i 1 C ) ] r,i t i Pr.{i W }. (31) Note that the actual recall quantity on SU i, i.e., C r,i,is determined by the channel recall scheme. Lemma 5.3: In the RMSA algorithm, the utility of SU i can only be ameliorated with the increase in stability factor θ i. Proof: Obviously, the utility of SU i has no monotonicity with the change of since the number of recalls on SU i, i.e., C r,i, also increases with the demand. However, C r,i will

9 YI AND CAI: MULTI-ITEM SPECTRUM AUCTION FOR CR NETWORKS WITH MULTIPLE HETEROGENEOUS SUs 789 decrease with the increase in θ i because of the proposed channel recall scheme. This means ( C r,i / θ i ) < 0. Thus G [ i = 1 C r (1 θ i ) C r,i + C ] r ( C r,i ) θ i θ i ) P t B log 2 (1 + > 0. (32) n 0 ( C r,i )B Although the compensation (C r,i / )t i would be monotonically decreased with θ i, this decrease is less than the increase in G i since the payment is always less than the gain to ensure nonnegative utility. Thus, U multiple i θ i 0. (33) Moreover, the increase in θ i will also result in the enhancement of Pr.{i W }. This means SU i with larger θ i has higher probability to win in the auction and the quantity of recall C r,i will decrease. In other words, the spectrum occupied by SU i with larger θ i is more stable. Based on Lemma 5.3, we summarize the benefit of RMSA algorithm in the following theorem. Theorem 5.2: The RMSA algorithm can provide economic incentive for all the SUs to participate in the auction since their utilities are nonnegative and the algorithm also ensures a fair spectrum allocation by considering the heterogeneous requirements of SUs. Proof: VCG mechanism employed in the algorithm is incentively compatible for all the players. Moreover, the payment scheme can also ensure nonnegative utility and maximize the social welfare [7]. According to our designed channel recall scheme, SU with larger stability factor is granted a more stable spectrum environment. The assumption of single-minded SUs and Lemma 5.3 demonstrate that SUs heterogeneous requirements can be satisfied with our auction model. VI. SIMULATIONS Here, we conduct simulations to evaluate our proposed RSSA and RMSA algorithms. With the use of recall-based PBS model, the performance in terms of auction revenue and SUs utilities are presented. A. Simulation Scenario Consider a CR network with a PBS and N heterogeneous SUs. PUs arrival rate λ = 2 and channel service rate μ = 0.1. Threshold γ is set to be s, and the PBS owns C = 36 channels to satisfy the inequality (8). The length of each frame T = 6 s; hence, the average number of PUs arrive in 1 min is 20 and the mean time of service for each PU is 60 s. These settings are commonly used in the design of the mobile base station [22]. Furthermore, B = 10 5 Hz, n 0 = W/Hz, and Pt = 0.01 W. Note that the number of SUs N, spectrum demands, and factors ρ i and θ i for each SU i are varied according to the evaluation scenarios. Fig. 2 shows the PBS s state information (i.e., number of idle and busy channels) at each frame. For each frame, the number of active PUs can be determined by the parameters Fig. 2. State information of the PBS in different auction frames. of the queuing system. Since the PBS auctions all the idle channels, the number of auctioned channels is increased when the number of PUs decreases. Moreover, the increase in recall quantity also leads to a decrease in the number of auctioned channels. Since the PBS is truthful in the long-term auction, it is shown that the announced number of maximum recalls is always larger than the number of actual recalls. All rest simulation results are based on the state information shown in this figure. B. Performance of the RSSA Algorithm In Fig. 3, the auction revenue of the PBS is compared between optimal winner determination, as described in (19), and our proposed RSSA algorithm. Intuitively, small-scale network has higher probability of coincidence that the optimal determination is the same as the decision made by our single-winner auction. Therefore, we consider a relatively large network with N = 50 SUs in this simulation. Moreover, the demand of each SU is selected randomly from integers 0 to 15, and risk factor is chosen randomly in [0, 1]. Fig. 3 shows that the curve of the PBS s auction revenue obtained by our proposed single-winner auction algorithm is close to the curve with optimal winner determination. It indicates that our algorithm can achieve close to optimal performance for the PBS. In order to demonstrate the superiority of enabling spectrum recall, we compare the utilities of PBS with and without recall. Here, the PBS s utility function is defined as [38] U PBS = R a + R s U punish { C Ra + ω s C s ω v C s p C = s, if C v C s C R a + ω s C v ω s C v (34) p C s, if C v <C s where R a denotes the auction revenue, R s denotes the revenue from its own users service, and U punish is a punishment term, which represents the loss due to excessive or insufficient channel reservation. ω s and ω p indicate average revenue per PU and the weight index of punishment, respectively. C v denotes the amount of channel reservation by the PBS before the auction,

10 790 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 2, FEBRUARY 2015 Fig. 3. Auction revenue of the PBS in RSSA (N = 50). Fig. 5. Utility of SU i with different spectrum demands in RSSA (N = 10). Fig. 4. Utility of the PBS in the recall-based system (N = 50). and C s denotes the actual demand of PBS s own users. In the simulation, we set ω s = ω p = 10 6.InFig.4,itisshownthat the PBS has lower and more fluctuating utility without recall, which clearly illustrates the improvement by using the recallbased PBS system. As proofs for Lemmas 5.1 and 5.2, we examine the impacts on a specific SU s utility by changing its spectrum demand and risk factor ρ i in Figs. 5 and 6, respectively. For simplicity, N = 10 SUs are considered, and we further assume that all SUs except SU i have a fixed spectrum demand 10 and risk factor 0.3. Fig. 5 demonstrates that larger spectrum demand cannot provide higher utility on SU i. Furthermore, it is also shown in the figure that larger leads to a higher probability of zero utility for SU i. This means that the winning probability will decrease when increases. However, SU i can benefit when its risk factor is larger. As shown in Fig. 6, when is fixed to 12, SU i s utility is monotonically increased with ρ i varied from 0.2 to 0.8. Moreover, larger ρ i also results in higher winning probability. In summary, the utility of SU i only depends on its risk factor. Fig. 6. Utility of SU i with different risk factors in RSSA (N = 10). C. Performance of the RMSA Algorithm Fig. 7 exhibits the comparison of channel utilization ratio between single-winner auction and multiwinner auction. The demand of each SU (N = 10) is selected randomly from integer 0 to 10, and ρ and θ are chosen randomly in [0, 1]. In addition, the ratio without auction is also presented, which is only determined by the number of active PUs. The figure shows that auction with multiple winners has higher spectrum utilization than single-winner auction, and this ratio almost reaches 100%. The percentage increase in min max fairness index by applying our proposed channel recall scheme is shown in Fig. 8. Compare our scheme with evenly distributed channel recall ratio, the fairness index is enhanced up to more than 40%. During some periods, e.g., T, there are no improvements of fairness. The reason is that the number of channel recalls is zero, as shown in Fig. 2. Thus, we can conclude that the proposed channel recall scheme in the RMSA algorithm is more fair and suitable for recall-based spectrum auction mechanism.

11 YI AND CAI: MULTI-ITEM SPECTRUM AUCTION FOR CR NETWORKS WITH MULTIPLE HETEROGENEOUS SUs 791 Fig. 7. Comparison of spectrum utilization between single-winner and multiple-winner auctions (N = 10). Fig. 9. Utility of SU i with different spectrum demands in RMSA. Fig. 8. Improvement on fairness with the proposed channel recall scheme. We further examine the relation between SU s utility and its private information in the multiple-winner case, as discussed in Lemma 5.3. For simplicity, we only consider N = 4SUs.We focus on the utility of a specific SU i by varying its spectrum demand and stability factor, while fixing other SUs with spectrum demand 5 and stability factor θ = 0.2. Fig. 9 shows that a larger spectrum demand cannot ensure a higher utility. Moreover, the utility is more unstable for the case with larger because the actual recall quantity on SU i will also increase. Furthermore, since the winner determination is a knapsack problem, smaller spectrum demand guarantees a higher winning probability. With = 5, we change the stability factor of SU i, i.e., θ i, to show its effect on SU i s utility in Fig. 10. When θ = 0.2, all the SUs in this auction are homogeneous with same spectrum demands and stability factor. Thus, the utility is highly fluctuated. Apparently, the utility can be more stable when θ i continues to increase. The reason is that larger θ i indicates higher payment for each channel; hence, the actual recall ratio on SU i will decrease because of our proposed channel recall scheme. Moreover, Fig. 10 justifies that the SU s utility is monotonically increased with its stability factor. Therefore, it provides incentive compatibility for heterogeneous SUs to Fig. 10. Utility of SU i with different stability factors in RMSA. participate in this multiwinner spectrum auction since their different QoS requirements can be satisfied. VII. CONCLUSION In this paper, a recall-based spectrum auction among multiple heterogeneous SUs has been discussed. Both single- and multiple-winner cases are considered. In order to meet SUs requirements on spectrum demands and stability, we have proposed new private valuation functions for single-minded SUs and designed RSSA and RMSA algorithms along with a new channel recall scheme. Theoretical and simulation results show that our proposed spectrum auction algorithm can improve the auction revenue of the PBS and can enhance spectrum efficiency by adopting multiple winners. Moreover, SUs heterogeneous QoS requirements can be satisfied, which provides economic incentive for all the users to participate in the spectrum auction. Our future works will consider the case that the amount of recalled spectrum as the strategy of the PBS, as well as possible spatial and temporal channel reuse among SUs.

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Wireless Commun., vol. 7, no. 11, pp , Nov Changyan Yi received the B.Sc. degree from Guilin University of Electronic Technology, Guilin, China, in He is currently working toward the M.Sc. degree in electrical and computer engineering with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada. His research interests include algorithmic game theory and optimization in wireless communications, dynamic spectrum management, and cognitive radio networks. Jun Cai (M 04 SM 14) received the B.Sc. and M.Sc. degrees from Xi an Jiaotong University, Xi an, China, in 1996 and 1999, respectively, and the Ph.D. degree from the University of Waterloo, ON, Canada, in 2004, all in electrical engineering. From June 2004 to April 2006, he was with Mc- Master University, Hamilton, ON, as a Natural Sciences and Engineering Research Council of Canada Postdoctoral Fellow. Since July 2006, he has been with the Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, MB, Canada, where he is currently an Associate Professor. His current research interests include energy-efficient and green communications, dynamic spectrum management and cognitive radio, radio resource management in wireless communications networks, and performance analysis. Dr. Cai served as the Technical Program Committee Co-Chair for the IEEE Vehicular Technology Conference 2012 Fall Wireless Applications and Services Track, the IEEE Global Communications Conference (Globecom) 2010 Wireless Communications Symposium, and International Wireless Communications and Mobile Computing (IWCMC) Conference 2008 General Symposium; the Publicity Co-Chair for IWCMn 2010, 2011, 2013, and 2014; and the Registration Chair for the First International Conference on Heterogeneous Networking for Quality, Reliability, Security and Robustness (QShine) in He also served on the editorial board of the Journal of Computer Systems, Networks, and Communications and as a Guest Editor of the special issue of the Association for Computing Machinery Mobile Networks and Applications. He received the Best Paper Award from Chinacom in 2013, the Rh Award for outstanding contributions to research in applied sciences in 2012 from the University of Manitoba, and the Outstanding Service Award from IEEE Globecom in 2010.