Spectrum Leasing for OFDM-Based Cognitive Radio ... - IEEE Xplore

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 5, JUNE 2013

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Spectrum Leasing for OFDM-Based Cognitive Radio Networks Seyed Mahdi Mousavi Toroujeni, Seyed Mohammad-Sajad Sadough, and Seyed Ali Ghorashi

Abstract—In this paper, we propose an Orthogonal Frequency Division Multiplexing (OFDM)-based cooperative communication scheme joint with spectrum leasing. In the proposed scheme, the Primary User (PU) leases some parts of its available resources (in both time and frequency) as a revenue of cooperation to the Secondary Users (SUs) who act as relays for the primary transmission. The optimization of parameters that control the amount of leased resources from the PU to the SU is investigated. An iterative framework is proposed for the PU to find its appropriate relays. Moreover, a dynamic noncooperative game is considered at the relays to compete among themselves for controlling their powers and the Nash Equilibrium (NE) is suggested as the solution of this competition. Analytical results show that the PU achieves a higher data rate by this cooperation process compared with the case where there is no cooperation. Index Terms—Cognitive network, cooperative communications, game theory, nash equilibrium, OFDM.

I. I NTRODUCTION

W

E PROPOSE joint cooperative diversity and competitive spectrum leasing for orthogonal frequency-division multiplexing (OFDM)-based Cognitive Radio (CR) systems. In the considered spectrum leasing scheme, the Primary User (PU) leases some parts of its allocated resources (i.e., frequency subcarriers and time-slots) to Secondary Users (SUs), and in exchange, SUs cooperate with the PU by relaying the PU’s data for improving the quality of the primary link. In other words, the secondary relays lease the primary spectrum in a fraction of time and use a part of the leased spectrum for relaying the source signal to destination and the rest for their own intrasecondary communications. In the proposed scheme, an iterative and negotiation-based approach is suggested for spectrum leasing in which the PU and the SUs update their parameters at each iteration, to obtain maximum profit. The PU tries to find the optimum amount of resources to lease to the SUs, whereas each SU’s goal is to find the optimum power level (based on the amount of leased resources from the primary), to maximize its own profit and dedicate it to the primary signal retransmission.

Manuscript received November 10, 2011; revised July 28, 2012 and October 4, 2012; accepted November 12, 2012. Date of publication December 3, 2012; date of current version June 12, 2013. The review of this paper was coordinated by Prof. C.-X. Wang. The authors are with Cognitive Telecommunication Research Group, Department of Electrical Engineering, Faculty of Electrical and Computer Engineering, Shahid Beheshti University G. C., 1983963113, Tehran, Iran (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2012.2231102

A noncooperative game is suggested for the SUs to find the optimum power consuming in the leasing process. Nash Equilibrium (NE) is considered as the solution for the SUs’ game, where each SU tries to maximize its profit function selfishly. Notice that, this leasing process is secondary driven and works as a monopoly market, in which the PU is the seller and makes the final decision. The aforementioned iterative interactions can be viewed as an auction [1], in which the SUs who act as relays are, in fact, the bidders, and the PU acts as an auctioneer. II. S TATE OF THE A RT AND R ELATED W ORKS A complete review of the CR technology and challenges related to its implementation can be found in [2]–[4]. Different widely-used economical concepts in CR are presented in [5]. Cooperative Cognitive Radio Networks (CCRNs) has attracted much attention for spectrum management over the past few years [6]–[11]. For instance in [6], a cooperation scheme between a PU and an ad hoc network acting as SUs is introduced, where Simeone et al. suggest the Stackelberg game in their considered system model. Based on the model introduced in [6], Yi et al. [7] studied a more general scenario for multiple PU and SUs. In the considered model, PUs and SUs communicate with their own access point, using time-division multiple access. Although in the scenario of [7] multiple users share the spectrum, Yi et al., however solved the problem by considering only one seller (i.e., primary access point) and one buyer (i.e., secondary access point) as a representative of other users. Other similar studies on spectrum leasing are provided in [8], using different revenue function for SUs. Both [7] and [8] did not consider the network topology and there is no power control in these works. In [10], spectrum management in multi channel CCRN composed of multiple PU and SUs is studied. Although in these works the authors considered both centralized and decentralized solutions, the time of spectrum sharing for cooperative communication is fixed and there is no mean to control this parameter. In [11], spectrum leasing for OFDM systems is studied and Toroujeni et al. considered that the secondary relays have a predefined power level for cooperation, whereas the solution of the problem is centralized in the primary transmitter. The main drawback in [6]–[8] and [10] is that all the analysis in the mentioned papers are based on an exact knowledge about channel state information (CSI), which cannot be made available anytime in CR systems, practically when fast decision making is necessary. Moreover, In most spectrum leasing studies considering CCRNs, the Stackelberg game is employed to solve the problems [6], [7] and [8]. In such a game, since the primary knows about the action of SUs,

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Fig. 2.

Fig. 1. Considered system model composed of a primary source and destination. Here, the total number of relays and the selected ones are set to N = 3 and K = 2, respectively. Solid arrows shows the primary data flow, and dashed arrows show the secondary intra-node data flow using the leased resources from the PU.

the whole leasing problem is solved by the PU and the SUs only need to follow the PU’s action. Based on the mentioned challenges in CR, the main contributions of this paper can be summarized as follows: • We assume that the spectrum owner wants to lease its resources with an incomplete knowledge of CSI. • We generalize the networks topology considered in [6] and [11] to a more realistic topology. The effect of SUs topology on final result of spectrum leasing will be evaluated in our simulation results. • We consider a scenario based on the concept of auction, instead of Stackelberg game, in which an iterative updating approach is considered among PU and SUs to reach the final result. III. S YSTEM M ODEL AND M AIN A SSUMPTIONS We consider a CR network composed of a pair of primary transmitter-receiver (also referred to as source (s)-destination (d)) located at a distance L from each other, and N pairs of secondary transmitter-receiver, randomly distributed in a circular area in which the PU wants to select K SUs as relay for cooperation (see Fig. 1). All users in this network use an OFDM with an equal number of subcarriers. Channel coefficients are assumed to be zero mean with slow variations in time, i.e., channel coefficients differ in frequency but are assumed constant within a frame of OFDM symbols. It is also assumed that only the variance of channel coefficients is known to users. Let us denote by ha,b the instantaneous channel gain between points a and b. We denote the variance of source-to-destination channel coefficients (i.e, 2 , and the variance of source-to-ith-relay chanE[|hs,d |2 ]) as σs,d 2 . The variance related nel coefficients (i.e., E[|hs,ri |2 ]) as σs,r i to other possible links follows a similar notation and is omitted for brevity.

Time and frequency allocation for different transmissions.

The PU divides each OFDM frame into two main parts for the leasing procedure. The first part, which contains Ns − α OFDM symbols, is allocated to the primary transmission, and the rest of the α symbols are allocated to the secondary transmissions. From these α symbols, the secondary relays use a number of γ subcarriers for their intralink communications and the remaining M − γ subcarriers are left for source signal retransmission to the primary link destination (see Fig. 2). We assume that the PU and SUs use a control channel for their negotiation. The PU announces its willingness about cooperation (leasing the spectrum) via a one-bit flag as a pilot in this channel. In the case that the PU wishes to cooperate, this flag can be turned to 1. We assume the PU knows the variance 2 , σr2i ,d and of channel coefficients in different links, i.e., σs,r i 2 2 σi,i (σi,i is related to ith secondary transmitter-receiver pair)1 . By gathering the information of the network and knowing the profit function2 of the SUs, the PU tries to find the optimum initial amount of resources it wishes to lease to the SUs (α and γ). After announcing this initial amount, each secondary relay selects the best power level for primary signal retransmission, i.e., Pri for i ∈ {1, . . . , N }, to maximize its own profit function, i.e., πi for i ∈ {1, . . . , N }. The negotiation between PU ans SUs continues until the convergence of parameters, i.e., α, γ and Pri (i ∈ {1, . . . , N }). Since each SU tries to maximize its own revenue selfishly, we model the SUs’ behavior as a noncooperative game to analyze their decision. More precisely, the NE is considered as a solution of this game with these properties: i) Players: i = 1, . . . , N , which presents the ith SU. ii) Action space: P = P1 × P2 × . . . PN where Pi = [0, Pr∗i ] shows the ith SUs’s action set in which Pr∗i is the point that the ith SU reaches its maximum profit, i.e, ∀p; πi (Pr∗i ) > πi (p). The action vector for all SUs is shown by P = [P1 , P2 , . . . , PN ] where Pi ∈ Pi . iii) Profit function: πi (Pri , P−ri , γ, α) which shows the profit function of the ith SU. This function is related to the resources that the PU may lease to the SUs, α and γ, and the action of the ith SU, Pri and other players except the ith one, P−ri . 1 We have assumed that the channel bandwidth used for negotiation and sending different information in this network is negligible. 2 The profit function of SUs is considered as the difference of achievable rate and power consumed in the cooperation process. This function will be defined and explained more precisely in Section IV-B.

TOROUJENI et al.: SPECTRUM LEASING FOR OFDM-BASED COGNITIVE RADIO NETWORKS

IV. S PECTRUM L EASING A NALYSIS A. Derivation of Achievable Rates for Different Links Involved in the Network The achievable rate3 in an OFDM system with M subcarriers for a transmission from source a to destination b is given by [12] ⎛ ⎞ (i) 2 M ρ h a a,b ⎟ ⎜ Ra,b = log2 ⎝1 + (1) ⎠ N0 i=1

If the source sends its data to its respective destination by using a group of best relays denoted by Gb , using Decode-andForward (DF) algorithm [13], the primary link data rate writes Rlease (α, γ, P(Gb )) = min{Rs,Gb (α, γ), RGb ,d (α, γ, P(Gb ))} (3) where P(Gb ), is a vector of length K that contains the bids of the relays group Gb , which is called the bidding profile. The PU should consider a number of K combinations from a set of N relays denoted by CN K to find the best K relays. Rs,Gb is the rate achieved in the s-to-Gb link within Ns − α OFDM symbols and can be written as for

i = 1, 2, . . . , K. (4)

It means since the relays use the DF algorithm for primary signal retransmission, the minimum data rate from the source to the secondary relay inside Gb , determines the rate in the s-toGb link, and it can be calculated similar to (2). For a fair comparison of the primary link final achievable rate, it is assumed that the total consumed power at the primary transmitter in the case of cooperation with SUs is equal to that in the case without cooperation. When α symbols are leased to the secondary relays, the power for a single OFDM symbol for primary transmission would be Ps = (Ns Ps0 )/(Ns − α). Thus, the primary pays an equal power in different transmission modes (cooperative or noncooperative). By employing space-time cooperative diversity [13] at relays to decode the source signal and forward it to the destination

3 In

this paper all rates are expressed in bit/sec.

in M − γ subcarriers, the rate in the Gb -to-d link, denoted by RGb ,d , writes

RGb ,d (α, γ, P(Gb )) = α(M −γ) log2 1+ SN Ri (5) i∈Gb

where SN Ri = (ρi σr2i ,d )/(N0 ). Consequently, when α symbols with γ subcarriers leased to the group Gb , the achievable rate for each relay can be written as: Rri ∈Gb (α, γ, P(Gb )) = αγ log2 (1 + SIN Ri )

(6)

in which

(i)

where ha,b is the channel coefficient in a-to-b link at subcarrier i; ρa = Pa /M where Pa is the total power that the transmitter a allocates to an OFDM symbol; and N0 is the single-sided power spectral density of the white Gaussian noise. For simplicity, we assume that the transmitters perform uniform power allocation for each subcarrier. If we replace the norm squared of the instantaneous channel gains by their expected values, we can rewrite (1) for the primary link (s to d) in a data frame containing Ns OFDM symbols as

2 ρs σs,d . (2) Rs,d = Ns M log2 1 + N0

Rs,Gb (α, γ) = min{(Ns −α)Rs,ri ∈Gb }

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SIN Ri =

2 ρi σi,i . 2 σ j∈Gb ,j=i j,i ρj + N0

(7)

B. Secondary Users’ Decision We now define the profit function for secondary relays from the revenue they can have and the power they consume in the cooperation process. This profit function is defined as

2 ρri σi,i −αPri (8) πi (Pri , P−ri , γ, α) = αωi γ log2 1+ N0 +Ii where ωi is the worth of the spectrum for the ith SU. Note on its that ωi can be different for each secondary relay, based 2 ρ σ location and its spectrum demand. Ii = N j=1,j=i j j,i is the interference imposed on the ith relay from other N − 1 relays 2 is the variance of the interference present in the game where σj,i channel from rj on ri ’s receiver. All of the N SUs, according to their profit function, choose by a noncooperative game their power to reach the maximum revenue. We consider the NE as the solution of this power control game. In our scenario, NE is a bidding profile P ∗ such that no SU can unilaterally increase its profit by changing its bid. i.e., we have

πi Pr∗i , P∗−ri , γ, α ≥ πi Pri , P∗−ri , γ, α ∀i ∈ N, Pri > 0. (9) The solution for each SU in this game can be defined as

BRi (Pri , P−ri , γ) = Pri = arg max πi (Pri , P−ri , γ, α) . Pri >0

(10) It can be shown that the profit function for each SU is concave and continuous4 . To get the optimum power level for each SU to be utilized in the leasing process, we derivate the profit function with respect to Pi . By doing so, we get

2 σi,i ∂πi Pr∗i , P∗−ri , γ, α = Cαωi γ 2 −α ∂Pri M (N0 + Ii ) + Pri σi,i (11) and consequently the solution function is obtained as BRi (Pri , P−ri , γ) = Cωi γ − 4 For

brevity, the proof is omitted.

M (N0 + Ii ) 2 σi,i

(12)

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where C = (ln 2)−1 . Therefore the power level for each SU dissipated in this cooperation process is equal to 1 Cωi γ − M (N20 +Ii ) if πi > 0 σi,i Pri = (13) M 0 otherwise. Note that the SU’s power is normalized with respect to the number of subcarriers, and hence, it does not depend on M anymore. We see from (13) that the parameter α has no effect on the secondary relays’ decision, because the channel coefficients are constant during an OFDM frame and any power for a specific OFDM symbol can be hold for other symbols. However, this power may change through the negotiation process. C. Primary User’s Decision To maximize its data rate, the source tries to find the best setting for leasing parameters, i.e., α and γ, by solving the following maximization problem: max

Rlease (α, γ, P(Gg ))

subject to :

M (Ns − α) = α(M − γ) γ ∈ [0, M ], α ∈ [0, Ns ]

g=1,2,...,CN K

(14)

where the first condition arises from the proposed cooperative model, in which the transmitted symbols for s-to-G should be equal to the number of symbols in the G-to-d link. More precisely, the multiplication of number of OFDM symbols and number of subcarriers for s-to-G should be equal to the G-to-d’s one. Therefore, the condition in (14) can be rewritten as α = Ns

M . 2M − γ

(15)

where Rs,Gg (γ, Gg ) and RGg ,d (γ, P(Gg )) are as (17) and (18), respectively. Thus M −γ = Ns M log2 2M −γ RGg ,d (γ, P(Gg )) M −γ = Ns M 2M ⎛ −γ

⎜ 2 σri ,d × log2 ⎝1+ i∈Gg

2

2M −γ Ps σwrg 1+ M −γ M N0

1 M

0 +Ii ) Cωi γ − M (N σ2 i,i

M N0

min{Rs,Gg , RGg ,d } determine the maximum achievable rate for the primary link. As shown in Fig. 3, since Rs,Gg is strictly descending and RGg ,d is concave and has one maximum (the proof is provided in Appendix A), the point where Rs,Gg equals to RGg ,d is the solution for (16). Therefore, by setting Rs,Gg equal to RGg ,d , we get M (N0 +Ii ) 1 2 Cω γ − 2 i M σi,i 2M − γ Ps σwrg = σr2i ,d , (19) M − γ M N0 M N0 i∈Gg

and after some algebraic manipulations, we obtain

By using the latter condition, we can calculate α as a function of γ. Therefore, the primary transmitter just needs to find γ and the best group of relays for cooperation, i.e., Gb . We consider a joint resource allocation and relay selection, for this purpose. To select K relays out of N , the PU solves (14) for different groups of secondary relays that contain K relays (CN K ). Equation (3) for the subset Gg containing K relays by using (2), (4), (5) and (15) can be rewritten as Rlease (γ, P(Gg )) = min Rs,Gg (γ, Gg ), RGg ,d (γ, P(Gg )) (16)

Rs,Gg (γ, Gg )

Fig. 3. An example of functions Rs,Gg and RGg ,d in (16). Case 1: Rs,Gg and RGg ,d have two common points, case 2: Rs,Gg and RGg ,d have no common point.

(17)

⎞ ⎟ ⎠ . (18)

2 2 is mini∈Gg {σs,r }, i.e., the relay in the group Gg where σwr g i which has the worst channel gain. Rs,Gg and RGg ,d inside

γ 2 − W γ + Z = 0,

(20)

where W and Z are given by (21) and (22), respectively as 2 + Ps σwr g W =M + M

i∈Gg (N0

i∈Gg 2 2Ps σwr + g Z = M2

σr2

i ,d

2 σi,i

Cωi σr2i ,d

i∈Gg (N0

i∈Gg

+ Ii )

+ Ii )

Cωi σr2i ,d

(21)

σr2

i ,d

2 σi,i

.

(22)

By solving (20), the PU can find the optimum value of γ for leasing to subset Gg . This equation may have at least one answer, i.e., the functions Rs,Gg and RGg ,d have at least one common point, (see Fig. 3). By defining Δ = W 2 − 4Z, we analyze the succeeding two cases. 1) Case I, Common Point: If Δ ≥ 0, Rs,Gg and RGg ,d have √ at least one common point γcp = {(W ± Δ)/2}. In this case, the value of γ, leading to a higher rate for the PU is (W − √ Δ)/2. 2) Case II, No Common Point: When Δ < 0, there is no common point in Rs,Gg and RGg ,d ; hence, we should evaluate the functions Rs,Gg and RGg ,d in the interval [0, M ]. Since RGg ,d is concave over this interval, the point which makes this equation maximum is the optimum answer for the PU.


Based on in Sections IV-C1 and IV-C2, the primary can find γ for the subset Gg , i.e., γg , as ⎧ √ ⎨ W− Δ Δ>0 2 γg = arg max RG ,d (γ, P(Gg )) Δ < 0. (23) g ⎩ γ∈[0,M ]

After calculating γg for different groups of relays, the primary can find the appropriate amount of spectrum to lease to the best subset of relays as {ˆ γ , Gb } = arg max

Rlease (γg , P(Gg )) .

(24)

g=1,2,...,CN K

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Announce γ ◦ WHILE |ˆ γ (t+1) − γˆ (t) | <

Secondary best response: The relays choose their best response based on (13), sequentially. N ◦ FOR g = 1 : CK Primary new γg : The PU updates the amount of leased resources based on (25) ◦ END FOR Winner: Calculate new γˆ and Gb based on (24) ◦ END WHILE Calculate α ˆ based on (15) ˆ. Cooperation: Cooperate with Gb , using γˆ and α

E. Existence of the NE and Stability Condition D. Iterative Updating After calculating γˆ and consequently α ˆ and determining the best subset of relays to cooperate with the PU at the first step, an iterative negotiation between PU and SUs is performed. The PU changes the amount of the leased spectrum, i.e., γ, based on the new power levels, according to the following formula: γˆg(t+1) = γˆg(t) − βe(t) g

Since the profit functions of the secondary relays are concave subject to P and closely bounded, there is at least one NE point for the considered game [14]. A necessary condition for the secondary relays’ power control game for reaching a unique NE point is to have the following condition for all secondary relays (for instance, for the ith one): N

(25)

where eg (t) = i∈Gg Pi (t) − Pi (t − 1), and β is the updating rate. Equation (25) shows that there is an inverse relation between the amount of spectrum for leasing and the relays’ power fluctuations. It means that if the power level of a selected group for cooperation decreases, the PU increases the amount of the leased spectrum to increase the relays’ powers by increasing their profit. This iterative process continues until convergence. Note that the selected group of relays, i.e., Gb , may change in different iterations based on their powers. The following Pseudo-code shows the proposed algorithm: Proposed Algorithm Channel model: Each user approximates the expected value of norm squared of the channel coefficients. (r-to-r, s-to-r, rto-d links). Gathering information: PU gathers the network information Initialization of γˆ and P: N ◦ FOR g = 1 : CK • Select the group Gg • Calculate W and Z based on (21) and (22) • Calculate Δ = W 2 − 4Z. ◦ IF Δ > 0 √ γg = (W − Δ)/2 ◦ ELSE Calculate Bg (γ, P(Gg )) based on (18) Let γg = arg maxγ∈[0,M ] Bg (γ, P(Gg )) ◦ End IF ◦ End FOR • Calculate γˆ (0) and the best relays based on (24) • Predict all relays’s bid: (P(0) ), by knowing the utility function of the SUs (Ii = 0)

j=1,j=i

2 σj,i 2 < 1. σi,i

(26)

The proof is provided in [15]. Denoting the final value of γ by γˆ , for a simple case with only two SUs in the network, the NE, i.e., P = [P1 , P2 ] can be obtained as

2 2 2 2 2 M N0 − Cω2 γˆ σj,j σj,j ω1 γˆ − M N0 σj,j + M σj,i Cσi,i Pi = 2 σ2 − M 2 σ2 σ2 σi,i j,j i,j j,i where

i, j ∈ {1, 2}, i = j.

(27)

Proposition: The equilibrium point in (27) is a fixed point of equation (25). With the proposed scheme, a fixed point always exists, and in the case of two SUs, the fixed point is stable if ω σ 2 σ 2 −M ω σ 2 σ 2 1 i i,i j,j j j,j j,i i, j ∈ {1, 2}, i = j. (28) C < 2 σ 2 −M 2 σ 2 σ 2 σi,i β j,j i,j j,i Proof: The proof is provided in Appendix B. V. S IMULATION R ESULTS AND D ISCUSSION A. Simulation Parameters Throughout our simulations, we consider a cognitive network including 6 pairs of SUs. The power allocated by the PU to an OFDM symbol without cooperation is set to Ps0 = 1, and the signal-to-noise (SNR) ratio in direct link transmission (i.e., without any cooperation) is assumed equal to SN R = Ps0 /N0 = 0 dB. Regarding the channel model, a path loss model is considered. The variance of the channel gains for different links of transmitters-receivers are set to: E[|hs,d |2 ] = 2 2 2 σs,d = 1/L2 , E[|hs,ri |2 ] = σs,r = 1/ls,r , where la,b is the i i distance between points a and b. The updating rate is set to β = 0.1, and ωi which is the worth of spectrum for the ith

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Fig. 4. The PU and SU achievable rates with different numbers of secondary relays in a network composed of six relays.

SU is considered as ωi = L/(L − li,i )2 . By using ωi derived from (12), the secondary transmitters, who are are located at a longer distance from their respective receivers, may suggest higher bid. A higher bid increases the chance of selecting an SU as relay and consequently increases the QoS for this user. The 2 = variance of the interference channel is E[|hrj ,ri |2 ] = σj,i 2 0.1/lrj ,ri . The distance between the source and the destination is normalized to L = 1. The stopping criteria of the iterative process are the difference in the value of γ in two subsequent iterations. This criteria are set to = 0.001, i.e., |γ (t+1) − γ (t) | < 0.001 ends the iterative process. Notice that in practical implementation of our proposed algorithm, for avoiding possible infinite loops, a maximum number of iterations can be considered if the stopping criteria defined previously are not met. Here, we set the maximum number of iterations to 50, and the final value of γ is obtained as (γ (tmax −1) + γ (tmax ) /2), where tmax is the maximum number of iterations. Without loss of generality, we consider the total bandwidth normalized to 1 and γ is set to 0 < γ < 1. Ns , which is the number of symbols in a frame of OFDM symbols is normalized to 1. In other words, a one by one block of resources is assumed to be shared between PU and SUs. We set the achievable rate for the primary link in the case where there is no cooperation as Rs,d = 1, by using (2). Our numerical results are averaged via Monte Carlo simulation over 104 different network spatial status. B. Discussion and Analysis Fig. 4 shows the final achievable rates for the PU and each SU after convergence of the algorithm when the SUs have reached the NE. As shown, by increasing the number of relays to more than 3, the achievable rate of the PU decreases, because increasing the number of relays, decreases the rate in the s-to-G link based on (4). When cooperation is performed by all users, the rate of cooperation, i.e., Rlease , is less than the rate of direct transmission, Rs,d . In this case, the primary may refuse to cooperate with secondary relays because it does not benefit from the cooperation process.

Fig. 5.

Leased resources to the SUs. The network consists of six relays.

Fig. 6.

Mean consumed power of each SU normalized to Ps .

The amount of spectrum leased to secondary relays is shown in Fig. 5. We can see that by increasing the number of active users, i.e., K, in the cooperation mode, γˆ and consequently α ˆ decreases. I addition, the amount of leased spectrum decreases because for a given group Gg , according to (16), RGg ,d (γ, P(Gg )) increases (i.e., the bidding profile P(Gg ) increases) and since Rs,Gg (γ, Gg ) is fixed, the optimal point which shows the best value for γg , decreases (as shown in Fig. 3). The area of leased resources, i.e., the product γˆ and α ˆ , is also shown in this figure. This area shows the resources that the SUs can use for their communication and decreases by increasing the number of active SUs similar to other two parameters. Fig. 6 shows the mean of the power consumed by each relay in the cooperation-leasing process. According to Fig. 6, as the number of cooperative secondary relays increases, the interference on each relay caused by other relays also increases; hence, the optimal value for the secondary powers, Pi tends to decrease to achieve the highest profit. Decreasing Pi results in decreasing the secondary relays achievable rates, as shown in Fig. 4. By comparing Figs. 4 and 6, we can see that in such a network, if the PU cooperates with two or three relays, the PU can reach


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Fig. 8. Trajectory toward the NE. The relays at the NE are satisfied and can achieve their highest profit with respect to the bids of other users.

Fig. 7. Probability density function of the location of the winner relays when the cooperation is performed (top) with one relay and (bottom) with three relays.

a better performance, and the relays can achieve a reasonable reward based on their consumed power. The number of relays leading to the highest cooperation performance depends on the number of total relays in the network. To have a better insight about the network and the winner relays in the leasing process, the probability density function (pdf) of the location of winner relays in the network is shown in Fig. 7 for two different cases. We can observe that when the PU cooperates with only one relay, the relays that are near to the destination are likely to be selected (Fig. 7—top part). By increasing the number of relays in the cooperation process, i.e., K, the rate in the G-to-d link, RG,d (α, γ, P(G)), increases based on (5) and the PU can select the winners that are closer to it to prevent the loss in the s-to-G link’s rate (Fig. 7—bottom part). Being far from the center of the network for the secondary relays increases the probability of being far from the respective receiver; therefore the spectrum is more valuable for these users, and based on the definition of ωi , these users wish to pay more (transmit with a higher level of power, Pi ) to be selected as the secondary relays, and this way they increase their QoS. Among the SUs that are far from the center of the network, as obvious from (3), the SUs who are approximately located at the middle path between source and destination have a higher chance to be selected as winners. This is the reason that there are two lobs in the sides of the network in the top part of Fig. 7.

Let us now consider the case where we have two SUs in the network and where the PU wants to cooperate with one of them. In the considered case, the transmitter of relays 1 and 2 are located at r1(T x) = (0.5, 0.4) and r2(T x) = (0.6, 0.4), and their respective receivers are located at r1(Rx) = (0.5, 0.85) and r2(Rx) = (0.6, 0.8), respectively. In addition, to see the iterative process in more detail, the updating rate β is set equal to 0.5. The convergence of the power for each SU to the NE, when there are two secondary relays in operation, is shown in Fig. 8. As shown in this figure, at the NE, each relay obtains its highest revenue based on the action of other relays. The final relays’ power for retransmitting the primary signal at the NE are P1 = 1.26 and P2 = 1. It is shown that the first relay can allocate more power to the cooperation process than the second relay, but in this case, the primary selects the second relay, since from the point of view of the primary transmitter, cooperation with the second relay has more benefit due to this relay’s location (see Fig. 7). It means that in contrast to classical auction frameworks such as English auction [16], in the proposed auction framework, the bidder who offers the highest bid is not necessarily the winner, and some other parameters such as the channel quality and the location of the relays may affect the PU’s decision. In this figure, we can also see that at the NE, each relay can reach to its highest revenue with respect to other SUs. The efficiency of a game can be evaluated by Pareto efficiency. A point is Pareto optimal in which no user can increase its profit without decreasing other users’ profit. In Fig. 9, we can observe that the NE is not Pareto optimal. In other words, the users can increase their profit without any harmful result on other users. This result can be generalized to games with multiple users in our considered scenario. In this special example, if the second user’s action is fix (for example at the NE), the first user can change its action and increase its profit from 0.68 to 0.85 to reach its maximum available profit without decreasing second user’s profit. Based on these two SUs’ location, at Pareto optimal point, the action of each user is: P1 = 0.81 and P2 = 0.44 for the first and second SU, respectively. It means that if the SUs have collision, they can not only pay less price but also reach to maximum profit in the cooperation process.

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2 )/ If we set r(γ) = (M − γ)/(2M − γ) and t = (Ps σs,wr g (M N0 ), (29) can be rewritten as t A (r(γ)) = CNs M r(γ) ln 1 + (30) r(γ)

where C = (ln 2)−1 . It can be shown that since 0 < γ < M , hence, 0 < r(γ) < 1/2 for different values of γ. Moreover, we can observe that ∂r(γ)/∂γ = −M/(2M − γ)2 < 0, which means that r(γ) is strictly descending with respect to γ. The derivative of A(r) with respect to r is equal to: ∂A(r) t t = CNs M ln 1 + + . (31) ∂r r r+t

Fig. 9. Pareto efficiency and Pareto optimal point of this game compared to the NE point.

Fig. 10. Averaged number of iterations necessary to converge to the final value of γ and α for two SUs. Averaging is performed over 104 realizations.

Fig. 10 shows the averaged number of iterations necessary to reach the optimum value for γ and α in the case where there are two SUs in the network. We observe that in this case, according to the stopping criteria previously defined, the optimum value of leased resources can be achieved after about 12 iterations, on average. The speed of convergence depends on the updating parameter and the location of SUs. Using the mentioned parameters, we can see that with two SUs, the PU tends to dedicate its resources more in time and less in frequency to the SUs. A PPENDIX A F UNCTIONS Rs,Gg AND RGg ,d Theorem 1: Equation Rs,Gg (γ, Gg ) is strictly descending in the interval [0, M ]. Proof: We write A(γ) as: A(γ) = Rs,Gg (γ, Gg ) = Ns M

M −γ log2 2M − γ

2

1+

2M − γ Ps σwrg M − γ M N0

Since r(γ) is descending with respect to γ, if we prove that A(r) is ascending with respect to r, we have then proved that A(r(γ)) is strictly descending with respect to γ. For proving that A(r) is ascending with respect to r, we first calculate the second derivative of A(r) with respect to r, i.e., 1 1 t ∂ 2 A(r) − = CNs M . (32) ∂r2 t+r t+r r Since ∂ 2 A(r)/∂r2 < 0, A(r) is strictly concave with respect to r, in the interval (0, 1/2). Therefore, A(r) is ascending if we show that the first derivative of A(r) at the end of interval (0, 1/2) is positive with respect to r, i.e., ∂A(r)/∂r|r=1/2 > 0. It can be proved that for different values of t (we always have t > 0), the first derivative of A(r) with respect to r, i.e., ∂A(r)/∂r|r=1/2 = CNs M (ln(1 + 2t) + 2t/(r + 2t)), is positive. Therefore, A(r) is strictly ascending in the interval (0, 1/2), and consequently, A(γ) is strictly descending in the interval (0, M ). The proof is then completed. Theorem 2: Equation RGg ,d (γ, P(Gg )) is concave in the interval (0, M ). Proof: If we show RGg ,d (γ, P(Gg )) as B(γ), the second order derivative of B(γ) with respect to γ is provided in (33) as: ∂ 2 B(γ) −2M nM = log2 (m) − C ∂γ 2 (2M − γ)3 m(2M − γ)2 nM n2 (M − γ) −C − 2 (33) m (2M − γ) m(2M − γ)2 2 where m = 1+ i∈Gg σr2i ,d (Cωi γ −(M (N0 +Ii )/σi,i )/(M 2N0 ) 2 and n = i∈Gg (Cωi σri ,d )/(M N0 ). Since m > 1 and n > 0, the second-order derivative of B(γ) in the interval (0, M ) is negative. It means that B(γ) is concave in the interval (0, M ). Moreover, we have B(γ)|γ=M = 0 and B(γ)|γ=0 = 0 (in this case, based on (13), the power of SUs and, consequently, B(γ) are zero). Therefore, from one hand, in the two sides of the interval (0, M ) the values of B(γ) are equal, and on the other hand, B(γ) is concave. Therefore, since B(γ) is a continues function over the interval (0, M ), it has one maximum point in this interval. A PPENDIX B S TABILITY C ONDITION

. (29)

Proof: If we rewrite (25) in detail for the first user, we get (t+1) (t) (t) (t−1) γ1 = γ 1 − β P1 − P1 . (34)


Since P (t) depends on γ (t) and γ (t−1) , we can express (34) as5 : γ (t+1) = γ (t) −β P γ (t) , γ (t−1) −P γ (t−1) , γ (t−2) . (35) The difference equation in (35) is of order 3. We can rewrite it composed of three simultaneous equations of order 1 as

⎧ (t+1) x = x(t) − β f x(t) , y (t) − f y (t) , z (t) ⎪

⎪ ⎨ ≡ F x(t) , y (t) , z (t)

(36) ⎪ y (t+1) = x(t) ≡ G x(t) , y (t) , z (t) ⎪

⎩ (t+1) z = y (t) ≡ H x(t) , y (t) , z (t) . A point (ˆ x, yˆ, zˆ) is a fixed point for this dynamic system if ⎧ ˆ=x ˆ − β (f (ˆ x, yˆ) − f (ˆ y , zˆ)) ⎨x (37) yˆ = x ˆ ⎩ zˆ = yˆ and therefore, x ˆ=x ˆ − β(f (ˆ x, x ˆ) − f (ˆ x, x ˆ)), which is always satisfied. It means that this dynamic system has a fixed point that depends on the primary transmitter decision. The stability of the fixed point, (i.e., the equilibrium point) is an important issue and must be also considered. A fixed point is stable if all eigenvalues of the Jacobian matrix of the system are strictly less than 1. The Jacobian of this system is as (38) ∂(F, G, H) ∂(x, y, z) 1−β ∂f (x, y) ∂x = 1 0 =β

∂f (y, z). ∂y

−β

∂f ∂y (x, y)

0 1

−

∂f ∂x (y, z)

β ∂f ∂y (y, z) 0 0 (38)

Therefore, if we set the absolute value of eigenvalues less than 1, the condition for stability of this system at point (ˆ x, yˆ, zˆ), based on equation (37) is written as ∂f β (ˆ < 1. (39) x , x ˆ ) ∂y Finally, based on (39), the stability condition of the fixed point which can be calculated for different relays (for instance,

5 We provide calculations by eliminating the index 1. These calculations can be easily and similarly formulated for the second relay.

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for relay number 1) is written as ω σ2 σ2 − M ω σ2 σ2 1 1 1,1 2,2 2 2,2 2,1 C < . 2 σ2 − M 2 σ2 σ2 σ1,1 β 2,2 1,2 2,1

(40)

The latter equation ends the proof. Similarly, we can obtain a stability condition in case of more than two SUs. R EFERENCES [1] O. Ileri, D. Samardzija, and N. B. Mandayam, “Demand responsive pricing and competitive spectrum allocation via a spectrum server,” in Proc. IEEE DYSPAN, Nov. 2005, pp. 194–202. [2] S. Ashley, “Cognitive radio,” Sci. Amer., vol. 294, no. 3, pp. 66–73, Mar. 2006. [3] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [4] Q. Zhao and B. M. Sadler, “A survey of dynamic spectrum access,” IEEE Signal Process. Mag., vol. 24, no. 3, pp. 79–89, May 2007. [5] D. Niyato and E. Hossain, “Spectrum trading in cognitive radio networks: A market-equilibrium-based approach,” IEEE Wireless Commun., vol. 15, no. 6, pp. 71–80, Dec. 2008. [6] O. Simeone, I. Stanojev, S. Savazzi, Y. Bar-Ness, U. Spagnolini, and R. Pickholtz, “Spectrum leasing to cooperating secondary ad hoc networks,” IEEE J. Sel. Areas Commun., vol. 26, no. 1, pp. 203–213, Jan. 2008. [7] Y. Yi, J. Zhang, Q. Zhang, T. Jiang, and J. Zhang, “Cooperative communication-aware spectrum leasing in cognitive radio networks,” in Proc. IEEE Symp. New Frontiers Dyn. Spectrum, 2010, pp. 1–11. [8] J. Zhang and Q. Zhang, “Stackelberg game for utility-based cooperative cognitive radio networks,” in Proc. 10th ACM Int. Symp. MobiHoc, 2009, pp. 23–32. [9] B. Wang, Z. Han, and K. J. R. Liu, “Distributed relay selection and power control for multiuser cooperative communication networks using buyer/seller game,” in Proc. IEEE Int. Conf. Comput. Commun., INFOCOM, 2007, pp. 544–552. [10] H. Xu and B. Li, “Efficient resource allocation with flexible channel cooperation in ofdma cognitive radio networks,” in Proc. IEEE Int. Conf. Comput. Commun., INFOCOM, 2010, pp. 1–9. [11] S. M. M. Toroujeni, S. M. S. Sadough, and S. A. Ghorashi, “On timefrequency resource leasing for OFDM-based cognitive networks,” Wireless Pers. Commun., vol. 65, no. 3, pp. 583–600, Aug. 2012. [12] H. Bolcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDMbased spatial multiplexing systems,” IEEE Trans. Commun., vol. 50, no. 2, pp. 225–234, Feb. 2002. [13] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004. [14] J. B. Rosen, “Existence and uniqueness of equilibrium points for concave n-person games,” Econometrica, vol. 33, no. 3, pp. 520–534, Jul. 1965. [15] R. Etkin, A. Parekh, and D. Tse, “Spectrum sharing for unlicensed bands,” IEEE J. Sel. Areas Commun., vol. 25, no. 3, pp. 517–528, Apr. 2007. [16] V. Krishna, Auction Theory. London, U.K.: Academic, 2002.

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