Spectrum Sharing Based on a Bertrand Game in

sensors Article

Spectrum Sharing Based on a Bertrand Game in Cognitive Radio Sensor Networks Biqing Zeng 1, *, Chi Zhang 2 , Pianpian Hu 1 and Shengyu Wang 1 1 2

*

School of Computing, South China Normal University, Guangzhou 510631, China; [email protected] (P.H.); [email protected] (S.W.) IFLYTEK Co., Ltd., Hefei 230000, China; [email protected] Correspondence: [email protected]; Tel.: +86-20-8521-3329

Academic Editors: Mianxiong Dong, Zhi Liu, Anfeng Liu and Didier El Baz Received: 28 October 2016; Accepted: 28 December 2016; Published: 7 January 2017

Abstract: In the study of power control and allocation based on pricing, the utility of secondary users is usually studied from the perspective of the signal to noise ratio. The study of secondary user utility from the perspective of communication demand can not only promote the secondary users to meet the maximum communication needs, but also to maximize the utilization of spectrum resources, however, research in this area is lacking, so from the viewpoint of meeting the demand of network communication, this paper designs a two stage model to solve spectrum leasing and allocation problem in cognitive radio sensor networks (CRSNs). In the first stage, the secondary base station collects the secondary network communication requirements, and rents spectrum resources from several primary base stations using the Bertrand game to model the transaction behavior of the primary base station and secondary base station. The second stage, the subcarriers and power allocation problem of secondary base stations is defined as a nonlinear programming problem to be solved based on Nash bargaining. The simulation results show that the proposed model can satisfy the communication requirements of each user in a fair and efficient way compared to other spectrum sharing schemes. Keywords: wireless sensor network; spectrum leasing; spectrum allocation; spectrum pricing; Bertrand game

1. Introduction In the field of Cyber-Physical Social Sensing, wireless sensor networks (WSNs) [1–3] play a very important role. WSNs are organized networks composed of some tiny sensor nodes that can perform information collection, processing and wireless transmission actions. Due to the large number of nodes in WSNs, a large amount of data needs to be collected [4–6], and these data also need to be transmitted, which leads to the problem of spectral tension and node limitations. In order to solve these problems, Cognitive Radio Sensor Networks (CRSNs) have become a research hotspot of many scholars at home and abroad. CRSNs make each sensor node have the ability of spectrum sensing and dynamic spectrum access by introducing cognitive radio (CR) [7,8] technology into WSNs, and is considered to be the most promising wireless sensor network of the next generation [9,10]. Dynamic spectrum sharing make cognitive user/secondary users that have CR technology share spectrum resources that can only be used by authorized user/primary users (PUs) by adaptively adjusting the transmission parameters in the CRSNs, which makes the spectrum resource utilization not be high in traditional static management mode and alleviates the shortage of spectrum resources. According to different wireless network access methods, dynamic spectrum sharing can be divided into underlay mode [11] and overlay mode [12,13]. In underlay mode, the secondary user is able to evaluate the interference temperature threshold of the primary network, by using Sensors 2017, 17, 101; doi:10.3390/s17010101

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spread spectrum technology and control transmission power below the interference temperature threshold, then can use a very wide frequency band to obtain a higher transfer rate. In overlay mode, the secondary users detect and use idle bandwidth that is currently not occupied by the PU, namely spectrum holes [14], when the primary user comes back to the spectrum section, secondary users switch to another spectrum band through spectrum migration. The two kinds of share mode lead to two main problems: power control and spectrum allocation. In addition, during the data transmission process, the uncertainty of spectrum resources is a big problem in dynamic spectrum sharing, which has prompted the birth of another way of spectrum sharing, whereby a spectrum manager allows secondary users to get a license at a certain price, so as to ensure the certainty of the available spectrum, this way is called spectrum transaction/spectrum leasing. Spectrum pricing is another important issue in the study of dynamic spectrum sharing. References [15,16] examined the problems of uplink transmission power allocation and power control. The base station sets the price per unit of transmission power, with the premise that every secondary user ensures its own transmission power is less than the interference temperature threshold, and the secondary user can complete up-link data transmission through paying the corresponding cost for the base station. For the problems of system spectrum pricing and allocation, references [17,18] studied the spectrum allocation problem of different market/topology structures based on the oligopoly market model. The primary user as the seller in market would determine the pricing per unit of spectrum bandwidth and lease it to the secondary users for maximum gains. The secondary users determine the quantity of purchase to maximize their own utility by the pricing and the QoS performance obtained. The game between the primary user and the secondary user could be stable in a state of balance of supply and demand in the market, namely market equilibrium. For multi-hop cognitive radio networks, [19] proposed the power control strategy based on time division multiplexing to achieve the maximum throughput from end to end. However, current studies rarely involve the spectrum pricing and allocation problem. The related research needs to ensure that the transmission power of secondary users is less than interference temperature threshold of the primary network, and the spectrum allocation of frequency and power dimension should be considered at the same time. In [20] a kind of hierarchical model to study the allocation problem of power and spectrum was proposed. However, the bandwidth demands that primary user pricing depends on is not from secondary user, and bandwidth and power had the same pricing, the power control problem was also neglected. Furthermore, the power control model based on pricing [15,16] is mainly based on a user utility function defined by the signal to noise ratio. In spectrum research based on pricing [17,18], it was generally considered that the secondary user utility function had a positive correlation with channel capacity or spectrum efficiency, but a negative correlation with rental costs. Only part of references [20,21] considered the secondary user utility from the point of view of the secondary network demand. Reference [20] defined a utility function reflecting QoS satisfaction degree based on a sigmoid function in service providers offering terminal users power allocation. For three kinds of network scenarios including delay-sensitive secondary users, [21] studied spectrum access control based on pricing strategy. This work didn’t consider the channel capacity demand of the secondary users. By defining a utility function based on channel utility or signal to noise ratio, it is possible to make the utility value greater when the secondary base station achieves more spectrum and power resources. However, on the one hand, excess spectrum beyond the need for user communication service can be considered as a waste of resources. On the other hand, excess transmission power not only increases system interference, but also shortens the battery life of mobile users. Reference [22] studied cooperative spectrum sharing of secondary users based on the communication demands of these secondary network users. Considering both QoS needs and power constraint, [23] studied power control and spectrum to reach maximum system throughput based on mixed-integer non-linear programming. However, these works did not study the secondary user utility from the perspective of communication requirements for the spectrum price problem. The study of secondary user utility

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from the perspective Sensors 2017, 17, 101

of communication demand can not only promote the secondary users to meet 3 ofthe 27 maximum communication needs, but also maximize the utilization of spectrum resources to reduce to waste of resources spectrumand resources excess power on the network environment caused by thereduce waste the of spectrum excess and power on the network environment caused by interference. interference. Thus, considered the communication of users, this paper proposes a kind of two Thus, considered the communication needs of users,needs this paper proposes a kind of two stage model of stage model of spectrum pricing to and allocation to study problem of spectrum sharing. main spectrum pricing and allocation study the problem of the spectrum sharing. The main workThe includes work includesthree the following the following aspects: three aspects:

(1) (1) In In the the first first stage, stage, designing designing aa secondary secondary user user utility utility function function reflecting reflecting the the needs needs facing facing the the secondary network spectrum. secondary network spectrum. (2) A spectrum rental model facing these needs was designed based on Bertrand game theory, and (2) A spectrum rental model facing these needs was designed based on Bertrand game theory, and a a spectral demand adjustment algorithm of secondary base stations was also designed. This spectral demand adjustment algorithm of secondary base stations was also designed. This made made the optimal spectrum rental quantities of secondary base stations just equal to the the optimal spectrum rental quantities of secondary base stations just equal to the secondary secondary network needs when the primary base station used Nash equilibrium pricing. network needs when the primary base station used Nash equilibrium pricing. (3) In the second stage, for the communication needs of secondary users, fairly and efficiently (3) In the second stage, for the communication needs of secondary users, fairly and efficiently allotting secondary base station spectrum resources based on a Nash bargaining scheme allotting secondary base station spectrum resources based on a Nash bargaining scheme considering frequency, power and time dimensions. considering frequency, power and time dimensions. 2. 2. System System Model Model and and Assumptions Assumptions 2.1. 2.1. Primary Primary and and Secondary Secondary Network Network Model Model Suppose thereisisa WSN, a WSN, M primary networks that can lease free spectrum to a Suppose there thatthat has has M primary networks that can lease free spectrum to a secondary secondary order meet the communication secondary users, the secondary network. Innetwork. order to In meet the to communication of secondaryofusers, the secondary network neednetwork to lease need to leasenetwork the primary network spectrum the basisthe of transmission controlling the transmission power of the primary spectrum on the basis of on controlling power of secondary users. secondary users. This is shown in Figure 1. This is shown in Figure 1. PU

PU ……

PU ni,λi

PBS 1

SU

nM,λM

Secondary Base Station

PBS M

PU

SU

SU

Figure 1. The model of multi-primary networks and a secondary network. Figure 1. The model of multi-primary networks and a secondary network.

Suppose we we have have aa secondary secondary network network comprising comprising N N secondary secondary users users and and each each secondary secondary user user Suppose i (i = = 1,2, 1, 2,… . .,.𝑁) , N )can can transmit transmit channel demand Ri to𝑅thetosecondary station to request 𝑖(𝑖 channelcapacity capacity demand the secondary station tospectrum request 𝑖 N spectrum resources for communication base user station collects user resources for communication services. The services. secondaryThe basesecondary station collects requirements ∑ Ri i requirements ∑𝑁 𝑖 𝑅𝑖 of all secondary users and decides the spectrum prices (𝜆1 , 𝜆2 , … , 𝜆M ) of of all secondary users and decides the spectrum prices (λ1 , λ2 , . . . , λM ) of different primary base different primary base stations, leasing spectrum (𝑞1 , 𝑞2 , … , 𝑞M ) from different primary base stations, leasing spectrum (q1 , q2 , . . . , qM ) from different primary base stations, wherein qi refers to the stations, wherein 𝑞i refers to the number of subcarriers, assuming that all subcarriers have the same number of subcarriers, assuming that all subcarriers have the same bandwidth w. In primary networks, bandwidth w. In primary networks, each primary base station adjusts its pricing 𝜆k continuously as each primary base station adjusts its pricing λk continuously as seller, until they can not further increase seller, until they can not further increase their own income, then the spectrum market is in an their own income, then the spectrum market is in an equilibrium state between the spectrum price for
equilibrium state between the spectrum price for primary base stations and ∗the ∗ spectrum primary base stations and the spectrum requirements is λ1 , λ2 , . . . , λ∗M ∗ ∗ for ∗secondary∗base ∗ stations ∗
requirements for secondary base stations is (𝜆1 , 𝜆2 , … , 𝜆𝑀 ) and (𝑞1 , 𝑞2 , … , 𝑞𝑀 ). After the secondary and q∗ , q2∗ , . . . , q∗M . After the secondary station pays for the spectrum resources, it would assign the station1pays for the spectrum resources, it would assign the corresponding spectrum resources and transmission power to each secondary user according to the needs of secondary users (𝑅1 , 𝑅2 , … , 𝑅𝑁 ) and the transmission power threshold 𝑝𝑇𝑂𝑇𝐴𝐿 .

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corresponding spectrum resources and transmission power to each secondary user according to the needs of secondary users ( R1 , R2 , . . . , R N ) and the transmission power threshold p TOTAL . 2.2. Oligopoly Market and Bertrand Game The network model that contains many primary base stations and a single secondary base station can use oligopolistic market theory to understand the interactions of spectrum leasing process for buyers and sellers. The oligopolistic market is between the monopolistic market and the perfectly competitive market, which refers to all the commodities in the market that are controlled by a small number of sellers, each seller is able to influence the spectrum markets that contains other sellers by changing strategies such as adjusting the price or volume of goods, and sellers’ decisions are visible to each other. The Bertrand game is a kind of game in the oligopoly market, which is used to model the price competition behavior in the market, so as to analyze and get the equilibrium pricing strategy of each seller. A Bertrand game basically includes the following three elements: (1) (2)

(3)

The player set N = {1, 2, 3, . . . , n}, which consists of the secondary users or the primary users participating in spectrum sharing; Strategy vector a, player i select a strategy ai from its own strategy set Si in each game, the strategy can be the number of purchased spectrum, or transmission power, strategy vector of all players a = ( a1 , a2 , . . . , an ); Utility function ui ( ai , a−i ) : S → R , which represents the degree of benefit or satisfaction that player i can obtain in strategy vector a = ( ai , a−i ). Usually, a−i = { a1 , a2 , . . . , ai−1 , ai+1 , . . . , an } denotes the strategy vector of other players except i, and u = {u1 , u2 , . . . , un } is called the utility vector.

In a Bertrand game based on an oligopoly market, the primary base stations of the primary network act as oligarchs, making decisions independently, and content of the decision is to change the price of goods. It is usually assumed that the goods of different sellers have their own characteristics, and the goods can’t be completely replaced. When the demand function of seller i is qk = Dk (λk , λ−k ), so i’s optimal pricing can be obtained by seeking extreme value income function formula (1) in the corresponding cost Ck ( Dk (λk , λ−k )): uk (λk , λ−k ) = λk Dk (λk , λ−k ) − Ck ( Dk (λk , λ−k ))

(1)

The primary and secondary user networks shown in Figure 1 are modeled based on the Bertrand game model. All of the primary base stations as oligarchs control the market spectrum, and each primary base station rationally adjusts the spectrum pricing in the pursuit of maximizing its own income; secondary base stations acts as buyers according to their own needs and the spectrum pricing of each primary base station to determine the amount of spectrum that they should lease from different sellers. In spectrum markets, primary and secondary base stations are competitive in the market by continuously adjusting pricing and spectrum requirements, ultimately stabilizing to a market equilibrium. The primary base stations also stabilize to a Nash equilibrium in the price competition of constantly adjusting prices. Since the primary base stations holds licensed spectrum, they have higher spectrum access priority, so primary base stations as “leaders” determine the spectrum price λ, while the secondary base station as “followers” determine the rent number q by the spectrum prices and their demand functions. Leaders know followers will react to their policy, thus they can take the followers’ reactions into account when determining the price. That is, before the primary base station rents spectrum to a secondary base station, it will forecast the rent amount qi of i when the price is λi , and then determining the price λi∗ that can make their own maximum income, under this pricing, the rental amount of i is qi∗ . For this transaction process where the primary base station determines the spectrum price and the secondary user decides the number of leases, we can use a reverse induction method to solve

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the spectrum pricing and spectrum requirements for the equilibrium point where assuming that 5 of 27 known the spectrum price λ, the supply of the primary base station is q = S(λ), and the spectrum ∗ needsAs of shown the secondary base is q =equilibrium D(λ). Usually, supply function is ademand non-decreasing price. in Figure 2, station the market (𝜆∗ , 𝑞the ) in the supply and curves function with respect to the price, and the demand function is a non-increasing function with respect junction, Equation (2) can be obtained. From Equation (2), we can see that the supply of the primary ∗ , q∗ ) in the supply * *and demand to the price. As shown in Figure 2, the market equilibrium λ ( base station equals the demand of all secondary base stations and D ( )  S ( ) , that is, (2) can be obtained. From Equation (2), we can see that theofsupply of the * * curves  S 1 (qjunction, )  S 1 (D Equation (* )) , so market equilibrium pricing and spectrum requirements secondary primary base station equals the demand of all secondary base stations and D(λ∗ ) = S(λ∗ ), that is, users are available: λ∗ = S−1 (q∗ ) = S−1 (D(λ∗ )), so market equilibrium pricing and spectrum requirements of secondary users are available:  q*  D   *  ( ∗ ∗ (2)  q* = D(λ * )  (2) q ∗ S   ∗ q = S( λ ) Sensors 2017, 17, 101

price Demand curve

λ*

0

Supply curve

Market equilibrium

q*

number

Figure2.2.The Thesupply supplyand anddemand demandcurves. curves. Figure

TheFirst FirstStage: Stage:Pricing PricingModel ModelBased Basedonona aBertrand BertrandGame Game 3.3.The Firstly,ininorder ordertotoquantify quantifywhether whetherthe theleased leasedspectrum spectrumsatisfies satisfiesthe thecommunication communicationneeds needsofof Firstly, secondary users, we need to design the utility function of the secondary base station. Secondary base secondary users, we need to design the utility function of the secondary base station. Secondary base stationscan candetermine determinethe thedemand demandfunction function𝒟(𝑹, D(R,𝝀)λof subcarriersininspecific specificcommunication communicationneeds needs ) ofsubcarriers stations and spectrum prices by this utility function. Subsequently, modeling the pricing game of primary base and spectrum prices by this utility function. Subsequently, modeling the pricing game of primary stations based on the Bertrand game, it is proved that the non-cooperative game can converge to a Nash base stations based on the Bertrand game, it is proved that the non-cooperative game can converge and the pricing the Nash equilibrium is used for primary stations. toequilibrium, a Nash equilibrium, and themethod pricingof method of the Nash equilibrium is the used for thebase primary base

stations. 3.1. The Utility Function and Demand Function of Secondary Base Stations 3.1. The Secondary Base(Stations InUtility orderFunction to meet and theDemand channelFunction capacityof requirements R1 , R2 , . . . , R N ) of all secondary users, the In number that a secondary base station needs a primary base station order of to subcarriers meet the channel capacity requirements (𝑅1 , 𝑅2to , …hire , 𝑅𝑁 from ) of all secondary users, theis TOTAL C when the primary network interference power threshold is p and the average subcarrier number of subcarriers that a secondary base station needs to hire from a primary base station is C channel is h,network assuming that all secondary users use p TOTAL /C in different when the gain primary interference power threshold is equal 𝑝𝑇𝑂𝑇𝐴𝐿 power and the average subcarrier subcarriers. According to Shannon’s theorem, C satisfies Equation (3), because the secondary users use 𝑇𝑂𝑇𝐴𝐿 ⁄𝐶 in different channel gain is h, assuming that all secondary users use equal power 𝑝 2 ·ln(5BER )). By solving Equation (3), M-quadrature amplitude modulation [24], so, η = − 1.5/(δ z ij subcarriers. According to Shannon’s theorem, 𝐶 satisfies Equation (3), because the secondary users we can obtain the total available subcarriers that a secondary base station needs,Equation indicated 2 use M-quadrature amplitude modulation [24], so, η = −1.5/(δz ·ln(5BERij)). By solving (3),by TOTAL ·|h|2 · η, W(·) is the Lambert-W function. Equation (4), where δ = p we can obtain the total available subcarriers that a secondary base station needs, indicated by Equation (4), where δ = pTOTAL·|h|N2·η, W(·) is the Lambert-W function. p TOTAL 2 R = ∑ Ri = C · w · log2 (1 + |h| η )(bit/s) C TOTAL Ni=1 p R=  R  C  w  log (1  |h|2 ) (bit / s) i 2 C δ C = −i  1 w · δ · W[−2 exp(− R/(δ · w)) · R/(δ · w)]/R + 1 C

 w    W[2exp( R / (  w))  R / (  w)] / R  1

So, the utility function of secondary base station is designed as follows:

(3)

(3) (4)

(4)

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So, the utility function of secondary base station is designed as follows: uSBS (q) = u1 + u2 + u3

(5)

2

u1 = µ − α(C − ∑kM=1 qk ) , u2 = − 12 (∑kM=1 q2k + 2v∑ jM6=k qk q j ), u3 = −2v∑kM=1 qk λk q = {q1 , q2 , . . . , q M } represents the number of subcarriers that rent from different primary base stations. The utility function consists of three parts: (1)

(2)

(3)

u1 represents the utility that leased spectrum can provide. When the rented spectrum is just equal to the spectrum requirements, u1 can obtain the maximum benefit. If the leased spectrum is less than the spectrum requirements, the quality that the communication of secondary networks needs can’t be guaranteed. If the leased spectrum is more than the spectrum requirements, this would result in excess spectrum and reduced resource utilization. The preference coefficient α reflects the desirability that a secondary base station meets its demand. In addition, to ensure that the utility value of this part is positive, setting µ(µ > 0) is the upper limit of its utility value. u2 describes the mutual substitutability of the spectrum sold by different primary base stations. The Bertrand game assumes the rental spectrum of different primary base stations can’t be completely replaced, therefore, a spectrum replacement rate v is introduced to consider the spectrum substitutability of different primary base stations. The meaning of replacement rate v ∈ [−1.0, 1.0], that is, when v = 0.0, the secondary user can’t switch between different primary base station spectra; when v = 1.0, the secondary user can switch between different spectra; when v < 0, the user spectrum sharing needa to use complementary spectrum, that is, when a secondary user uses a spectrum of the primary base station, it needs to lease one or more other spectra of primary base stations together. u3 represents the cost that must be paid for the subcarriers {q1 , q2 , . . . , q M } of the primary base station. When given the price of spectrum λ = (λ1 , λ2 , . . . , λ M ), the following conclusions are established:

Theorem 1. When the utility function of the secondary base station is given by Equation (5), the corresponding demand function that the subcarriers share is: Dk (λ) = D1 · λk + D2 (λ−k ), k = 1, 2, ..., M D1 = D2 (λ−k ) =

(6)

1 2α + v [ − 1], 1 − v 2α · M + 1 + v · ( M − 1)

2α · M · C − ∑ j6=k λ j 1 · [2α · C − (2α + v) ] 1−v 2α · M + 1 + v( M − 1)

∂2 uSBS (q) < 0, we just need to get ∂qi 2 the maximum value of the function to get qk = D1 λk + D2 (λ−k ), (k = 1, 2, ..., k), which proves the theorem. 

Proof. Since the utility function (5) is a concave function, namely,

3.2. The Utility of the Primary Base Station and the Bertrand Game For the network model which has several or more primary networks and only one secondary network, we know that the supply of the primary base station k is equal to the market demand which we called Dk (λ) when the market is proportionate, so the corresponding utility is: ukPBS (λk , λ−k ) = λk · Dk (λk , λ−k ) − β k · Dk (λk , λ−k )

(7)

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The loss coefficient of β k stands for the loss caused by the primary base station k in the process of renting. In the market of this spectrum, spectrum pricing of the primary base station k is called λk ; it will not only be influenced by other primary base stations’ pricing, but it also affects the pricing of other primary base stations. To deal with this problem, we can build a Bertrand competition to study the price competition among different primary base stations. For the corresponding Bertrand competition Γ = h M, S, ui, let: (1) (2) (3)

M represents the set of game players. This set is composed of primary base stations, such as {1, 2, . . . , M}. λk ∈ Sk represents a player’s strategy. The primary base station sets the renting price  Sk = λk |λk > β k , λk ∈ R+ , k ∈ M of subcarriers of per unit. µk (λk , λ−k ) represents a payoff function. The primary base station k determines the available utility that spectrum pricing λk could acquire, calculated by the formula µkPBS (λk , λ−k ).

The primary base station k, as a player in the Bertrand games constantly adjusts its pricing, in satisfying certain conditions, it can eventually be stable in a state of equilibrium, which is called a Nash equilibrium (NE). Definition 1 (Nash Equilibrium). The Nash equilibrium of the strategic form game is a strategy vector
λ∗ = λ∗k , λ∗−k , λ∗ satisfies:

ui λ∗k , λ∗−k ≥ uk λk , λ∗−k , ∀k ∈ M, ∀λk ∈ Sk , λ∗−k ∈ S−k

(8)

This definition shows that no player can unilaterally change its own strategy to obtain a greater incentive in a Nash equilibrium, so, when other players choose a Nash equilibrium strategy, the best strategy of the player can choose is the Nash equilibrium strategy, that is: λ∗k ∈ Bk (λ∗−k ) = argmaxak ∈Sk uk (λk , λ∗−k ), ∀k ∈ M

(9)

Herein the function Bk (·) is called the optimal response function. However, a Nash equilibrium doesn’t always exist in any form of strategy. The existence condition of a Nash equilibrium is given in [25]. Lemma 1. For any player k ∈ M in the strategic form of game Γ = h M, S, ui, if the policy set Sk is a convex compact subset of the Euclidean space, the utility function uk is to be concave and continuous on the policy set Sk then the Nash equilibrium of this game exists. Theorem 2. For Bertrand game Γ = h M, S, ui there exists a Nash equilibrium. Proof. A Bertrand competition in normal form is a structure Γ = h M, S, ui, where a player’s strategy  is a nonempty convex subset λk |λk i β k , λk ∈ R+ . We can prove that µkPBS is a concave function by Equations (6) and (7). By Theorem 2 and Equation (9), we can find that there is a Nash equilibrium
λ1∗ , λ2∗ , . . . , λ∗M in this game, and λ∗k = argmaxλk ∈Sk ukPBS (λk , λ∗−k ), ∀k ∈ M, which proves the theorem.  This means that when all parameters are known in the network, we can get a Nash equilibrium pricing strategy (Equation (11)) of a primary base station by Equation (6) and linear Equation (10) and the amount of subcarrier that each secondary base station rents from the primary base station could be solved by its demand function (Equation (6)): ∂ukPBS (λ) = 0, k = 1, 2, ..., M ∂λk

(10)

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λ∗k = argmax ukPBS (λk , λ∗−k ) = β k /2 − D2 (λ∗−k )/2D1 , ∀k = 1, 2, ..., M

(11)

λ k ∈ Sk

For example, in the situation of two primary base stations, the corresponding linear equation is:    

2α · M · C − λ2 2λ − β 1 2α + v 1 [2α · C − (2α + v) ]+ 1 [ − 1] = 0 1−v 2α · M + 1 + v( M − 1) 1 − v 2α · M + 1 + v · ( M − 1)

  

1 2α · M · C − λ1 2λ − β 2 2α + v [2α · C − (2α + v) ]+ 2 [ − 1] = 0 1−v 2α · M + 1 + v( M − 1) 1 − v 2α · M + 1 + v · ( M − 1)

(12)

3.3. Dynamic Bertrand Game Different primary base stations in different primary networks can’t monitor the earnings that each other has acquired and also know nothing about the current strategies of others, so they need to obtain optimal pricing by designing a corresponding distributed program. We provide a distributed program based on dynamic Bertrand competition, which iterates the Bertrand competition process repeatedly to reach optimal pricing. For every iteration named t and each primary base station named i, when the spectrum pricing of i is λk [t], other primary base stations’ pricing will be λk [t]. When we enter the next t + 1 iteration, the primary base station k adjusts the spectrum pricing of this iteration with a different update strategy to approach Nash equilibrium closer and closer using the different information acquired, where: (1)

The update strategy of pricing can use the following formula if the primary base station can obtain other players’ historical strategy during the last iteration: λk [t + 1] = Bk (λ−k [t]) = argmax ukPBS (λk , λ−k [t]), ∀k ∈ M

(13)

λ k ∈ Sk

(2)

The following learning strategy will be used when the primary base station can only study local information, where σk means the step size of each update: λk [t + 1] = λk [t] + σk · (

∂ukPBS ) ∂λk

(14)

PBS The marginal revenue ∂u∂λ of Equation (14) can be calculated by Equation (15). When setting a suitable threshold τ and Equation (16) is established, with the increase of the number of iterations and when the utility value of the primary user is close to stable, we can argue that the dynamic Bertrand game can reach the equilibrium state. Moreover, it also can use Equation (17) to calculate the marginal revenue by observing the margin of their utility’s change when a minute pricing adjustment occurs. Let ε represent the margin of each minute adjustment, it will stop updating when the conditions of Equation (18) are satisfied:

∂ukPBS (λ) ∂Dk (λ) = Dk ( λ ) + ( λ k − β k ) · = Dk (λ) + (λk − β k ) · D1 ∂λk ∂λk

(15)

Dk (λ) + (λk − β k ) · D1 < τ

(16)

∂u PBS u (λ[t] + ε) − u PBS (λ[t] − ε) ≈ PBS ∂λ 2ε

(17)

u PBS (λ[t] + ε) − u PBS (λ[t] − ε) < τ

(18)

3.4. Stability and Complexity Analysis of Dynamic Game By analysis of the stability and complexity of a dynamic game, we can definite whether this dynamic game could reach a Nash equilibrium when it converges to a stable state gradually and how

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to converge to stable state quickly. We can see the two update strategies in Equations (13) and (14) as two different self-mapping functions, so we can learn from the Routh-Hurwitz criterion that this self-mapping function is stable if and only if all its eigenvalues ξ i are located in the unit circle of a complex plane, that means |ξ i | 1). It can be seen from the above stability analysis that the dynamic game update strategy of (13) can converge to the Nash equilibrium according to the optimal response value. The convergence rate of the dynamic game update strategy in (14) is highly dependent on the update step, and the complexity of the model depends on the convergence rate and number of iterations of the dynamic game, so the update step is the key to the complexity of the model. In the later simulation experiments, we will further discuss the effect of the update step on the convergence speed of the dynamic game to ensure the lowest model complexity.

of the dynamic game update strategy in (14) is highly dependent on the update step, and the complexity of the model depends on the convergence rate and number of iterations of the dynamic game, so the update step is the key to the complexity of the model, In the later simulation experiments, we will further discuss the effect of the update step on the convergence speed of the dynamic game to ensure the lowest model complexity. Sensors 2017, 17, 101 10 of 27 3.5. Spectrum Requirement Adjustment and System Scheduling 3.5. Spectrum Requirement Adjustment and System Scheduling In the model of spectrum pricing and spectrum allocation which are both based on Bertrand competition, whenof thespectrum subcarrier rent inand the spectrum state of market equilibrium is lower theondemand of In the model pricing allocation which are both than based Bertrand subcarriers inwhen secondary networks,rent rental subcarriers will notequilibrium satisfy the spectrum requirements of the competition, the subcarrier in the state of market is lower than the demand of secondary network. subcarriers in secondary networks, rental subcarriers will not satisfy the spectrum requirements of the From Figure 3, we can learn that when we have set preference factor, replacement rate and loss secondary network. coefficient, then 3, both real spectrum and the optimal rent total rate amount From Figure we the can learn that when requirements we have set preference factor, replacement and M * * loss coefficient, thenlinearly both the real spectrum requirements optimal rentsecondary total amount q  D ( λ ) will increase, so we can do the nextand twothe steps. Firstly, base k 1 k q∗ = ∑kM=1 Dk (λ∗ ) will linearly increase, so we can do the next two steps. Firstly, secondary base stations could present their actual spectrum requirements to the primary base station in advance and stations could present their actual spectrum requirements to the primary base station in advance and then by the optimal pricing that the primary network has formulated, we could define the then by the optimal pricing that the primary network has formulated, we could define the relationship relationship between the actual spectrum requirements and the amount of rental subcarriers. between the actual spectrum requirements and the amount of rental subcarriers. Secondly, in order to Secondly, in order to acquire spectrum resources that could fulfill the demands of the secondary acquire spectrum resources that could fulfill the demands of the secondary networks, we can adjust networks, we can adjust the spectrum requirements again based on the first step. That means, the the spectrum requirements again based on the first step. That means, the secondary base stations send secondary base stations send actual requirements 𝐶 + 𝜀 ′ and 𝐶 − 𝜀 ′ , which have undergone a actual requirements C + ε0 and C − ε0 , which have undergone a minute adjustment to each primary minute adjustment to each primary base station, and then calculate the corresponding total ∗ amount 0 base station, and then∗ calculate′ the corresponding ∗ ′ total amount of rental subcarriers q total (C + ε ) of rental subcarriers 𝑞 (𝐶 + 𝜀 ) and 𝑞 (𝐶 − 𝜀 ) by the requirement function (6) after each base 𝑡𝑜𝑡𝑎𝑙 𝑡𝑜𝑡𝑎𝑙 ∗ 0 and q each base station has returned the optimal (C − ε ) by the requirement function∗ (6) after stationtotalhas returned the optimal pricing 𝜆 (𝐶 + 𝜀 ′ ) and 𝜆∗ (𝐶 − 𝜀 ′ ) under Nash Equilibrium pricing λ∗ (C + ε0 ) and λ∗ (C − ε0 ) under Nash Equilibrium conditions. Secondary base stations find conditions. Secondary base stations find the solution set of slopes which indicate the linear the solution set of slopes which indicate the linear relationship between their requirements and the relationship between their requirements and the amount of rent subcarriers by Equation (22), and then amount of rent subcarriers by Equation (22), and then determine which virtual spectrum requirement determine which virtual spectrum requirement 𝐶’ to send to the primary base station using Equation C 0 to send to the primary base station using Equation (23). Through lots of experiments, we can know (23). Through lots of experiments, we can know that the amount of spectrum rental subcarriers that the amount of spectrum rental subcarriers based on the subcarrier requirements could satisfy the based on the subcarrier requirements could satisfy the spectrum requirements of the secondary spectrum requirements of the secondary networks: networks: q∗ * (C + ε0') − q*∗sum (C −' ε0 ) q (C   )  qsum (C   ) slope ≈ sum slope  sum 2ε0 2 ' C 0 =' (C − q∗sum (C − ε'0 ))/slope + C −' ε0 * C  (C  qsum (C   )) / slope  C  

(22) (22) (23) (23)

Figure 3. The relationship between spectrum demand and optimal rent number. Figure 3. The relationship between spectrum demand and optimal rent number.

In summary, when the primary base station is based on the local information of dynamic spectrum pricing, the scheduling process of first stage is as shown in Figure 4.

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summary, SensorsIn2017, 17, 101

when the primary base station is based on the local information of dynamic 11 of 27 spectrum pricing, the scheduling process of first stage is as shown in Figure 4. Primary base station k(k=1,2,…,M) Secondary network user i submit their own channel capacity demand Ri

The price of primary base station k:  k [t ]

Secondary base station determines the

C  

Secondary base station calculate

'

 i ( )

real demand of subcarriers

and

D1

according to formula

(6)

t=t+1; Adjust pricing according to (14)

Secondary base station calculate the * * rental amount: q s u m ( C   ) and q s u m ( C   )

λ (C   ) *

'

Primary base station calculate according to (10)

u

PBS



NO Secondary base station determine the virtual demand according to (22), (23)

C

Judging whether (16) is established

'

YES

Secondary base station determine the optimal amount of renting:

 (C ) *

'

Determine pricing:  k*   k [ t ]

 k (  ), ( k  1, 2 , ..., M )

Figure Figure 4. 4. Spectrum Spectrum pricing pricing and and allocation allocation scheduling. scheduling.

4. The 4. The Second Second Stage: Stage: Cooperation Cooperation Spectrum Spectrum Sharing Sharing M D ( λ) by Secondary base SC of of subcarriers subcarriers in total number number ∑ Secondary base station station can can obtain obtain the the set set SC in aa total k =k 11Dkk ( λ ) by paying of primary basebase stations. This isThis usedisinused the communication services ofservices secondary paying for fora anumber number of primary stations. in the communication of users in the secondary network, so secondary base stations need to solve how to assign a set of secondary users in the secondary network, so secondary base stations need to solve how to assign a subcarriers for each user based on theon channel capacity requirements , . 1. ,.𝑅, R ( R1 , R2(𝑅 set of subcarriers forsecondary each secondary user based the channel capacity requirements …), and 𝑅𝑁 ) 2,N the transmission power p of each subcarrier j, and ensure not causing interference for the primary and the transmission power 𝑝𝑖𝑗 of each subcarrier j, and ensure not causing interference for the ij network communication. According to the characteristics of the problem, we can model and solveand the primary network communication. According to the characteristics of the problem, we can model problem using the bargaining game model of cooperative game theory. solve theby problem by using the bargaining game model of cooperative game theory. M

4.1. Spectrum Sharing Sharing Based Based on on aa Bargaining 4.1. Spectrum Bargaining Game Game Secondary users in in a Secondary users a secondary secondary network network can can fairly fairly and and efficiently efficiently share share available available spectrum spectrum resources through cooperative games. Bargaining games, as a kind of a cooperative game, can can offer offer resources through cooperative games. Bargaining games, as a kind of a cooperative game, game players a win-win and fair solution: game players a win-win and fair solution:
Definition N, 𝑺, S, 𝑼, U,𝒖u00) is is called called the Definition 22 (Bargaining (Bargaininggame gameand andbargaining bargainingsolution). solution).The Thegame gameΓ𝛤== (𝑁, the N , and there is at least one feasible payoff vector, for any I, bargaining game, if U is a closed convex subset of R bargaining game, if U is a closed convex subset of RN, and there is at least one feasible payoff vector, for any I, ui ≥ ≥ 𝑢u00i that exists. A bargaining scheme is a function that maps the bargaining problem to the only feasible 𝑢
bargaining∗ scheme is a function that maps the bargaining problem to the only feasible 𝑖 𝑖 that exists. A 0 payoff U, u𝒖0 ):: 𝑼U→→𝒖∗u. . payoff vector: vector: f𝑓(𝑼, According to the we can can regard According to the bargaining bargaining game game definition, definition, we regard secondary secondary users users in in secondary secondary networks as game players that reach a cooperative agreement, and we can use a bargaining game to set networks as game players that reach a cooperative agreement, and we can use a bargaining game to up a module for the secondary base station spectrum resource allocation problem. The corresponding set up a module for the secondary base
station spectrum resource allocation problem. The bargaining game formula isgame Γ = formula N, S, U, is u0 Γ: = (N, 𝐒, 𝐔, 𝒖0 ): corresponding bargaining (1) The The set set of of game game players: players: secondary secondary user user set set {1, {1, 2,…, 2, . . .i,…, , i, . N}; . . , N};

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(2)

(3)

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Player strategy ai = (SCi , pi ) ∈ Si : the secondary user i asks the secondary base station for  communication subcarrier set SCi . The power transmission of every subcarrier is pi = pij , ( j ∈ SCi ); Payoff function uk ( ak , a−k ): The user i uses power pij to transport information in the corresponding subcarrier of the subcarrier set SCi . It can obtain channel capacity  2 ∑ w·log2 1 + pij hij ·η . j∈SCi

(4)

Payoff vector is u0 when it doesn’t reach a cooperation agreement: The demand of channel capacity of the secondary user is (R1 , R2 , . . . , RN );

Thus, the problem of subcarriers and power allocation of secondary base station can be transformed into a feasible payoff vector problem in cooperative spectrum communion based on a bargaining game. The bargaining solution is considered a kind of selection criteria of the optimal feasible payoff vector. The Nash bargaining solution (NBS) is a kind of general resource sharing bargaining solution. Not only can it satisfy the secondary user’s needs, but also it can ensure the fairness of resource allocation to the greatest extent. Therefore, according to the Nash bargaining solution, we can define the allocation problem of subcarrier set SCi and power transmission pij as the nonlinear programming problem shown in Equation (24) in order to fairly assign spectrum resources in the  and efficiently 2  secondary network. In the equation, rij = w·log2 1 + pij hij ·η represents the fact that secondary user i can obtain channel capacity by using subcarrier j to transfer. The constraint (1) makes it possible for every the secondary user to get higher channel capacity than demanded. In order to avoid affecting the primary network communication, the condition (2) controls that the power transmission of the secondary user network is less than the interference power threshold p TOTAL . The condition (3) ensures that subcarriers assigned for all the secondary users will be in the subcarrier set rented by the secondary base station: N max ∏ ( ∑ w · log2 (1 + pij hij 2 · η ) − Ri ) i =1 j∈SCi

s.t.(1) ∑ w · log2 (1 + pij hij 2 · η ) ≥ Ri j∈SCi N

(2) ∑ ∑ pij ≤ p TOTAL

(24)

i =1 j∈SCi N

(3) ∑ SCi ≤ SC i =1

(4)SCi , pij ∈ R+ , ∀i = 1, 2, ..., N, j ∈ SCi 4.2. The Optimal Allocation of Subcarriers, Power and Time For nonlinear programming Equation (24), the general solution is to list all subcarrier sets and transmission power levels. However, for communication systems that allow a time sharing strategy, the exhaustive method will lead to inestimable computing resource consumption because the full range extends from the finite set to the infinite set. We can reduce the complexity of solving the problem through introducing time allocation scheduling. Namely, the secondary base station used all subcarriers for a communication service for a certain user. Therefore, the allocation problem of subcarrier set and transmission power (24) can be translated into the allocation problem of transmission time and transmission power (25). In the equation, ti represents that the time which all subcarriers are assigned by the secondary base station is used for the secondary user i is the proportion accounting for the entire scheduling cycle. Consequently, the actual transmission time of the secondary user i is ti · T, and T is the total time of scheduling period:

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N max ∏ (ti · ∑ w · log2 (1 + pij · hij 2 · η ) − Ri ) j∈SC

i =1

s.t.(1)ti · ∑ w · log2 (1 + pij · hij 2 · η ) ≥ Ri j∈SC

(2) ∑ pij ≤ p TOTAL , ∀i = 1, 2, ..., N

(25)

j∈SC N

(3) ∑ t i ≤ 1 i =1

(4) pij ∈ R+ , ∀i = 1, 2, ..., N, j ∈ SC Theorem 3. A feasible solution for the nonlinear programming problem (25) exists only when ∑iN=1 Ri /ri∗ ≤ 1 In the formula, the optimal power allocation strategy and time allocation strategy are: " pij∗ =

1 1 1 · ( p TOTAL + ∑ ) − h · η |SC | | | hij 2 · η il 2 l ∈SC ti∗ =

#+ , i = 1, 2, ..., N, j ∈ SC

N 1 R R · (1 − ∑ ∗l ) + ∗i , i = 1, 2, ..., N N r ri l =1 i

(26)

(27)

2 where [x]+ represents max{x,0}, and ri∗ = ∑ j∈SC w · log2 (1 + pij∗ · hij · η ). Proof. The optimal power allocation is considered firstly. For the feasible time allocation strategy ti∗ given p = ( pij∗ ) (i = 1, 2, .., N, j ∈ SC ) is the optimal power allocation strategy for (25) only when pij∗ is also the optimal solution for programming problem (28): max ∑ w · log2 (1 + pij · hij 2 · η ) j∈SC

(28)

s.t. ∑ pij ≤ p TOTAL , ∀i = 1, 2, · · · , N j∈SC

So, the optimal power allocation problem for (25) can be obtained by the Lagrange Multiplier Approach solving programming problem (27). The corresponding Lagrange function is: L(p, µ) =

∑

N

j∈SC

w · log2 (1 + pij · |hij |2 · η ) − ∑ µi ( i =1

∑

pij − p TOTAL )

(29)

j∈SC

In the equation, µ = (µ1 , µ2 , . . . , µ N ) is Lagrange multiplier, so: w· hij 2 · η ∂L(p, λ) = − µi ∂pij ln 2 · (1 + pij · hij 2 · η ) ∂L(p, µ) = ∂µi



∑

j∈SC

pij − p TOTAL

(30)

(31)

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We can apply the KKT condition to get Equations (32) and (33). The optimal power allocation strategy (26) can be achieved by substituting Equation (33) into Equation (32): pij∗

1 w − ∗ ln 2 · µi hij 2 · η

=

ln 2 1 = µi∗ w·|SC |

( p

TOTAL

+

∑

l ∈SC

!+ (32)

1 |hil |2 · η

) (33)

Then, the existence of the feasible solution should be considered. The existence of the optimal power allocation strategy doesn’t mean that the feasible solution for the programming problem (25) exists. Based on the optimal allocation strategy (26), the feasible solution for (25) exists when the constraint conditions (1) and (3) of (25) are satisfied. Therefore, through putting the optimal power allocation strategy into (1) and (3), we can get the necessary and sufficient condition for the existence of feasible solution ∑iN=1 Rr∗i ≤ 1: i

N

max ∏ (ti · ri∗ − Ri ) i =1

(34)

N

s.t. ∑ ti ≤ 1 i =1 N

max ∑ ln(ti · ri∗ − Ri ) i =1

(35)

N

s.t. ∑ ti ≤ 1 i =1

Lastly, based on the optimal allocation, we consider the optimal time allocation. 2 Let ri∗ = ∑ j∈SC w · log2 1 + pij∗ · hij · η , and we can achieve the programming problem (34) by putting p = ( pij∗ ) into Equation (25). We can solve the programming problem (35) through logarithmic transformation of Equation (34) because f (x) = ln(x) is a monotonically increasing concave function in domain. Meanwhile, the Lagrange Multiplier Approach is used for solving the problem. The corresponding Lagrange function is: L(t, ϕ) =

N

N

i =1

i =1

∑ ln(ti · ri∗ − Ri ) − ϕ · ( ∑ ti − 1)

(36)

Here ϕ is the Lagrange multiplier. By applying the KKT condition, we can get the equations: ri∗ ∂L(t, ϕ) = −ϕ=0 ∂ti ti · ri∗ − Ri

(37)

N ∂L(t, ϕ) = ∑ ti − 1 = 0 ∂ϕ i =1

(38)

The optimal time allocation can be achieved through solving Equations (37) and (38). Therefore, the corresponding subcarriers and power optimal allocation process can be expressed as shown in Figure 5.

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For each subcarrier, the secondary base station carry out the optimal power allocation for each user according to (26),and obtain p *

N

Judging  i 1

Ri ri

*

1

YES

NO

Time allocation according (27), and t * obtain

Figure5.5.Spectrum Spectrumpricing pricingand andallocation allocationscheduling. scheduling. Figure

Simulation 5.5.Simulation The simulation simulation uses uses two two primary primary networks, networks, setting setting the the primary primary network network interference interference power power The thresholdas aspTOTAL pTOTAL 50 mW. bandwidth of subcarrier each subcarrier w = 25 kHz, noise thermal threshold = 50= mW. The The bandwidth of each w = 25 kHz, thermal levelnoise δZ2 = level 10−11 − 2 2 − 11 δZ Desired = 10 BER W, desired BERthe = 10 , and the simulation channel modeled by Rayleigh and W, = 10−2, and simulation channel is modeled by is Rayleigh fading and the fading loss factor loss factor is 3. Suppose thatusers two secondary are located at m a distance of range 200 mof within the isthe 3, Suppose that two secondary are locatedusers at a distance of 200 within the the base range of stationnetwork, in the secondary andchannel their respective capacity station inthe thebase secondary and theirnetwork, respective capacity channel requirements arerequirements R1 = 2 Mb/s, = 2 Mb/s, R2 = 3 Mb/s. Considering path loss of boundary the channel at thethe boundary the Rare 2 =R 31Mb/s. Considering the path loss of thethe channel at the (when distance(when from the distance from them), base station is 200 m), demand when the subcarrier demand is enumeration, 144 by Equation (4) or base station is 200 when the subcarrier is 144 by Equation (4) or a channel enumeration, channel capacity of 5.0 Mb/s can be provided. capacity of 5.0 aMb/s can be provided. 5.1.Spectrum SpectrumPricing Pricingand andAllocation Allocation 5.1. 5.1.1.Preference PreferenceCoefficient Coefficient and and Rate Rate of of Substitution Substitution 5.1.1. After setting settingthe thespectrum spectrumpricing pricingofoftwo twoprimary primarybase basestation stationasasp1p=1 460, = 460, = 440, effects After p2 p=2440, thethe effects of of different preference coefficients α and substitution rates v on the rent number of subcarriers inthe the different preference coefficients α and substitution rates v on the rent number of subcarriers in secondary networks networks are are studied. number of secondary studied. As As shown shown in in Figure Figure6,6,the thesecondary secondarybase basestation stationrents rentsthe the number subcarriers whose price is significantly lower than thethe higher price. If the degree of preference of the of subcarriers whose price is significantly lower than higher price. If the degree of preference of secondary base station to satisfy the demand is greater, it will tend to hire more subcarriers to meet the secondary base station to satisfy the demand is greater, it will tend to hire more subcarriers to the needs of secondary networks. However, when thewhen number subcarriers meets the demand, the meet the needs of secondary networks. However, theofnumber of subcarriers meets the preference coefficient increase that influences the number of subcarriers rented becomes gradually demand, the preference coefficient increase that influences the number of subcarriers rented becomes smaller. Insmaller. addition, 6 shows that substitution rate is smaller, the switching difficulty that gradually In Figure addition, Figure 6 shows that substitution rateand is smaller, and the switching secondarythat users in the spectrum different primary base stations experience will be bigger, thus difficulty secondary users inofthe spectrum of different primary base stations experience willthey be have to rent more expensive subcarriers; when the substitution rate is higher, secondary users are bigger, thus they have to rent more expensive subcarriers; when the substitution rate is higher, more likelyusers to rent cheap subcarriers. secondary arerelatively more likely to rent relatively cheap subcarriers.

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Figure 6. 6. Influence Influence of of preference preference coefficient on spectrum demand. Figure

5.1.2. Pricing 5.1.2. The The Influence Influence of of other other Primary Primary Base Base Stations Stations on on Spectrum Spectrum Pricing According to the results in Section 5.1.2, we set the preference coefficient α α = 12, and the replacement and setset different spectrum pricing values of the base base station 2: λ2 replacement rate rateof ofvv= =0.40.4 and different spectrum pricing values of primary the primary station =2:460, 480, 520, 550, and then we examine the relationship between the spectrum pricing and revenue λ2 = 460, 480, 520, 550, and then we examine the relationship between the spectrum pricing and of the primary station 1, As shown 7, in when the 7, spectrum pricing of thepricing primaryofbase revenue of thebase primary base station 1. in AsFigure shown Figure when the spectrum the station gradually in a increases certain range, because a high price can obtain more the primary1 base stationincreases 1 gradually in a certain range, because a high price can revenue, obtain more income however, when the profit value exceeds a peak point, because thebecause secondary revenue,also the increases; income also increases; however, when the profit value exceeds a peak point, the network will choose a cheaper primary station to rent subcarriers, it resulting in a secondary network will choose a cheaper primary station to rent subcarriers, it resulting in a reduced reduced subcarrier requirementfor forthe theprimary primarybase basestation station revenue of the primary station 1 subcarrier requirement 1, 1, soso thethe revenue of the primary basebase station 1 will will be reduced. In addition, with the increase in the price of another primary base station 2, the be reduced. In addition, with the increase in the price of another primary base station 2, the demand demand of the secondary network forstation the base station 1 alsotherefore, increases,the therefore, of the secondary network for the base 1 also increases, primary the baseprimary station 1base can station 1 can set a higher price to get greater benefits. set a higher price to get greater benefits.

Figure 7. Utility Utility function function of a primary base station.

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5.1.3. Optimal Response Function and Nash Equilibrium 5.1.3. Optimal Response Function and Nash Equilibrium Setting different substitution rates v1 = 0.4, v2 = 0.6 and different primary base station loss Setting different substitution rates v1 = 0.4, v2 = 0.6 and different primary base station loss coefficients β1,2 = 420, β1,2 = 400 (β1,2 represents β1 = β2), according to the optimal response function, coefficients β1,2 = 420, β1,2PBS= 400 (β 1,2 represents β1 = β2 ), according to the optimal response function, * that is  (k , λ*k ),(∗k  M ) , Figure 8 can be drawn. The Nash equilibrium is ∗ k  arg max k Sk uk PBS that is λk = argmaxλk ∈Sk uk (λk , λ−k ), (∀k ∈ M ), Figure 8 can be drawn. The Nash equilibrium is located at the junction of the optimal response function of the different primary base stations, located at the junction of the optimal response function of the different primary base stations, namely, namely, the set (12). Theloss lesscoefficient the loss coefficient of abase primary base station, the the the equationequation set (12). The less the of a primary station, the smaller thesmaller rental per rental per unit bandwidth by loss, and the base primary baseisstation is moreto willing lower unit bandwidth caused bycaused loss, and the primary station more willing lowerto the pricethe of price of spectrum resources, therefore, the Nash equilibrium price is lower. In addition, the spectrum resources, therefore, the Nash equilibrium price is lower. In addition, the substitution rate substitution rate also affects the location of the pricing primaryand base pricing andThe thegreater Nash also affects the location of the primary base station the station Nash equilibrium. equilibrium, The greater the replacement rate, the more difficult it is for the secondary network to the replacement rate, the more difficult it is for the secondary network to switch from the spectrum switch fromprimary the spectrum of different base stations, and base to stations of different base stations, andprimary secondary base stations aresecondary more inclined lower are the more price inclined to lower the price of rental spectrum resources, therefore, the price of the primary base of rental spectrum resources, therefore, the price of the primary base station is lower when a Nash station is lower when a Nash equilibrium exists. equilibrium exists.

Figure Figure 8. 8. Optimal Optimal Response Response Function and Nash Equilibrium.

5.1.4. 5.1.4. Dynamic Dynamic Game Game We set setaasecondary secondarynetwork networkspectrum spectrum substitution rate = 0.4 coefficients of two the We substitution rate v =v0.4 andand thethe lossloss coefficients of the two primary base stations are β 1 = 420, β2 = 380. According to whether the primary base station can be primary base stations are β1 = 420, β2 = 380. According to whether the primary base station can be observed in the thehistory historyofofthethe policy of other players, we study the process the primary base observed in policy of other players, we study the process of the of primary base station station a dynamic game. Fortwo these twothe cases, the primary base station is respectively pricing pricing throughthrough a dynamic game. For these cases, primary base station is respectively based based on the optimal response strategy (Equation (13)) and learning strategy (Equation (14)) to on the optimal response strategy (Equation (13)) and learning strategy (Equation (14)) to dynamically dynamically update its spectrum pricing. Wherein, theconditions sufficient of conditions of game a dynamic game for update its spectrum pricing. Wherein, the sufficient a dynamic for stability is (1−𝑣)(2𝑀−2)

2M(0, −2) (1−vσ)(∈ stability ), we the learning updating = σ2 and = 0.25 1 = σ2 = σ ∈ (0, is 2M ), 2𝑀−1 we select theselect learning strategy strategy updating steps σ1steps = σ2 σ =10.25 σ1and = σσ 2 = 0.35, −1 0.35, wethe setstability the stability threshold = 0.01 and initial pricingofofthe thetwo twoprimary primary base base stations as we set threshold τ = τ0.01 and thethe initial pricing λ 2(0) == β2,βthen thethe dynamic game process is shown in Figure 9. As the primary base station λ11(0) (0) ==ββ1,1λ ,λ dynamic game process is shown in Figure 9. As the primary base 2 (0) 2 , then is able is toable observe the historical policies of the other primary base station, station to observe the historical policies of the other primary base station,a adynamic dynamicgame game can quickly converge convergetotoa aNash Nash equilibrium based onoptimal an optimal response, but when a primary base equilibrium based on an response, but when a primary base station station on the information demand information fed back by secondary networks to can onlycan be only basedbeonbased the demand fed back locally bylocally secondary networks to update its update pricing strategy, the rate of convergence of the dynamic game is largelyon dependent onstep the pricing its strategy, the rate of convergence of the dynamic game is largely dependent the update update step size, and shown Figure when the replacement rate lower, the dynamic game size, and as shown in as Figure 10,inwhen the10, replacement rate is lower, theisdynamic game with larger with larger step convergence has a faster speed. convergence When therate replacement is higher, a update step update has a faster When speed. the replacement is higher, arate smaller update smaller update step should be set to ensure the convergence rate. In addition, because of the high

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18 of 27 to ensure the convergence rate. In addition, because of the high cost of the primary base loss2017, coefficient Sensors 17, 101 of high rental per unit bandwidth, the optimal pricing of the equilibrium is 18 higher of 27 cost the primary base loss coefficient high rental per unit bandwidth, the optimal pricing of the thanof the primary base station of low lossofcoefficient. cost of the is primary of high rental equilibrium higherbase thanloss thecoefficient primary base station of per lowunit lossbandwidth, coefficient.the optimal pricing of the equilibrium is higher than the primary base station of low loss coefficient.

Figure and Nash Equilibrium. Dynamic game and andNash NashEquilibrium. Figure9.9.Dynamic Dynamic game game

Figure10. 10.The Therelationship relationshipbetween between the the update iterations. Figure update step stepsize sizeand andthe thenumber numberofof iterations. Figure 10. The relationship between the update step size and the number of iterations.

5.1.5. Loss Coefficient on Primary Base Station Revenue 5.1.5. Loss Coefficient on Primary Base Station Revenue 5.1.5. Loss Coefficient on Primary Base Station Revenue Figure 11 shows the main effect of the loss coefficient of the primary base station 2 for its own Figure 11 shows the main effect of the loss coefficient of the primary base station 2 for its own and and the revenue of the themain primary station 1 for different replacement rates. When theitsloss Figure 11 shows effectbase of the loss coefficient of the primary base station 2 for own the revenue of the primary base station 1 for different replacement rates. When the loss coefficient coefficient of β 2 gradually increases, the price of the primary base station 2 will be increased at a and the revenue of the primary base station 1 for different replacement rates. When the loss of β2 gradually increases, the price of the primary base station 2 will be increased at a smaller rate smaller rate than that of the β 2 , due to the fact the spectrum can be replaced between different base coefficient of β2 gradually increases, the price of the primary base station 2 will be increased at a than that of the the spectrum can be replaced between different base stations, 2 , due to the fact stations, the βthat corresponding spectrum also decreases, the revenue of basedifferent stations is smaller ratesothan of the β2, due to the rental fact the spectrum canand be replaced between base so gradually the corresponding spectrum rental also decreases, and the revenue of base stations is gradually PBS reduced by u 2 (λ) = (λ 2 − β2)· 𝒟 2(λ), On the other hand, the increase of β2 will increase the stations, so thePBS corresponding spectrum rental also decreases, and the revenue of base stations is reduced u2base(λ) = (λ2 −the β2rent )· D 2 (λ). Onisthe other hand, so thethe increase ofincreases. β2 will increase the price price ofby the station revenue Not increase only that, gradually reduced by u2PBS1,(λ) = (λ2 −amount β2)· 𝒟2(λ),also Onincreased, the other hand, the increase of β2 will the of but the the base station 1, the rent amount is also increased,base so the revenue increases. only that, will affect revenue. For theNot same price of thereplacement base stationrate 1, the rentalso amount isthe alsoprimary increased, sostation the revenue increases. Not onlyloss that, butcoefficient, the replacement rate the willreplacement also affect the primary base station revenue.pricing For theofsame loss coefficient, increasing rate will decrease the optimal all primary base but the replacement rate will also affect the primary base station revenue. For the same loss increasing the replacement rate will decrease the optimal pricing of all primary base stations in the stations in the system, resulting in a decline in revenue. coefficient, increasing the replacement rate will decrease the optimal pricing of all primary base system, resulting in a decline in revenue. stations in the system, resulting in a decline in revenue.

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Figure 11. Maximum revenue of the the primarybase base station. Figure Figure 11. 11. Maximum Maximum revenue of of the primary primary base station. station.

5.1.6. Spectrum Requirement Adjustment 5.1.6. Spectrum 5.1.6. SpectrumRequirement RequirementAdjustment Adjustment Setting the loss coefficient of as β β rate Setting the primarybase basestations stations = 420, β2 =and 380replacement and replacement Setting theloss losscoefficient coefficient of of two primary primary base stations as as β11 ==β1420, 420, β22 == 380 380 and replacement rate v = 0.4, we search for a particular preference factor, the relationship of the real needs of the rate 0.4, search particular preference factor, relationship needs v =v = 0.4, wewe search forfor a aparticular preference factor, thethe relationship of of thethe realreal needs of of thethe subcarriers of secondary base stations and the final amount of the rental spectrum. As shown in subcarriers amount of of the therental rentalspectrum. spectrum.AsAs shown subcarriersofofsecondary secondarybase basestations stations and and the the final amount shown in in Figure 12, when the spectrum need is the same, as the preference coefficient increases, the degree of Figure whenthe thespectrum spectrumneed needis is the the same, same, as the thethe degree of of Figure 12,12, when the preference preferencecoefficient coefficientincreases, increases, degree preference the secondary base station for hire more subcarriers, preference of thesecondary secondarybase basestation station for for the the need need isisgreater, greater, so they tend toto hire more subcarriers, preference ofof the need is greater,so sothey theytend tendto hire more subcarriers, thus increasing the spectrum demand of the secondary network, so the amount of rent subcarriers thus increasingthe thespectrum spectrumdemand demand of of the the secondary secondary network, subcarriers thus increasing network,sosothe theamount amountofofrent rent subcarriers increases, and linear fitting for with different requirements and its corresponding 𝑞∗∗ (𝐶), so increases, and linearfitting fittingfor forC 𝐶 𝐶with with differentrequirements requirementsand andits itscorresponding corresponding rental rental increases, and linear different rental q𝑞∗ ((𝐶), C ), so so C ∗ ∗ (𝐶) show an approximately linear correlation. C and 𝑞 ∗ C and 𝑞 (𝐶) show an approximately linear correlation. and q (C ) show an approximately linear correlation.

Figure 12. Secondary base station spectrum optimal rent number. Figure12. 12.Secondary Secondary base station Figure station spectrum spectrumoptimal optimalrent rentnumber. number.

Subsequently, Subsequently, according according to to the the adjustment adjustment strategy strategy introduced introduced in in Section Section 3.5, 3.5, we we can can determine determine Subsequently, according to the adjustment strategy introduced in Section 3.5, we ′ ∗ ′can determine′ the ′ and calculating the corresponding hire 𝑞 ∗ (𝐶 ′ ) based on 𝐶 ′ . the virtual demand for subcarrier 𝐶 the virtual demand for subcarrier 𝐶 and calculating the corresponding hire 𝑞 (𝐶 ) based on 𝐶 . virtual demand for subcarrier C 0 and calculating the corresponding hire q∗ (C 0 ) based on C 0 . Figure 13 describes the relationship between the virtual spectrum requirement C 0 , the real spectrum needs C and

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Figure 13 describes the relationship between the virtual spectrum requirement 𝐶 ′ , the real spectrum ∗ ′ spectrum 𝑖 ∗ (Cspectrum 0 ) when Figure 13 and describes theamount relationship the virtual 𝐶 ′ , It the real 𝐶 the of hiring (𝐶144, ) when 𝐶 requirement = It144, =seen 0.5. can bespectrum seen that theneeds total amount oftotal hiring spectrum qbetween C𝑞= εi = 0.5. canεbe that the adjustment ′ ∗ (𝐶 ′ ) when 𝐶 = 144, ε𝑖 = 0.5. ∗ that 0 ∗ 0 needs 𝐶 and the total amount of hiring spectrum 𝑞 It can be seen of the virtual spectrum demand 𝐶 corresponds to the leased of meet just 𝑞the (𝐶 ′real ) of the the adjustment virtual spectrum demand C corresponds to the leased number of justnumber q (C ) to ′ ∗ ′ the adjustment of the virtual spectrum demand 𝐶 corresponds to the leased number of just 𝑞 (𝐶 ) to meet the real needs of the network spectrum 𝐶. needs of the network spectrum C. to meet the real needs of the network spectrum 𝐶.

Figure 13. Spectrum demand and spectrum rent. Figure demand and andspectrum spectrumrent. rent. Figure13. 13. Spectrum Spectrum demand

5.2. Cooperation Spectrum Sharing 5.2.5.2. Cooperation CooperationSpectrum SpectrumSharing Sharing 5.2.1. The Degree of Satisfaction for the Secondary Network Communication Demand 5.2.1. The Degree of Satisfaction for the Secondary Network Communication Demand 5.2.1. The Degree of Satisfaction for the Secondary Network Communication Demand When calculating the spectrum demand in the secondary network, if the secondary base station When calculating the spectrum demand in secondarynetwork, network,ififthe thesecondary secondary base station When theof spectrum demand in the the boundary, secondary base station is based on calculating the path loss the secondary network we calculate the channel gain (case I). is based on the path loss of the secondary network boundary, we calculate the channel gain (case I). is based on the path loss of the secondary network boundary, we calculate the channel gain (case I). iIf the channel capacity requirement of the network is 5 Mb/s, the secondary base station needs to If the channel capacity requirement of the network is 5isMb/s, themsecondary base station needs toto hire iIf the channel capacity requirement of the 5 Mb/s, secondary base station needs hire 144 subcarriers. Suppose secondary usernetwork 𝑆𝑈1 is located 100 the from the secondary base position, 144hire subcarriers. Suppose secondary user SU is located 100 m from the secondary base position, and 144 subcarriers. Suppose secondary user 𝑆𝑈 is located 100 m from the secondary base position, 1 and the distance between 𝑆𝑈2 and secondary 1base stations changes between 110 m and 200 m,the distance SU secondary basescheme, stations changes between 110 m 200 m, through and thebetween distance between 𝑆𝑈2 and secondary base stations changes 110 m and 200 m, a 2 and through a cooperative spectrum sharing each secondary the between user canand obtain the channel cooperative scheme, each secondary the user can obtain the channel through cooperative spectrum sharing scheme, each secondary the user can thecapacity channel capacity aasspectrum shown in sharing Figure 14, where the horizontal dotted line in the graph is obtain the secondary user as capacity as shown in Figure 14, where the horizontal dotted line in the graph is the secondary user shown in Figure 14, where the horizontal dotted line in the graph is the secondary user channel channel capacity requirement, when the distance 𝐷2 between the user 𝑆𝑈2 and the base station channel capacity requirement, distance 𝐷2 between user and base increases, station capacity requirement, when distance D the user the SU2resulting and𝑆𝑈 the base station 2 in increases, the channel gain the ℎ𝑖𝑗 2when and the 𝐷23 is negative correlation, a the decrease of 𝑆𝑈2 2abetween 2 3 2 3 increases, the channel gain ℎ and 𝐷 is a negative correlation, resulting in a decrease 𝑆𝑈2 thechannel channelcapacity. gain hij and D2 is a𝑖𝑗 negative2 correlation, resulting in a decrease of SU2 channelof capacity. channel capacity.

Figure 14. Each secondary user channel capacity. Figure userchannel channelcapacity. capacity. Figure14. 14. Each Each secondary secondary user

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of 27 To obtain sufficient channel capacity for the secondary user SU2 , the spectrum sharing21scheme To obtain sufficient channel forsubcarrier the secondary user 𝑆𝑈the , the spectrumtime sharing scheme can be used to reduce the time SUcapacity to increase occupancy of SU 2 , so as 1 is using To obtain sufficient channel capacity for the secondary user 𝑆𝑈22, the spectrum sharing scheme can be used to reduce the time 𝑆𝑈 is using subcarrier to increase the occupancy time of 𝑆𝑈 , so as to meet the channel capacity needs1 of SU2 , as shown in Figure 15. Thus, the cooperative spectrum can be used to reduce the time 𝑆𝑈1 is using subcarrier to increase the occupancy time of 𝑆𝑈22, so as to meet the channel capacity needs of 𝑆𝑈 , as shown can in Figure 15. Thus, theresources cooperative spectrum sharing strategy based capacity on the Nash bargaining allocate more equitably. to meet the channel needs of 𝑆𝑈22, asscheme shown in Figure 15.spectrum Thus, the cooperative spectrum sharing strategy based on theofNash bargaining scheme can allocate spectrum resources more In sharing addition, when the distance the SU distance is more than 20 m, the total channel capacity 2 strategy based on the Nash bargaining scheme can allocate spectrum resources more is equitably. In addition, when the distance of the SU 2 distance is more than 20 m, the total channel higher than the demand,when whenthe D2distance > 180 m, loss is increased, and20the channel equitably. In addition, of the the path SU2 distance is more than m, simulation the total channel N𝐷2 R>i 180 m, the path loss is increased, and the simulation capacity is higher thanofthe demand, when can’t meet the increase the number of ≤ 1. ∑ ∗ capacity is higher than the demand, when i=𝐷12 r>i 180𝑁m,𝑅𝑖the path loss is increased, and the simulation channel can’t meet the increase of the number of ∑𝑖=1 𝑅𝑖∗ ≤ 1. 𝑟 channel can’t meet the increase of the number of ∑𝑁 𝑖=1 𝑟𝑖∗ ≤ 1. 𝑖

Figure15. 15.The Theoptimal optimal time time assign assign of Figure of secondary secondaryusers users. Figure 15. The optimal time assign of secondary users

When thesecondary secondary basestation station calculates the network spectrum demand, it considers the When the network networkspectrum spectrumdemand, demand, considers Whenthe the secondarybase base station calculates calculates the it it considers thethe secondary user’s path loss of the secondary base station at the farthest distance (case II), so with the secondary user’s base station stationatatthe thefarthest farthestdistance distance (case with secondary user’spath pathloss lossof ofthe thesecondary secondary base (case II),II), so so with thethe increase of the distance between the 𝑆𝑈2 and secondary base station, the spectrum leasing number increase ofof the basestation, station,the thespectrum spectrum leasing number increase thedistance distancebetween between the the SU 𝑆𝑈2 and and secondary secondary base leasing number that the channel capacity of 5 MB/s needs increases, as shown in Figure 16. that thethe channel capacity increases,as asshown shownininFigure Figure16. 16. that channel capacityofof55MB/s MB/s needs increases,

Figure 16. Spectrum requirements for secondary network of two cases. Figure16. 16.Spectrum Spectrum requirements requirements for Figure for secondary secondarynetwork networkofoftwo twocases. cases.

Through cooperative spectrum sharing algorithms, two secondary users were able to get the Through cooperative spectrum sharing algorithms, two secondary users were able to get the channel capacity shown spectrum in Figure 17. As can be seen from figure, 𝑆𝑈 and 𝑆𝑈 able can to getget onthe Through cooperative sharing algorithms, two the secondary users were channel capacity shown in Figure 17. As can be seen from the figure, 𝑆𝑈11 and 𝑆𝑈22 can get on average 13% and 5% higher demand for channel capacity, respectively, and with 𝑆𝑈 away from channel capacity shown in Figure 17. As can be seen from the figure, SU SU can get on average 2 2 𝑆𝑈 away from average 13% and 5% higher demand for channel capacity, respectively,1 and with 2 theand base station, the amount of spectrum that secondary base stations rent increases, to the 13% 5% higher demand for channel capacity, respectively, and with SU away from to thethe base 2 the base station, the amount of spectrum that secondary base stations rent increases, transmission time that 𝑆𝑈 uses for all the subcarriers to communicate is reduced, and the 1 station, the amount of spectrum that secondary base stations rent increases, to the transmission time transmission time that 𝑆𝑈1 uses for all the subcarriers to communicate is reduced, and the transmission time ofthe 𝑆𝑈subcarriers 18), but 𝑆𝑈2is isreduced, still ableand to maintain a higher demand for 2 increases (Figure that SU uses for all to communicate the transmission time of SU2 1 transmission time of 𝑆𝑈2 increases (Figure 18), but 𝑆𝑈2 is still able to maintain a higher demand for channel capacity. Moreover, from Figure 17 we can know that when the distance between the 𝑆𝑈 increases 18),Moreover, but SU2 isfrom still Figure able to17 maintain higher demand for distance channel between capacity. the Moreover, channel(Figure capacity. we can aknow that when the 𝑆𝑈2 and boundary is greater than 20 m, the probability of the secondary base station through the2 and boundary is greater than 20 m, the probability of the secondary base station through the

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Sensors 2017, 17,we 101 22 of 27 fromSensors Figure 2017,17 17, 101 can know that when the distance between the SU2 and boundary is greater 22 of 27than 20 m, the probability of the secondary base is station the cooperative spectrum sharing algorithm cooperative spectrum sharing algorithm about through 80%, Compared Compared to the the case case the probability probability of the the cooperative spectrum sharing algorithm is about 80%, to I,I, the of cooperative spectrum sharing algorithm is about 80%, Compared to the case I, the probability of the is about 80%. Compared tosituation the casehas I, the probability of theofcase II probability of the situation case II probability of the declined, but the cost the secondary base station is able to has case II probability of the situation has declined, but the cost of the secondary base station is able to case II60%, probability of the situation hasbase declined, butisthe cost of the 60%, secondary base station is able to save as shown in Figure 19. declined, but the cost of the secondary station able to save as shown in Figure 19. save 60%, as shown in Figure 19.

save 60%, as shown in Figure 19.

Figure 17. Each secondary user’s channel capacity.

Figure 17.Each Each secondary user’s channel capacity. Figure user’schannel channel capacity. Figure17. 17. Each secondary secondary user’s capacity.

Figure 18. The optimal time assignments of secondary users. Figure 18. The optimal time assignments of secondary users.

Figure 18. assignmentsofofsecondary secondary users. Figure 18.The Theoptimal optimal time time assignments users.

Figure 19. The cost of two secondary network leased spectrum cases. Figure 19. The cost of two secondary network leased spectrum cases.

Figure Thecost costof oftwo two secondary secondary network cases. Figure 19.19.The networkleased leasedspectrum spectrum cases.

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5.2.2. The Fairness of the Cooperative Spectrum Sharing Scheme Based on NBS 5.2.2. The Fairness of the Cooperative Spectrum Sharing Scheme Based on NBS 5.2.2. The Fairness the Cooperative Spectrum Sharing Scheme Assuming two of secondary user’s spectrum needs are equal,Based bothon areNBS 2.5 Mb/s, the distance of Assuming two secondary user’s spectrum needs are equal, both are 2.5 Mb/s, the distance of SU1 andAssuming the secondary base station is 100 m, and the distance of SU and the secondary base station 2 are 2.5 Mb/s, two secondary user’s spectrum needs are equal, both the distance of 𝑆𝑈1 and the is 100 m, and the distance of (4) 𝑆𝑈2 and the secondary base station is between 110secondary m and 200base m. station Under the same constraint ((1) to Equation (25)), the cooperative 𝑆𝑈1 and the secondary base station is 100 m, and the distance of 𝑆𝑈in and the secondary base station 2 is between 110 mscheme and 200 m. Under thewith same constraint ((1) to (4) in Equation (25)), the cooperative spectrum sharing compared the spectrum allocation scheme based maximum is between 110 m and 200ism. Under the same constraint ((1) to (4) in Equation (25)), on the the cooperative spectrum sharing scheme is compared with the spectrum allocation scheme based on the maximum global rate [26] and the spectrum allocation scheme [27] based on the maximum degree of flatness. spectrum sharing scheme is compared with the spectrum allocation scheme based on the maximum global rate [26] and the spectrum allocation scheme [27] based on the maximum degree of flatness, global rate [26] and the spectrum allocation schemesharing [27] based on theare: maximum degree of flatness, The objective function of the two kinds of spectrum schemes The objective function of the two kinds of spectrum sharing schemes are: The objective function of the two kinds of spectrum sharing schemes are: N

N

N r Max max ∑ Maxrate rate :: max ri i Max rate : max   ii =1 1ri i 1

(39)(39) (39)

Max fairness: :maxminr max min r , i  1, 2, L2,, N Max f airness (40)(40) Max fairness : max min riii,,ii=1, 1, 2, L , L, NN (40) 2 where where r =  j ww· log 1 +ppij ij| hhijij| 2 2· )η.) .  log22((1 ∈SC jiSC w  log wherei rrii ∑ .  2 (1  pij | hij |  ) jSC For two cases I and II stationbased basedon onthe themaximum maximum coverage For two cases I and II(that (thatis, is,aasecondary secondary base base station coverage of of thethe Forortwo cases Ithe anddistance II (that is, a secondary baseremote stationusers basedand on the maximum coverage of thethe network, based on between the most the base station, to calculate network, or based on the distance between the most remote users and the base station, to calculate network, ortobased on the the distance between the most remote requirements), users and the base to calculate channel gain determine secondary network spectrum for station, these the channel gain to determine the secondary network spectrum requirements), forthree thesespectrum three the channel gain to determine the secondary network spectrum requirements), for these three allocation schemes, the secondary user SU , SU channel capacity can be obtained as shown 2 𝑆𝑈 , 𝑆𝑈 channel capacity can be obtained as in 1 user spectrum allocation schemes, the secondary spectrum allocation schemes, the secondary user 𝑆𝑈11 , 𝑆𝑈22 channel capacity can be obtained as Figures 21. 20 and 21. shown20inand Figures shown in Figures 20 and 21. i

i

Figure 20. Secondary user channel capacity comparison I. Figure capacitycomparison comparisonI.I. Figure20. 20.Secondary Secondary user user channel channel capacity

Figure 21. Secondary user channel capacity comparison II. Figure21. 21.Secondary Secondary user user channel channel capacity Figure capacitycomparison comparisonII.II.

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The corresponding total withFigures Figures2222and and23.23. Defining the fair totalthroughput throughput is is compared compared with Defining the fair SensorsThe 2017,corresponding 17, 101 24 of 27 coefficient is the ratio of the secondary user channel capacity, that is r /r , so the fairness of the three The corresponding totalsecondary throughput compared with Figures 23.fairness Defining thethree fair coefficient is the ratio of the userischannel capacity, that is r1122 /r2, 2and so the of the kinds of The spectrum scheme is asshown shown Figures 24 25. coefficient is theallocation ratio of the secondary userischannel capacity, that is24r1and /r so fairness of the kinds of spectrum allocation scheme is obtained obtained as inin Figures 25.the corresponding total throughput compared with Figures 222, and and 23. Defining thethree fair kinds of spectrum allocation is obtained as shown in Figures 25.the fairness of the three coefficient is the ratio of the scheme secondary user channel capacity, that is24r1and /r2, so kinds of spectrum allocation scheme is obtained as shown in Figures 24 and 25.

Figure 22. The comparison of total throughput for the secondary network I.

Figure 22.22. The throughputfor forthe thesecondary secondary network Figure Thecomparison comparisonof of total total throughput network I. I. Figure 22. The comparison of total throughput for the secondary network I.

Figure 23. The comparison of total throughput for the secondary network II. Figure 23. The comparison of total throughput for the secondary network II.

Figure 23. The comparison of total throughput for the secondary network II. Figure 23. The comparison of total throughput for the secondary network II.

Figure 24. The comparison of fairness I. Figure 24. The comparison of fairness I. Figure24. 24. The The comparison comparison ofoffairness Figure fairnessI. I.

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Figure25. 25.The The comparison comparison of Figure offairness fairnessII.II.

In case I and case II, for the program based on the maximum global rate, the spectrum will In case I and case II, for the program based on the maximum global rate, the spectrum will allocated as much as possible to the secondary base station that is close to 𝑆𝑈1 while ensuring 𝑆𝑈2 allocated as much as possiblethat to the secondary base station is close 1 while just satisfy the constraints, is, the secondary users whosethat channel gaintoisSU larger will ensuring obtain the SU2 just maximum satisfy thepossible constraints, that is, the secondary users whose channel is larger will obtain the secondary network total throughput; the aim of thegain maximum fair spectrum maximum possible secondary network total throughput; the aim of the maximum fair spectrum allocation scheme is to reduce the total throughput of the secondary network, using the maximum allocation scheme is to reduce the total throughput of the secondary network, using the maximum fairness to ensure that the channel capacity obtained by 𝑆𝑈1 and 𝑆𝑈2 is the least. Relatively fairness to ensure thatspectrum the channel capacity obtained SU1 and SU2between is the least. Relatively speaking, speaking, the NBS sharing scheme is able by to compromise the fair and spectrum efficiency, it can not only ensure the fair allocation ofbetween spectrum but also enhance theit can the NBS spectrum sharing scheme is able to compromise theresources, fair and spectrum efficiency, spectrum utilization rate, finally obtaining a larger total network throughput. not only ensure the fair allocation of spectrum resources, but also enhance the spectrum utilization

rate, finally obtaining a larger total network throughput. 6. Conclusions

6. Conclusions With the popularity of the Internet of Things technology [28,29], the improvement of the communication efficiencyofand of CRSNs has become topic, and With the popularity thespectrum Internetutilization of Things technology [28,29],a hot the research improvement of the dynamic spectrum sharing is the main countermeasure to improve spectrum utilization and alleviate communication efficiency and spectrum utilization of CRSNs has become a hot research topic, the shortage of spectrum resources. However, due to the complexity of the application and the and dynamic spectrum sharing is the main countermeasure to improve spectrum utilization and spectrum environment, most of current research on spectrum sharing is based on a large number of alleviate the shortage of spectrum resources. However, due to the complexity of the application assumptions or lacks consideration of actual application scenarios. This paper presents a spectrum andpricing the spectrum environment, most of research ontheory. spectrum sharing is based on a large and allocation model based on current the Bertrand game Spectrum pricing, subcarrier number of assumptions or lacks consideration application scenarios. This paper presents allocation, power control, and allocation issues of areactual studied by synthesizing practical scenarios such a spectrum pricing and allocation based It onprovides the Bertrand gameand theory. Spectrum as the frequency, power and timemodel dimensions. a practical effective solution pricing, for subcarrier allocation, power control, and allocation are scheme studiedcan by reflect synthesizing practical spectrum sharing, and theory and experiments proveissues that the the spectrum requirements thefrequency, secondary base stations benefit of the base station, and scenarios such asofthe power and and timethe dimensions. It primary provides a practical andrealize effective efficient provides a better foundation and practical basis for the solution for spectrum spectrumsharing. sharing,It and theory and experiments prove that the scheme canpractical reflect the application of spectrum and provides theoretical for the development of spectrum requirements of sharing, the secondary base stations andsupport the benefit offurther the primary base station, CRSNs. and realize efficient spectrum sharing. It provides a better foundation and practical basis for the The simulation results also show that the model can reflect the demand of the spectrum of the practical application of spectrum sharing, and provides theoretical support for the further development secondary network. Primary base station spectrum pricing through a dynamic game of setting of CRSNs. specific parameters, eventually converges to a Nash equilibrium; the adjusted spectrum requirements of The simulation results also show that the model can reflect the demand of the spectrum of the secondary base station obtain the spectrum rent amount under market equilibrium, just to meet the secondary network. Primary basecapacity, station while spectrum pricing through a dynamic game ofbase setting specific needs of the network channel in the secondary network, the secondary stations parameters, converges to asharing Nash scheme, equilibrium; spectrum requirements are based eventually on a cooperative spectrum whichthe canadjusted allocate spectrum resources fairly of secondary base station obtain the spectrum rent amount under market equilibrium, just to meet the and efficiently.

needs of the network channel capacity, while in the secondary network, the secondary base stations Acknowledgments: This work is supported by the scheme, National Natural Foundation of Chinaresources under Grant are based on a cooperative spectrum sharing whichScience can allocate spectrum fairly No. 71272144 and Grant No. 61503143. and efficiently. Acknowledgments: This work is supported by the National Natural Science Foundation of China under Grant No. 71272144 and Grant No. 61503143.

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Author Contributions: Biqing Zeng designed the first stage of the model, completed the optimal spectrum pricing and subcarrier allocation of the primary network and carried out relevant experiments. Chi Zhang designed the second stage of the model, completed the fair and efficient allocation scheme of subcarrier and power based on Nash bargaining scheme and carried out relevant experiments. Pianpian Hu wrote the paper. Shengyu Wang reviewed and verified the paper. All authors read and approved the final manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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