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Abstract-Cognitive radio is a new technology for improving frequency spectrum usage in wireless networks. A pivotal issue in cognitive radio is spectrum sharing, which allows primary (licensed) users to share their licensed radio frequency spectrum with secondary (unlicensed) users. In this paper, we will model the cognitive radio environment as a quality based competition. We consider the spectrum sharing problem between multiple primary users and only one secondary user. The novelty of this paper is that we formulate this problem as an oligopoly market competition and use the Cournot game (quality-based competition) to obtain the optimal spectrum allocation for the secondary user. Nash equilibrium is considered as the solution of this game. We will consider two different cases: static game in which each user can observe the adopted strategies and the payoff of others and dynamic game in which the strategy of each user is selected based on only the information obtained during the game. At the end, the stability condition for the dynamic case of spectrum sharing is investigated. Performance evaluation shows that our model results in more profits for primary users with lower offered prices for the secondary user and larger shared spectrum sizes at the expense of some stability decrease.
Keywords- Spectrum Sharing, Cognitive Radio, Game Theory, Cournot Game, Nash Equilibrium, Stability Analysis.
Cognitive radio technology is introduced for improving the utilization of radio frequency spectrum . For this purpose, cognitive radio has the capability of determining its surrounding environment and adapting its parameters according to the information obtained from surrounding environment. Hence, it is able to sense spectrum holes and use them when primary users don't use these parts of spectrum and therefore, the spectrum efficiency will be improved.
The issue of pricing in dynamic spectrum access is considered in the literature [2-7]. In [2-3] a competitive price-based model under Bertrand game for spectrum sharing is introduced and investigated. In , an integrated pricing, allocation, and billing system for cognitive radio is proposed. In this system, the problem of pricing negotiation between the operator and the services is formulated as a multi-unit sealed-bid auction. In , optimal bidding, pricing, and service differentiation for code division multiple access (CDMA) systems are analyzed. However, the issue of equilibrium among multiple operators and the stability of the bidding are ignored. The competition among spectrum owners is modeled as a game in [6-7]. However, substitutability factor is not considered, and also the stability of the strategy adaptation is not investigated.
In this paper, the problem of dynamic spectrum sharing is modeled as a dynamic game, which is based on a quality (spectrum size) competition. This problem is modeled as an oligopoly market in which primary users compete with each other to sell a part of their accessible frequency spectrum to maximize their profits. The novelty of this paper is that in our proposed method, we first use the Cournot game to model the competition-based spectrum sharing problem according to the quality (spectrum size) for an environment consisting of multiple primary users and only one secondary user. After determining the price of shared spectrum size by the Cournot game, we will define each primary user's profit by considering its Quality of Service (QoS) degradation. Furthermore, since each primary user is not aware of other users' performances and their strategies in practice, we propose a dynamic game in this case. Meanwhile, its stability and convergence to the Nash equilibrium (which is the best response of one user, when the strategies of others are known) are investigated.
The rest of the paper is organized as follows. In Section II, system model is introduced and well described. In Section III, the Cournot game is introduced and it is used to model spectrum sharing competition. Proposed price by primary users and their achievable profits are calculated in two different situations (static and dynamic game). In Section IV, stability of proposed dynamic game is investigated. In Section V, we evaluate our system operation and simulation results are shown. Finally, this paper is concluded in Section VI.
We consider a wireless system with multiple primary users (say N) operating on Fi and only one secondary user. In this case, primary user i serves Mi local connections, each of which intending to sell a part of its accessible spectrum with unit price of frequency spectrum (pi). Secondary user uses adaptive modulation for its transmission in which transmission rate is adjusted dynamically according to Signal to Noise Ratio (SNR). For the un-coded quadrature amplitude modulation (QAM) with square constellation (e.g. 4-QAM or 16-QAM) spectrum efficiency is defined as follows :
where is the SNR at the receiver and is Bit Error Rate (BER).
In this paper, dynamic spectrum sharing problem, which is based on competition, is modeled as an oligopoly market, which operates based on quality competition. In this kind of market (oligopoly market), all firms operate non-coordinately, as well as independently to maximize their profits by controlling the quality and the price of their products. Here firms are primary users, which compete non-cooperatively to sell their free frequency spectrum. The Cournot game is a well-known economical model, which models quality-based competitions. Performance of each firm affects on those of others. Here, the proposed price for unit of shared frequency spectrum is determined by the Cournot game and then the profit of each primary user is defined according to the amount of degradation of its QoS performance. Each primary user adjusts the size of its spectrum, which is going to share, in order to maximize its own profit.
Quality-based Competition and Solutions
In this sub-section, we propose a quality-based competition for spectrum sharing using game theory concept. First, we use the Cournot game to calculate the proposed prices for unit of shared frequency spectrum of each primary user. Then, according to the amount of the QoS performance degradation of each local connection we define the profit function. According to the Cournot game, the price for unit of shared spectrum frequency can be defined as :
where is the size of shared spectrum offered by N primary users. is the size of the shared spectrum by other primary users . Also, is the spectral efficiency of wireless transmission for the secondary user. This secondary user uses the frequency spectrum Fi, which belongs to primary user i. is the substitution factor where means that secondary user is not allowed to switch between frequency spectra, while means that it can switch freely. If , spectrum sharing by the secondary user is complementary (i.e. secondary user can buy frequency spectrum and sell it).is the proposed price for unit of spectrum by primary user i.
Profit Function for Primary users
The QoS performance of primary user is a dominant parameter for defining profit function. Sharing some parts of primary users' spectrum with secondary user leads some degradation in QoS performance of primary users, and hence this affects the selling price of unit of frequency spectrum offered by primary users. The profit of each primary user is expressed as follows (similarly to the profit function defined in ):
where represents the required bandwidth of a primary connection, denotes the shared spectrum size of primary user i, is the number of its primary connections and Wi is its total spectrum size. Meanwhile, is spectral efficiency for the primary user i. and are some constant terms.
Static Game Model
In the static game each user can observe the adopted strategies and the payoff of others. Nash equilibrium is the best strategy of a given player when other players' strategies are known. Hence, each primary user's profit will be maximized with the shared frequency spectrum size obtained by the Nash equilibrium. Nash equilibrium is defined as follows:
where is the Nash equilibrium. The mathematical approach for obtaining this equilibrium is to derivate the profit function with respect to and put it equal to zero. In this case, we should replace spectrum price of unit of frequency spectrum with the equation of the Cournot price ( in (2)). Then, we have:
By solving the above N linear equations, the Nash equilibrium for N primary users are achieved.
In practice, primary users are unaware of the strategies of other users and they cannot see their profits. Each primary user obtains some information about others' performance by the game history; it means that each primary user uses a distributed algorithm for adjusting shared spectrum size, which moves toward the Nash equilibrium slowly. Suppose is the proposed shared spectrum size by primary user i at iteration t of the game. Primary users can select their strategies in a way to maximize their profits.
Relation of the present and future strategies of a primary user is defined as :
is the learning rate (or adjusting rate).
Stability Analysis of Dynamic Game
In dynamic algorithms, stability means being sure about the convergence of the game toward the Nash equilibrium which is significantly related to learning rate. By using Jacobin matrix and the concept of Eigen values, we can investigate the stability of proposed dynamic game. When all Eigen values () are inside the unit circle (), game is stable. Jacobin matrix is defined as:
For the case of two primary users, the values of Jacobin elements are:
For stability we should have, so:
Since each is a function of , M, and channel quality, changing these parameters affect on the stability region of the game.
Consider a cognitive radio environment with two primary users and only one secondary user with total spectrum size of 20 MHz for each primary user (). Number of connections in each primary user is 10 (), bandwidth required in each connection of primary user is , and .
The best response functions of primary users are shown in Fig. 1 under different channel qualities and substitution factor. Nash equilibrium is the point that best response functions collide each other. When channel quality is better, we expect higher transmission rates and consequently, the sizes of shared frequency spectrum increase. So, in this situation primary users obtain more profits. Besides, by increasing the value of substitution factor, the absolute slope of the best response functions increases which can change the place of Nash equilibrium for the shared spectrum sizes.
In Fig. 2, convergence of dynamic game towards the Nash equilibrium has been investigated. Fast convergence is expected, when other users' strategies are known. However, when they are unknown, each primary user only observes the spectrum demand of secondary user. In this figure, ten iterations of the game are shown. Increasing learning rate (Î±) from 0.1 to 0.25 causes fluctuation in the amount of spectrum, which is going to be shared. It is notable that if Î±=0.25, the size of spectrum, which can be shared, is not stable after ten iterations, while if Î±=0.1, as it is seen in Fig.2, after five iterations, the bandwidth, which can be shared is converged to a fixed value. So we conclude that convergence speed to the Nash equilibrium depends on learning factor.
Best response functions and Nash equilibrium
Convergence to the Nash equilibrium
Fig. 3 and Fig. 4, respectively show shared spectrum sizes and the prices of the unit of frequency spectrum in the Nash equilibrium under different channel qualities. Shared spectrum sizes increase by increasing channel quality and therefore prices also increase. In these figures, we have considered a fixed channel quality for channel two (=15 dB) while the quality of channel one changes from 8 dB to 22 dB. When the quality of channel one is less than 15 dB, the size of shared spectrum proposed by the second primary user is higher than that of the first primary user. By improving the quality of channel one we arrive to the point that shared spectrum sizes of both channels are equal and after that shared spectrum sizes offered by the first primary user are higher.
Profits of primary users with respect to channel quality are shown in Fig. 5 in which it is shown that by improving channel quality, more profits are obtained.
Nash equilibrium of spectrum sharing under different channel qualities
Spectrum price under different channel qualities
Spectrum profit under different channel qualities
By comparing our results with the results obtained in  it is seen that at the Nash equilibrium, in our proposed dynamic game, primary users share more spectrum sizes with lower price for unit of shared frequency spectrum to the secondary user while they achieve higher profits.
As stated before, stable region is defined by eigen values; when they are all in the unit circle (), our dynamic game is stable. Fig. 6 shows the variation of stability region of our proposed dynamic game by changing learning rates. As mentioned earlier, higher learning rates lead instability. Furthermore, increasing and the number of connections for each primary user will cause instability (according to equations (9) and (10) ).
Fig. 7 shows stability region of our dynamic game under different channel qualities. So, as we have concluded before, stability of our dynamic game is related to substitution factor, number of local connections of each primary user and channel quality.
In this paper, we proposed a dynamic game which is based on quality (shared spectrum size) competition for the spectrum sharing problem in cognitive radio networks.
We considered a situation of existence of two primary users, which compete with each other to sell the free parts of their spectrum to only one secondary user. We used the Cournot game model to determine the price for unit of shared frequency spectrum and then profit earned by each primary user is defined according to the amount of degradation of its QoS performance. Effects of channel quality and substitution factor were investigated in our proposed method. For the case in which users are not aware of each others' strategies, we proposed a dynamic game, whose stability was investigated with respect to learning rate, substitution factor and number of local connections.
By comparing our results with the results of , in which Bertrand game model (a price-based competition) is used for modeling spectrum sharing problem, we can see that, under the same circumstances considered in , the profits obtained by primary users in our model (the Cournot game which is a quality-based competition) are improved extensively while the offered prices for the secondary user are lower and the shared spectrum sizes are greater. Although, it is notable that in our proposed game, there is some reduction in the stability region of the dynamic game (e.g. for the case of and , there is approximately 30% reduction (the worst situation) in the stability region of dynamic game compared to that of  ).
Stability region under different learning rates
Stability region under different channel qualities
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