Abstract-Cognitive wireless is a new engineering for bettering frequence spectrum use in radio webs. A polar issue in cognitive wireless is spectrum sharing, which allows primary ( licensed ) users to portion their accredited wireless frequence spectrum with secondary ( unaccredited ) users. In this paper, we will pattern the cognitive wireless environment as a quality based competition. We consider the spectrum sharing job between multiple primary users and merely one secondary user. The freshness of this paper is that we formulate this job as an oligopoly market competition and utilize the Cournot game ( quality-based competition ) to obtain the optimum spectrum allotment for the secondary user.
Nash equilibrium is considered as the solution of this game. We will see two different instances: inactive game in which each user can detect the adoptive schemes and the final payment of others and dynamic game in which the scheme of each user is selected based on merely the information obtained during the game. At the terminal, the stableness status for the dynamic instance of spectrum sharing is investigated.
Performance rating shows that our theoretical account consequences in more net incomes for primary users with lower offered monetary values for the secondary user and larger shared spectrum sizes at the disbursal of some stableness lessening.
Cognitive wireless engineering is introduced for bettering the use of wireless frequence spectrum [ 1 ] . For this intent, cognitive wireless has the capableness of finding its environing environment and accommodating its parametric quantities harmonizing to the information obtained from environing environment.
Hence, it is able to feel spectrum holes and utilize them when primary users do n’t utilize these parts of spectrum and hence, the spectrum efficiency will be improved.
The issue of pricing in dynamic spectrum entree is considered in the literature [ 2-7 ] . In [ 2-3 ] a competitory price-based theoretical account under Bertrand game for spectrum sharing is introduced and investigated. In [ 4 ] , an incorporate pricing, allotment, and charge system for cognitive wireless is proposed. In this system, the job of pricing dialogue between the operator and the services is formulated as a multi-unit sealed-bid auction. In [ 5 ] , optimum command, pricing, and service distinction for codification division multiple entree ( CDMA ) systems are analyzed. However, the issue of equilibrium among multiple operators and the stableness of the command are ignored. The competition among spectrum proprietors is modeled as a game in [ 6-7 ] . However, replaceability factor is non considered, and besides the stableness of the scheme version is non investigated.
In this paper, the job of dynamic spectrum sharing is modeled as a dynamic game, which is based on a quality ( spectrum size ) competition. This job is modeled as an oligopoly market in which primary users compete with each other to sell a portion of their accessible frequence spectrum to maximise their net incomes. The freshness of this paper is that in our proposed method, we foremost use the Cournot game to pattern the competition-based spectrum sharing job harmonizing to the quality ( spectrum size ) for an environment consisting of multiple primary users and merely one secondary user. After finding the monetary value of shared spectrum size by the Cournot game, we will specify each primary user ‘s net income by sing its Quality of Service ( QoS ) debasement. Furthermore, since each primary user is non cognizant of other users ‘ public presentations and their schemes in pattern, we propose a dynamic game in this instance. Meanwhile, its stableness and convergence to the Nash equilibrium ( which is the best response of one user, when the schemes of others are known ) are investigated.
The remainder of the paper is organized as follows. In Section II, system theoretical account is introduced and good described. In Section III, the Cournot game is introduced and it is used to pattern spectrum sharing competition. Proposed monetary value by primary users and their accomplishable net incomes are calculated in two different state of affairss ( inactive and dynamic game ) . In Section IV, stableness of proposed dynamic game is investigated. In Section V, we evaluate our system operation and simulation consequences are shown. Finally, this paper is concluded in Section VI.
System theoretical account
We consider a radio system with multiple primary users ( state N ) runing on Fi and merely one secondary user. In this instance, primary user I serves Mi local connexions, each of which meaning to sell a portion of its accessible spectrum with unit monetary value of frequence spectrum ( pi ) . Secondary user uses adaptative transition for its transmittal in which transmittal rate is adjusted dynamically harmonizing to Signal to Noise Ratio ( SNR ) . For the un-coded quadrature amplitude transition ( QAM ) with square configuration ( e.g. 4-QAM or 16-QAM ) spectrum efficiency is defined as follows [ 2 ] :
( 1 )
where is the SNR at the receiving system and is Bit Error Rate ( BER ) .
In this paper, dynamic spectrum sharing job, which is based on competition, is modeled as an oligopoly market, which operates based on quality competition. In this sort of market ( oligopoly market ) , all houses operate non-coordinately, every bit good as independently to maximise their net incomes by commanding the quality and the monetary value of their merchandises. Here houses are primary users, which compete non-cooperatively to sell their free frequence spectrum. The Cournot game is a well-known economical theoretical account, which theoretical accounts quality-based competitions. Performance of each house affects on those of others. Here, the proposed monetary value for unit of shared frequence spectrum is determined by the Cournot game and so the net income of each primary user is defined harmonizing to the sum of debasement of its QoS public presentation. Each primary user adjusts the size of its spectrum, which is traveling to portion, in order to maximise its ain net income.
Quality-based Competition and Solutions
In this sub-section, we propose a quality-based competition for spectrum sharing utilizing game theory construct. First, we use the Cournot game to cipher the proposed monetary values for unit of shared frequence spectrum of each primary user. Then, harmonizing to the sum of the QoS public presentation debasement of each local connexion we define the net income map. Harmonizing to the Cournot game, the monetary value for unit of shared spectrum frequence can be defined as [ 8 ] :
( 2 )
where is the size of shared spectrum offered by N primary users. is the size of the shared spectrum by other primary users. Besides, is the spectral efficiency of wireless transmittal for the secondary user. This secondary user uses the frequence spectrum Fi, which belongs to primary user i. is the permutation factor where means that secondary user is non allowed to exchange between frequence spectra, while agencies that it can exchange freely. If, spectrum sharing by the secondary user is complementary ( i.e. secondary user can purchase frequence spectrum and sell it ) .is the proposed monetary value for unit of spectrum by primary user I.
Net income Function for Primary users
The QoS public presentation of primary user is a dominant parametric quantity for specifying net income map. Sharing some parts of primary users ‘ spectrum with secondary user leads some debasement in QoS public presentation of primary users, and therefore this affects the merchandising monetary value of unit of frequence spectrum offered by primary users. The net income of each primary user is expressed as follows ( likewise to the net income map defined in [ 3 ] ) :
( 3 )
where represents the needed bandwidth of a primary connexion, denotes the shared spectrum size of primary user I, is the figure of its primary connexions and Wi is its entire spectrum size. Meanwhile, is spectral efficiency for the primary user i. and are some changeless footings.
Inactive Game Model
In the inactive game each user can detect the adoptive schemes and the final payment of others. Nash equilibrium is the best scheme of a given participant when other participants ‘ schemes are known. Hence, each primary user ‘s net income will be maximized with the shared frequence spectrum size obtained by the Nash equilibrium. Nash equilibrium is defined as follows:
( 4 )
where is the Nash equilibrium. The mathematical attack for obtaining this equilibrium is to derivate the net income map with regard to and set it equal to zero. In this instance, we should replace spectrum monetary value of unit of frequence spectrum with the equation of the Cournot monetary value ( in ( 2 ) ) . Then, we have:
( 5 )
( 6 )
By work outing the above N additive equations, the Nash equilibrium for N primary users are achieved.
In pattern, primary users are incognizant of the schemes of other users and they can non see their net incomes. Each primary user obtains some information about others ‘ public presentation by the game history ; it means that each primary user uses a distributed algorithm for seting shared spectrum size, which moves toward the Nash equilibrium easy. Suppose is the proposed shared spectrum size by primary user I at loop T of the game. Primary users can choose their schemes in a manner to maximise their net incomes.
Relation of the present and future schemes of a primary user is defined as [ 9 ] :
. ( 7 )
is the larning rate ( or seting rate ) .
Stability Analysis of Dynamic Game
In dynamic algorithms, stableness means being certain about the convergence of the game toward the Nash equilibrium which is significantly related to larning rate. By utilizing Jacobin matrix and the construct of Eigen values, we can look into the stableness of proposed dynamic game. When all Eigen values ( ) are inside the unit circle ( ) , game is stable. Jacobin matrix is defined as:
( 8 )
For the instance of two primary users, the values of Jacobin elements are:
( 9 )
( 10 )
For stableness we should hold, so:
. ( 11 )
Since each is a map of, M, and channel quality, altering these parametric quantities affect on the stableness part of the game.
See a cognitive wireless environment with two primary users and merely one secondary user with entire spectrum size of 20 MHz for each primary user ( ) . Number of connexions in each primary user is 10 ( ) , bandwidth required in each connexion of primary user is, and.
The best response maps of primary users are shown in Fig. 1 under different channel qualities and permutation factor. Nash equilibrium is the point that best response maps collide each other. When channel quality is better, we expect higher transmittal rates and accordingly, the sizes of shared frequence spectrum addition. So, in this state of affairs primary users obtain more net incomes. Besides, by increasing the value of permutation factor, the absolute incline of the best response maps increases which can alter the topographic point 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 ‘ schemes are known. However, when they are unknown, each primary user merely observes the spectrum demand of secondary user. In this figure, 10 loops of the game are shown. Increasing larning rate ( I± ) from 0.1 to 0.25 causes fluctuation in the sum of spectrum, which is traveling to be shared. It is noteworthy that if I±=0.25, the size of spectrum, which can be shared, is non stable after 10 loops, while if I±=0.1, as it is seen in Fig.2, after five loops, the bandwidth, which can be shared is converged to a fixed value. So we conclude that convergence velocity to the Nash equilibrium depends on larning factor.
Best response maps and Nash equilibrium
Convergence to the Nash equilibrium
Fig. 3 and Fig. 4, severally show shared spectrum sizes and the monetary values of the unit of frequence spectrum in the Nash equilibrium under different channel qualities. Shared spectrum sizes addition by increasing channel quality and therefore monetary values besides increase. In these figures, we have considered a fixed channel quality for channel two ( =15 dubnium ) while the quality of channel one changes from 8 dubnium to 22 dubniums. When the quality of channel one is less than 15 dubnium, the size of shared spectrum proposed by the 2nd primary user is higher than that of the first primary user. By bettering 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.
Net incomes of primary users with regard to impart quality are shown in Fig. 5 in which it is shown that by bettering channel quality, more net incomes are obtained.
Nash equilibrium of spectrum sharing under different channel qualities
Spectrum monetary value under different channel qualities
Spectrum net income under different channel qualities
By comparing our consequences with the consequences obtained in [ 3 ] it is seen that at the Nash equilibrium, in our proposed dynamic game, primary users portion more spectrum sizes with lower monetary value for unit of shared frequence spectrum to the secondary user while they achieve higher net incomes.
As stated before, stable part is defined by Eigen values ; when they are all in the unit circle ( ) , our dynamic game is stable. Fig. 6 shows the fluctuation of stableness part of our proposed dynamic game by altering larning rates. As mentioned earlier, higher acquisition rates lead instability. Furthermore, increasing and the figure of connexions for each primary user will do instability ( harmonizing to equations ( 9 ) and ( 10 ) ) .
Fig. 7 shows stability part of our dynamic game under different channel qualities. So, as we have concluded before, stableness of our dynamic game is related to permutation factor, figure of local connexions 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 job in cognitive wireless webs.
We considered a state of affairs of being of two primary users, which compete with each other to sell the free parts of their spectrum to merely one secondary user. We used the Cournot game theoretical account to find the monetary value for unit of shared frequence spectrum and so net income earned by each primary user is defined harmonizing to the sum of debasement of its QoS public presentation. Effectss of channel quality and permutation factor were investigated in our proposed method. For the instance in which users are non cognizant of each others ‘ schemes, we proposed a dynamic game, whose stableness was investigated with regard to larning rate, permutation factor and figure of local connexions.
By comparing our consequences with the consequences of [ 3 ] , in which Bertrand game theoretical account ( a price-based competition ) is used for patterning spectrum sharing job, we can see that, under the same fortunes considered in [ 3 ] , the net incomes obtained by primary users in our theoretical account ( the Cournot game which is a quality-based competition ) are improved extensively while the offered monetary values for the secondary user are lower and the shared spectrum sizes are greater. Although, it is noteworthy that in our proposed game, there is some decrease in the stableness part of the dynamic game ( e.g. for the instance of and, there is about 30 % decrease ( the worst state of affairs ) in the stableness part of dynamic game compared to that of [ 3 ] ) .
Stability part under different acquisition rates
Stability part under different channel qualities
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