CN102404747B - Efficient and fair dynamic spectrum allocation method - Google Patents

Abstract

The invention discloses an efficient and fair dynamic spectrum allocation method which is applicable to interactive cognitive radio networks and comprises the following steps: on the basis of cooperative differential game theory, dividing the interactive cognitive radio network into different alliances and establishing an alliance-competition type architecture; setting that the interactive cognitive radio network has a plurality of cognitive users as well as the objective function and the payoff function of each cognitive user; and solving the objective function and the payoff function. The invention can effectively relieve the shortage of spectrum, improve the utilization ratio of spectrum, and take account of the fairness and the efficiency of spectrum use.

Description

Efficient and fair method for allocating dynamic frequency spectrums

Technical field

The present invention relates to a kind of efficient and fair method for allocating dynamic frequency spectrums that solves the performance and fairness problem of cognition wireless network dynamic frequency spectrum deployment, belong to the dynamic resource partitioning technology field of cognition radio communication network.

Background technology

Dynamic frequency spectrum deployment is the problem of dynamic game in many ways distributed, the multi-user.Each cognitive user is in the situation that resource-constrained, and the process that takies of frequency spectrum be can be regarded as to a game process, and this game comprises competition successively of call duration time, the competition of communication frequency etc.Each cognitive user, when using frequency spectrum, all wishes at utmost to meet the communication requirement of oneself, and still, for whole system, unordered competition may cause the decline of overall performance on the contrary.Therefore, in order to improve the spectrum allocation may efficiency of a plurality of cognitive user, just must allow each cognitive user use frequency spectrum according to certain rule or algorithm.

All adopt at present game theoretic theory, model and algorithm, the spectrum allocation may between the research multi-user, solve the frequency spectrum resource race problem between a plurality of cognitive user.

Dynamic Spectrum Management in interactive cognition wireless network need to be considered the problem of four aspects: the interference problem of (1) cognitive user to the primary user; (2) race problem between cognitive user; (3) the optimum spectrum allocation may problem of frequency spectrum time-varying characteristics; (4) efficiency of cognitive radio system and the fairness problem between the user.For (1), (2) two problems, usually adopt model and the method for Static Game opinion.(3) individual problem, particularly the optimum spectrum allocation may problem of continuous time varying spectrum, be the process of time dependent dynamic game and cooperation.(4) individual problem belongs to cooperation with collaborative, by coordinating to realize cooperation, and the chief and the advantage of performance each side, the collaborative situation of creating win-win.

Document 1 (Z.Ji, and K.J.R.Liu.Multi-Stage Pricing Game for Collusion-Resistant Dynamic Spectrum Allocation.IEEE Journal on Selected Areas in Communications, 2008,26 (1): 182-191.) dynamic frequency spectrum deployment is modeled as to a multistage Pricing Game, and propose a kind of suppress the conflict the Dynamic Pricing method, use optimum reservation price conflict removal, and maximize user's utility level.Simultaneously, use and to receive assorted bargaining solution and analyze the boundary condition of this scheme, the boundary constraint needed is lower.Simulation result shows, under different user's conflict situations, all can obtain the higher availability of frequency spectrum.Document 2 (Dusit Niyato, Ekram Hossain.Competitive Pricing for Spectrum Sharing in Cognitive Radio Networks:Dynamic Game, Inefficiency of Nash Equilibrium, and Collusion.IEEE Journal on Selected Areas In Communications, 2008, 26 (1): the Bertrand model that 192-202.) proposes price of spectrum game between a plurality of main systems, but suppose to participate in the main system full symmetric of game, without any difference, this is the special circumstances of cognition wireless network.

Visible, most of technical scheme in the cognition wireless network technology is all launched based on Static Game opinion and repeated game at present, although above these methods, realizing having obtained some progress aspect efficient and fair dynamic frequency spectrum deployment, can not reflect use scenes truly.In fact, dynamic frequency spectrum deployment is a process of dynamic interaction in time, and the optimum spectrum allocation schemes of previous moment may no longer keep at lower a moment its optimality.That is to say, in cognition wireless network, radio environment is along with the variation in time and space has different characteristics, and the band informations such as operating frequency and bandwidth also have different characteristics; Simultaneously, because user in cognition wireless network changes at any time to the demand of bandwidth, quantity, situation and the position of available channel.Therefore, frequency spectrum distributing method seems most important flexibly and effectively.Cognition wireless network is in order to meet user's QoS demand, must the time select best frequency band in the available band that becomes.

Summary of the invention

The object of the present invention is to provide the efficient and fair method for allocating dynamic frequency spectrums of a kind of interactive cognition wireless network based on the game of cooperation differential, the problem of varying firing rate environment in the time of still can not reflecting truly to solve current method for allocating dynamic frequency spectrums.

For solving the problems of the technologies described above, efficient and fair method for allocating dynamic frequency spectrums provided by the invention comprises the following steps: based on cooperation differential theory of games, interactive cognition wireless network is divided into to different alliances, sets up interactive cognition wireless network " alliance-competition " type architecture; Set interactive cognition wireless network and there is n cognitive user, and target function or the pay off function of each cognitive user i ∈ N are: $\underset{u_i}{\max }{\∫}_0^{\∞}\left[\frac{x\left(s\right)-u_i\left(s\right)}{x\left(s\right)}{u}_i\left(s\right)\right]\exp (-\mathrm{rs})\mathrm{ds},i\∈N=\{\mathrm{1,2},...,n\},s\∈[0,\∞)---\left(1\right);$ In formula (1), s means that N means the cognitive user set constantly, and state variable x (s) means the percentage of interactive cognition wireless network at moment s usable spectrum, control variables u _i(s) mean the frequency spectrum access rate of each cognitive user i ∈ N at moment s, normal number r means the discount rate of interactive cognition wireless network, and the percentage x (s) of usable spectrum meets following dynamical system: $\stackrel{\·}{x}\left(s\right)=x\left(s\right)-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{u}_i\left(s\right);x\left(0\right)=x_0---\left(2\right)$

Formula (1) is solved.

The present invention is by combining " game of cooperation differential " with " interactive cognition wireless network ", can be based on cooperation differential theory of games, to interactive cognition wireless network dynamic frequency spectrum deployment modeling and simulation, solve dynamic and the cooperation issue of frequency spectrum access, simultaneously, can also take into account fairness and the efficiency that frequency spectrum is used.Therefore, the rare of frequency spectrum resource can be effectively alleviated in the present invention, improves the utilance of frequency spectrum, to manager, attendant, manufacturer and terminal use, all will produce far-reaching influence.For spectrum management person, can effectively improve the utilance of frequency spectrum, for new communications applications provides how available frequency spectrum.For the attendant, can utilize frequency spectrum by maximizing, can maximum gain; Can realize the seamless switching of heterogeneous network; Can effectively reduce O&M cost.For manufacturer, can reduce costs, and new demand is made to quick response.For the terminal use, can use any equipment to access at an easy rate arbitrary network at any time and any place, realize wireless roaming.

The accompanying drawing explanation

In order to be illustrated more clearly in the embodiment of the present invention or technical scheme of the prior art, below will the accompanying drawing of required use in embodiment or description of the Prior Art be briefly described, apparently, accompanying drawing in the following describes is only some embodiments of the present invention, for those of ordinary skills, under the prerequisite of not paying creative work, can also obtain according to these accompanying drawings other accompanying drawing.

The schematic flow sheet that Fig. 1 is the embodiment of the present invention;

Fig. 2 is model solution and the analytic process schematic diagram based on the game of cooperation differential.

Embodiment

Below in conjunction with accompanying drawing of the present invention, technical scheme of the present invention is clearly and completely described, obviously, described embodiment is only the present invention's part embodiment, rather than whole embodiment.Embodiment based in the present invention, those of ordinary skills, not making under the creative work prerequisite the every other embodiment obtained, belong to the scope of protection of the invention.

Fig. 1 schematically shows the step of the embodiment of the present invention, as shown in it, in order to improve the efficiency of interactive cognition wireless network dynamic frequency spectrum deployment, at first in step S11, propose and set up a kind of interactive cognition wireless network " alliance-competition " type architecture based on cooperation differential theory of games, and preferably according to position, frequency range, frequency is used dense degree and transmitting power, cognition network is divided into to different alliances, there is cooperation each alliance inside, between each alliance, competition is arranged, slice-of-life cognitive radio system so better, there is better using value.

Then, as step S12, set interactive cognition wireless network and have n cognitive user, consider that this has the frequency spectrum share scene of n cognitive user, wherein target function or the pay off function of each cognitive user i ∈ N are as follows:

$\underset{u_i}{\max }{\∫}_0^{\∞}\left[\frac{x\left(s\right)-u_i\left(s\right)}{x\left(s\right)}{u}_i\left(s\right)\right]\exp (-\mathrm{rs})\mathrm{ds},i\∈N=\{\mathrm{1,2},...,n\},$

s∈[0，∞) (1)。

In above formula (1), s means that N means the cognitive user set constantly, and state variable x (s) means the percentage of interactive cognition wireless network at moment s usable spectrum, control variables u _i(s) mean the frequency spectrum access rate of each cognitive user i ∈ N at moment s, normal number r means the discount rate of interactive cognition wireless network, preferably establishes r=1.5.

In interactive cognition wireless network, the percentage of usable spectrum meets following dynamical system:

$\stackrel{\·}{x}\left(s\right)=x\left(s\right)-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{u}_i\left(s\right),x\left(0\right)=x_0;---\left(2\right).$

As step S13, solve above formula (1), and preferably, the solution procedure of cooperation differential game comprises the following steps:

(1) calculate the maximum return of total alliance;

(2) calculate the Nash Equilibrium Solution of each member in noncooperative situation in alliance;

(3) calculate except total alliance the income of alliance likely;

(4) determine characteristic equation;

(5) calculate Sharp's profit value;

(6) determine a distribution of income program (Income Distribution Procedure is called for short IDP) with time consistency.

Specifically solve with analytic process as shown in Figure 2, the differential game is the theory of processing both sides or in many ways dynamically resisting continuously problem of game, at present mainly towards military, economic dispatch field, not yet be applied to communication resource sharing problem, its Major Mathematics basis is " mathematical theory of optimal process " of Bel's graceful " Dynamic Programming " and Pang Te lia king.Optimized method is very extensive in the application of the communications field, as graph theory, genetic algorithm, neural net etc.As long as find target function and the state equation (group) of describing cognition wireless network corresponding model and algorithm, can be converted into a decision-making problem of multi-objective, finally solve corresponding Bernoulli equation group or in block and put forward (Riccati) equation group, can try to achieve optimal solution, and then analyze uniqueness of solution etc.And no matter these equation group are numerical solution or analytic solutions, all can separate under certain condition.

In order to make those skilled in the art further understand the present invention, one embodiment of the invention specifically provide a kind of method that solves formula (1), specific as follows:

(1) calculate the maximum return of total alliance.

Be exactly, total calculate the dynamic programming problems that the maximum return of alliance is separated a standard, target function is to maximize all cognitive user frequency spectrum utilization rates, constraints is above-mentioned formula (2), obtains:

$\underset{u_1,u_2,...u_n}{\max }\underset{i\∈N}{\mathrm{\Σ}}{\∫}_t^{\∞}\left[\frac{x\left(s\right)-u_i\left(s\right)}{x\left(s\right)}{u}_i\left(s\right)\right]\exp [-r(s-t)]\mathrm{ds};---\left(3\right)$

$\stackrel{\·}{x}\left(t\right)=x\left(t\right)-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{u}_i\left(t\right),x\left(0\right)=x_0,x\left(t\right)=x^N\left(t\right);---\left(4\right)$

In order to solve the feedback Nash Equilibrium Solution of game (3)-(4), can obtain Bellman equation as follows:

$\mathrm{rW}(N,x,t)=\underset{u_1,u_2,...u_n}{\max }\left\{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\frac{x-u_i}{x}{u}_i{+W}_x(N,x,t)(x-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{u}_i)\right\};---\left(5\right)$

Wherein, W(N, x, t) mean the graceful value function of Bel of above-mentioned dynamic programming problems, to the both sides of formula (5) respectively to u _iask partial derivative, can obtain:

$u_i^N=[1-W_x(N,x,t)]x/2,i\∈N$

Will

bring formula (5) into and solve, can obtain:

$W(N,x,t)=\frac{n+1-\sqrt{2n+1}}{n}x;---\left(6\right)$

$u_i^N=\frac{\sqrt{2n+1}-1}{2n}x,i\∈N;---\left(7\right)$

Formula (7) has shown frequency spectrum access rate under total alliance

relation with the percentage x of usable spectrum.Although not explicitly depend on time t, but x becomes while being, so the radio frequency environment become when it has reflected truly.

The percentage of the usable spectrum of interactive cognition wireless network optimum is as follows:

$x^N\left(t\right)=x_0\exp \left(\frac{3-\sqrt{2n+1}}{2}t\right);---\left(8\right)$

(2) calculate the Nash Equilibrium Solution of each member in noncooperative situation in alliance.

The feedback Nash Equilibrium Solution of non-cooperative game (1)-(2) must meet following condition:

$r{V}^i(t,x)=\underset{u_i}{\max }\{\frac{x-u_i}{x}{u}_i+V_x^i(t,x)(x-u_i-\underset{j=1,j\≠i}{\overset{n}{\mathrm{\Σ}}}{u}_i\left)\right\};---\left(9\right)$

Wherein, V ⁱ(t, x) means the graceful value function of the Bel of i cognitive user; To the both sides of formula (9) respectively to u _iask partial derivative, can obtain:

$u_i^{*}=[1-V_x^i(t,x)]x/2,i\∈N;---\left(10\right)$

Formula (10) is brought into to formula (9) and is solved, can obtain:

$V^i(t,x)=\frac{n+1-\sqrt{n^2+2}}{2n-1}x,i\∈N;---\left(11\right)$

The feedback Nash Equilibrium Solution of the frequency spectrum access rate of each cognitive user i ∈ N solves as follows:

$u_i^{*}=\frac{n-2+\sqrt{n^2+2}}{2(2n-1)}x,i\∈N;---\left(12\right)$

(3) calculate except total alliance the income of alliance likely.

Calculate the income except the game of total alliance and single cognitive user composition, such alliance has 2 ⁿ-n-2.Use W (K, x, t) to mean the graceful value function of Bel of the K of alliance, W (K, x, t) must meet following Bellman equation:

$\mathrm{rW}(K,x,t)=\underset{u_i,i\∈K}{\max }\{\underset{i\∈K}{\mathrm{\Σ}}\frac{x-u_i}{x}{u}_i+W_x(K,x,t)[x-\underset{i\∈K}{\mathrm{\Σ}}{u}_i-\underset{j\∉K}{\mathrm{\Σ}}{u}_j^{*}\left]\right\};---\left(13\right)$

To the both sides of formula (13) respectively to u _iask partial derivative, can obtain:

$u_i^K=[1-W_x(K,x,t)]x/2,i\∈K.$

Will

bring formula (13) into and solve, can obtain:

$W(K,x,t)=\frac{A-\sqrt{A^2-k^2}}{k}x;---\left(14\right)$

$u_i^K=\frac{k-A+\sqrt{A^2-k^2}}{2k}x,i\∈K;---\left(15\right)$

Wherein: $A=n+1-(n-k)\·\frac{n+1-\sqrt{n^2+2}}{2n-1}.$

(4) defined feature equation.

Characteristic function v (the K of this cooperative game; X, t) be defined as follows:

v({i}；x，t)＝V ⁱ(t，x)；

v(K；x，t)＝W(K，x，t)，

Wherein, the characteristic function value is given by following formula:

$v\left(\right\{i\};x,t)=V^i(t,x)=\frac{n+1-\sqrt{n^2+2}}{2n-1}x,i\∈N;$

$v(K;x,t)=W(K,x,t)=\frac{A-\sqrt{A^2-k^2}}{k}x;$

$A=n+1-(n-k)\·\frac{n+1-\sqrt{n^2+2}}{2n-1},K\⊆N.$

(5) calculate Sharp's profit value.

Sharp's profit value of this cooperative game is as follows:

${\mathrm{\Φ}}_i^v(x,t)=\underset{i\∈K}{\mathrm{\Σ}}\frac{(k-1)!(n-k)!}{n!}[v(K;x,t)-v(K\backslash \{i\};x,t)]$

i＝1，....，n ；(16)

When n=3, have:

${\mathrm{\Φ}}_i^v(x,t)=\frac{4-\sqrt{7}}{9}x,i\∈\left\{\mathrm{1,2,3}\right\};---\left(17\right)$

Sharp's profit value has provided concrete spectrum allocation schemes by the mode of probability, has shown fairness and uniqueness, and each cognitive user can obtain optimum frequency spectrum access chance.

(6) determine a distribution of income program with time consistency.

The percentage x of the usable spectrum of the interactive cognition wireless network optimum meaned along formula (8) ⁿ(t), for Sharp's profit is worth

an instantaneous distribution is provided, can uses following partition function:

$B_i\left(t\right)=r{\mathrm{\Φ}}_i^v(x^N\left(t\right),t)-\frac{d}{\mathrm{dt}}{\mathrm{\Φ}}_i^v(x^N\left(t\right),t);---\left(18\right)$

Directly calculate:

$B_i\left(t\right)=\frac{4\sqrt{7}-7}{18}{x}^N\left(t\right);---\left(19\right)$

In same alliance inside, each cognitive user does not have otherness to the use of frequency spectrum, has shown the fairness of model, can improve the utilance of frequency spectrum simultaneously, the radio frequency environment become while reflecting truly.

To sum up, the present invention is for the dynamic that solves the cognition wireless network dynamic frequency spectrum deployment and the problem of cooperative aspect, based on cooperation stochastic differential game theory, new thinking and direction have been proposed, for the Tactic selection problem of dynamic scene provides Mathematics Proof, also make the searching steady state solution become possibility, thereby, can effectively utilize frequency spectrum resource, meet the demand of different user to frequency spectrum resource, increase power system capacity, improve the availability of frequency spectrum.Therefore, the game of cooperation differential is introduced to cognition wireless network and can solve cognition wireless network dynamic frequency spectrum deployment problem, effectively alleviate the rare of frequency spectrum resource, improve the utilance of frequency spectrum.

The above; be only the specific embodiment of the present invention, but protection scope of the present invention is not limited to this, anyly is familiar with those skilled in the art in the technical scope that the present invention discloses; can expect easily changing or replacing, within all should being encompassed in protection scope of the present invention.Therefore, protection scope of the present invention should be as the criterion by the described protection range with claim.

Claims (4)

1. an efficient and fair method for allocating dynamic frequency spectrums, be applicable to interactive cognition wireless network, it is characterized in that, comprises the following steps:

Based on cooperation differential theory of games, interactive cognition wireless network is divided into to different alliances, set up interactive cognition wireless network " alliance-competition " type architecture;

Set interactive cognition wireless network and there is n cognitive user, and target function or the pay off function of each cognitive user i ∈ N are:

$\underset{u_i}{\max }{\∫}_0^{\∞}\left[\frac{x\left(s\right)-u_i\left(s\right)}{x\left(s\right)}{u}_i\left(s\right)\right]\exp (-\mathrm{rs})\mathrm{ds},i\∈N=\{\mathrm{1,2},...,n\},s\∈[0,\∞);---\left(1\right)$

In formula (1), s means that N means the cognitive user set constantly, and state variable x (s) means the percentage of interactive cognition wireless network at moment s usable spectrum, control variables u _i(s) mean the frequency spectrum access rate of each cognitive user i ∈ N at moment s, normal number r means the discount rate of interactive cognition wireless network, and the percentage x (s) of usable spectrum meets following dynamical system:

$\stackrel{\·}{x}\left(s\right)=x\left(s\right)-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{u}_i\left(s\right)$

x(0)＝x ₀ ；(2)

Formula (1) is solved.

2. efficient and fair method for allocating dynamic frequency spectrums according to claim 1, is characterized in that, the basis that interactive cognition wireless network is divided into to different alliances is used dense degree and transmitting power for position, frequency range, frequency.

3. efficient and fair method for allocating dynamic frequency spectrums according to claim 1, is characterized in that, establishes normal number r=1.5.

4. efficient and fair method for allocating dynamic frequency spectrums according to claim 1, is characterized in that, to formula (1), solves further comprising the steps:

Calculate the maximum return of total alliance;

In this step, the maximum return of calculating total alliance is considered as solving to the dynamic programming problems of a standard, maximizing the target function of all cognitive user frequency spectrum utilization rates or the constraints of pay off function is described formula (2), obtains whereby:

$\underset{u_1,u_2,...u_n}{\max }\underset{i\∈N}{\mathrm{\Σ}}{\∫}_t^{\∞}\left[\frac{x\left(s\right)-u_i\left(s\right)}{x\left(s\right)}{u}_i\left(s\right)\right]\exp [-r(s-t)]\mathrm{ds};---\left(3\right)$

$\stackrel{\·}{x}\left(t\right)=x\left(t\right)-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{u}_i\left(t\right),x\left(0\right)=x_0,x\left(t\right)=x^N\left(t\right);---\left(4\right)$

Calculate Bellman equation as shown in the formula (5), to solve the feedback Nash Equilibrium Solution of formula (3)-(4):

$\mathrm{rW}(N,x,t)=\underset{u_1,u_2,...u_n}{\max }\{\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\frac{x-u_i}{x}{u}_i+W_x(N,x,t)(x-\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{u}_i)\};---\left(5\right)$

Wherein, W (N, x, t) means the graceful value function of Bel of described dynamic programming problems, to the both sides of formula (5) respectively to u _iask partial derivative, can obtain:

$u_i^N=[1-W_x(N,x,t)]x/2,i\∈N;$

Will

bring formula (5) into and solve, can obtain:

$W(N,x,t)=\frac{n+1-\sqrt{2n+1}}{n}x;---\left(6\right)$

$u_i^N=\frac{\sqrt{2n+1}-1}{2n}x,i\∈N;---\left(7\right)$

Formula (7) is presented at frequency spectrum access rate under total alliance

with the relation of the percentage x of usable spectrum, and the percentage of the usable spectrum of interactive cognition wireless network optimum as shown in the formula:

$x^N\left(t\right)=x_0\exp \left(\frac{3-\sqrt{2n+1}}{2}t\right);---\left(8\right)$

Calculate the Nash Equilibrium Solution of each member in noncooperative situation in alliance;

In this step, the feedback Nash Equilibrium Solution of formula (1)-(2) meets following formula (9):

${\mathrm{rV}}^i(t,x)=\underset{u_i}{\max }\{\frac{x-u_i}{x}{u}_i+V_x^i(t,x)(x-u_i-\underset{j=1,j\≠i}{\overset{n}{\mathrm{\Σ}}}{u}_j)\};---\left(9\right)$

Wherein, V ⁱ(t, x) means the graceful value function of the Bel of i cognitive user; To the both sides of formula (9) respectively to u _iask partial derivative, can obtain:

$u_i^{*}=[1-V_x^i(t,x)]x/2,i\∈N;---\left(10\right)$

Formula (10) is brought into to formula (9) and is solved, can obtain:

$V^i(t,x)=\frac{n+1-\sqrt{n^2+2}}{2n-1}x,i\∈N;---\left(11\right)$

The feedback Nash Equilibrium Solution of the frequency spectrum access rate of each cognitive user i ∈ N solves as follows:

$u_i^{*}=\frac{n-2+\sqrt{n^2+2}}{2(2n-1)}x,i\∈N;---\left(12\right)$

Calculate except total alliance the income of alliance likely;

In this step, calculate the income except the game of total alliance and single cognitive user composition, such alliance has 2 ⁿ-n-2.The graceful value function of Bel that means the K of alliance with W (K, x, t), W (K, x, t) meets following Bellman equation:

$\mathrm{rW}(K,x,t)=\underset{u_i,i\∈K}{\max }\{\underset{i\∈K}{\mathrm{\Σ}}\frac{x-u_i}{x}{u}_i+W_x(K,x,t)[x-\underset{i\∈K}{\mathrm{\Σ}}{u}_i-\underset{j\∉K}{\mathrm{\Σ}}{u}_j^{*}\left]\right\};---\left(13\right)$

To the both sides of formula (13) respectively to u _iask partial derivative, can obtain:

$u_i^K=[1-W_x(K,x,t)]x/2,i\∈K;$

Will

bring formula (13) into and solve, can obtain:

$W=(K,x,t)=\frac{A-\sqrt{A^2-k^2}}{k}x;---\left(14\right)$

$u_i^K=\frac{k-A+\sqrt{A^2-k^2}}{2k}x,i\∈K;---\left(15\right)$

Wherein: $A=n+1-(n-k)\·\frac{n+1-\sqrt{n^2+2}}{2n-1};$

Determine characteristic equation;

In this step, by the characteristic function v (K of described cooperative game; X, t) be defined as follows:

v({i}；x，t)＝V ⁱ(t，x)；

v(K；x，t)＝W(K，x，t)，

Wherein, the characteristic function value is given by following formula:

$v\left(\right\{i\};x,t)=V^i(t,x)=\frac{n+1-\sqrt{n^2+2}}{2n-1}x,i\∈N;$

$v(K;x,t)=W(K,x,t)=\frac{A-\sqrt{A^2-k^2}}{k}x;$

$A=n+1-(n-k)\·\frac{n+1-\sqrt{n^2+2}}{2n-1},K\⊆N;$

Calculate Sharp's profit value;

In this step, the Sharp of this cooperative game profit value is as follows:

${\mathrm{\Φ}}_i^v(x,t)=\underset{i\∈K}{\mathrm{\Σ}}\frac{(k-1)!(n-k)!}{n!}[v(K;x,t)-v(K\backslash \{i\};x,t)]$

i＝1，...，n； (16)

When n=3, have:

${\mathrm{\Φ}}_i^v(x,t)=\frac{4-\sqrt{7}}{9}x,i\∈\{1,2,3\};---\left(17\right)$

Determine a distribution of income program with time consistency.

The percentage x of the usable spectrum of the interactive cognition wireless network optimum meaned according to formula (8) in this step, ⁿ(t), utilize the partition function of following formula (18), to Sharp's profit value

an instantaneous distribution is provided:

$B_i\left(t\right)={\mathrm{r\Φ}}_i^v(x^N\left(t\right),t)-\frac{d}{\mathrm{dt}}{\mathrm{\Φ}}_i^v(x^N\left(t\right),t);---\left(18\right)$

Directly calculate:

$B_i\left(t\right)=\frac{4\sqrt{7}-7}{18}{x}^N\left(t\right).---\left(19\right)$

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