Comparison between Different Approaches

In the Fig. 4.10 withus = 8, we compare the performance between proposed approach and other approaches, which are specified as follows:

1. No cooperation approach: each PU only uses direct link to transmit signals to PBS without SU’s help. Hence, SUs have no opportunity to access the channels.

2. Equal Power approach: there is no power control in the scenario, both PUs and SUs transmit with fixed power level.

3. Proposed algorithm: proposed algorithm based on coalitional game with consid-ering power control and time allocation problems.

Fig. 4.10 shows performance in different approaches. First, we can see that SUs’

payoffs are zero in the no cooperation approach. This is due to no channel for SU’s accessing in this approach. Therefore, each PU only uses direct link to transmit signals to PBS without SU’s assistance. The payoffs of PU 1, PU 2, and PU 3 in this approach are 14.28, 12.87, and 11.44, respectively. On the other hand, players can obtain higher payoffs in the equal power approach. Each player’s payoff in the equal power approach is better than acting alone, i.e. no cooperation approach. Hence, the payoff allocation in the equal power approach is incentive for players, which implies that players have incentives to join the coalition.

Clearly, each SU’s payoff is improved significantly in the proposed algorithm. The rationale behind this is that SUs allocate more power on accessing, so SUs’ payoffs in-crease dramatically. However, PU 2’s payoff is 18.63 in the equal power approach better than 14.67 in the proposed algorithm. The reason is that PU 2’s rate is determined by the SUs to PBS part, while proposed algorithm allocates less power on SU’s relaying. Hence, PU 2’s rate is reduced in the proposed algorithm. Nevertheless, the coalition value is improved significantly by the proposed algorithm, so players are more incentive for the proposed algorithm. The coalition value is 170.30, whereas in the equal power approach, the coalition value is 122.49. Therefore, the payoff allocation of the proposed algorithm is stable.

PU 1 PU 2 PU 3

Figure 4.10: Comparison the performance between different approaches: (a) The payoff of PUs (b) The payoff of SUs.

No. 1 No. 2 No. 3 0

50 100 150 200 250

Different approaches

Sum payoff

Figure 4.11: Comparison the sum payoff between different approaches.

Fig. 4.11 shows the comparison of the sum payoff between different approaches, where No. 1 is the no cooperation approach and No. 2 is the equal power approach; No. 3 is the proposed algorithm. The sum payoff of No. 1, No. 2, and No. 3 are 38.59, 122.49, and 170.30, respectively. The sum payoff is the lowest in the no cooperation approach among approaches, because each PU only uses direct link to transmit signals to PBS without SUs’ help. Whereas, SUs have no chance to access channels to transmit their own traffic. Hence, both PUs and SUs perform poor in the no cooperation approach. For other two approaches, the sum payoff is also called as coalition value. In theses two approaches, all the players form the grand coalition. According to the definition of the v(S), the coalition value is the summation of players’ payoffs in the coalition. However, the sum payoff in the no cooperation approach cannot be called as coalition value, because players just act alone and they do not form any coalition. The coalition value in the proposed algorithm increases 39.03% comparing with the equal power approach. The main reason is that SUs allocate more power on accessing, so SUs’ payoffs improve significantly in the proposed algorithm. While, the equal power approach sets the same power levels to SU’s relaying and SU’s accessing. Hence, SUs are beneficial in the proposed algorithm.

As a result, proposed algorithm can increase the coalition value dramatically.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.12: Proposed algorithm converges to the core

4.2.4 Algorithm Converges to the Core

Fig. 4.12 illustrates that the algorithm converges to the core in the aspect of time allo-cation. At first, the time coefficients are initialized at(α, β)=(0.06, 0.03). Then, at each allocated time point, we conduct optimal power control to maximize system utility. The time coefficients ofα and β are updated iteratively according to (3.15) and (3.16), re-spectively. In the figure, we can see that as the iteration increases, the updating step size decreases. Eventually, the algorithm converges to the equilibrium at (0.642, 0.217) with maximum coalition value 170.30. Then, we examine the payoff allocation obtained at this equilibrium with the definition of the core, which has two conditions. The first condition is guaranteed by the optimization problem’s objective function, so we only need to exam-ine the second condition of the definition. In the second condition, we have to show that players benefit most in the grand coalition. For example, if PU 1 acts alone, the payoff is 14.28; after joining the coalition with SU 1, the payoff increases to 17.01. Whereas, SU 1’s payoff is zero before joining any coalition; after joining the coalition with PU 1, SU 1’s payoff increases to 0.44. If PU 2 joins the coalition with PU 1 and SU 1, PU 2’s payoff increases to 14.28. The more PUs joining the coalition, SUs can access the more chan-nels, so SUs’ payoffs increase. Also, the more SUs in the coalition, PUs’ transmission

can be assisted by the more SUs. After comparing with different coalitions, we discover that player’s payoff in the grand coalition is the most beneficial. Therefore, the payoff allocation achieved by the algorithm lies in the core. This implies that all the players have incentives for the proposed payoff allocation. As a result, proposed algorithm converges to the core.

Chapter 5

Conclusion and Future Work

5.1 Conclusion

We have applied a coalitional game to model the problem in the CCRN scenario. We consider the problem formulation in two cases, i.e. power control, and power control and time allocation case. In the power control case, the main purpose is to allocate the SU’s power levels for relaying and accessing in order to maximize the system utility. On the other hand, in the power control and time allocation case, the problem is not guaranteed to be convex, so we proposed a novel algorithm to solve the problem iteratively in two steps.

In the first step, we allocate the time coefficients and then, we conduct power control optimization int the second step. The proposed algorithm guarantees the problem solved in convex procedure. In addition, we have studied the convergence of the algorithm and the solution achieved by the algorithm lies in the core. Our problem formulation satisfies the time-sharing condition, which guarantees the problem’s zero duality gap. We apply the time-sharing condition to proof that the core is nonempty. In the simulations, we have shown that the PUs’ and SUs’ payoffs lie in the core. We also discuss the relation-ship between time coefficients and coalition value. While, comparing between different approaches, our proposed algorithm can achieve a stable payoff allocation. Finally, the proposed algorithm converges to the equilibrium in the core.