## 6 Simulation Results
However, the optimal transmit power of selected SSNs by using the proposed OTPA algorithm entirely depends upon maximum transmit power of selected SSNs and pricing factors in differential game model. We can also observe that the optimal transmit power of selected SSNs of the proposed OTPA algorithm for the CoPC problem under discount factor \(r=0.5\) is considerably lower than that of the NoCoPC problem. That is, the proposed OTPA algorithm for the CoPC problem outperforms that of the NoCoPC problem. This can be intuitively explained by the fact that there exists an operation of summing with respect to \(e^{\delta_{i}\cdot D_{m}(\mathcal{P}_{d}\|\mathcal{P}_{r})}\) in the denominator of the analytical cooperative optimal transmit power \(p_{m}^{C}(t)\). Moreover, the increase of the slot \(\tau_{rp}\) will generate lower transmit power for both the NoCoPC problem and CoPC problem with discount factor \(r=0.5\). This is because based on Equations (28) and (34), the optimal transmit power for both the NoCoPC problem and CoPC problem is inversely proportional to the slot \(\tau_{rp}\). This observation emphasizes the importance of selecting the proper time interval of slot \(\tau_{rp}\) on the optimal transmit power. Figure 8: Optimal transmit power vs. discount factor \(r\) (\(\tau_{rp}=12\) ms, \(l=250\) iterations). NoCoPC: noncooperative power control; CoPC: cooperative power control; PB: power balancing. In Figure 10, we look at the performance of the transmitted data sequence of SSR from selected SSNs to bottlenecked SU \(b\) during the slot \(\tau_{rp}=12\) ms with the evolution of discount factor \(r\) from 0.1 to 0.9. From the results, we can see the transmitted data sequence of SSR for selected SSNs gradually increase with the growth of discount factor \(r\). Meanwhile, the transmitted data sequence of SSR for PB algorithm with 250 iterations and our proposed distributed power control framework under the condition of the slot \(\tau_{rp}=12\) ms tends to be close when discount factor \(r=0.9\). The reason for this is that the transmitted data sequence of SSR can be approximately expressed as a function of channel capacity according to Equation (38), which is in direct proportion to the optimal transmit power of selected SSNs. It has also been shown that an increased discount factor from 0.1 to 0.9 will increase the optimal transmit power of selected SSNs under the proposed OTPA algorithm for both the NoCoPC problem and CoPC problem with the slot \(\tau_{rp}=12\) ms. Consequently, higher optimal transmit power yields more transmitted data sequence of SSR. In Figure 11, we examine the impact of discount factor \(r\) from 0.1 to 0.9 on the reduction of data sequence of SSR with respect to selected SSNs in the internal buffer of bottleneck SU \(b\). It can be observed from the figure that the reduction of data sequence of SSR by our proposed congestion mitigation approach under the slot \(\tau_{rp}=12\) ms will gradually decrease with the growth of discount factor \(r\), except that the reduction result by PB algorithm with 250 iterations appears to a fixed constant value. This trend is the result of the inverse relationship between the reduction of the data sequence of SSR \(\Delta\theta_{b}^{I}(\tau_{rp})\) and the optimal transmit power of selected SSNs \(p_{m}^{NC}(t)\) for the NoCoPC problem or \(p_{m}^{C}(t)\) for the CoPC problem according to Equations (42) and (47). Additionally, in Figure 11, we can also observe that the proposed congestion mitigation approach by using the OTPA algorithm for the CoPC problem with the slot \(\tau_{rp}=12\) ms achieves the higher reduction of the data sequence of SSR compared with those of the NoCoPC problem and PB algorithm with 250 iterations. In other words, through cooperation among all IT and MT SSNs, the reduction of the data sequence of SSR in the internal buffer of bottlenecked SU \(b\) can be further improved. This observation has verified the analytical derivation of the proposed congestion mitigation approach.