# Game-Theoretic Power and Rate Control in IEEE 802.11\(p\) Wireless Networks
## 4 Numerical Results
### Evaluation Results  
Let us assume that our algorithm operates in a road network, which is interconnected operating with \(802.11p\). The bandwidth is set to \(10\) MHz, and and we set a delay of \(0.05\) s. We also set the arrival rate of the packets to \(10\) packets per second and the \(PRR_{w}=0.9\)since we have very reliable links due to the nature of the problem. Note that there will be interference from other vehicles that want to receive a message from another as well as interference from other vehicles that reside nearby. Initially, we take the case where two vehicles want to exchange messages with two respective vehicles. We assume that the Carrier Sense Multiple Access (CSMA) is disabled in order to ensure that simultaneous transmission to a vehicle is allowed. We duty cycle the two vehicles into two time slots; hence, the utility of each vehicle can reach the maximum when the maximum transmission power is used due to the increase of the SINR from the transmission power increase, as we do not experience interference. The reader can see the results of the plots of Equations (1) and (17). For the latter, we included the cases with no interference and duty cycled interference in Figure 5. Here, with the red color, we see the utility when the constraints are satisfied. The best case Nash equilibrium is the one with the highest SINR. Moreover, we plotted the curve of Equation (1) and the plot for the duty cycled interference from Equation (17). For the latter, we see that with the increase of the SINR, the PRR increases from the grey area to the maximum and then the rate is maximised after the PRR being 1. This takes place since the PRR can be one with lower than the maximum SINR. Furthermore, the zoom of the plot can be seen directly below the figure to see the change in the \(PRR^{phy}\). The value of the rate starts increasing when the \(PRR_{w}=0.9\), which is when the SINR is 9 dB, as it can be seen with the red line of the plot. Note that for the case of no interference, the utility rapidly increases, since the SINR exceeds the reception threshold and the \(PRR^{phy}\) jumps from 0 to 0.9. When the SINR reception  
Figure 5: Utility as a function of SINR \(\gamma_{ij}\). the PRR becomes 1, as we can see from Equation (1) and Figure 2; thus, the PRR jumps to a high value. The \(PRR^{phy}\) is 0.9 since, due to the low transmission power, a packet may be lost due to the coverage or other problems, such as an obstacle between the sender and the receiver. For the duty-cycled case, the utility appears to be like a step function, since the duty cycling mechanism is applied and the \(PRR^{phy}\) jumps from 0.5 to a final 1.0. We constructed the experiment in such a way in order to show the duty-cycled mechanism when it is applied to the transmission of the packets. In practical scenarios, the PRR fluctuates when interference is experienced, and the PRR falls into the grey area, especially when the signal is not strong enough from the transmission power perspective. Once the duty cycling mechanism is activated, the signal can go through and the reception threshold at the received is exceeded, meaning the successful transmission of a packet. This means that the PRR will be increased dramatically. In a similar manner, the rate increases with the increase in SINR, coming from the transmission power becoming higher, as we can see in Figure 6. Since in this work, we duty cycle vehicles to stabilise or diminish interference, the SINR changes with the change of the transmission power. The reader can see that in Equation (2), whereby the denominator remains the same and the nominator increases. In terms of packets lost, we take the case where there is one interferer in the communication of two vehicles. For one second, we need a maximum of 20 time slots and a minimum of 10 slots (if the vehicles are in a sparse place whereby the packets simultaneously go through) for the packets to be successfully transmitted for the interferer and the sender in one second.