# Game-Theoretic Power and Rate Control in IEEE 802.11\(p\) Wireless Networks
## 4 Numerical Results
### Evaluation Results
If we do not apply duty cycling and have a simultaneous transmission and allow three re-transmissions per packet, we can have a maximum of 60 transmission slots for this case and a minimum of 10 time slots, if the message will go through with one re-transmission. The reader can see the results of this case in Table 1.  
For different values of delay, the times of packets' transmission can be given in Table 1 as well. As it can be seen, the number of seconds delay for the duty-cycled approach is a minimum of 0.5 s delay and maximum 1 s delay on top of the programmed transmission. For the not duty-cycled approach, we see a delay of minimum 0.5 s and maximum of 3 s on top of the transmission. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Min Slots** & **Max Slots** & **Min Time (s)** & **Max Time (s)** \\ \hline Duty Cycled & 10 & 20 & 0.5 & 1 \\ No Duty Cycle & 10 & 60 & 0.5 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of slots and seconds for duty-cycled and not duty-cycled cases. Figure 6: Normalised rate with SINR. From the second constraint in (5), it can be derived that the increase of the delay provides a smaller value of the rate. Moreover, the transmission power increases the rate as well, as we can see from [PERSON]'s formula; however, a large transmission power by a vehicle can increase interference to other vehicles, thus increasing the delay in the case that we allow the re-transmission of packets. With our work, the duty cycling mechanism will not mimic re-transmission, albeit it is similar. The difference is that we duty cycle the nodes to stabilise or diminish interference and not re-send the same packet a number of times. Since the potential function is essentially the sum of the sigmoidal functions when using Equation (1) (utility functions, there exists an algorithm to solve this optimisation problem, namely the branch and bound method, which is described in [26]). We further altered the SINR coming from the transmission power to exceed the value 9.0, which we consider a threshold that can ensure a high PRR. This alteration gives us another property as to make the utility function strictly concave. This will ensure that a local maximum of the function is a global maximum as well. We can see the dashed line which separates the utility function into two parts in Figure 5. The PRR in this setting is 1.0 when the constraints are satisfied and the rate increases, as we can see in Figure 6. Note that the constraint has been set to the normalised value of the rate when the delay is 0.05 s.  
The natural question that arises is that the delay will increase with the duty cycling of the vehicles. This can be also the case with the CSMA, since there will be a backoff for the attempt of a new transmission. One of the main problems of the IEEE 802.11\(p\) protocol is its functionality in dense environments [27]. Moreover, using proximity sensors, the vehicles may instruct the passing or coming behind vehicles to transmit using a beacon. The beacon will include the unique ID of the vehicle and notify that one will transmit, further limiting the interference. For our scenario, the delay of transmission will be 0.04 s. For our future work, we will investigate a denser scenario. Moreover, the utility function \(u\) was plotted against the transmission power levels in \(\mu\)W in Figure 7. The purpose of this experiment is to indicate the difference with the duty-cycled and not duty-cycled case. As the reader can see in Figure 7, there are two cases: the one being with duty-cycled vehicles and the other without the duty-cycled mechanism. The interference in this example is initially set to 300 \(\mu\)W. In the duty-cycled case, the interference remains stable throughout the experiment, and the PRR goes from 0 to 0.9 and then to 1.0. On the other hand, in the not duty-cycled case, there is an increase in interference reaching 500 \(\mu\)W at the 800 \(\mu\)W transmission power utilised.