# Game-Theoretic Power and Rate Control in IEEE 802.11\(p\) Wireless Networks
## 4 Numerical Results
### Practical Considerations  
We know from practical scenarios that the PRR can be calculated at the link layer when a message is successfully received at the receiver, which then sends an acknowledgment to the sender. Thus, the sender has knowledge of whether the message has been sent, and it does not re-transmit it. The MAC layer is responsible for the re-transmission of the packet or the acknowledgment. In this paper, we push the PRR down the wireless stack, and we measure it at the physical layer. We aim to eliminate acknowledgments on both sides. Instead, the receiver will send a message of non-ack to the sender to inform it about a unsuccessful transmission. We do this to reduce messages in the network, since our objective is to optimise throughput via link quality. Certain works in the literature suggest metrics of link estimators on the PHY layer [22; 23; 24]. In summary these works employ the packet decoding process to compute the link estimator. They take into account the packet drop in the link estimation algorithms even though they analyse packets at the bit level as well. Our technique attempts to reduce messages, while link estimation is not compromised. The IEEE 802.11\(p\) PHY layer employs the OFDM method to perform encoding and decoding processes. It performs signal processing at the bit level and it separates decoding into specific stages [25]. * _StartReceive_: This essentially is the reception of the first packet bit, whose signal strength must exceed the reception threshold for successful reception. * _EndPreamble_: This event check is the preamble reception process. It includes specific conditions for the preamble's reception to be initiated. The state of reception should not be busy; also, the signal strength of the incoming signal should have an SINR larger than the reception threshold to schedule the _EndHeader_ decoding event. * _EndHeader_: This decoding event should be aware of the modulation, coding length and parity check of the incoming data. The header information needs to be justifiable to initiate the _EndRx_ decoding event. Figure 4: Messages Received with Duty Cyclin ©2022 IEEE. Reprinted, with permission, from [PERSON], & [PERSON] (2021, June). On the Potential Game of Packet Reception Ratio Utility Functions in Wireless Sensor Networks. In 2021 29 th Mediterranean Conference on Control and Automation (MED) (pp. 880–885). IEEE. * _EndRx_: During this event, data are decoded by applying signal demodulation and error correction. Moreover, channel estimation is performed to the OFDM samples, and when data are decoded successfully, the _RxPhyOk_ event takes place. Errors in the aforementioned stages that result in packet drop include the errors of _StartReceive_, _EndPreamble_, and _EndHeader_ due to channel interference as well as for decoding reasons. When a packet is successfully received on the PHY layer, the _RxPhyOk_ occurs [22]. As such, we transform the PRR definition to the PHY layer for a time window of \(k\) packets as follows:  
\[PRR_{i,j}^{\textit{Phy}}=\frac{\sum\limits_{s=1}^{k}RxPhyOk_{i,j}}{SN_{i,j}}, \tag{17}\]  
where \(RxPhyOk_{i,j}\) is a packet successfully received by the receiver \(j\) from sender \(i\), and \(SN_{i,j}\) is the packet sequence number coming from the sender in the header. Note that PRR can be given by two formulas, the ones being in Equations (1) and (17), and the other  
\[PRR_{i,j}=\frac{P_{i,j}^{\textit{recv}}}{P_{i,j}^{\textit{ sent}}} \tag{18}\]  
over a window of \(W\) transmissions, where \(P_{i,j}^{\textit{recv}}\) is the number of packets received by node \(j\) from node \(i\) and \(P_{i,j}^{\textit{sent}}\) is the overall numbers sent from node \(i\) to node \(j\). For practical considerations, we utilize (17) and (18) to estimate the utility function. The packets that are received can be calculated in two ways. For the first, we can use the PxPhyOK and for the latter, we can use the Bit-Error Rate (BER).