# PROFEE: A Probabilistic-Feedback Based Speed Rate Adaption for IEEE 802.11 bc
## 3 Estimation of the Number of STAs Receiving Broadcast Messages Correctly/Incorrectly  
This section presents an algorithm that allows for the AP to estimate the number of STAs that are receiving broadcast messages correctly (\(n_{ACK}\)) or incorrectly (\(n_{NACK}\)). The solution uses probabilistic ACKs/NACKs as feedback from STAs. When an STA correctly decodes the preamble and the corresponding message payload, it sends an ACK to the AP with probability \(p_{ACK}\). Similarly, when an STA correctly decodes the preamble of a message but incorrectly decodes the rest of the payload, it sends a NACK to the AP with probability \(p_{NACK}\). This scheme is aided by the preambles being always sent using BPSK with a \(1/2\) coding rate [12] (the lowest MCS) to maximize the probability of preamble reception. STAs know precisely which values of \(p_{NACK}\) and \(p_{ACK}\) to use because the AP includes those probabilities within each message it transmits. In this algorithm, the AP performs an active role in selecting the best values of \(p_{NACK}\) and \(p_{ACK}\) that will result in the best estimation of \(n_{ACK}\) and \(n_{NACK}\). On the other hand, STAs play a passive role as they simply react to sending feedback messages according to the probabilities included in each message. To avoid ACK and NACK colliding in the AP after each transmitted message, NACKs are only sent in response to the odd-sequenced AP messages, and ACKs are sent only in response to even-sequenced AP messages. This work assumes no previous association of STAs with the AP, as this is one requirement of the new 802.11 bc standard. This condition makes the problem harder, as the AP does not know the total number of listening STAs and their distribution over the network area. However, in those scenarios in which the number of STAs is known in advance, it will simplify the protocol's operation as it is only necessary to estimate either \(n_{ACK}\) or \(n_{NACK}\) to calculate the other. Figure 1 illustrates the operation of the proposed algorithm with three STAs, where transmitted NACKs and ACKs are represented as red and blue rectangles, respectively. After sending message \(M_{1}\), \(STA_{1}\) and \(STA_{3}\) received it incorrectly, while \(STA_{2}\) received it correctly. \(STA_{1}\) and \(STA_{3}\) will transmit a NACK with probability \(p_{NACK}\). In this example, \begin{table}
\begin{tabular}{l l l l l l l l l l l l l l l l l l} \hline \hline  & [2] & [3] & [6] & [4] & [5] & [7] & [8] & [9] & [15] & [16] & [20] & [18] & [13] & [14] & PROFEE \\ \hline Avoid collisions of fb & X & X & X & X & X & X & X & X & X & X & X & X & X & X & X & - \\ Decrease number of fb messages & X & X & - & - & - & - & - & - & - & - & - & X & X & X & X \\ Make the fb collision resistant & - & - & - & X & X & - & - & - & - & - & - & - & - & - & - \\ Decrease the size of the fb & - & - & X & X & - & - & - & - & - & - & - & - & - & - & - \\ Turns the broadcast into unicast & - & - & - & - & - & X & X & X & - & - & - & - & - & - & - \\ Indirect fb from unicast packets & - & - & - & - & - & - & - & - & - & X & X & X & - & - & - & - \\ Based on clusters & X & X & - & - & - & - & - & - & - & - & - & X & - & - & - \\ Consider positive fb & - & - & X & - & X & X & X & X & X & X & X & X & - & - & X & X \\ Consider negative fb & X & X & - & X & X & - & - & - & - & - & - & - & - & - & - & X \\ Considers silence fb & - & - & - & - & - & - & - & - & - & - & - & - & - & X & \\ Seeks 100\% receptions & X & - & X & X & X & X & X & - & - & - & - & - & - & - & - & - \\ Random-delayed fb & X & X & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\ Time slotted pre-assigned fb & - & - & X & - & - & - & X & X & - & - & - & - & - & - & - & - \\ Use RTS/CTS & X & - & - & X & X & X & X & - & X & X & - & - & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the state-of-the-art algorithms (fb: feedback). only \(STA_{1}\) sends the NACK, which is received in the AP.