# Conservative Interference Injection to Minimize Wi-Fi Sensing Privacy Risks and Bandwidth Loss
## 5 Proposed Contention Model for Defence Against Wi-Fi Sensing
### Throughput of CSMA/CA WLANs  
The CSMA/CA process described above has been represented using a Markovian model in a prominent work by [PERSON] around the time of its advent [9]. Subsequent work [26] has built upon [PERSON]'s model to express the collision probabilities of 802.11 stations. This can be boiled down to two important expressions:  
\[p=1-(1-\pi)^{N-1} \tag{4}\]  
\[\pi=\frac{2(1-2p)}{(1-2p)(W+1)+pW(1-(2p)^{m})} \tag{5}\]  
where \(\pi\) is the probability that a station will transmit during a slot, and then \(p\) is the probability that there is a collision in that slot. The probability \(\pi\) is considered to be equal for all stations, operating under the assumption that they are all attempting to perpetually transmit packets. This is formally referred to as the saturation condition by [PERSON] [9]. These two expressions can be solved simultaneously for \(\pi\) and \(p\) as a function of the number of stations (\(N\)). We use the following set of parameters for 802.11n networks:Subsequently, the parameters in Table 2 can be used to solve Equations (4) and (5) simultaneously. We solve for \(N=[2,25]\) devices and illustrate the collision probabilities below. In Figure 7, we illustrate the collision probability (\(p\)) and the transmission probability (\(\pi\)) as functions of the number of competing stations in 802.11n networks. The x-axis represents the number of 802.11n devices in the network, and the y-axis represents probabilities ranging from 0 to 1. Then, the blue curve represents the probability of a collision occurring in the channel and the red curve represents the probability that any station will choose to transmit at a given time. We observe that the probability of collisions within the channel grows logarithmically as more users enter the channel. This growth is particularly aggressive when the patronage increases from 2 to 5, with the probability of collision increasing from 0.11 to 0.28 (254% increase). We are subsequently interested in the effect this increased collision probability has on the overall throughput for all devices in the channel. Building on Equations (4) and (5), it has been further shown in prior work [26] that the throughput of a CSMA/CA network can be expressed as follows:  
\[\tau=\frac{E[\text{Payload Transmitted in a slot}]}{E[\text{Duration of Slot Time}]}\]  
\[\tau=\frac{P_{\text{s}}P_{\text{tr}}L}{P_{\text{s}}P_{\text{tr}}T_{\text{s }}+P_{\text{tr}}(1-P_{\text{s}})T_{\text{c}}+(1-P_{\text{tr}})T_{\text{id}}} \tag{6}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Slot Time & 20 \(\upmu\)s \\ DIFS & 50 \(\upmu\)s \\ Short Inter-Frame Space (SIFS) & 10 \(\upmu\)s \\ Acknowledgement (ACK) Time & 304 \(\upmu\)s \\ \(T_{\text{Header}}\) & 400 \(\upmu\)s \\ \(T_{\text{Payload}}\) & 3.4 ms \\ Minimum Contention Window (\(W\)) & 16 slots \\ \hline \hline \end{tabular}
\end{table} 
Table 2: 802.11n CSMA/CA parameters. Figure 7: Collision probability of 802.11n networks with increasing users. for a packet of length \(L\) and a slot time \(T_{\rm id}\). Here, the probability of transmission (\(P_{\rm tr}\)) and probability of that transmission being successful (\(P_{\rm s}\)) are further defined as follows:  
\[P_{\rm tr}=(1-(1-\pi)^{N}),P_{s}=\frac{N\pi(1-\pi)^{N-1}}{1-(1-\pi)^{N}} \tag{7}\]  
and the average transmission time (\(T_{\rm s}\)) and average time spent in collision (\(T_{\rm c}\)) are expressed as follows:  
\[T_{\rm s}=T_{\rm Header}+T_{\rm Payload}+\rm{SIFS}+\sigma+\rm{ACK}+\rm{DIFS}+\sigma \tag{8}\]  
\[T_{c}=T_{\rm Header}+T_{\rm Payload}+\rm{DIFS}+\sigma \tag{9}\]  
where \(\sigma\) is the propagation delay. We solve Equation (6) for \(N=[1,10]\) devices, and vary the length of the payload between 700 bits (roughly the size of an ICMP ping packet) to 1900 bits. In Figure 8, we illustrate the analytical throughput for 802.11n networks. The x-axis represents the number of Wi-Fi stations in proximity, and the y-axis represents the throughput (bits/sec) of a station.