# Real-Time Station Grouping under Dynamic Traffic for IEEE 802.11 ah
## 5 Optimal Input Parameter Derivation
### Maximum Number of Packet Transmissions in One Beacon Interval  
Let \(S_{max}\) denote the throughput that can be achieved by assigning \(\sigma^{r}_{opt}\) stations in one RAW slot as shown in Algorithm 2, the maximum number of packet transmissions in one beacon interval \(\pi^{b}_{max}\) is then calculated as follows:  
\[\pi^{b}_{max}=\frac{S_{max}\times t^{b}}{L}, \tag{4}\]  
where \(L\) is the average packet payload size. When the channel is fully utilized (i.e., \(\pi^{b}_{max}\) packets scheduled to be transmitted in one beacon interval), the duration \(t^{r}_{avg}\) of a RAW slot \(r\) equaling the time needed on average to finish all packet transmissions assigned to \(r\), can be calculated as:  
\[t^{r}_{avg}=\frac{\pi^{r}\times t^{b}}{\pi^{b}_{max}}. \tag{5}\]  
However, as 802.11 ah employs EDCA/DCF inside RAW slots, these calculations are based on average back-off times, and only guarantee successful transmission up to a certain probability. The transmission success probability can be evaluated by the model proposed by [PERSON] et al. [23], which assumes each station only has one packet to transmit during their assigned RAW slot. Given the sensor use case considered in this article, the assumption that each station will at most transmit one packet per beacon interval (generally set to 100 ms) can generally be assumed to hold. Based on this model, Figure 5 depicts how the transmission success probability changes over time using a data rate of 0.6 Mbps with packet sizes 64 and 256 bytes, respectively. For a packet size of 64 bytes, \(\sigma^{r}_{opt}\) is 5 and \(t^{r}_{avg}\) is 16.53 ms, while \(\sigma^{r}_{opt}\) is 3 and \(t^{r}_{avg}\) is 17.59 ms for a packet size of 256 bytes. The figure clearly shows that the transmission success probability is only 50% and 82% respectively when \(t^{r}_{avg}\) is used as RAW slot duration.  
The solution is to increase \(t^{r}_{avg}\) to get a higher transmission success probability. However, \(\pi^{b}_{max}\) will also become lower and more channel time will be wasted, which, in turn, degrades performance. However, the model of [PERSON] et al. [23] considers cross slot boundary transmission as forbidden. As the cross slot boundary feature of 802.11 ah does allow transmissions to continue after the current RAW slot ends, and our algorithm only estimates traffic intensity based on transmission success at the  
Figure 5: Transmission success probability as a function of time for \(\sigma^{r}_{opt}=5,3\), which are used for data rate 0.6 Mbps with packet size 64 and 256 bytes, respectively.  
end of the beacon interval (and not at the end of the slot), the actual transmission success probability when using the optimally calculated \(\pi_{max}^{b}\) is much higher in reality than depicted in Figure 5. As such, we propose the use of the cross slot boundary feature in combination with the slot duration calculated in Equation (5) when the channel is fully utilized. Moreover, we assume all slots are RAW capable, and therefore the entire channel time is occupied by RAW. When the channel is not fully utilized, \(\pi^{b}<\pi_{max}^{b}\) packet transmissions are allowed in one beacon. The RAW slot \(r\) then has the following duration:  
\[t^{r}=\frac{\pi^{r}\times t^{b}}{\pi^{b}}>t_{\text{avg}}^{r}. \tag{6}\]  
As such, the transmission success probability is improved. Therefore, instead of looking for an optimal \(\pi_{max}^{b}\) and \(t^{r}\) that can balance transmission success probability and wasted channel time, we simply obtain them with Equations (4) and (5) by taking advantage of the cross-slot boundary feature and allowing the entire channel time to be occupied by RAW slots.