## 3 Omnidirectional and Directional Antennas Issues vs. Round-Robin MAC
### Round-Robin MAC Scheduling Issues
The number of sectors in the plane varies from 3 to 8, while the queue size is evaluated based on the following equation:  
\[Queue\,size=Q\times[N+w_{Q}(N_{s})]\qquad w_{Q}\in\overline{W_{Q}}. \tag{1}\]  
The queue size is a function of \(N\) that is the number of nodes in the network, and \(N_{s}\) denotes the number of sectors in the plane. Q is a value chosen in the interval \(0.1\leq Q\leq 0.25\) in order to maximize the packet delivery ratio performance of [24]. The number of sectors is weighted by the term \(w_{Q}\), which varies from 1 to 6 as the sector number value increases. In particular, given the sector number array \(\overline{S}=[3,4,5,6,7,8]\), the weight vector \(\overline{W_{Q}}\) is expressed by  
\[\overline{W_{Q}}=[1,2,3,4,5,6]. \tag{2}\]  
In order to verify the impact of the two simulation scenarios relating to the queue issues, we compare the waiting queue time of the two running configurations in function of the number of sectors and queue size. Figure 5 illustrates the average waiting queue time in function of the number of sectors. The waiting queue time is averaged by varying the \(Q\) parameter from 0.1 to 0.25. In Figure 5a, it can be highlighted that the waiting queue time in the case of non-uniform distributed nodes remains higher than the uniform case independently from the sector number value; in particular, the gap between the waiting queue time of the two considered cases seeks to grow for sector number values higher than 5. The curves of Figure 5b are plotted in function of the \(Q\) parameter, which represents a proper index of the queue size; as can be deduced from Equation (1), the higher the \(Q\) value is, the higher the queue size is. As shown in Figure 5a, the waiting queue time in the case of non-uniform  
Figure 4: Residual capacity progression. traffic lies above the uniform case curve independently from the queue size. However, as opposed to Figure 5a, the difference between curves has remained almost constant as the queue size increases. In the uniform traffic case, it can be noted how the waiting time slightly decreases for the highest values of \(Q\), while, in the non-uniform case, the decrease corresponding to the same values is closely negligible. In summary, as the size of the queue increases, the waiting time in the non-uniform case does not tend to decrease, or rather, the decreasing related to high queue size values is not significant. The trends evaluated in Figure 5 are certainly justified by one of the most common issues implied by a round-robin algorithm scheduling: the time slice problem [44; 45]; basically, the time slice represents the quantum assigned to each sector (the sector time) in equal portions; therefore, the communications are handled in a circular way among sectors without priority. In the case of non-uniform distributed traffic if most mobile nodes are focused in a specific sector of the plane, the communications related to those nodes are enqueued until the beam is allowed in the specific sector; as a consequence, the larger the quantum assigned to each sector is, the larger the waiting queue time is. In the same way, as observed in [24], if the quantum assigned to each sector is too low, the system will provide bad performance in terms of overall throughput; a similar behavior is produced if the quantum assigned to each sector is averaged in a specific interval as the number of sectors increases, which results in the highest waiting queue time values.