## 4 The Proposed Model  
### Comparison with RR-MAC and Motivations  
In order to explain the relevance of using a round-robin based scheduling algorithm for directional MAC protocols let us consider the beamforming scenario shown in Figure 2a in which the round-robin approach is applied together with the plane sectorization of Figure 3. This process can be modeled as a Markov chain (Figure 6) defined by the following parameters:  
\(\bullet\)**state:** the set of the directions associated with a sector (in other terms, a state is a sector);  
\(\bullet\)**transition:** the switching process from a sector to another sector. Figure 5: Queue issues with round-robin scheduling. (**a**) Average waiting queue time vs. number of sectors. (**b**) Average waiting queue time vs. queue size. The transitions can be expressed as  
\[b_{i,k}=\frac{T_{i}-k}{T_{i}}b_{i,0},\quad 0\leq i\leq N_{s},\quad k\in[0,T_{i}]. \tag{3}\]  
In Equation (3), \(b_{i,0}\) represents the transition from the state \(i\) to the state \(0\); \(T_{i}\) is the duration of the sector \(i\) (_sector time_), while \(N_{s}\) is the maximum number of sectors. Figure 3 illustrates the Markov model applied to our case study. Assuming a beamfoming context, considering a period \(T\) with \(0\leq T\leq T_{i}\), we make the following definitions:  
* \(\mathbf{P}_{sec}=\text{prob. }\) that a node \(x\) attempts for a transmission in a randomly chosen sector. \(P_{sec}=2/\left(T_{i}+1\right)\)
* \(\mathbf{P}_{tr}=\text{prob. }\) that there is at least a transmission in a certain sector. \(P_{tr}=1-\left(1-P_{sec}\right)^{N_{s}}\)
* \(\mathbf{P}_{nb}=\text{prob. }\) that at least one of the neighbors of \(x\) attempts for a transmission. \(P_{nb}=\left(N_{s}-1\right)\left(1-P_{tr}\right)\)
* \(\mathbf{P}_{B}=\text{prob. }\) that \(x\) is beamformed from at least one of its neighbors. \(P_{B}=\left(1/N_{s}\right)+w\left(T_{i}\right)\)  
The term \(P_{B}\) depends on the number of sectors and on a weight assigned to each sector \(w\); in the case of the RR-MAC, for example, the weight is the same for each sector. We can express the probability that the node \(x\) experiences a collision in the considered period as follows:  
\[P_{c}=P_{B}(T_{i})P_{tr}(N_{s}-1)P_{nb}(1-P_{nb})^{N_{b}-1}. \tag{4}\]  
Note that \(P_{B}\) is a function of the _sector time_ that also represents the beamforming time. As is clear from Equation (4), the larger the beamforming time is, the higher the number of frame collisions becomes. However, limiting the beamforming time of nodes is not a sufficient adjustment if the objective is to design an energy-efficient directional MAC approach. As explained in Section 3, the main challenges related to the RR-MAC approach are referred to the size of the queues and to the delay produced by static time slices that cannot be modeled in function of the traffic in the network. However, as observed in [23, 24], the quantum of time assigned for the beamforming process is the same for each sector as well as the amplitude of the width. Therefore, in the work [24], a proper set of the sector time value affects the overall performance more than the sector width choice; this implies that the time slot assigned to sectors needs to be carefully assigned in order to improve system efficiency. For these reasons, we propose an approach that modifies the original round-robin algorithm formulation relative to the evaluation of the sector time, while keeping unchanged the sectorization of the plane and thus the width of the sectors. In order to understand the modifications introduced with respect to the round-robin MAC algorithm, we briefly recall the mathematical formulation of this latter:  
\[ \begin{split}\alpha_{i}=\alpha_{j}=\ldots=\alpha\\ T_{i}=T_{j}=\ldots=T_{s}\\ \alpha_{i}(T_{i})=\alpha_{j}(T_{j})=\ldots=\alpha(T)\\ \forall i,j=1,2,\ldots N_{s},\quad i\
eq j.\end{split}  \tag{5}\]  
From the formulation given by Equation (5), it is easy to observe that the width is equal for all sectors as well as the sector time.