## 5 Discussions and Implications  
We summarize our results in Table 1. We also compare our results with omni-directional cases [12]. Note that we assume \(R_{i}=1\) is fixed and \(R_{t}\) is adjustable. The coverage angle \(\beta\) is given in Equation (1), which decreases with the increased interference ratio \(r\). When \(r=2\), the angle is denoted by \(\beta_{1}\). When \(r=4\), the angle is denoted by \(\beta_{2}\). Thus, we have \(\beta_{2}<\beta_{1}\).  
When interference ratio \(r=1\), the upper bound for the network with directional antennas is \(2D-1\) and the upper bound for the network with omni-directional antennas is \(2D_{o}-3\). Different from the node density \(D\) with directional antennas, \(D_{o}\) is defined as the maximum number of nodes within a circle of interference range. Generally, we have \(D\
eq D_{o}\).  
When \(r=2\), the upper bound on the number of channels is \(\frac{3}{2}D\) for directional antennas and \(\frac{3}{2}D_{o}\) for omni-directional antennas. Similarly, when \(r=4\), the upper bound is reduced to \(D\) for directional antennas and \(D_{o}\) for omni-directional antennas. From those results, we have found that the number of channels needed for a collision-free transmission scales linearly with the node density \(D\), and is non-increasing as the interference ratio \(r\) increases.  
When the interference ratio \(r\) approximates the infinity, then \(R_{t}\) approximates 0. This means all links have length 0. So, any link will conflict with at most \(\frac{1}{2}D-1\) links in the directional case and \(\frac{1}{2}D_{o}-1\) links in the omni-directional case.  
From Table 1, we also observe that upper bounds derived from the omni-directional antennas have almost the same coefficients as those derived from the directional antennas except for the case when \(r=1\) (although they have different node densities, _i.e._, \(D\) and \(D_{o}\)). Both the upper bounds derived from the omni-directional case and those derived from the directional case heavily depend on the node density and the interference ratio. An interesting question is whether the upper bounds are independent of the actual radiation patterns of antennas.  
Our derived theoretical results can be applied to solve many practical problems. For example, given a wireless network with a number of wireless nodes, our derived bounds can be used to estimate the number of channels required to ensure a collision-free communication. For another example, when the number of channels is given (e.g., there are 14 channels but only three orthogonal channels available in IEEE 802.11), our results can be used to offer suggestions on the node density in the node deployment or suggestions on the channel assignment for a given network.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline  & _OMN_ Networks [12] & _DIR_ Networks \\ \hline \(r\) & Upper bounds & Lower constraints & Upper bounds & Lower constraints \\ \hline
1 & \(2D_{o}-3\) & \(D_{o}-1\) & \(2D-1\) & \(D-1\) \\ \hline
2 & \(\frac{3}{2}D_{o}\) & \(\frac{2}{3}(D_{o}-1)\) & \(\frac{3}{2}D\) & \(\lfloor\frac{\beta_{1}}{\theta_{m}}\frac{D}{2}\rfloor+1\) \\ \hline
4 & \(D_{o}\) & \(\frac{1}{2}D_{o}\) & \(D\) & \(\lfloor\frac{\beta_{2}}{\theta_{m}}\frac{D}{2}\rfloor+1\) \\ \hline \(\infty\) & \(\frac{1}{2}D_{o}\) & \(\frac{1}{2}D_{o}\) & \(\frac{1}{2}D\) & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparisons of _OMN_ networks and _DIR_ networks.