## 6 Resource Reservation Schemes: Static vs. Dynamic  
The considered system consists of a set of wireless clusters, as illustrated in Figure 4. The Call Arrival Time (CAT) follows a Poisson distribution, while the \(CHT\) is exponentially distributed. In order to evaluate the number of cells \(C_{e}\) visited by the mobile host during its \(CHT\), the \(CST\) of mobile hosts has been obtained by a preliminary set of simulations: it follows a Gaussian distribution under the CityMob generator, with different values of maximum acceleration and maximum speed [29; 31]. \(C_{e}\) can be evaluated as in [31], but without any information about user mobility pattern (directional preferences), \(C_{e}\) can be used only to make circular reservations (Figure 4), around the active cell. In this way, following the same approach of [48; 49], the number of required passive reservations \(C_{r}\) for NSIS-MIP services increases with polynomial trend, as indicated in Equation (12). Table 1 shows the numerical values for Equation (12) and for the directional reservation policy P, shown later:  
\[C_{r}=3\times C_{e}\times(C_{e}-1). \tag{12}\]For example, Figure 4 shows how for \(C_{e}=2\), 3 or 4 a higher number of \(C_{r}\) passive reservations must be made, through a fixed circular cluster of \(C_{r}=6\), 18 or 36 cells with a radius of \(C_{e}\) cells (including the active one). This introduces huge resource wasstages, due to the enormous amount of passive pre-reserved bandwidth over \(C_{r}\) cells, which increases for longer calls or for higher values of \(v_{max}\), for fixed values of \(CHT\) and \(v_{max}\), respectively. If additional information about the mobility directional behavior is available, the issues above can be avoided, with a lower \(C_{r}\) (near or equal to \(C_{e}\)). This last opportunity is provided by our proposal, and the obtained values of \(C_{r}\) depend on the adopted reservation thresholds and policies. A generic cell can be well approached by an \(n\)-edge regular polygon as depicted in Figure 4 (\(n\) can be considered as an input control parameter and it is commonly equal to 6). For higher values of \(n\), better approximations can be reached. A set \(S_{ho}\) (hand-off directions set) of \(n\) possible movement directions (i.e., hand-off directions) can be obtained: \(S_{ho}=\{d_{1}\...\ d_{n}\}\), with \(d_{j}=\theta(2j-1)/2\) rad., \(\theta=2\pi/n\) rad., \(j=1\...\ n\) and \(\mid S_{ho}\mid=n\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mobility Parameters**} & \multicolumn{2}{c}{**P(1-1-1)**} & \multicolumn{2}{c}{**P(1-2-3)**} & \multicolumn{2}{c}{**P(3-3-3)**} \\  & **Cr(P)** & **Cr** & **Cr(P)** & **Cr** & **Cr(P)** & **Cr(P)** & **Cr** \\ \hline \(a_{\max}=1.4\ m/s^{2}\), \(a_{\min}=-2\ m/s^{2}\), \(\tau=0.3\) & & & & & & \\ \(\mu=29.26\), \(s\), \(\sigma=0.45729\) s & 5 & 60 & 22 & 60 & 52 & 60 \\ \(a_{\max}=1\ m/s^{2}\), \(a_{\min}=-2.5\ m/s^{2}\), \(\tau=0.4\) & & & & & & \\ \(\mu=62.22\ s\), \(\sigma=5.09547\) s & 2 & 6 & 3 & 6 & 4 & 6 \\ \(a_{\max}=1.3\ m/s^{2}\), \(a_{\min}=-2\ m/s^{2}\), \(\tau=0.2\) & & & & & & \\ \(\mu=32.46\ s\), \(\sigma=8.18356\) s & 4 & 36 & 16 & 36 & 21 & 36 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of cells involved in the bandwidth reservation phase, for different policies. Figure 4: Simulated topology with circular/directional reservation policies (hexagonal coverage). The conditional probability that a mobile host will hand-out to direction \(y\in S_{ho}\) after \(CST\) amount of time (Gaussian distributed), having handed-in the cell from direction \(x\in S_{ho}\) can be defined as:  
\[M(x,y)=p_{x,y}=p_{CNSIS-MIP}(x,y)=p(\text{outtoy}\in S_{ho}\ t=t_{0}+CST/\text{ infnorm}x\in S_{ho}\ t=t_{0}), \tag{13}\]  
where \(t_{0}\) is the time at which the user hands-in the considered cell. Once \(n\) and \(S_{ho}\) have been defined, a square \(nxn\) HDP matrix \(M\) is defined as: \(M(x,y)=p_{x,y}=p_{CNSIS-MIP}(x,y)\). To be noticed that \(CST\sim N\) (\(\mu_{CST},\sigma^{2}_{CST}\)).