## 5 Performance Evaluation
### Simulation Results  
Figure 3 compares the different multicast transmission schemes in terms of the average bandwidth by varying \(|\Omega_{B}|\) when \(|\Omega|=6\). In the conventional multicast transmission scheme operating in a single cell, the required bandwidth, \(B_{c}\), is constant and given as  
\[B_{c}=\frac{R_{T}}{log_{2}(1+\gamma_{o})} \tag{10}\]  
where \(\gamma_{o}\) is a threshold satisfying \(\Pr\bigl{(}\gamma_{j}<\gamma_{o}\bigr{)}=0.05\). The single-cell multicast scheme with only a single multicast channel is denoted as BMCH-only and its bandwidth can be determined as \(\frac{R_{T}}{log_{2}\bigl{(}1+\gamma(\Omega_{B})\bigr{)}}\). The scheme proposed in Reference [8] is denoted as Multi + Unis. Since the required bandwidth varies according to channel quality, its average value is used as a performance measure except for the conventional multicast transmission scheme. As expected, the average bandwidth remains constant regardless of \(|\Omega_{B}|\) for BMCH-only and Multi + Unis since only a single multicast channel is used. In the scheme in Reference [14], with \(|\Omega_{B}|=0\) or \(6\), only a single multicast channel exists and hence, its performance is the same as that of BMCH only. And the minimum average bandwidth can be obtained at \(|\Omega_{B}|=3\). The proposed scheme, however, can obtain the minimum average bandwidth with \(|\Omega_{B}|=0\). As seen from (7) and Figure 2, the optimal bandwidth is sought in the range \(B_{B}\geq B_{|\Omega_{B}|}\) and therefore, as \(|\Omega_{B}|\) increases, the search range shrinks and the required bandwidth increases. Note that the users in \(\Omega_{B}\) need to send channel quality feedback for the BMCH, while feedback from the users in \(\Omega\backslash\Omega_{B}\) is required for both the BMCH and the SMCH. In addition, the algorithm works with more \(f_{j}(B_{B})\)'s with a smaller \(|\Omega_{B}|\). Thus, there is a compromise between the performance and feedback overhead/computational complexity; as \(|\Omega_{B}|\) increases, the feedback overhead and computational complexity can be reduced at the cost of performance degradation. The minimum average bandwidth can be reduced by approximately \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Path-loss constant & \(-24\) dB \\ Path-loss exponent & \(3.4\) \\ Fading & Rayleigh \\ Neighboring BS density, \(\lambda\) & \(3.54\) [BSs/km\({}^{2}\)] \\ Cell radius, \(R\) & \(300\) m \\ Minimum distance to the neighboring BS, \(D_{min}\) & \(600\) m \\ Transmission power spectral density, \(p_{0}\) & \(43\) dBm/10 MHz \\ Noise spectral density, \(\sigma_{0}^{2}\) & \(-174\) dBm/Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. 20% and 40% by applying the proposed multicast transmission scheme instead of the scheme in Reference [14] and BMCH-only, respectively. To better understand the operation of the proposed multicast transmission scheme, snapshots of the bandwidths, \(B_{B}\), \(B_{S}\) and \(B_{B}+B_{S}\) of the proposed scheme as well as that of BMCH-only, are shown in Figure 4 when \(|\Omega_{B}|=0\) and \(|\Omega|=6\). It is interesting to note that at sample indexes 1, 12, 15 and 18, only BMCH is allocated, while at sample indexes 2, 3, 7 and 20, only SMCH is allocated. In the other sample indexes, both the BMCH and the SMCH are allocated. In order to explain the operation of the proposed scheme in more detail, the channel quality of the users at sample indexes 1, 11 and 20 are given in Table 2.  
Figure 4: Snapshots of the required bandwidth (\(|\Omega|=6,\ |\Omega_{B}|=0\))  
Figure 3: Average bandwidth versus \(|\Omega_{B}|\) (\(|\Omega|=6\)). First, see sample index 20, because \(|\Omega_{B}|=0\), \(B_{B}^{(0)}=0\). Then, since user 4 has the worst channel quality on the SMCH, \(j^{(0)}=4\). Note that user 4 also experiences worse channel quality on the BMCH, that is, \(\alpha_{j^{(0)}}>0\). Therefore, the proposed algorithm stops and only the SMCH is allocated.