# Resource-Efficient Multicast URLLC Service in 5G Systems
## 4 Performance Evaluation
### Analysis of Precoder Selection Algorithms  
Let us start with the performance comparison of precoder selection algorithms. For this, we sampled the values of the channel matrix \(H_{i,j}\) in all RBs for 20 UEs using the NS-3 channel model. For each matrix, we construct the precoder with algorithms from Section 3.4. The X-axis in Figures 2 and 3 corresponds to the average difference between the minimal SINR \(SINR_{j}\) obtained with the considered algorithm and the upper bound obtained with the SDR approach. The Y-axis is the mean time needed to construct a single precoder with a 3.3 GHz Intel Core i3-2120 processor [39]. Note that the gNB can implement two approaches to compute the precoder: (i) offline, in which the precoder is calculated in advance for each multicast group when the gNB receives the corresponding SRSs, and (ii) online, in which the precoder is calculated each time the gNB schedules transmission to a particular multicast group. For the offline approach, the precoder computation time should be much lower than \(T_{SRR}\), while for the online approach, the precoder computation time should be much lower than \(D^{OoS}\). Since in our experiments, \(T_{SRR}\) = 20 ms and \(D^{OoS}\) = 10 ms, we consider 10 ms as the reference value which limits the precoder computation time. If the precoder construction algorithm has a computational time greater than 10 ms, we consider it unacceptable for mURLLC applications. Figure 2 shows the results for the M-MIMO case (i.e., 64 antennas at the gNB) for both LOS and NLOS channel models. Let us start with an analysis of the SCA-ADMM algorithm. For this algorithm [35], we check different convergence thresholds \(\varepsilon\). We can see that the SINR difference for the SCA-ADMM algorithm reduces for a stricter convergence threshold \(\varepsilon\). However, even for the highest convergence threshold \(\varepsilon=10^{-1}\), the corresponding precoder construction time is \(\approx\) 0.1 s, which is not acceptable for mURLLC applications. The FF-C2 and RC-C2 algorithms [33], which search a multicast precoder over UE pairs, provide reasonable construction time \(1-10\) ms. However, as each precoder in a search space takes into account only two considered UEs, their SINR is much lower than the SDR upper bound. Multicast adaptations of linear beamforming algorithms from [31], such as mMRT, mZF, and mMMSE, provide the lowest precoder construction time. However, they provide the worst SINRs among considered solutions. The IU and MU algorithms, which \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Carrier frequency & 3.6 GHz \\ Bandwidth & 100 MHz \\ TDD structure & \(k_{dl}=8\), \(k_{ul}=2\), \(T_{slot}=0.5\) ms \\ Number of RBs & 256 \\ Channel model & 3 GPP TR 38.901, Urban Macro \\ gNB/UE antennas & 64 or 4 \\ UE antennas & 1 \\ gNB/UE TX power & 33/23 dBm \\ Traffic & CBR: 400 bytes, 1 ms period \\ QoS requirements & \(PLR^{QoS}=10^{-5}\), \(D^{OoS}=10\) ms \\ Simulation time & 1000 s, 40 runs \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main simulation parameters. iteratively construct precoders using simple linear algebra operations, show much better performance. As mentioned in Section 3.4, the MU algorithm optimizes the proportional fair objective function from Problem (5) instead of the max-min SINR objective function from Problem (2). Thus, the IU algorithm provides better performance than the MU algorithm in terms of minimal SINR over RBs. Finally, the SBFC and QR algorithms, which construct the precoder as a linear combination of UE channels, provide the SINR closest to the upper bound with reasonable construction time. Let us analyze how the number of antennas influences precoder construction algorithm performance and complexity. For that, in Figure 3, we consider the case of four antennas at the gNB, corresponding to 4G systems.