# Pilot Sequence Allocation Schemes in Massive MIMO Systems Using Heuristic Approaches
## 4 Proposed Solutions
### Variable Neighbourhood Search
In relation to the average maximum spectral efficiency, the results are presented in Table 6. It is clear that RA, BPSO and GA algorithms have a worse performance than the other ones. In fact, the average maximum spectral efficiency more than doubles when the same cell radius is compared. Figure 8: Empirical CDF for different cell sizes with \(K=20\) users, \(G_{\text{max}}=50\) generations and \(I=50\) iterations. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{3}{*}{**Parameters**} & \multicolumn{5}{c}{**Minimum Spectral Efficiency per User (in bps/Hz)**} \\ \cline{2-6}  & **BPSO/GA** & \multicolumn{2}{c}{**SPV–PSO**} & \multicolumn{2}{c}{**SPVG–PSO**} \\ \cline{3-6}  & **w/o VNS** & **with VNS** & **w/o VNS** & **with VNS** \\ \hline \(K=20,R=1\) Km & 7.56 \(\times 10^{-5}\) & 1.4143 & 1.3762 & 1.3897 & 1.3191 \\ \(K=20,R=500\) m & 7.85 \(\times 10^{-5}\) & 1.4233 & **1.402** & **1.4093** & 1.3561 \\ \(K=20,R=250\) m & 7.90 \(\times 10^{-5}\) & 1.4142 & 1.3858 & 1.3782 & **1.3873** \\ \(K=20,R=100\) m & **8.03**\(\times 10^{-5}\) & **1.4414** & 1.3605 & 1.3986 & 1.3424 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Minimum spectral efficiency per user in different cell sizes. The best value for each algorithm is marked in bold type. Finally, in terms of cell size, it is clear that for all the results presented here, the impact in terms of CDF, average maximum and minimum spectral efficiency is marginal which corroborates the argument that the system is not interference bounded. ### System Loading Impact on Performance

Figure 9 describes the empirical CDFs for different system loading scenarios: \(K=20\), \(K=40\) and \(K=60\). These values represent three system loading scenarios which represent low, medium and high user density. The main objective here is to determine whether the higher density of users in the system implies more interference among them. It is evident that RA, BPSOand GA suffer an impact in performance as the system loading increases for the same cell size. However, that significant difference in performance is not observed for the other algorithms, meaning that they can handle user density better than the former three algorithms. Numerically, there is an 11 dB CDF shift from \(K=20\) users to \(K=40\) and a 7 dB shift from \(K=40\) to \(K=60\). Meanwhile, the performance difference between \(K=20\) and \(K=60\) is less than a 0.1 dB shift. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{4}{c}{**Maximum Spectral Efficiency per User (in bps/Hz)**} \\ \cline{2-6}  & \multirow{2}{*}{**BPSO/GA**} & \multicolumn{2}{c}{**SPV–PSO**} & \multicolumn{2}{c}{**SPVG–PSO**} \\ \cline{3-6}  & & **w/o VNS** & **with VNS** & **w/o VNS** & **with VNS** \\ \hline \(K=20,R=1\) Km & \(13.4015\) & \(29.3362\) & \(29.3362\) & \(29.3378\) & **29.3515** \\ \(K=20,R=500\) m & \(12.9367\) & \(28.8354\) & \(28.9150\) & \(28.8825\) & **28.9445** \\ \(K=20,R=250\) m & \(12.7704\) & \(28.7173\) & \(\mathbf{28.8393}\) & \(28.7896\) & \(28.7380\) \\ \(K=20,R=100\) m & \(13.1714\) & \(29.0950\) & \(29.0974\) & \(29.1499\) & **29.2260** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Maximum spectral efficiency per user in different cell sizes. The best value for each algorithm is are marked in bold type. Figure 9: Empirical CDF for \(K=20\), \(K=40\) and \(K=60\) users in an \(R=1000\) m cell size scenario with the maximum number for iterations set to 50 for all algorithms. ## 6 Conclusions

This work presented six different heuristics to solve the pilot sequence allocation problem in Massive MIMO systems. We optimized the heuristics parameters to achieve the best performance in the discussed problem while still considering the computational time constraints inherent to the problem's nature. We analyzed the impact of different scenario configurations on the performances of the algorithms.