# Pilot Sequence Allocation Schemes in Massive MIMO Systems Using Heuristic Approaches
## 4 Proposed Solutions
### Variable Neighbourhood Search
Even though the performance differences of the tested parameters values are marginal, the best values were used to evaluate other aspects of the proposed solutions. The second parameter tested in the PSO-based algorithms was the maximum velocity \((v_{\text{max}})\). Values from one to four were used, and the best results were different for each algorithm: \(v_{\text{max}}=4\) for the BPSO, \(v_{\text{max}}=3\) to SPV-PSO and \(v_{\text{max}}=2\) to SPVG-PSO. The tests results are shown in Figure 5. As it was observed with the inertia coefficient, the increment in system performance were less than 0.1% when compared to the random allocation process. Figure 4: Results of tests to find the better parameter to inertia (\(K=20\), \(R=1000\) m and \(v_{\text{max}}=1\)). The third and forth parameters tested were the local and global solution weights (\(c_{1}\) and \(c_{2}\)). These parameters are not found in the SPVGs variants, so they were only tested for BPSO and SPV-PSO. All the previous analysis considered that \(c_{1}=c_{2}=1\), so we compared the results with those solutions. Increasing the local solution weight \(c_{1}=2\) and maintaining the global weight \(c_{2}=1\) downgrades the BPSO performance in \(0,001\%\), while increasing the SPV-PSO performance in \(0.02\%\). Further increasing the local solution \(c_{1}=5\) and \(c_{2}=1\) made the BPSO algorithm's performance around \(0.004\%\) higher than the baseline (\(c_{1}=c_{2}=1\)), while an increase of \(0.03\%\) was found for the SPV-PSO when compared to the baseline. Finally, changing the parameters to \(c_{1}=1\) and \(c_{2}=5\) increased the BPSO performance in \(0.005\%\) while decreased the SPV-PSO performance in \(0.02\%\)

This test showed that \(c_{1}\) and \(c_{2}\) values are not significant in terms of performance since their impact are in the one thousandth of a bit/s/Hz mark. The last PSO-based parameters optimization was the VNS rate. We tested four different configurations, and the results are presented in Figure 6. They demonstrate that a rate of \(0.9\) is the best configuration for both SPV-PSO-VNS and SPVG-PSO-VNS. Finally, the GA algorithm parameter was tested. Figure 7 shows the GA's performance for different mutation rates, and the \(T_{m}=0.9\) was found to be the best parameter value. Figure 5: Result of tests to find the better parameter to \(v_{\max}\) (\(K=20\), \(R=1000\) m and \((\omega)=0.25\)). ### Cell Size Impact on Performance

We conducted an analysis of the impact of cell size on all of the algorithms performance. We set up a scenario with \(K=20\) users per cell and the maximum number of iterations per algorithm to 50. To evaluate performance, Figure 8 shows the cumulative distribution function using our dataset. As one may see, the impact of cell size on the CDF is really marginal, which may indicate that the system is not interference bounded. This is corroborated with two facts. Figure 6: Results of tests to find the better parameters to VNS (\(K=20\), \(R=1000\) m, \((\omega)=0.25\) and \(v_{\max}=1\)). Figure 7: Results of tests to find the better parameters to \(T_{m}\) (\(K=20\), \(R=1000\) m). No intracell interference exists due to the orthogonality of the pilot sequences within a cell. The path-loss and shadowing effects deteriorate the interfering signals from other cells. Although the optimization problem aims to maximize the system spectral efficiency, we analysed the minimum spectral efficiency among all users in the system which may be seen as a fairness measure in the system. Table 5 presents the numerical values of the average lowest spectral efficiency in the system considering the 1000 instances in the dataset. The difference between the results for the RA, BPSO and GA algorithms were marginal, so they are listed in the same column. An important remark to make here is: for a cell with a 100 m radius, the SPV-PSO has an average lowest spectral efficiency 1.7% higher than the RA, BPSO and GA algorithms.