# Resource Assignment Algorithms for Autonomous Mobile Robots with Task Offloading
## 5 Simulation Results and Discussion
### 6/7/4_Fixed Scenario  
Scenario 6/7/4_fixed_ is simple, as it is characterized by \(N_{\text{AV}}=6\), \(N_{\text{BS}}=7\), and \(N_{\text{DC}}=4\). The service in each DC has typical latency \(l_{m}=33\) ms and typical load \(\gamma_{m}=1/4\) CPU, \(m=1,2,4\), and \(\gamma_{3}=1/8\) CPU. The internal service latency is \(\ell_{i}=111\) ms, while the bit rates produced by AVs are 32 Mbit/s, 22 Mbit/s, 35 Mbit/s, 24 Mbit/s, 31 Mbit/s and 17 Mbit/s. The connection matrix is seen in the graph of Figure 6, while access bit rates and latency values are shown on the AV-BS and BS-DC edges, respectively. In this scenario, BS7 is a 5G station connected to a cloud server (DC3) with a slow link, while all other BS are IEEE 802.11 ad connected to edge servers with fast links. The number of possible configurations is \(N_{\text{all}}=5.9\times 10^{8}\), of which \(N_{\text{val}}=3.6\times 10^{3}\) are valid. For the RSAM algorithm, we chose \(N_{\text{call}}=5\times 10^{4}\gg N_{\text{val}}\); thus, it has a high probability of finding the optimal solution. Indeed, RSAM may generate a configuration that is already generated; hence, \(N_{\text{call}}=N_{\text{val}}\) does not guarantee the optimal solution. VEX is simulated since \(N_{\text{val}}\) is a tractable number of configurations, while for GA, we have set a maximum of \(N_{\text{call}}=2\times 10^{6}\) calls. Finally, SM has \(\vartheta=2\). The KPIs for the found solutions are reported in Table 3. For all the KPIs, all algorithms except MLAT attain the minimum value: 301.0 ms of total latency, 50.2 ms of average latency, 1% of average radio load without any radio overload, 31% of average CPU load, and no DC overloads. As for the computation cost, GA is the most complex, with about \(10^{6}\) actual function calls and a 17.8 s run time, followed by RSAM with a 0.4 s run time. Then, VEX and SM have very low complexity, with 3600 and 65 calls, respectively, which are performed in about 15 ms. Finally, MLAT is the fastest, with just 7 ms of run time (note that MLAT does not make any call, but its complexity is roughly equivalent to that of 1 call). The configurations obtained by the simulated algorithms produce the connection graphs shown in Figure 7a-e. The optimal configurations indicate that AV4 has been assigned to the internal service and that, generally, BS4 is preferred for the radio connection and DC1 for the task offloading. Note that in this case VEX and SM produce the same (optimal) solution. Figure 6: Connection graph of the 6/7/4_fixed_ scenario. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **RSAM** & **VEX** & **MLAT** & **GA** & **SM** \\ \hline \(\sum_{i}L_{i}\) (ms) & 301.0 & 301.0 & 438.0 & 301.0 & 301.0 \\ \(\frac{1}{N_{\text{val}}}\sum_{i}L_{i}\) (ms) & 50.2 & 50.2 & 73.0 & 50.2 & 50.2 \\ \(\frac{1}{N_{\text{BS}}}\sum_{k}U_{k}\) & 0.01 & 0.01 & 0.87 & 0.01 & 0.01 \\ \(V_{\text{BS}}\) & 0 & 0 & 1 & 0 & 0 \\ \(\frac{1}{N_{\text{DC}}}\sum_{m}\Gamma_{m}\) & 0.31 & 0.31 & 0.34 & 0.31 & 0.31 \\ \(V_{\text{DC}}\) & 0 & 0 & 0 & 0 & 0 \\ \(N_{\text{call}}\) & 50,000 & 3600 & 1 & 953,440 & 65 \\ Run time (s) & 0.418 & 0.013 & 0.007 & 17.830 & 0.015 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: KPIs for the solutions to the _6/7/4_fixed_ scenario. The configuration summaries printed in Figure 8a-e describe the plots of Figure 7a-e: a the first line reports the algorithm and the found optimal KPI value, while the remaining lines detail each AV by reporting the connection BS and offloading DC, together with their respective loads, closed by the obtained latency value for the single AV. The results highlight that MLAT loads BS7 at 600%, thus resulting in a total latency of 438.0 ms, which is 46% higher than the minimum obtainable value. Note that this underperformance is due to MLAT minimizing each AV route to the DC independently from that of other AVs. Figure 7: Paths in the graph corresponding to the found configurations for the _6/7/4_fixed_ scenario: (**a**) RSAM solution. (**b**) VEX solution. (**c**) MLAT solution. (**d**) GA solution.