# Routing Strategies for Isochronal-Evolution Random Matching Network
## 4 Two Routing Strategies
### Planning Algorithm for the LHPMD
When \(R>R_{\text{C}}(6)\), the APPSLs and the APPTLs of Scenario 5 are larger than those of Scenario 6 in the beginning; afterwards, Scenario 6 overtakes Scenario 5 as \(R\) increases. 3. When \(R\leq R_{\text{C}}(g_{2})\) (\(g_{2}\in\{3,7\}\)), the gap between Scenario \(g_{2}+1\) and Scenario \(g_{2}\) in terms of the APPSLs becomes a little smaller, and the APPSLs of Scenario \(g_{2}+1\) outnumber those of Scenario \(g_{2}\), so the APPTLs do too. When the IERMN is congested, in terms of the average posterior path length, a path with more vertices or fewer vertices with a poor delivery capability will have a shorter average posterior path length for the packets delivered by the LHPMD routing strategy than those delivered by the LDPMH routing strategy; if a path has fewer vertices with a strong delivery capability, it will have a shorter average posterior path length for the delivered packets with the LDPMH routing strategy than with the LHPMD routing strategy. The reason why the LHPMD routing strategy leads to a smaller APPTL in some cases is also due to the different planning costs of an LDPMH and an LHPMD. As mentioned in Section 5.2.1, the LDPMH should contain more vertices than the LHPMD. Within the limits of the FIFO, PIA, and \(C\), the more vertices a packet visits, the more frequently the packet may be held. Therefore, it may take more time to transmit the packet with LDPMHs than with LHPMDs. Figure 12: Statistics on the APPTL s for Scenarios 1–8. The ordinate scale uses a logarithmic scale. ### Comparison with the Benchmark Method

In this section, the performance of the proposed routing strategies is compared with a benchmark method called the flooding routing strategy, which means that each vertex always sends \(C\) packets to its successor vertex until the packet has reached its destination. We designed 4 scenarios using the flooding routing strategy, as tabulated in Table 2. The simulations for Scenarios 9-12 were designed in the same way as Scenarios 1-8, as presented in Section 5.1. A comparison of the order parameter \(H\) versus \(R\) for eight scenarios is illustrated in Figure 4:0 By comparing Figure 15 with Figures 6 and 7, we find that \(R_{\text{C}}\) in a scenario using the flooding routing strategy is more than in a scenario using the LHPMD or \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenario Number** & **Routing Strategy** & \(\boldsymbol{\Theta}\) & \(\boldsymbol{C}\) \\ \hline
9 & Flooding & 30 & 1 \\
10 & Flooding & 400 & 1 \\
11 & Flooding & 30 & 10 \\
12 & Flooding & 400 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Setup in scenarios using the flooding routing strategy. Figure 14: Average posterior path of the temporal length versus \(R\). Figure 13: Average posterior path of the spatial length versus \(R\). ## 6 Discussion and Conclusions

In this paper, we put forward a new network model called an IERMN and two routing strategies for the IERMN: the LDPMH routing strategy and the LHPMD routing strategy. We drew some conclusions about the two proposed routing strategies by the traffic dynamics simulations, which are described as follows:

1. In terms of the maximum traffic capability of the IERMN, the LHPMD routing strategy performs better than the LDPMH routing strategy. 2. In terms of the capability of delivering packets to their destinations, when the IERMN is in the free-flow state, the performance of the LDPMH routing strategy nearly equals that of the LHPMD routing strategy; when the IERMN is congested, the LHPMD routing strategy overwhelms the LDPMH routing strategy. 3. In terms of the average posterior path length of the delivered packets, when the IERMN is in the free-flow state, the LHPMD routing strategy is better than the LDPMH routing strategy; when the IERMN is in the congestion state, the LHPMD routing strategy is better in some cases, and the LDPMH routing strategy is better in other cases.