# Routing Strategies for Isochronal-Evolution Random Matching Network
## 4 Two Routing Strategies
### Planning Algorithm for the LHPMD
The numbers of delivered packets in the 8 scenarios are illustrated in Figure 8. Obviously, a scenario using the LHPMD routing strategy will have more delivered packets than a scenario using the LDPMH routing strategy under the same conditions. We analyzed the number of delivered packets in each scenario. Figure 9 shows the relationship between the number of delivered packets and \(R\). If we compare the curves of Scenario \(g_{1}\) and Scenario \(g_{1}+1\) (\(g_{1}\in\{1,3,5,7\}\)) presented in Figure 9, we find that:

1. When \(R\leq R_{C}(g_{1})\), the IERMNs in Scenario \(g_{1}\) and Scenario \(g_{1}+1\) are in the free-flow state and there are tiny differences in the number of delivered packets between them. 2. When \(R>R_{C}(g_{1})\), the IERMNs in Scenario \(g_{1}\) and Scenario \(g_{1}+1\) are in the congestion state and there are large gaps in the number of delivered packets between them. Figure 10 shows the relationship between the packet delivery ratio and \(R\). The packet delivery ratio [38] is the proportion of delivered packets from all packets. If we compare the curves of Scenario \(g_{1}\) and Scenario \(g_{1}+1\) (\(g_{1}\in\{1,3,5,7\}\)) in Figure 10, we find that when \(R>R_{C}\), i.e., the IERMNs become more congested with an increase in \(R\), the packet delivery ratios in a scenario using the LDPMH routing strategy decrease more rapidly than those in a scenario using the LHPMD routing strategy under the same conditions. Therefore, when in the free-flow state, an IERMN using the LDPMH routing strategy and one using the LHPMD routing strategy have almost the same number of delivered packets; when in the congestion state, an IERMN using the LHPMD routing strategy has more delivered packets than the IERMN using the LDPMH routing strategy. Figure 8: The numbers of delivered packets in the 8 scenarios. Figure 9: The number of delivered packets versus \(R\). #### 5.2.3 Simulation Analyses of the Posterior Path Length

In this section, we present the analysis of the average posterior path lengths of 2,419,434 delivered packets. Figures 11 and 12 illustrate the statistics on the Average Posterior Path Spatial Lengths (APPSLs) and the Average Posterior Path Temporal Lengths (APPTLs). The eight scenarios have the same minimum APPSL; they also have the same minimum APPTL. Scenario \(g_{1}\) has larger means, maximums, and standard deviations for the APPSL and APPTL than Scenario \(g_{1}+1\) (\(g_{1}\in\{1,3,5,7\}\)). In general, a scenario using the LHPMD routing strategy has smaller posterior path lengths than a scenario using the LDPMH routing strategy under the same conditions. We analyzed the average posterior path lengths of the delivered packets in each scenario. Figure 11: Statistics on the APPSLs for Scenarios 1–8. The ordinate scale uses a logarithmic scale. Figure 10: The packet delivery ratio versus \(R\). Figure 13 illustrates the relationships between the APPSLs of the delivered packets and \(R\). Figure 14 illustrates the relationships between the APPTLs of the delivered packets and \(R\). If we compare the curves of Scenario \(g_{1}\) and Scenario \(g_{1}+1\) (\(g_{1}\in\{1,3,5,7\}\)) presented in Figures 13 and 14, we find that: when \(R\leq R_{\text{C}}(g_{1})\), the APPSLs in a scenario using the LDPMH routing strategy overwhelm those in a scenario using the LHPMD routing strategy under the same conditions, while the difference in their APPTLs is quite small. Hence, when an IERMN is in the free-flow state, it has a smaller average posterior path length for the delivered packets when the LHPMD routing strategy is used compared with the LDPMH routing strategy. Nevertheless, when the IERMN is in the congestion state, there are different performances in different scenarios, which can be described as follows. 1. When \(R>R_{\text{C}}(2)\), the gap between Scenario 1 and Scenario 2 in the APPSL is narrowed with an increase in \(R\), and the APPSLs of Scenario 1 are always larger than those of Scenario 2, so the APPTLs are too. 2.