# Fine-Grained Boundary Conditions in Field-Based Routing
## 6 Performance Evaluation
### Simulation Results  
#### 6.3.1 Outer Boundary Conditions  
We begin by examining the results for the grid regular network topology with holes, which represents the simplest scenario, as shown in Figure 3. Note that the areas labeled as hole in the figure are just one example. In actual simulations, these holes were randomly selected. Simulations are conducted with a 30% offered traffic load generated by randomly selecting 10 mesh nodes. Figure 4 shows the overall packet delivery ratio under different outer boundary conditions with respect to the broken link ratio. Here, the broken links are defined as the links which are not normally connected to neighbors due to the holes. Note that our performance evaluation is based on the broken link ratio rather than the number of failed nodes. This approach allows us to fairly consider various scenarios, including simultaneous failures of neighboring nodes and distant nodes. When neighboring nodes fail simultaneously, there are relatively fewer broken links, resulting in minimal disruption to packet transmission. However, when distant nodes fail simultaneously, more broken links may occur, presenting more significant challenges in packet delivery. Considering these factors, measuring performance in terms of the broken link ratio can provide a more fair assessment in the evaluation. When considering the simplest scenario of the grid regular topology, we focus solely on the overall packet delivery ratio results, without including additional metrics such as throughput. This approach allows us to provide a clear and concise illustration of the impact of network holes as the first demonstration of our concept. Finally, the performance of the proposed ALFA by Robin boundary conditions (referred to as Robin BC) is significantly better than the conventional ALFA with Dirichlet-based boundary conditions (referred to as Dirichlet BC) as the ratio of broken links increases, with a maximum difference of 48% in terms of the overall packet delivery ratio. Specifically, up to a broken link ratio of 15%, there is little difference in performance between the two schemes. However, the performance gap begins to widen: at 20%, it is 7%; at 25%, it is 8%; and at 30%, it increases to 48%. The reason is that a routing field is not normally converged due to the absence of a sophisticated method to calculate potential values at the vacant regions by the existing simple Dirichlet-based boundary conditions to assign a fixed potential value to boundary nodes. On the other hand, there is no such limitation imposed by the proposed boundary conditions, as all the boundary nodes can better adapt to various situations by establishing a stable routing field in any scenario. In other words, ALFA with the proposed boundary conditions is more robust for the networks with holes. Next, we conduct simulations in a non-grid regular network topology with several holes to evaluate the proposed outer boundary conditions, similar to the grid scenario. In detail, we consider a non-grid regular network consisting of 154 mesh nodes and two gateways distributed across two sub-domains, with each sub-domain having a gateway deployed, as shown in Figure 5. These sub-domains consist of one small domain (referred to as sub-domain 1) and one large domain (referred to as sub-domain 2). The holes within the network are randomly generated with a broken link ratio of 30% per simulation run. We randomly select 10 mesh nodes at the intersections (highlighted in grey) of the sub-domains to create a congested hot spot by generating highly loaded traffic with an 80% offered load. Additionally, background traffic with a 10% offered load is generated by randomly selected 20 nodes, excluding the 10 mesh nodes chosen for generating highly loaded traffic at the intersections of sub-domains. Figure 3: Grid regular topology with holes.