## 5 Numerical Analysis
### Scenario B--Analysis of Fog Resources Variation  
Figure 5 presents the results of observing the variation of fog resources capabilities. It is essential to mention that the cloud was isolated and not considered in this scenario. The MRT obtained by fog resources variation (see Figure 5a) was significantly distinct from the one obtained with cloud variation. First of all, the results here were higher than in the cloud variation scenario. Considering five nodes, the highest MRT was nearly 160 ms with the fog, whereas the cloud obtained 97 ms with the same number of nodes. This is explained by the cloud capacity--in terms of service time--being more efficient than the fog. The difference between 10 and 15 nodes in the cloud variation is more significant. Choosing 10 or 15 nodes carefully does matter when having only the fog as an available resource. The point where the MRT growth stagnates remains the same after 0.3 j/ms, meaning that after this point, the MRT will be the same, independent from the AR value. Ultimately, the hypothetical SLA of 70 ms (green line) would not be satisfied, not even with 15 fog nodes. Therefore, the designer should be aware that the users would not be delighted only with the fog. Figure 5b shows the use of a gateway in the considered three scenarios. As the AR increases, the use of gateway grows exponentially. The gateway has the same level of usage for all configurations of fog. When the AR is equal to 0.1 j/ms, the gateway's use is approximately 16% for the three scenarios. However, after the AR exceeds 0.6 j/ms, the gateway starts to use approximately 90% of its capacity. When the AR reaches 1.0 j/ms, the gateway uses approximately 97% of its capacity for all fog configurations. The gateway utilization growth tends to stagnate from AR = 0.6 j/ms until reaching its maximum utilization of approximately 97%. Considering the extreme point of most significant demand (AR = 1.0 j/ms), the gateway utilization did not reach 100%. However, it becomes impossible to meet more requests at this stage of processing without data loss. Figure 5c presents the fog utilization. In the cloud's previous scenario, the utilization has taken more time to reach high utilization levels. However, here in the fog scenario, with five and 10 nodes, the utilization achieved 100% with just 0.3 j/ms of AR. It is essential to mention, as well, that 15 nodes did not achieved the 100% utilization level. Another difference to the cloud scenario is that the fog results were close to each other almost in all AR points, meaning that it does not make much difference to choose one or another number of nodes after 0.6 j/ms of AR. Figure 5d,e presents the utilization to the output components, family member, and emergency. Again, both results are presented together due to their similarities. The only difference between graphs is that the family member results are slightly higher than the emergency one. This difference may be explained by the service times. The family member is configured with a higher service time (see Table 2). More equidistant lines are observed comparing the cloud results, about 20% of the interval between the lines. Meanwhile, in the cloud, the results for 10 and 15 nodes were almost the same. Figure 5: Results for analysis of the fog resources variation. Figure 5f depicts the number of jobs in the system. The number of jobs within the system increases as the AR increases, as jobs are queued for resource constraints. A slightly different result from the other two scenarios is that of five nodes. When the AR is lower, the system's number is higher for five nodes than the other scenarios. When the AR becomes higher (\(>\)0.35 j/ms), called the \"changing point\", the number of jobs in the system is lower for five nodes than in the other scenarios. In the cloud scenario, this changing point happened later, around 0.6 j/ms. The authors believe that the higher drop rate levels explain this difference in the fog scenarios.