# Minimum-Cost-Based Neighbour Node Discovery Scheme for Fault Tolerance under IoT-Fog Networks
## 3 Materials and Methods
### Problem Formulation and Objective Function
The obtained results of the proposed methods show that they achieved a higher performance than existing methods in terms of the following: (i) the active number of Fog access nodes, with an average improvement of 41.2%, 37.82%, and 28.57% compared to the existing methods without FT, NFT-WOA, and DFTLA, respectively, using the minimum number of active Fog access nodes for task-offloading; (ii) average response time with a response time that was shorter by 0.9 ms, 0.3 ms, and 0.1 ms than the HBI-LB, Throttled, and Round Robin methods, respectively; (iii) the completion rate, with a 62.5% higher performance than the RPMFT method; (iv) acceptance ratio, with an improvement of 24%, and completion time, which outperforms the method without FT by 16.27% and the method without replicas by 29.84%; and (v) achieved energy savings, which are improved by 1.55 J compared to the system without replicas and 0.85 J for the system without an FT method in terms of residual energy. The higher performance of the proposed results was only achieved because prioritized tasks were assigned to the Fog access nodes with the most efficient residual energy for task-processing. The prior availability of candidate Fog access nodes was ensured at the time the fault occurred, using RPF to track source nodes' information and to create a dynamic

Figure 11: Residual energy vs. simulation time. network with scalability. Furthermore, in the future, malicious nodes will be identified in a dynamic Fog networking environment, and those nodes will be removed for efficient task-offloading. Additionally, an efficient trust-management scheme can be proposed for use in dynamic Fog-IoT environments to securely offload the task. Conceptualization, P.P. and P.B. ; methodology and software, P.B. ; formal analysis and investigations, P.P. and P.B. ; article drafting, P.B. ; review, editing and supervision, P.P. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The data presented in this study are available on request from the corresponding author. The authors declare no conflicts of interest. ## References

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