## 5 Results and Discussion
### Coverage and Connectivity  
A concept associated with coverage is the k-coverage, which is related to the number of sensors sensing a specific area or point. The k-coverage depends on the application and, in some cases, can be dynamically configured [38]. The maximum value of k is called the coverage degree [52]. Fault tolerance in WSNs increases network costs while enhancing robustness. However, sensing coverage would only be worthwhile with communication paths between the sink and the source sensors. In other words, network connectivity is essential. Similar to the concept of k-coverage, there exists k-connectivity. A network is said to have k-connectivity if removing k \(-\) 1 nodes does not result in lost communication between them. k-coverage and k-connectivity are used to guarantee fault tolerance. This work also determined the behavior of the exponential, uniform, and normal distributions concerning the requirements of k-coverage and k-connectivity. For analysis, the tool was executed 500 times for each of the prototype mines and each probability function. Figure 10a-c show the maximum value for k-coverage and k-connectivity for exponential, normal, and uniform distributions in small-sized mines. In exponential distribution, it obtained k-coverage = 3 and k-connectivity = 2. This means that one point can be sensing up to three sensors and that for 54.2% of the time, it is possible to have k = 2, for 37.2% of the time, k = 1, and for 8.6% of the time, k = 3. In addition, normal and uniform distributions, for 76.2% and 74.8% of the time, a value of k = 1 was obtained. Figure 10: Cont. Regarding connectivity, again, in Figure 10, the exponential distribution has the maximum value with k = 2 (2-connectivity); in 72.8% of the cases, it is possible to have two routes through which the data can be routed. On the other hand, in normal and uniform distributions, k = 1 (1-connectivity) was obtained. Now, the results obtained in the medium-sized mine will be analyzed. Figure 11a-c show the maximum value for k-coverage and k-connectivity for exponential, normal, and uniform distributions in medium-sized mines. The three distributions function shows the same values for k-coverage k = 2, although for the highest percentage of times (53%) it is the exponent function, a value close to that obtained in the small mine for the exact value of k. In the case of the normal and uniform functions, it is observed that although we have the same values of k = 2, the percentage of times that it is repeated increases. Likewise, in regard to k-connectivity, the same behavior is shown. The exponential distribution has k = 2, while in the normal and uniform distribution functions, k = 1. Figure 11: _Cont._  
Figure 10: k-coverage and k-connectivity in small-sized mines by distribution function. (**a**) Coverage and connectivity percentage for small-sized mine using an exponential function. (**b**) Coverage and connectivity percentage for small-sized mine using a normal function. (**c**) Coverage and connectivity percentage for small-sized mine using a uniform function. The analysis of the degrees of coverage and connectivity reveals that the exponential distribution achieves the highest values, facilitating a fault-tolerant network. This is because it deploys a more significant number of nodes compared to the other distributions. However, this increased number of nodes increases network implementation and maintenance costs. Therefore, it is vital to find a balance between the number of nodes and coverage and connectivity requirements versus the cost of deployment. The selection of the distribution must be aligned with the specific objective of the deployment, and it is necessary to perform a trade-off analysis between the goals, that is, choose a distribution that meets all the requirements in a balanced way. Few studies address the problem of k-coverage in underground mines.