## 4 Geometric Distribution Approach  
To tackle challenges arising from the biased node distribution, a novel algorithm designed to optimize the SF assignment for each node called the geometric distribution (GD) algorithm is proposed here. The geometric distribution is basically known as a discrete probability distribution used to model events with two outcomes in repeated trials until success is achieved. The Probability Mass Function (PMF) is a measure providing probabilities for possible values of a random variable. In an attempt to distribute the biased number of SF nodes, the probability of success rate (\(p\)-value) of the PMF is the key feature.  
The PMF is presented in Equation (15). Notably, the PMF exhibits a distinct characteristic--the probability of success is the highest on the first trial (\(x=1\)) and decreases in subsequent trials. The \(p\)-value is a crucial parameter ranging from 0 to 1, influencing the likelihood of success and the overall distribution of success rates. For example, if the \(p\)-value is less than 1, the probability of success will always be less than 1 regardless of the number of trials.  
\[P(x)=\left(1-p\right)^{x-1}p,\ x=1,\ 2,\ 3,\ \ldots \tag{15}\]  
In an attempt to apply the GD to distribute the biased SF nodes, we defined two outcomes that are success and failure in distributing biased SF nodes to higher SF values. The \(p\)-value can be flexibly chosen to represent the desired probability of success. Since both node patterns shown in Table 2 are biased with SF7, the probability of success in the first trial, which is the highest, is certainly the assignment to SF7 nodes. Subsequent trials (\(x=2\)-6) were assigned to SF8 through SF12, where the probability of success in distributing SF7 nodes to these higher SFs decreases monotonically at the fixed \(p\)-value. Figure 5 presents a visual representation of the probability of success derived from the GD PMF at \(p=0.2\), 0.6, and 0.8.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Coverage Area** & **Pattern** & **\%SF7** & **\%SF8** & **\%SF9** & **\%SF10** & **\%SF11** & **\%SF12** & **DER Value** \\ \hline \multirow{3}{*}{2 km} & Uniform & \multirow{2}{*}{100.00} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{0.514} \\  & random & & & & & & & & \\ \cline{2-8}  & Fixed & 100.00 & - & - & - & - & - & 0.510 \\ \hline \multirow{3}{*}{3 km} & Uniform & \multirow{2}{*}{100.00} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{0.512} \\  & random & & & & & & & & \\ \cline{2-8}  & Fixed & 100.00 & - & - & - & - & - & 0.514 \\ \hline \multirow{3}{*}{4 km} & Uniform & \multirow{2}{*}{87.60} & \multirow{2}{*}{12.40} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{0.600} \\  & random & & & & & & & \\ \cline{2-8}  & Fixed & 96.53 & 3.47 & - & - & - & - & 0.538 \\ \hline \multirow{3}{*}{5 km} & Uniform & \multirow{3}{*}{70.33} & \multirow{3}{*}{15.40} & \multirow{3}{*}{14.27} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{-} & \multirow{2}{*}{0.676} \\  & random & & & & & & & \\ \cline{2-8}  & Fixed & 89.53 & 4.93 & 5.53 & - & - & - & 0.589 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The DER of the original ADR for the uniform random and fixed node patterns.