## 4 Proposed Scheme
### Direct vs. Multi-Hop Inter-Cluster Communication  
Let '\(\mathcal{K}\)' be the variable representing the possible number of hops available between the outer-most cluster head (CH) and the base station (BS). Due to unequal cluster sizes, the distances between these hops vary and are denoted as \(r_{1},r_{2},r_{3},\ldots,r_{\mathcal{K}}\). The energy consumed by the outermost CH in transmitting a message of size '\(l\)' bits directly to the base station can be worked using the first-order radio model [15], as discussed earlier. The energy consumed during direct communication is given by:  
\[E_{Direct}=E_{TX}(l,d=r_{1}+r_{2}+r_{3}\ldots+r_{\mathcal{K}})\]  
where:  
\[E_{Direct}=\left\{ \begin{matrix}E_{ELEC}\cdot l+\epsilon_{fs}\cdot l\cdot d^{2 }\ if\ d\leq d_{o}\\ E_{ELEC}\cdot l+\epsilon_{mp}\cdot l\cdot d^{4}\ otherwise\end{matrix} \right\}\]  
Similarly, energy consumed during multi-hop inter-cluster communication can be computed as:  
\[E_{Multiop}=\mathcal{K}*E_{TX}\left(l,d={r_{1}}^{2}+{r_{2}}^{2}+\ldots+{r_{ \mathcal{K}}}^{2}\right)+(\mathcal{K}-1)E_{RX}\]  
To determine when inter-cluster multi-hop routing is required, the following two cases are considered. Case 1: \(d\leq d_{o}\)  
If the overall distance '\(d\)' between a cluster head and the base station is less than or equal to the threshold distance '\(d_{o}\)' as defined by the radio model, then energy consumed by direct communication is:  
\[E_{Direct}=E_{ELEC}*l+\epsilon_{fs}*l*d^{2}\]  
Direct communication is preferred if the following condition is satisfied:  
\[E_{Direct}<E_{Multiop}\]  
\[E_{ELEC}*l+\epsilon_{fs}*l*d^{2}<l\left[\left[\mathcal{K}E_{ELEC}+\mathcal{K} \epsilon_{fs}\left({r_{1}}^{2}+{r_{2}}^{2}+\ldots+{r_{\mathcal{K}}}^{2} \right)\right]+\left[\mathcal{K}E_{ELEC}-E_{ELEC}\right]\right]\]  
\[d^{2}<2\frac{E_{ELEC}(\mathcal{K}-1)}{\epsilon_{fs}}+\mathcal{K}\left({r_{1}} ^{2}+{r_{2}}^{2}+\ldots+{r_{\mathcal{K}}}^{2}\right)\]  
Case 2: \(d>d_{o}\)  
If the overall distance between the cluster head and the base station exceeds the threshold distance '\(d_{o}\)', the energy consumed by direct communication is computed as:  
\[E_{Direct}=E_{ELEC}*l+\epsilon_{mp}*l*d^{4}\]  
Direct communication is preferred if:  
\[E_{Direct}<E_{Multiop}\]  
\[E_{ELEC}*l+\epsilon_{mp}*l*d^{4}<l\left[\left[\mathcal{K}E_{ELEC}+\mathcal{K} \epsilon_{fs}\left({r_{1}}^{2}+{r_{2}}^{2}+\ldots+{r_{\mathcal{K}}}^{2} \right)\right]+\left[\mathcal{K}E_{ELEC}-E_{ELEC}\right]\right]\]  
\[d^{4}<2(\mathcal{K}-1)\frac{E_{ELEC}}{\epsilon_{mp}}+\mathcal{K}\left({r_{1}} ^{2}+{r_{2}}^{2}+\ldots+{r_{\mathcal{K}}}^{2}\right)\frac{\epsilon_{fs}}{ \epsilon_{mp}}\]  
Energy Balanced Inter-Cluster Multi-Hop Communication  
If the network size and configuration favour multi-hop routing as the more energy-efficient option, an issue of imbalance load among cluster heads may arise, as illustrated in Figure 7. Cluster heads that are located closer to the base station (BS) experience a heavierrelaying load, particularly in large-scale networks, as they are responsible for forwarding data from distant cluster heads to the BS. This uneven energy consumption can lead to premature failure of the nearer cluster heads, reducing the overall network lifespan. The proposed scheme explores the condition to balance the load among cluster heads, thereby increasing network lifetime and stability. The following mathematical model illustrates this:  
Let the cluster heads \(CH_{a}\) and \(CH_{b}\) be located at the same level and have the same number of nodes '\(a\)', and the same data rate on them, denoted by '\(l\)'. The total energy consumption for \(CH_{a}\) is expressed as:  
\[E_{Direct}=E_{ELEC}*l+\epsilon_{fs}*l*d^{2}\]0  
Assuming that the energy consumption for \(CH_{a}\) and \(CH_{b}\) is equal, then  
\[E_{Direct}=E_{ELEC}*l+\epsilon_{fs}*l*d^{2}\]1  
Let '\(e\)' be the number of nodes in cluster \(CH_{e}\).