# Lifetime Maximization via Hole Alleviation in IoT Enabling Heterogeneous Wireless Sensor Networks
## 3 LiMha
### Throughput Maximization  
In this section, we maximize the network performance through efficient bandwidth allocation to super nodes. During the process, normal nodes send data to their respective super nodes in their transmission range. For the selection of \(SN\) as a data forwarder maximum bandwidth is required such that to forward data maximum amount of data. For the selection of \(SN\), it is necessary that \(SN\) has high bandwidth in order to make sure that congestion is avoided and data is delivered to the sink  
Figure 3: Feasible Region: Energy.  
successfully. Based on this, an objective function under linear constraints has been defined as follows:  
\[maximize\sum_{r=1}^{r_{max}}Th(r)\qquad\forall r\in r_{max} \tag{20}\]  
where, \(Th(r)\) is computed like given in Equation (21)  
\[Th(r)=\sum_{r=1}^{r_{max}}Pkt(r)\qquad\forall r\in r_{max} \tag{21}\]  
**Constraints:**  
\[Con_{1}: E_{n}\leq\epsilon_{o}\qquad\forall n\in N \tag{22}\] \[Con_{2}: E_{F}\leq E_{F}^{min}\qquad\forall n\in N\] (23) \[Con_{3}: TR_{n}\leq TR_{n}^{max}\qquad\forall n\in N\] (24) \[Con_{5}: Max\sum_{sn=1}^{sn_{max}}B_{SN}\qquad\forall sn\in SN \tag{25}\]  
Objective function in Equation (20) is used to maximize the throughput of the network. Constraint \(Con_{1}\) and \(Con_{2}\) illustrate that the energy consumption of nodes must be minimum. Moreover from Equation (23) it is clear that energy of forwarder node is less than the node with minimum energy in a specific region. Equation (24) states that transmission range of sensor nodes is within the maximum transmission range of sensor nodes. Equation (25) depicts the allocation of maximum bandwidth to \(SN\) and remaining is divided equally between \(NN\).  
Let \(B\) denotes the total bandwidth allocated to the network i.e.,:  
\[B=B_{NN}+B_{SN} \tag{26}\]  
where \(B_{NN}\) is the \(B\) allocated to \(NN\) and \(B_{SN}\) is the \(B\) allocated to \(SN\). The network area is divided into \(N\) coronas' and further divided into \(S\) sectors. So the allocation of bandwidth in \(S\) can maximize the network performance. Let \(B_{n}\) be the \(B\) allocated to a node in \(C_{i}\) then:  
\[B_{NN}=B_{n}\times N_{nn} \tag{27}\]  
Let assume in a network \(N_{nn}\) ranges from \(\{2,3,....,\frac{N_{nn}}{N}\}\) in \(N\) coronas network. For efficient routing, we must have at least two nodes in a sector and maximum of \(\frac{N_{nn}}{N}\) nodes. Exceeding this value results in decreasing number of nodes in other sectors.  
Also we have  
\[B_{n}=\frac{ATF}{t} \tag{28}\]  
where ATF denotes average traffic load on sensor nodes which can be calculated from Equation (11).  
\[B_{SN}=B-B_{NN} \tag{29}\]  
**Graphical Analysis:** Let assume total \(N_{nn}\) = 110 and \(N\) = 10, \(B\) = 500 kHz to 1 MHz, \(R\) = 500 m, \(r\) = 60 m, \(\zeta\) = 3 m, \(x\) = 5 m, \(t\) = 10 ms. For \(N_{nn}\) = 2, ATF = 466.12 bits, \(B_{NN}\) = 93.2 kHz and \(B_{SN}\) = 406.8 kHz. For \(N_{nn}\) = 11, \(B_{NN}\) = 512 kHz and \(B_{SN}\) = 488 kHz. Now Equations (26), (28) and (29) can be represented as:  
\[93.2\leq B_{NN}\leq 512 \tag{30}\] \[406.8\leq B_{SN}\leq 488 \tag{31}\]\[500\leq B_{NN}+B_{SN}\leq 1000 \tag{32}\]  
The bounded region from the aforementioned equations is drawn in Figure 4 which is the feasible region of this bounded region.  
\(P_{1}\) (93.2, 406.8) = 500 kHz  
\(P_{2}\) (93.2, 488) = 581.2 kHz  
\(P_{3}\) (512, 406.8) = 918.8 kHz  
\(P_{4}\) (512, 488) = 1 MHz  
These points validate a valid solution for lifetime maximization. Hence selecting any value from the bounded region for \(B\) allocation always results in increasing the lifetime of forwarder nodes.