# Retransmission Avoidance for Reliable Data Delivery in Underwater WSNs
## 4 Linear Optimization
### Throughput Maximization  
In this section, we are using linear programming to maximize network throughput. In FLMPC-One and FLMPC-Two, multiple copies of a data packet are relayed towards the destination. Throughput is the number of distinct packets successfully received at the sink. The objective function is defined as follows:  
\[Max\sum_{t=1}^{t_{max}}P(t)\ \ \forall\ t\ \in\ R \tag{24}\]  
Figure 7: Energy Consumption Feasible Region.  
where,  
\[P(t)=\sum_{t=1}^{t_{max}}pt\times c\ \ \forall\ c\ \in\ P \tag{25}\]  
and,  
\[pt=\begin{cases}1&if\ distinct\ packet\\ 0&if\ duplicate\ packet\end{cases} \tag{26}\]  
\[L_{1}:E_{t}\leq E_{o} \tag{27a}\] \[L_{2}:DR\leq DR_{pr}\] (27b) \[L_{3}:E_{s}\geq E_{tx}\] (27c) \[L_{4}:PER<PER_{req}\] (27d) \[L_{5}:Min\sum_{d=d_{o}}^{d_{max}}No_{th}\ \forall d\ \in\ R\] (27e) \[L_{6}:Min\sum_{d=d_{o}}^{d_{max}}No_{s}\ \forall d\ \in\ R \tag{27f}\]  
In Equation (24), our objective is to maximize the number of distinct packets \(P\) received at the sink in aggregated time (\(t_{max}\)). In Equation (25), \(pt\) is the packet type and \(c\) is the counter. Counter \(c\) increments by one if \(pt\) is 1 in Equation (26). It ensures that the data packet is not a duplicate one. Equations (27a)-(27c) are same as explained in previous section without them data packet can not be transmitted. Constraint Equation (27d) ensures that \(PER\) must be less than or equal to required level (\(PER_{req}\)). Equations (27e) and (27f) show the thermal noise (\(No_{th}\)) and shipping noise (\(No_{s}\)), respectively. Where \(d\) is the distance from source node to surface gateways. If noise intensity is high, data packet becomes erroneous. If \(PER\) increases the required level, data packet is dropped at gateway. In order to increase throughput, noises should be minimum. Total noise is given by:  
\[No_{total}=No_{th}+No_{s}\ \ \ \forall\ No_{th},No_{s}\in\ R \tag{28}\]  
where \(No_{th}\)[23] and \(No_{s}\)[24] are given by:  
\[No_{th}=\frac{15+20 log(f)}{D} \tag{29}\]  
\[No_{s}=\frac{NL-20 log(\frac{f}{100})}{D} \tag{30}\]  
where \(NL\) is noise level and it is based upon shipping density.  
**Graphical Analysis:** Let \(f\) = 1000 Hz, \(d\) = 100 m to 2000 m and \(NL\) = 60 dB. The calculated value of \(No_{th}\) is 0.0375 dB to 0.75 dB and \(No_{s}\) lies in the range of 0.02 dB to 0.4 dB.  
\[0.0375+0.02\leq No_{th}+No_{s}\leq 0.75+0.4 \tag{31}\]  
\[0.0375\leq No_{th}\leq 0.75 \tag{32}\]  
\[0.02\leq No_{s}\leq 0.4 \tag{33}\]  
In Figure 8, intersection of lines results in feasible region.  
All points in this region provide a valid solution. Now testing corner points to validate the results of Equations (32) and (33). Calculating values at corner points:  
* \(P_{1}(0.0375,0.02)=0.0375+0.02=0.0575\) dB
* \(P_{2}(0.0375,0.4)=0.0375+0.4=0.438\) dB
* \(P_{3}(0.75,0.4)=0.75+0.4=1.15\) dB
* \(P_{4}(0.75,0.02)=0.75+0.02=0.77\) dB  
The lesser the noise is, the higher are the chances of successful packet reception at gateways. If a data packet faces noise value from this feasible region, the error is less in data packet and it is acceptable at gateways.