# On Reliable and Efficient Data Gathering Based Routing in Underwater Wireless Sensor Networks
## 9 RE-AEDG: Optimization Models
### Delay Minimization Model  
Our proposed RE-AEDG implements cooperative communication to improve reliability while paying the cost of energy consumption and data latency. After energy consumption minimization model in the previous sub section, we present delay minimization model in this portion of paper.  
\[Min\sum_{i=1}^{n}Dy_{i}\qquad\forall\;i\epsilon N \tag{19}\]  
Subject to:  
\[d_{th}\;\leq\;d_{i}^{h_{i}}\;\leq\;T_{xR}\;\;\;\forall\;i\; \epsilon\;(MNs||GWs) \tag{19a}\] \[0\;\leq\;h_{i}\;\leq\;h_{max}\;\;\;\forall\;i\;\epsilon\;MNs\] (19b) \[d_{i}^{h_{i}}\;\;\geq\;\;d_{i}^{h_{i}}\] (19c) \[t_{min}\;\leq\;S_{t}\;\leq\;t_{max}\] (19d) \[0\;\leq\;\rho_{h}^{i}\;\leq\;\rho_{th}\;\;\;\forall\;i\; \epsilon\;MNs\] (19e) \[0\;\leq\;H_{i}\;\leq\;H_{max}\;\;\;\forall\;i\;\epsilon\;(MNs||GWs)\; whereas\qquad H\;=\;\frac{1}{R_{e}\times d_{i}^{h_{i}}}\] (19f) \[P_{i}(t)\;\leq\;C_{cap}\;\forall\;i\;\epsilon\;(MNs||GW)\] (19g) \[M_{AUV}\;\geq\;\frac{1}{3}\;W_{net} \tag{19h}\]  
Our objective in Equation (19) is to minimize the time delay created in data transmission process. Equation (19a) defines that depth difference between \(i\) and its next hop \(h_{i}\) must remains in between depth threshold and predefined transmission range. Whereas depth threshold is kept to a value, such that sender node may not be able to define its priority node at an extremely near vicinity. Applicability of this law causes larger depth difference between transmitter and receiver which in turn results in greater energy consumption and lowered delay. This can also be written as \(D\propto No.ofib\)s. Equation (19b) indicates that the total no. of hops a packet requires to reach destination must not exceed than the maximum allowable no. of hops. Implementation of this clause prevents extra per packet network energy consumption. Equation (19c) depicts that depth difference in between \(i\) and \(h_{i}\) must be maximum out of Eligible Neighbors set. This rule in incorporated because transmission at lower distance node increases packet transmission reception rate. Equation (19d) states that sojourn stop time for an AUV must be an optimum duration. Failing to implement this constraint causes GWs to hold packets for larger time which causes extensive delays. Equation (19e) shows that if a node has lesser number of predecessor nodes then queuing delay for packet sufficiently reduces which in turn leads to lowered processing delays. For the purpose of delay optimization Equation (19f) suggests that holding time of a packet must be an optimum value. Moreover, it also states that nodes with higher residual energy have lower holding time. Therefore, lowered energy nodes also prevents them from data transmission by holding this specified clause. Equation (19g) indicates that packets being generated by \(i\) at time \(t\) must not exceeds the channel capacity. This clause mitigates the channel congestion possibilities and lowers overall packets drop rate. Implication of this law lowers data re-transmissions probability. Equation (19h) defines that major axis of sink trajectory must cover at least one third portion of the network width, this causes reduced transmission distance between source and destination nodes.