## 3 Proposed System Model
### DAWPC-MAC Model Overview
This dynamic adjustment of power allows for efficient energy management in the network. The condition is expressed as follows:  
\[P_{\text{WS}}= \begin{cases}P_{\text{min}}&\text{(initial low power)}\\ P_{\text{min}}+k\cdot\Delta P+L(d)&\text{(increased power for extended coverage)}\\ 0&\text{(sleep mode with zero power).}\end{cases} \]  
In this context, \(P_{\text{min}}\) represents the minimum wake-up signal power, while \(k\) is the iteration count used to incrementally increase the power to enhance communication range. The power step for this increment is denoted as \(\Delta P\), and \(L(d)\) refers to the path loss function,  
Figure 3: Space-time uncertainty in UWSNs: (**a**) Scheduled Time Slot Network with Collision Occurrence: A scenario where nodes are assigned specific time slots, but a collision occurs due to overlapping transmission slots. (**b**) Scheduled Time Slot Network without Collision: A scenario where slots are allocated without collisions, ensuring interference-free communication. (**c**) Collision–Free Adaptive Slot Scheduling Network: A dynamic slot scheduling method that reduces the likelihood of collisions by adjusting time slot assignments and transmission power based on network conditions. which is influenced by the distance between the nodes and environmental factors such as water salinity and temperature as depicted in Equation (2): Additionally, \(P_{\text{sleep}}\) represents the minimal transmission power during sleep mode. In sleep mode, the power is set to zero to ensure that no power is consumed when the node is not actively communicating, thereby optimizing energy consumption and maximizing efficiency in the network. The current model defines path loss as \(L(d)\), a function of distance \(d\) and environmental factors like water properties (salinity \(w\), temperature \(T\)). \[L(d)=d\cdot\alpha(f)+L(w,T), \tag{2}\]  
while the attenuation, as defined by using [PERSON]'s model, can be computed for a given frequency \(f\) and included in \(L(d)\). Where, \(\alpha(f)\) is the \"Attenuation per kilometer\" as defined by [PERSON]'s formula. Propagation path-loss \(P_{L}\) for acoustic wave in underwater, expressed in dB, can be given by [38] as depicted in Equation (3):  
\[P_{L}=20\log(r)+\alpha r\times 10^{-3}, \tag{3}\]  
where \(\alpha\) represents the absorption coefficient in dB/km and \(r\) is the transmission range. The absorption coefficient \(\alpha\) can be calculated using [PERSON]'s expression at frequencies above a few hundred Hz as below. Where, \(\alpha(f)\) : Attenuation per km as defined by [PERSON]'s formula [39] in Equation (4):  
\[\alpha(f)=\frac{0.11\cdot f^{2}}{1+f^{2}}+\frac{44\cdot f^{2}}{4100+f^{2}}+2.7 5\times 10^{-4}\cdot f^{2}+0.003. \tag{4}\]  
To conserve energy, nodes in UWSNs initially operate at low transmission power, reducing their coverage area and preventing unnecessary energy expenditure. If no response is received from nearby nodes, the system adapts by increasing the transmission power to expand the coverage area and enhance the chances of establishing communication.The minimum necessary transmission power level, \(P_{\text{required}}\), for the data packet can be calculated by node \(T_{x}\) based on the received power level \(P_{r}\), the transmitted power level \(P_{0}\), and the noise level at the receiver \(P_{n}\). The calculation is given by [40] in Equation (5):  
\[P_{\text{required}}=\frac{P_{0}\cdot G\cdot\text{SIR}_{\text{thresh}}}{P_{r}}. \tag{5}\]  
Once \(P_{\text{required}}\) is determined, it is specified by node \(j\) in the Clear-To-Send (CTS) message to node \(i\). After the CTS is received, the data packet is transmitted by node \(i\) using the power level \(P_{\text{required}}\). Since the signal-to-noise ratio at the receiver \(j\) is considered, a more accurate estimation of the appropriate transmission power level for the data is ensured by this method.