## 3 System Model
The transmission probability is given by the following:  
\[P_{\text{trans}}=1-(1-p)^{W},\]  
where  
* \(p\) is the probability of a node attempting to transmit in a given time slot,
* \(W\) is the contention window size. The probability of successful transmission, denoted as \(P_{\text{success}}\), is defined as the likelihood that no competing node initiates transmission within the same time slot. This probability can be mathematically expressed as follows:  
\[P_{\text{success}}=P_{\text{trans}}\times\prod_{i=1}^{N-1}(1-P_{\text{trans}}).\]  
Here, \(N\) denotes the total number of nodes attempting to transmit within the same time slot. In the TDMA protocol, the communication timeline is partitioned into discrete time slots, with each node assigned a dedicated slot for data transmission. Let \(T_{\text{frame}}\) denote the duration of a complete frame, and let \(T_{\text{slot}}\) slot the duration of an individual time slot. The total number of slots per frame, represented by \(S\), can then be calculated as follows:  
\[S=\frac{T_{\text{frame}}}{T_{\text{slot}}},\]  
where  
* \(T_{\text{frame}}\) frame represents the total duration of a complete TDMA frame. * \(T_{\text{slot}}\) slot denotes the duration allocated to each individual time slot within the frame. In this scenario, each node is assigned a time slot such that no collisions occur, and the transmission within each time slot is guaranteed to be collision-free. The probability of a successful transmission in TDMA, \(P_{\text{TDMA success}}\), is as follows:  
\[P_{\text{TDMA success}}=1-\sum_{i=1}^{N}P_{\text{collision}}(i),\]  
where  
* \(P_{\text{collision}}(i)\) is the probability of collision for node \(i\), which is zero in a well-synchronized TDMA system where each node has a dedicated time slot. Figure 3: Proposed hybrid TDMA and CSMA-CA MAC architecture for UWSNs. The mathematical modeling of the hybrid architecture switches between CSMA/CA and TDMA depending on the network load. Let \(L\) denote the network load, where \(L\) is the ratio of the total count of currently active nodes to the total number of nodes. If \(L<L_{\text{threshold}}\), CSMA/CA is used, and, if \(L\geq L_{\text{threshold}}\), TDMA is used. The switching condition is given by the following:  
\[L_{\text{current}}=\frac{N_{\text{active}}}{N},\]  
where  
* \(N_{\text{active}}\) is the number of active nodes. * \(N\) is the total number of nodes in the network. When \(L_{\text{current}}\) crosses the threshold, the system switches from CSMA/CA to TDMA, ensuring optimal performance for both low and high network loads. To mathematically formulate the optimization model of energy efficiency, each node adjusts its transmission power based on environmental conditions, such as path loss, depth, and temperature. The transmission power \(P_{\text{tx}}\) of a node is modeled as a function of the distance \(d\) to the receiver, the path loss exponent \(\alpha\), and the environmental factors \(E_{\text{env}}\):  
\[P_{\text{tx}}(d)=P_{\text{base}}\cdot\left(\frac{d}{d_{0}}\right)^{-\alpha} \cdot E_{\text{env}},\]  
where  
* \(P_{\text{base}}\) is the base transmission power. * \(d_{0}\) is the reference distance. * \(\alpha\) is the path loss exponent. * \(E_{\text{env}}\) represents environmental conditions affecting signal strength. By dynamically adjusting the transmission power, the system can minimize energy consumption while ensuring reliable communication. Network architecture model formulation was the subsequent step to then give due consideration. To elaborate on this specific aspect, the initial considered scenario (Figure 1) needed to be briefly outlined again. The UWSN consists of sensor nodes positioned at different underwater sites, creating a multihop communication topology. The system time is divided into superframes, each composed of a beacon frame; a control period (CP), which is referred to as a BEACON; and a data transmission period (DTP).