## 4 Problem Definition  
As with TDMA-based and round-robin methods in clustered sensor networks, each transmission round is divided into frames, f. In each frame, cluster members send a fixed-length (L) data packet to their cluster head. We consider a data backlogging system where nodes capture measurements for long-term monitoring applications and always have data to transmit when it is their turn. In long-term monitoring applications, we consider a clustered network, assuming bi-directional communication in cluster heads to operate control signals and data aggregation. Additionally, to evaluate energy efficiency in these applications, low latency and high reliability are considered secondary concerns. This approach allows for the sustainability of the network over long periods, prioritizing energy efficiency over quality of service (QoS) and ensuring data integrity at the cluster head level. All nodes have the same data rate and are allowed to send the same number of packets in each frame. We use transmission power control (TPC) so that each transmission is held with a transmission power of nodes to satisfy a predetermined SNR for successful packet reception [34]. We consider a network of non-uniformly dispersed 100 nodes in varied depths underwater. Nodes maintain channel access in TDMA fashion, communicating with cluster heads over their scheduled time slots per frame. Cluster heads determine the time intervals and notify cluster members of their time slots for data transmission. We denote the set of \(N\) acoustic sensor nodes by \(S=\{s_{1},s_{2},\ldots,s_{N}\}\). We assume stochastic energy harvesting processes modeled by Poisson distribution at each node. The remaining energy and the amount of harvested energy of a sensor node \(s_{i}\) in a time slot \(t\) denoted as \(E_{i}(t)\) and \(E_{i}^{h}(t)\), respectively. \(E_{i}^{h}(TS)\) represents total harvested energy per frame. Considering frames consist of \(TS\) many transmission slots, the same as the number of nodes in clusters, the transmission slot set is represented as \(TS=\{t_{1},t_{2},\ldots,t_{N}\}\), the total amount of harvested energy in cluster within time steps from 1 to TS in frame, under a policy \(\pi\) where \(t_{tx_{i}}\) is the designated time slot index of sensor i, \(\mathbf{1}_{\{t_{tx_{i}}\}}\in\{0,1\}\) is the indicator function, and harvested energy at time slot \(t_{m}\) can be written as \(E_{i}^{h}(t_{m})=E_{i(t_{m-1})}^{h}\mathbf{1}_{\{t_{tx_{i}}\}}\). Overall harvested energy under a policy is determined as  
\[V_{i}^{\pi}(TS)=\sum_{i=1}^{N}V_{i}^{\pi}(TS)=\sum_{i=1}^{N}\sum_{t=1}^{TS}E_{ i}^{h}(t) \tag{14}\]  
where \(V_{i}^{\pi}(TS)\) is the sum of harvested energy under policy \(\pi\) in the cluster from starting time slot 1 through \(TS\)  
\[E_{i}^{h}(TS)=\sum_{t=1}^{TS}\left[E_{i}^{h}(t)-\sum_{t=t_{tx_{i}}}^{t_{tx_{i }}+1}E_{i}^{h}(t)\mathbf{1}_{\{t_{tx_{i}}\in TS\}}\right] \tag{15}\]  
Our objective is to maximize the total amount of harvested energy by each node over a finite time horizon,  
\[\max_{\{\pi(t)\}_{t=1}^{T}}\frac{1}{T}\mathbb{E}[V_{i}^{\pi}(T)] \tag{16}\]  
In clustered networks, optimal time slot scheduling algorithms employ cluster heads to schedule the transmission slots of each node. However, the cluster head may not select the optimal transmission slot without incurring communication overhead to gather knowledge on the energy harvesting states of nodes. Even when this information is provided to the cluster head, its timeliness might be compromised due to costs associated with complexity and computational delays due to slow convergence. Adopting a global optimal solution approach is not feasible. Therefore, instead of relying on centralized methods, delegating the transmission slot allocation task to cluster members not only delegates autonomy but also unveils the harnessing of ambient energy through adaptive transitions between transmission and idle energy harvesting modes.