## 6 Optimization Scenarios for 6G MCNs
### Task Offloading to Edge Nodes in MCNs  
Many maritime nodes such as ships, UAVs, AUVs, and buoys are equipped with on-board sensors and computing units that continuously generate data (e.g., from video feeds, radar, sonar, or environmental monitoring). However, processing these data-intensive workloads locally can lead to high energy consumption, increased latency, or degraded performance, especially in resource-constrained platforms. Sixth-generation introduces a continuum of edge, aerial, and satellite computing nodes, enabling these maritime devices to offload computation to more powerful or better-situated processing units [39; 98; 99]. The goal is to optimize the offloading decision such that total latency and energy consumption are minimized while respecting network and compute constraints. **Optimization Problem Formulation:** Let \(\mathcal{U}\) denote the set of task-generating nodes (e.g., ships), and \(\mathcal{E}\) denote the set of candidate computing edge nodes (e.g., UAVs, satellites, Sea BSs). We consider a per-node decision \(D_{i}=1\) when the task is to be offloaded. If offloading is decided, each task offloading decision can be represented by \(x_{i,n}\in\{0,1\}\), a binary variable that is \(x_{i,n}=1\) when node \(i\in\mathcal{U}\) offloads its task to edge node \(n\in\mathcal{E}\) (otherwise 0). When the task is locally processed in node \(i\), we denote the processing delay as \(T_{i}^{local}\) and the energy consumed as \(E_{i}^{local}\). With a known number of CPU cycles \(c_{i}\) needed for task \(i\) and local CPU frequency \(f_{i}\) (in Hz) of node \(i\), the time to compute the task locally is [39; 98]:  
\[T_{i}^{\text{local}}=\frac{c_{i}}{f_{i}} \tag{6}\]  
Similarly, when the task of node \(i\) is offloaded to edge node \(n\), we let \(T_{i,n}^{offload}\) be the offloading delay and \(E_{i,n}^{offload}\) be the respective energy consumption. The computational capacity of edge node \(n\) is \(C_{n}\) (in Hz). Note that, \(T_{i,n}^{offload}\) is the total delay to transmit the task towards node \(n\) (denoted as \(T_{i,n}^{tx}\)), plus the delay to process it at node \(n\) (denoted as \(T_{i,n}^{proc}\)), which can be written as:  
\[T_{i,n}^{\text{offload}}=T_{i,n}^{\text{tx}}+T_{i,n}^{\text{proc}} \tag{7}\]  
The goal is to minimize the total weighted cost of delay and energy across all tasks and offloading options, as reflected by the optimization problem:  
\[ \begin{split}\underset{D_{i}\
eq i,n}{\min}&\sum \limits_{i\in\mathcal{U}}(\begin{array}{c}D_{i}\big{(}\alpha T_{i}^{\text{ local}}+\beta E_{i}^{\text{local}}\big{)}+(1-D_{i})\sum\limits_{n\in \mathcal{E}}x_{i,n}\Big{(}\alpha T_{i,n}^{\text{offload}}+\beta E_{i}^{\text{ offload}}\Big{)}\end{array})\\ \text{s.t.}&\text{(C1)}\;D_{i}+\sum\limits_{n\in\mathcal{E}}x_{i,n}\leq 1,\;\forall i\in\mathcal{U}\\ &\text{(C2)}\sum\limits_{i\in\mathcal{U}}x_{i,n}\cdot c_{i}\leq C_{n},\; \forall n\in\mathcal{E}\\ &\text{(C3)}\,x_{i,n}\cdot\frac{d_{i}}{B_{i,n}}\leq T_{max},\; \forall(i,n)\in\mathcal{U}\times\mathcal{E}\end{split}  \tag{8}\]  
where \(\alpha\) and \(\beta\) are application-specific constant weighting factors (e.g., emphasizing latency over energy or vice versa). Evidently, the objective function represents a weighted optimization of delay and energy consumption for both local computation (first term) and edge computation upon offloading (second term). The goal is to find the decision variables \((D_{i},x_{i,n})\) that minimize the overall delay and power consumption, while respecting constraints (C1)-(C3). The first constraint (C1) ensures that each task is executed to at most 1 node. The second constraint (C2) guarantees that the total processing demand of all tasks offloaded to edge node \(n\) does not exceed its computational budget \(C_{n}\).