## 3 Network Architecture and System Model
### Computing Resources Using a Fussy Set Model  
Realistically, the computational resources of each node can be greatly influenced by external factors. [PERSON] et al. [4] considered the resource uncertainty of MECN in a traditional environment. In this paper, to account for the computing resource uncertainty of ECNs, we propose a realistic scenario-based set with a definite probability distribution to represent the possible range of uncertain resources. Based on a fuzzy set of random parameters \(\xi\), we will make the one-stage offloading decision and the two-stage cloud resource request. [PERSON] et al. [39] considered a set of uncertainties with specific boundaries and moments, derived from real-world data, i.e.,  
\[\mathcal{D}\big{(}\xi,\sum\mu_{0},\gamma_{1},\gamma_{2}\big{)}\] (19) \[=\left\{\begin{aligned} P\big{\{}\xi\in\mathcal{M} \big{\}}=1\\ P\  
Above all, we express (18) in terms of matrix form variables and parameters. We consider the set \(\mathcal{D}=\mathcal{B}\cup\mathcal{U}\cup\mathcal{S}\) and \(M=B+U+S\). The formulated problem is written as follows:  
\[\min_{X}\ \mathbb{N}^{compT}\times X\times\delta \tag{20}\] \[s.t.\ \ X\times O_{m}=O_{n},\] (20a) \[X^{T}\times\mathbb{N}^{up}\leq L^{cache},\] (20b) \[X^{T}\times F\leq L^{comp},\] (20c) \[X\in R^{N\times M},\delta\in R^{N},\mathbb{N}^{comp}\in R^{M}, \mathbb{N}^{up}\in R^{N},\] \[f\in R^{N},L^{cache}\in R^{M},L^{comp}\in R^{M} \tag{20d}\]  
where the offloading strategy matrix variable \(X:=\left\{x_{nm}\right\}_{n\in\mathcal{N},m\in\mathcal{D}}\), the upper bound of the computing resource vector parameter \(L^{comp}:=\left\{L^{comp}_{m}\right\}_{m\in\mathcal{D}}\), the upper bound of the cache ceiling vector parameter \(L^{cache}:=\left\{L^{cache}_{m}\right\}_{m\in\mathcal{D}}\), the computing resource vector parameter \(F:=\left\{f_{i}\right\}_{i\in\mathcal{N}}\), the ratio between the cost and the data size vector parameter \(\delta:=\left\{\delta_{i}\right\}_{i\in\mathcal{N}}\), and the all-ones vector parameter is \(\mathcal{O}\). Thus, given the uncertainty of the parameter \(L^{comp}\), a two-stage offloading strategy model is created using the distribution set proposed above. We constructed a fuzzy set based on the historical data of the computing resources of the offload nodes, expressed as follows: \(\xi=\hat{L}^{comp}\). \[\min_{X}\ \mathbb{N}^{compT}\times X\times\delta+\mathbb{E}_{\xi}[\theta(X, \xi)], \tag{21}\]  
\[s.t.((\ref{eq:20a})-(\ref{eq:20d})), \tag{21a}\] \[\left\{P\left\{\xi\in\mathcal{M}^{\xi}\right\}=1\right.\] \[=\left\{\begin{aligned} P\big{\{}\xi\in\mathcal{M} \big{\}}=1\\ P\  
Above all, we express (18) in terms of matrix form variables and parameters. We consider the set \(\mathcal{D}=\mathcal{B}\cup\mathcal{U}\cup\mathcal{S}\) and \(M=B+U+S\). The formulated problem is written as follows:  
\[\min_{X}\ \mathbb{N}^{compT}\times X\times\delta \tag{20}\]0 \[\text{with}\ \theta(X,\xi)=\left\{ \begin{aligned} &\min_{Y}\ \ \varphi^{T}\times Y,\\ &(X\otimes F^{req})^{T}\times O_{n}\leq\xi+Y,\\ & 0\leq Y,\end{aligned} \right. \tag{21b}\]  
\(\theta(X,\xi)\) represent the additional cost of uploading tasks to the cloud when the real-time computing resources change in a phase of decision task offloading and are insufficient to meet demand. Vector \(\varphi\) indicates the ratio of additional costs to required computational power. Instead, we further consider the mean-CVaR (conditional value at risk) criterion in the model, which can be expressed in the following form:\[VaR_{\alpha}(\zeta) =\inf\{v\in R:\mathbb{F}(v)\geq\alpha\} \tag{22a}\] \[CVaR_{\alpha}(\zeta) =\mathbb{E}\{\zeta|\zeta\geq VaR_{\alpha}(\zeta)\}\] \[=\min_{v\in R}\{v+\frac{1}{1-\alpha}\mathbb{E}[(\zeta-v)_{+}]\} \tag{22b}\]  
where \(\mathbb{F}\) is the cumulative distribution function of the function used to solve the random variable \(\zeta\), and a is the given confidence level. \((a)_{+}\) means \(\max(a,0)\).