## 3 Motivational Scenario and System Model
### System Model  
The key is to collect information from all SBS-MEC servers, and, according to the fuzzy rules, it decides the task handling location. Therefore, we design the architecture of the fuzzy based collaborative task offloading scheme for MEC-enabled SCN. Figure 4 shows our designed architecture: mobile device users, SBS with MEC server and remote cloud. The first layer consists of different end users who want to compute their tasks either locally or offload to the local SBS. The second layer is made up of a number of SBSs in which a single MEC server is deployed to each BS. Virtualization technology, such as virtual machine (VM), is used by the MEC server to create execution environments. The distributed SBS-MEC server is connected to the fuzzy-based edge orchestrator. The fuzzy edge orchestrator module acts as a coordination point of the system to collect information from all SBS-MEC servers and, according to the fuzzy rules, it decides the task handling location. Finally, the core network is attached to the remote cloud for providing centralized cloud computing services.  
According to above consideration, we define \(N=\{1,2,3,\ldots,N\}\), which represents the number of SBS-MEC servers deployed to the small-cell network. Let \(M=\{1,2,3,\ldots,M\}\) be the number of mobile device users, who are covered by \(N\) SBS. We assume that the computation task, \(v_{i}^{t}\), is generated from  
Figure 4: Proposed collaborative architecture.  
each mobile device user \(i,i\in M\) at each time slot \(t\) following the Poisson processing. The total arrival computation tasks in one SBS in time slot \(t\), \(v^{t}\), is given as:  
\[v^{t}=\{v^{t}_{m}\}_{m\in M} \tag{1}\]  
Moreover, each mobile device user can execute a task at time \(t\). The local-executed computation task is defined as:  
\[\Im_{i}(t)=\{d_{i},\lambda_{i},\zeta^{max}_{i}\} \tag{2}\]  
where \(d_{i}\) denotes the size of the input data for computation, \(\lambda_{i}\) represents the number of CPU cycles that need to process the task \(\Im_{i}(t)\), and \(\zeta^{max}_{i}\) describes the highest allowable latency for accomplishing the computation task.  
Considering each SBS, let \(M_{i}\) be the number of mobile device users in the SBS-MEC server \(i\). The total arrival computation task, \(\Upsilon^{t}_{i}\), will be offloaded to the SBS-MEC server, \(i\), and it can be computed as follows:  
\[\Upsilon^{t}_{i}=\sum_{m\in M_{i}}(v^{t}_{m}-\Im_{m}(t)) \tag{3}\]  
The number of computation tasks cannot be executed in SBS-MEC server \(i\) and is given as shown below:  
\[\delta_{i}=\psi^{max}_{i}-\Upsilon^{t}_{i} \tag{4}\]  
where \(\psi^{max}_{i}\) is the maximum task arrival rate that can be handled by the SBS-MEC server, \(i\). If \(\delta_{i}<0\), the SCN needs collaboration to execute all remaining computation tasks for minimization of the total processing time. Otherwise, the system can run according to the baseline scheme.  
The total computation tasks arriving at all SBS-MEC servers in the SCN at time slot \(t\), \(\Upsilon^{t}\) is computed as:  
\[\Upsilon^{t}=\{\Upsilon^{t}_{i}\}_{i\in N} \tag{5}\]  
The problem under consideration is: how to design decisions for all tasks \(\Upsilon^{t}\) which are executed at local SBS-MEC servers or offloaded to neighboring SBS-MEC servers. Thanks to the fuzzy-based edge orchestrator, each task will be decided based on fuzzy-logic approach, which will be described in the next section.