## 3 Motivational Scenario and System Model
### Collaborative Execution Model  
To solve the limited capacity and the overload problems, a mobile device user can offload their tasks to the target server for processing. We assume that mobile device users always offload their computation tasks to their corresponding local SBS-MEC server, but if a set of SBS-MEC servers works collaboratively, user devices can experience better edge services. Otherwise, the computation task will be further offloaded to the remote cloud. According to the connections between mobile device user, SBS-MEC server, and remote cloud, there are three types of collaboration models: mobile device user to SBS-MEC server, SBS-MEC server to remote cloud, and SBS-MEC server to SBS-MEC server. The notations used throughout this paper are summarized in Table 2.  
Let \(\psi_{1}^{\textit{sbs}},\psi_{2}^{\textit{sbs}},\ldots,\psi_{n}^{\textit{sbs}}\) represent the SBS-MEC server's computational capacity of small-cell base station \(1,2,\ldots,n\), respectively. Let \(\omega_{1},\omega_{2},\ldots,\omega_{m}\) be the amount of received mobile workload from various mobile device users \(1,2,\ldots,m\), respectively. Note that the collaboration model occurs if and only if a mobile device user cannot execute the task(s) locally. For user #1, we consider the workload \(\omega_{1}\) which cannot be processed locally by itself. By comparing \(\psi_{1}^{\textit{sbs}}\) and \(\omega_{1}\), there are three possible ways to process as follows:  
1. Case 1: \(\omega_{1}\) is less than \(\psi_{1}^{\textit{sbs}}\). This computation task will be executed in SBS-MEC 1, as shown in Figure 2a, where some of the tasks will be computed locally by themselves and the remaining tasks will be offloaded to the SBS-MEC 1 server for execution. 2. Case 2: Figure 2b shows \(\omega_{1}\) is greater than \(\psi_{1}^{\textit{sbs}}\) and there are no collaborations between neighboring SBS-MEC servers or the total processing time is minimized if there are collaborations between neighboring SBS-MEC servers. In this case, this offloaded task \(\omega_{1}\) will be divided according to \(\psi_{1}^{\textit{sbs}}\) into two parts: one is processed in the SBS-MEC server 1; the remaining task, \(\tau_{1}=\omega_{1}-\psi_{1}^{\textit{sbs}}\), will be sent to the remote cloud for processing. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Meaning** \\ \hline \(M\) & Number of mobile device users \\ \(N\) & Number of SBS-MEC servers \\ \(T\) & Number of tasks \\ \(\psi^{\textit{sbs}}\) & SBS-MEC server’s computational capacity \\ \(\omega_{i}\) & Amount of received mobile workload from mobile user \(i\) \\ \(v_{i}^{t}\) & Computation task is generated from each mobile device user \(i\) \\ \(\Im_{i}(t)\) & Local-executed computation task \\ \(d\) & Size of the input data for computation \\ \(\lambda\) & Number of CPU cycles needed to process the task \\ \(\zeta_{i}^{\textit{max}}\) & Highest allowable latency for accomplishing the task \\ \(\psi_{i}^{\textit{max}}\) & Maximum task arrival rate that can be handled by the SBS-MEC server \(i\) \\ \(\dot{\gamma}^{t}\) & Total computation tasks arriving in SBS-MEC server at time slot \(t\) \\ \(\alpha\) & Task size \\ \(\beta\) & Delay sensitivity of task \\ \(\gamma\) & Local SBS-MEC VM utilization \\ \(\delta\) & Network delay \\ \(\theta\) & Neighboring SBS-MEC VM utilization \\ \(\chi^{*}\) & X-coordinate of the center of gravity \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations and their meanings. Case 3: \(\omega_{1}\) is greater than \(\psi_{1}^{sbs}\) and there are collaborations between neighboring SBS-MEC servers. The minimized total processing time will elapse if this task is processed in neighboring SBS-MEC servers, as shown in Figure 3. Similar to case 2, this offloaded task \(\omega_{1}\) will be divided according to \(\psi_{1}^{sbs}\) into two parts: one is processed in SBS-MEC server 1; the remaining task, \(\tau_{1}=\omega_{1}-\psi_{1}^{sbs}\) needs to be helped.