## 3 System Model and Problem Formulation
### System Model  
Different devices run tasks that cause different delays. This paper considers two cases: the local execution of the current task and it is offloaded to the MEC server. Since the strategy is to make offloading decisions for tasks with dependencies, the finish time of the previous tasks has an impact on the subsequent tasks. Therefore, this section will discuss the calculation delay, transmission delay, and task finish time calculation. Let \(T_{i}^{loc}\) denote the local computation delay and \(T_{i}^{s}\) denote the MEC server computation delay of the task \(v_{i}\). In addition, \(f_{m}^{s}\) and \(f_{i}^{loc}\) are the CPU clock speed of the MEC server numbered m and the mobile device where task \(v_{i}\) is located, respectively. \(\varpi_{i}\) is the number of clock cycles required to process each bit of data of task \(v_{i}\), and \(C_{i}\) is the total number of clock cycles required by task \(v_{i}\). \[T_{i}^{loc}\ =\ \frac{C_{i}}{f_{i}^{loc}}\ =\ \frac{d_{i}\ \times\ \varpi_{i}}{f_{i}^{loc}},\ T_{i}^{s}\ =\ \frac{C_{i}}{f_{m}^{s}}\ =\ \frac{d_{i}\ \times\ \varpi_{i}}{f_{m}^{s}}. \tag{1}\]  
Before calculating the delay of data offloading on mobile devices, the upload rate should be calculated according to [PERSON]'s formula, which can be defined as:  
\[R_{i}(\omega)\ =\ B\ \times\ \log_{2}\{1\ +\ \frac{[\omega P_{send}\ +\ (1\ -\ \omega)P_{ rec}]\ \times\ H_{i}^{t}}{\sigma^{2}}\} \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline
**Notations** & **Definition** \\ \hline \(a_{i}\) & The offloading action of task \(v_{i}\) \\ \(M\) & MEC server collection \\ \(T_{i}^{loc}\), \(T_{i}^{s}\) & The local and MEC server computation delay of the task \(v_{i}\) \\ \(f_{m}^{s}\) & The CPU clock speed of the MEC server numbered m \\ \(f_{i}^{loc}\) & The CPU clock speed of the mobile device where task \(v_{i}\) is located \\ \(R_{i}(\omega)\) & Sending rate or receiving rate \\ \(C_{m}^{MEC}\) & Total computing resources of the MEC server numbered m \\ \(d_{i}\) & The data size of task \(v_{i}\) \\ \(C_{i}\) & The total number of clock cycles required by task \(v_{i}\) \\ \(F_{insh}\_T_{i}\) & The finish time of task \(v_{i}\) \\ \(F_{insh}\_T_{i}^{loc}\) & Local finish time of task \(v_{i}\) \\ \(F_{insh}\_T_{i}^{trans}(\omega)\) & The finish time of upload and download task \(v_{i}\) \\ \(F_{insh}\_T_{pre}\) & The finish time of the previous task \\ \(F_{insh}\_T_{i,m}^{ser}\) & The finish time of task \(v_{i}\) on the MEC server numbered m \\ \(A\)_Avail\_T_{i}^{loc}\) & The CPU idle time of the earliest available task on the local processor \\ \(A\)_Avail\_T_{i,m}^{trans}(\omega)\) & The earliest available time of the transmission link \\ \(A\)_Avail\_T_{i,m}^{ser}\) & The earliest available CPU idle time of the MEC server \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._where \(B\) is the wireless channel bandwidth between the mobile device and the edge cloud, and \(P_{send}\) and \(P_{rcc}\) are the transmitted and received power, respectively. \(\omega\)\(\in\){0, 1}, where, when \(\omega\) = 1, \(R_{i}\)(\(\omega\)) represents the sending rate, and when \(\omega\) = 0, \(R_{i}\)(\(\omega\)) represents the receiving rate. \(H_{i}^{t}\) is the wireless channel gain between the device and the base station in time slot \(t\), and \(\sigma^{2}\) is the noise power consumption. Therefore, the value of \(\omega\) in \(T_{i}^{trans}\)(\(\omega\)) is 1 and 0, which, respectively, represent the upload delay or download delay between the mobile device and the MEC server. The transmission delay can be defined as:  
\[T_{i}^{trans}(\omega)\ =\ \frac{d_{i}}{R_{i}(\omega)} \tag{3}\]  
where \(M\) is the set of MEC server numbers, and the size of this set depends on the number of servers. Let O\({}_{1:\ n}\) = [\(a_{1}\), \(a_{2}\),..., \(a_{k}\),..., \(a_{n}\)] denote the offloading decision for all tasks, \(a_{k}\)\(\in\)[0, M]. Specially, \(a_{k}\) = 0 means that task \(v_{k}\) is scheduled to the local processor.