## 2 System Architecture and Problem Formulation
### Problem Formulation
\[\mathrm{t_{m,j}^{tran}}=\frac{\mathrm{D_{m,j}}}{\mathrm{R_{m,j}}}=\frac{\sum_ {k=1}^{k_{m,j}}\mathrm{O_{m,j,k}}\mathrm{d_{m,j,k}}}{\mathrm{R_{m,j}}}, \tag{5}\]  
In the above equation, the transmission rate (\(\mathrm{R_{m,j}}\)) is determined by the Shannon Theorem, with B indicating the channel bandwidth, \(\mathrm{P_{m,j}}\) representing the device's transmission power, and \(\mathrm{G_{m,j}}\) indicating the channel gain. \(\mathrm{\sigma^{2}}\) stands for noise, while \(\mathrm{I_{m,j}}\) denotes the interference on this channel. \(\mathrm{R_{m,j}}\) is defined by Equation (6). \[\mathrm{R_{m,j}}=\mathrm{B}\log_{2}\left(1+\frac{\mathrm{P_{m,j}}\mathrm{G_{m,j}}}{\mathrm{\sigma^{2}}+\mathrm{I_{m,j}}}\right), \tag{6}\]  
The channel gain (\(\mathrm{G_{m,j}}\)) from the device to the small base station equals the gain (\(\mathrm{g_{0}}\)) divided by the square of the distance between the location (\(\mathrm{x_{SMD}}\), \(\mathrm{y_{SMD}}\)) of the smart mobile device (SMD) and the location of the small base station (\(\mathrm{SeNB}\)) (\(\mathrm{x_{SeNB}}\), \(\mathrm{y_{SeNB}}\)) plus the square of the base station's fixed height (H) [13]. In this context, \(\mathrm{g_{0}}\) denotes the channel gain at a reference distance of 1 m and a transmission power of 1 watt:  
\[\mathrm{G_{m,j}}=\frac{\mathrm{g_{0}}}{\mathrm{d}}=\frac{\mathrm{g_{0}}}{ \left(\mathrm{x_{SMD}}-\mathrm{x_{SeNB}}\right)^{2}+\left(\mathrm{y_{SMD}}- \mathrm{y_{SeNB}}\right)^{2}+\mathrm{H}^{2}}, \tag{7}\]The product of the device's transmission power (\(\rm{p_{i,j}}\)) and the channel gain (\(\rm{G_{m,j,i}}\)) from the connected small base station determines the total devices (\(\rm{U_{i,j}}\)) in adjacent cells employing an identical transmission channel to the smart device (\(\rm{U_{m,j}}\)), leading to the evaluation of channel interference (\(\rm{I_{m,j}}\)). The channel decision (\(\rm{a_{j,i,m,j}}\)) depends on whether devices (\(\rm{U_{i,j}}\)) in neighboring cells utilize the same transmission channel as the device (\(\rm{U_{m,j}}\)); a value of 1 signifies channel congruence, whereas 0 denotes disparate channels. \[\rm{I_{m,j}}=\sum\
olimits_{l=1,l\
eq m}^{M}\sum\
olimits_{i=1}^{U_{l}}\rm{a_ {l,i,m,j}}\rm{p_{i,j}}\rm{G_{m,j,i}}, \tag{8}\]  
We multiply the aggregate data size of transmitted tasks (\(\rm{D_{m,j}}\)) by the transmission power (\(\rm{p_{m,j}}\)) of the device and subsequently divide by the transmission rate (\(\rm{R_{m,j}}\)) from the device to the small base station [2]. The energy consumption (\(\rm{e_{m,j}^{\rm{tran}}}\)) follows the equation  
\[\rm{e_{m,j}^{\rm{tran}}}=\rm{P_{m,j}}\frac{\rm{D_{m,j}}}{R_{m,j}}=\frac{\rm{P_ {m,j}}\sum\
olimits_{k=1}^{K_{m,j}}\rm{O_{m,j,k}}\rm{d_{m,j,k}}}{\rm{R_{m,j}}}, \tag{9}\]  
The total computational capacity needed for all remote computing tasks is calculated by multiplying the computational capacity required for each task (\(\rm{c_{m,j,k}}\)) by the offloading decision (\(\rm{O_{m,j,k}}\)) and summing the results. Subsequently, this total is divided by the computational capacity of the edge server (F):  
Figure 3:0  
The total network latency (T) encompasses the duration of local computation (\(\rm{t_{m,j}^{\rm{local}}}\)) for all smart device tasks, the transmission time for remote computation (\(\rm{t_{m,j}^{\rm{tran}}}\)), and the task computation time at the edge server (\(\rm{t_{m,j}^{\rm{mec}}}\)):  
Figure 3:1  
The total energy consumption (E) comprises the energy used for local computation (\(\rm{e_{m,j}^{\rm{local}}}\)) and the transmission energy for remote computation (\(\rm{e_{m,j}^{\rm{tran}}}\)) of all smart device tasks. Figure 3:2  
This study considers three constraint violations (CVs): the combined energy of local computation (\(\rm{t_{m,j}^{\rm{local}}}\)) and remote computation transmission (\(\rm{t_{m,j}^{\rm{tran}}}\)) must not exceed the device's remaining energy (\(\rm{e_{m,j}}\)). Figure 3:3  
Smart devices must possess computational capabilities that surpass the requirements for local computations.