## 3 Proposed Model
### Application Energy Consumption  
The total power consumption of workflow application is a merger of computation time and communication time. Since computation cost could include location and remote execution after application partitioning. The communication cost is determined by the weight of data transport and available network bandwidth. The average power consumption of workflow application due to offloading is expressed as follows:  
\[EG_{total}=\sum_{v\in V}Pr_{v}\cdot EG_{v}^{loc}+\sum_{v\in V}(1-Pr_{v})\cdot EG _{v}^{cl}+\sum_{e(v_{i}v_{j})\in E}Pr_{e}\cdot T_{e}^{trans} \tag{2}\]  
whereas Equation (2) describes the average consumption of the workflow application, which is the sum of local and remote computation cost and communication cost. In the same way, the vector \(Y=\left\{y_{ij}:v\in V,j\in M\right\}\) with variable \(x_{ij}\) indicates either task offload or not, namely:  
\[x_{ij}=\left\{ \begin{array}{ll}1,&\text{if }v_{i}\in v_{loc}\\ 0,&\text{if }v_{i}\in v_{cl}\end{array} \right. \tag{3}\]  
According to Equation (4), the communication cost is determined by \(E_{cut}=1\), otherwise same location tasks have no communication cost. \[P_{r_{e}}=\left\{ \begin{array}{ll}1,&\text{if }e\in E_{cut}\\ 0,&\text{if }e\
otin E_{cut}\end{array} \right. \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline N & Number of tasks v \\ \(\Omega\lambda\) & Application partitioning factor during offloading \\ \(D_{i}\) & Deadline of the application \\ \(\zeta u_{i}\) & Speed rate of jth virtual machine VM \\ \(\zeta u_{m}\) & Speed rate of mobile processor \(v_{i}\) \\ \(C_{i}^{e}\) & Execution cost of the task \(v_{i}\) on cloudlet k \\ \(M_{i}^{e}\) & Execution cost of the task \(v_{i}\) on mobile \\ \(P_{cw}\) & Power consumption rate at cloudlet virtual machine VM \\ \(P_{mw}\) & Power consumption rate at the mobile device \\ \(P_{ij}\) & Assignment of task \(v_{i}\) on virtual machine VM, j \\ \(B_{i}\) & Begin time of the task \(v_{i}\) \\ \(F_{i}\) & Finish time of the task \(v_{i}\) \\ G (V, E) & DAG call graph \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mathematical notation. The considered problem is mathematically modeled as below:  
\[minZ=\sum_{i=1}^{N}\sum_{m=1}^{\psi}\sum_{j=1}^{M}x_{ij}\lambda_{i}\times\zeta_{ \mu j}\times P_{cw}\times C_{i}^{e}+T_{i}^{trans}+x_{ij}\lambda_{i}\times\zeta_{ \mu m}\times P_{cm}\times M_{i}^{e} \tag{5}\]  
\[\sum_{v\in V}Pr_{v}\cdot E{C}_{v}^{loc}=\ M_{i}^{e}=\sum_{i=1}^{N}x_{ij} \lambda_{i}\times\frac{W_{i}}{\zeta_{\mu m}} \tag{6}\]  
\[\sum_{v\in V}(1-Pr_{v})\cdot E{C}_{v}^{cl}=\ C_{i}^{e}=\sum_{i=1}^{N}x_{ij} \lambda_{i}\times\frac{W_{i}}{\zeta_{\mu j}} \tag{7}\]  
\[\sum_{e(v_{i,p})\in E}Pr_{e}\cdot T_{e}^{trans}=\ w\Big{(}e\Big{(}v_{i,v}j \Big{)}\Big{)}=\frac{in_{ij}}{B_{U}}+\frac{out_{ji}}{B_{D}} \tag{8}\]  
\[T_{j,0}=0, \tag{9}\]  
\[T_{i,j}=T_{i-1,j}+\sum_{i=1}^{N}x_{ij}\lambda_{i}C_{i}^{e}, \tag{10}\]  
\[C_{i}^{e}=\sum_{i=1}^{V_{e}}x_{ij}\lambda_{i}\times\frac{W_{i}}{\zeta_{\mu j}} \,M_{i}^{e}=\sum_{i=1}^{V_{loc}}x_{ij}\lambda_{i}\times\frac{W_{i}}{\zeta_{ \mu m}}, \tag{11}\]  
\[F_{i}=\sum_{j=1}^{M}T_{i,j}x_{ij}, \tag{12}\]  
\[\sum_{i=1}^{N}x_{ij}=1,\ \sum_{j=1}^{M}x_{ij}\lambda=1,\ \sum_{m=1}^{\psi}x_{ij} \lambda_{0}=1, \tag{13}\]  
\[minZ=\sum_{i=1}^{N}\sum_{m=1}^{\psi}\sum_{j=1}^{M}x_{ij}\lambda_{i}\times\zeta_{ \mu j}\times P_{cw}\times C_{i}^{e}+T_{i}^{trans}+x_{ij}\lambda_{i}\times\zeta_{ \mu m}\times P_{cm}\times M_{i}^{e} \tag{5}\]0  
Equation (5) is proposed for the measurement of power consumption of all tasks from start to end on clouds and local nodes on execution. Equation (6) is given for power consumption of mobile devices that how much power consumed during the execution of a particular task. Energy consumptions at the cloud side can be calculated by using Equation (7) and the same way for the cost of communication in power consumption Equation (8) is given. According to Equation (9) initial time for any VM machine is assumed to be zero.