## 3 Problem Formulation
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Notation** & **Description** & **Notation** & **Description** \\ \hline D & Datacenter sites & \(\tau_{ij}\) & Pheromone value between \(VM_{j}\) and \(PM_{i}\) \\ \hline \(PM\) & List of servers in a datacenter & \(\tau_{0}\) & Initial pheromone value \\ \hline \(PM_{i}^{CPU}\) & Total CPU of \(PM_{i}\) & \(\rho\) & Pheromone decay parameter (\(0<\rho<1\)) \\ \hline \(PM_{i,current}^{CPU}\) & \(PM_{i}\) current \(CPU\) utilization & \(\Delta\tau_{ij}\) & Pheromone reinforcement value \\ \hline \(PM_{i}^{RAM}\) & Total RAM of \(PM_{i}\) & \(S^{Best}\) & Current generation’s best solution \\ \hline \(PM_{i}^{BW}\) & Total bandwidth of \(PM_{i}\) & \(N^{ants}\) & Number of ants \\ \hline \(PM_{i,t}^{power}\) & \(PM_{i}\) power consumption at time t & \(q_{0}\) & Exploration and exploitation behavior of ants \\ \hline \(PM_{i,current}^{BW}\) & Current \(PM_{i}\) bandwidth & \(q\) & Random number [0, 1] \\ \hline \(PM_{i,available}^{BW}\) & \(PM_{i}\) available network bandwidth & \(\eta_{ij}\) & Heuristic info between \(VM_{j}\) and \(PM_{i}\) \\ \hline \(PM_{ij}^{Power}\) & Estimated power consumption of \(VM_{j}\) & \(\beta\) & Importance of heuristic information \\  & after placing on \(PM_{i}\) & \(\beta\) & Probability of placing \(VM_{j}\) and \(PM_{i}\) \\ \hline \(PM_{j}^{available}\) & Set of available \(PMs\) for placement & \(P_{ij}\) & Probability of placing \(VM_{j}\) and \(PM_{i}\) \\ \hline \(PM_{i,min}^{Power}\) & \(PM_{i}\) idle power & \(PM_{i,t+}^{overall}\) & Overhead power consumption of server at time t \\ \hline \(PM_{i,max}^{Power}\) & Peak power of \(PM_{i}\) & \(H_{t}\) & Datacenter outside temperature at time t \\ \hline \(VM_{j,estimated}^{BW}\) & Estimated bandwidth usage of \(VM_{j}\) & \(C_{T}\) & The cost of running all \(VM\)s at different sites \\ \hline \(PM_{i,t+}^{power}\) & Power consumption of \(PM_{i}\) at time t after placing new \(VM\) & \(C_{ij}^{mission}\) & Estimated carbon emissions of placing \(VM_{j}\) \\ \hline \(VM\) & List of running \(VM\)s & \(R_{d}^{Tax}\) & Carbon tax in datacenter d \\ \hline \(VM_{new}\) & List of \(VM\)s to be (re)placed & \(R_{d}^{Carbon}\) & Carbon footprint in datacenter d \\ \hline \(VM_{j}^{CPU}\) & Required CPU for \(VM_{j}\) & \(C_{EC}\) & Energy consumption cost \\ \hline \(VM_{j}^{RAM}\) & Required RAM for \(VM_{j}\) & \(C_{CF}\) & Carbon footprint cost \\ \hline \(VM_{j}^{BW}\) & Required bandwidth for \(VM_{j}\) & \(Th^{under}\) & Underutilization threshold \\ \hline \(VM_{j,current}^{CPU}\) & \(VM_{j}\) current \(CPU\) utilization & \(x_{ij}\) & Matrix element to show VM-to-PM mapping \\ \hline \(Tlx^{over}\) & Overutilization threshold & \\ \hline \(S_{a}\) & \(VM\) assignment solution & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations and symbols. Figure 1: System model for geographically distributed green cloud computing environment. The assignment is represented by a zero-one adjacency matrix \(X\), as described in Equation (1), where the element \(x_{ij}\) indicates whether \(VM_{j}\) is assigned to \(PM_{i}\). If \(VM_{j}\) is placed on server \(PM_{i}\), then \(x_{ij}=1\); otherwise, \(x_{ij}=0\). \[\sum_{i=1}^{M}x_{ij}=1,\forall j\epsilon VM, \tag{1}\]  
It is important to ensure that each server has sufficient resources to meet the demand of all VMs assigned to it. It is assumed that no VM requires more resources than the capacity of a single server, and VMs cannot be assigned across servers. Equations (2)-(4) define the _CPU_, _RAM_, and \(BW\) constraints for server capacity. \[\sum_{j=1}^{N}VM_{j}^{CPU}.x_{ij}\leq PM_{i}^{CPU},\forall iePM \tag{2}\]  
\[\sum_{j=1}^{N}VM_{j}^{RAM}.x_{ij}\leq PM_{i}^{RAM},\forall iePM \tag{3}\]  
\[\sum_{j=1}^{N}VM_{j}^{BW}.x_{ij}\leq PM_{i}^{BW},\forall iePM \tag{4}\]