## 4 Workload and Data Center Modeling
### Analytical Modeling Approach
#### 4.1.2 Data Center Modeling  
For data center modeling, the final output is the provisioned resources \(PR\) that aims to match the workload demand \(W\). As mentioned in Section 3, there are four machine types in the machine list \(M\) that are provisioned to serve the demanded resources vector \(R\). Machine types are classified based on their available resource size, where the resource unit is normalized to the maximum machine resource size. The four machine types are now classified into five classes as a ratio of maximum machine capability \(MC\). Class \(A\) machine type has a maximum resource value of \(A=1.0\times MC\), followed by class \(B=0.75\times MC\), class \(C=0.50\times MC\), class \(D=0.25\times MC\) and class \(E=0\times MC\). Class \(E\) represents a machine type that has no available resources. The average requested resource vector is also normalized to \(MC\), where \(R_{ij}=0.08\times MC\). This value is considered for the optimization of the provisioned resources by selecting the optimal machine sub-list from list \(M\). The data center model depends on machine states that, in turn, depend on the transition probability between classes. Google data center is provisioned gradually, as depicted in Figure 8, where demands fluctuate in shape, but the provisioned data center resources increase smoothly and gradually. This means the machine transition between classes follows a contiguous \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \(T_{ret}\) & total number of resubmitted tasks during time window \(T_{w}\) \\ \(RR_{t}\) & re-requested resources vector at time \(t\) during time window \(T_{w}\) \\ \(RD_{t}\) & total demanded resources at time \(t\) during time window \(T_{w}\) \\ \(CT\) & current discrete time (window counts) \\ \(W_{t}\) & workload demands at specific time \(t\) \\ \(P(J_{st}(j))\) & probability of submitted job \(j\) during time window \(T_{w}\) \\ \(P(T_{st}(i))\) & probability of submitted task \(i\) during time window \(T_{w}\) \\ \(\mu\) & the mean of Gaussian random variable \\ \(\sigma^{2}\) & the variance of Gaussian random variable \\ \(\overline{x}\) & statistical sampled mean \\ \(s^{2}\) & statistical sampled variance \\ \(\lambda\) & Gamma distribution scale factor \\ \(r\) & Gamma distribution shape factor \\ \(P(T_{et})\) & probability of ended tasks during time window \(T_{w}\) \\ \(X\) & logs array for one \(T_{w}\) \\ \(W_{tp}\) & predicted workload for next time window \\ \hline \hline \multicolumn{3}{c}{Data center Modeling Symbols} \\ \hline \(M\) & data center machine list \\ \(MC\) & machine capability \\ \(A\) & resources class \(A=1.00\times MC\) \\ \(B\) & resources class \(B=0.75\times MC\) \\ \(C\) & resources class \(C=0.50\times MC\) \\ \(D\) & resources class \(D=0.25\times MC\) \\ \(E\) & resources class \(E=0.00\times MC\) \\ \(NM\) & new machine state \\ \(FM\) & failed machine state \\ \(UM\) & upgrade machine state \\ \(NM_{t}\) & number of new machines in new machine state during time window \(T_{w}\) \\ \(P(NM_{t})\) & probability of adding \(NM_{t}\) machines at time \(t\) \\ \(T_{X}\) & time of machine staying in the same state, \(X\) means any states in a Bayesian network \\ \(A_{t}\) & number of new machines in any state \(X\), in Bayesian network \\ \(\lambda_{X}\) & average arriving rate in Poisson and exponential random distributions of any machine \\  & state \(X\) in a Bayesian network, there will be \(\lambda_{X}\) and \(X\) means any states in Bayesian network \\ \(S\) & Bayesian network matrix \\ \(MList\) & data center machine list capabilities \\ \(MS\) & data center machines states statistics \\ \(Trans\) & active scaling transition matrix \\ \(PTrans\) & predicted active scaling transition matrix \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\)sequence, which can be modeled using a Bayesian network.