## 3 Cost Model and Problem Description
### Cost Model
#### 3.1.1 Cost of Host Energy Consumption
* **Disk energy consumption model** The power consumption of disk \(p_{i,k}^{disk}\) is directly related to its read and write throughput, we have: \[p_{i,k}^{disk}=\alpha_{i}^{r}\times u_{i,k}^{r}+\alpha_{i}^{w}\times u_{i,k}^{ w}+p_{i,tile,i}^{disk},\] (4) where \(u_{i,k}^{r}\) and \(u_{i,k}^{w}\) are the disk read and write number of bytes of \(h_{i}\) in any time in \(t_{k}\), respectively; \(\alpha_{i}^{r}\) and \(\alpha_{i}^{w}\) are the disk read and write fit index, respectively; and \(p_{i,tile,i}^{disk}\) is the disk power in idle state. Here, we assume that the disk read and write number of bytes at any time in \(t_{k}\) are equal. [PERSON] et al. [22] discovered that the difference between disk read power and write power is negligible. Hence, Equation (4) can be updated as: \[p_{i,k}^{disk}=p_{i,tile,i}^{disk}+\alpha_{i}^{disk}\times u_{i,k}^{disk},\] (5)where \(u_{ijk}^{disk}=u_{i,k}^{r}+u_{i,k}^{w}\), and \(\alpha_{i}^{disk}\) is the fit index. In the actual operation of host, the power consumption of the disk is related to random read and write and sequential read and write. Different read and write modes generate different read and write power consumption. Hence, \(\alpha_{i}^{disk}\) is divided into \(\alpha_{i}^{seq}\) (sequential read and write) and \(\alpha_{i}^{cmd}\) (random read and write). In practice, it is impractical to monitor and detect the read/write mode of the disk. Therefore, for convenience, we can assume that the probability of sequential reads and writes caused by all users is equal to that of random reads and writes. We obtain:  
\[\alpha_{i}^{disk}=(\alpha_{i}^{seq}+\alpha_{i}^{cmd})/2. \tag{6}\]  
[PERSON] et al. [24] found that setting \(\alpha_{i}^{seq}=0.07\)\(\mathrm{W}/\mathrm{MB}/\mathrm{s}\) and \(\alpha_{i}^{cmd}=0.22\)\(\mathrm{W}/\mathrm{MB}/\mathrm{s}\) can more accurately estimate the power consumption of the disk. * **Network power model** We mainly discuss the power of NICs. Under the framework of the TCP/IP protocol, we discuss the power NIC \(p_{i,k}^{NIC}\), which is mainly based on the number of IP packets (the size of an IP packet is 500 bytes, denoted as _ip_size_) that it can handle per second. We have:  
\[p_{i,k}^{NIC}=p_{i,i,k}^{NIC}+\alpha_{i}^{NIC}\times u_{i,k}^{NIC}, \tag{7}\]  
where \(\alpha_{i}^{NIC}\) is the power consumption of each IP packet processed by \(h_{i}\)'s NIC, \(u_{ijk}^{NIC}=s_{ijk}^{int}/ip\_size\) is the number of IP packets processed at any time in \(t_{k}\), and \(p_{i,i,k}^{NIC}\) is the power of NIC in idle state. Here, we assume that the number of IP packets processed by the NIC at any time in \(t_{k}\) is equal. [PERSON] et al. [47] found that the value of \(\alpha_{i}^{NIC}\) should be \(1.26\times 10^{-5}\) joule/packet. Hence, combining with Equations (2), (3), (5) and (7), we obtain:  
\[p_{h_{i}}=p_{i,k}^{cpu}+p_{i,k}^{mem}+p_{i,k}^{disk}+p_{i,k}^{NIC}. \tag{8}\]  
Hence, combing with Equation (1), we obtain the energy consumption cost of all hosts \(C_{H}\):  
\[C_{H}=\sum_{i=1}^{N}C_{h_{i}}=\sum_{i=1}^{N}\sum_{k=1}^{L}[EP\times(p_{i,k}^{ cpu}+p_{i,k}^{mem}+p_{i,k}^{disk}+p_{i,k}^{NIC})\times T\times\lambda_{i,k}]. \tag{9}\]