## 3 Performance Analysis
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description** \\ \hline \(n\) & number of UEs \\ \(m\) & number of SBSs \\ \(u_{i}\) & a UE \(i\) \\ \(B_{j}\) & a BS \(j\) \\ \(C_{h}\) & capacity needed to serve a high rate data traffic service \\ \(C_{l}\) & capacity needed to serve a low rate data traffic service \\ \(C_{S^{max}}^{max}\) & maximum capacity of an SBS \\ \(C_{M}^{max}\) & maximum capacity of an MBS \\ \(\rho_{S}\) & slope of the load-dependent power consumption in an SBS \\ \(\rho_{M}\) & slope of the load-dependent power consumption in an MBS \\ \(P_{S}^{f}\) & fixed power consumption of an SBS in _on_ state \\ \(P_{S}^{g}\) & fixed power consumption of an SBS in _sleep_ state \\ \(P_{M}^{f}\) & fixed power consumption of an MBS in _on_ state \\ \(P_{S}^{max}\) & maximum power consumption of an SBS \\ \(P_{M}^{max}\) & maximum power consumption of an MBS \\ \(P_{on-off}\) & power consumption for a switching between _on_ and _off_ states in an SBS \\ \(P_{on-sleep}\) & power consumption for a switching between _on_ and _sleep_ states in an SBS \\ \(T_{on-off}\) & delay for a switching between _on_ state and _off_ state in an SBS \\ \(T_{on-sleep}\) & delay for a switching between _on_ state and _sleep_ state in an SBS \\ \(T_{c}\) & delay for a new connection initiation between a UE and a BS \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notations. The type of required data traffic of UE \(i\) is denoted as \(r_{i}\) and the value of it is defined as in Equation (4):  
\[r_{i}= \begin{cases}1,&\text{If $u_{i}$ required high rate data service}\\ 0,&\text{If $u_{i}$ required high low data service}\end{cases} ,\ (0\leq i<n), \tag{4}\]  
In Equation (5), \(w_{j}\) is related with whether the BS \(j\) is in _on_ state or not and \(e_{j}\) is related with whether the BS \(j\) is in _sleep_ state or not in Equation (6):  
\[w_{j}= \begin{cases}1,&\text{If $B_{j}$ is in on state}\\ 0,&\text{otherwise}\end{cases} ,\ (0\leq j<m+1), \tag{5}\]  
\[e_{j}= \begin{cases}1,&\text{If $B_{j}$ is in sleep state}\\ 0,&\text{otherwise}\end{cases} ,\ (0\leq j<m), \tag{6}\]  
The notations related with state transitions of an SBS are \(f_{j}\) and \(s_{j}\), and the values of them are defined as in Equations (7) and (8):  
\[f_{j}= \begin{cases}1,&\text{If $B_{j}$ switches between on state and off state}\\ 0,&\text{otherwise}\end{cases} ,(0\leq j<m), \tag{7}\]  
\[s_{j}= \begin{cases}1,&\text{If $B_{j}$ switches between on state and sleep state}\\ 0,&\text{otherwise}\end{cases} ,\ (0\leq j<m), \tag{8}\]  
Using the above equations, the power consumption of an SBS \(j\), \(P(j)_{small}\), is obtained as in Equation (9):  
\[P(j)_{small}=w_{j}\times\left(1-e_{j}\right)\times\left(P_{S}^{f}+\rho_{S} \times P_{S}^{tx}(j)\right)+e_{j}\times\left(1-w_{j}\right)\times P_{S}^{e}, \tag{9}\]  
where if an SBS is in _on_ state, the power consumption is calculated as the sum of fixed power consumption and load-dependent power consumption, and if an SBS is in _sleep_ state, then the power consumption for _sleep_ state is used for the calculation. Regarding load dependent power consumption, \(P_{S}^{tx}(j)\) is defined as in Equation (10):  
\[P_{S}^{tx}(j)=P_{S}^{tx,max}\times\sum_{i=0}^{n-1}\left(d_{i}\times a_{j,i} \times\frac{(r_{i}\times C_{h}+(1-r_{i})\times C_{l})}{C_{S}^{max}}\right) \tag{10}\]  
Likewise, the power consumption of MBS, \(P_{macro}\), is obtained as in Equation (11) as the sum of fixed power consumption and load-dependent power consumption.