## 3 Performance Analysis
\[P_{macro}=P_{M}^{f}+\rho_{M}\times P_{M}^{tx}, \tag{11}\]  
where \(P_{M}^{tx}\) is defined as as in Equation (12):  
\[P_{M}^{tx}=P_{M}^{tx,max}\times\sum_{i=0}^{n-1}\left(d_{i}\times a_{j,i}\times \frac{(r_{i}\times C_{h}+(1-r_{i})\times C_{l})}{C_{M}^{max}}\right), \tag{12}\]  
Also, the power consumption of state transitions of SBS, \(P_{switching}\), is obtained as in Equation (13):  
\[P_{switching}=\sum_{j=0}^{m-1}\left(f_{j}\times\left(1-s_{j}\right)\times P_{ on-off}+s_{j}\times\left(1-f_{j}\right)\times P_{on-sleep}\right), \tag{13}\]It is assumed that the maximum power consumption of an SBS is \(P_{S}^{max}\), and thus, the power consumption of an SBS \(j\) should be smaller than or equal to \(P_{S}^{max}\):  
\[P(j)_{small}\leq P_{S}^{max},(0\leq j\leq m-1), \tag{14}\]  
Likewise, it is assumed that the maximum power consumption of an MBS is \(P_{M}^{max}\), and thus, the power consumption of an MBS should be smaller than or equal to \(P_{M}^{max}\):  
\[P_{macro}\leq P_{M}^{max}, \tag{15}\]  
If the SBS \(j\) is on state, UE \(i\) can have the connection with SBS \(j\). However, if the SBS \(j\) is not on state, UE \(i\) cannot have the connection with SBS \(j\). This relationship is constrained as follows:  
\[a_{j,i}\leq w_{j}, \tag{16}\]  
The type of required data traffic of UE \(i\) is defined as either 1 or 0 depending on the type of the required data service and this is smaller than or equal to the existence of data traffic demand of UE \(i\):  
\[r_{i}\leq d_{i}, \tag{17}\]  
Since UE \(i\) can have at most one connection with either an MBS or any of SBSs within an MBS, the constraint is expressed as follows:  
\[\sum_{j=0}^{m}a_{j,i}\leq 1, \tag{18}\]  
Since an SBS \(j\) can be in one of on, sleep, and off states, the sum of \(w_{j}\) for on state and \(e_{j}\) for off state should be smaller than or equal to 1:  
\[P_{S}^{tx}(j)=P_{S}^{tx,max}\times\sum_{i=0}^{n-1}\left(d_{i}\times a_{j,i} \times\frac{(r_{i}\times C_{h}+(1-r_{i})\times C_{l})}{C_{S}^{max}}\right) \tag{10}\]0  
Since an SBS \(j\) can have only one state transition at any time, i.e., either between on state and off state or between on state and _sleep_ state, the sum of \(f_{j}\) for the state transition between on state and off state and \(s_{j}\) for the state transition between on state and off state should be smaller than or equal to 1:  
\[P_{S}^{tx}(j)=P_{S}^{tx,max}\times\sum_{i=0}^{n-1}\left(d_{i}\times a_{j,i} \times\frac{(r_{i}\times C_{h}+(1-r_{i})\times C_{l})}{C_{S}^{max}}\right) \tag{10}\]1  
The maximum number of UEs in an MBS is \(n\), the maximum number of SBSs in an MBS is \(m\), the number of MBS is 1, and they are constrained as follows:  
\[P_{S}^{tx}(j)=P_{S}^{tx,max}\times\sum_{i=0}^{n-1}\left(d_{i}\times a_{j,i} \times\frac{(r_{i}\times C_{h}+(1-r_{i})\times C_{l})}{C_{S}^{max}}\right) \tag{10}\]2  
Using the above constraints, the optimization problem of the proposed energy saving scheme can be formulated as follows:  
\[P_{S}^{tx}(j)=P_{S}^{tx,max}\times\sum_{i=0}^{n-1}\left(d_{i}\times a_{j,i} \times\frac{(r_{i}\times C_{h}+(1-r_{i})\times C_{l})}{C_{S}^{max}}\right) \tag{10}\]3  
Subject to: (14)-(21). In this paper, we also analyze the aggregate delay of the proposed energy saving scheme. To this end, the \(T_{aggregate}\), which is defined as the time needed to get service from a BS by a UE, is additionally introduced as a performance measure as in Equation (23), where the \(T_{switching}\) is defined as average time needed to make a transition between states of an SBS and the \(T_{connect}\) is defined as average connection initiation time between a UE and a BS. \[P_{S}^{tx}(j)=P_{S}^{tx,max}\times\sum_{i=0}^{n-1}\left(d_{i}\times a_{j,i} \times\frac{(r_{i}\times C_{h}+(1-r_{i})\times C_{l})}{C_{S}^{max}}\right) \tag{10}\]4The \(T_{switching}\) can be obtained as in Equation (24), where the \(T_{on-off}\) is the time needed to make a transition from \(off\) state to \(on\) state by an SBS and the \(T_{on-slcap}\) is the time needed to make a transition from _sleep_ state to \(on\) state.