## 3 System Model and Problem Formulation  
We consider a single-cell architecture in H-CRAN consisting of a BBU pool connected to a single MBS and set of RRHs, as shown in Figure 1. The MBS provides the voice services and transmits control signaling to UEs; therefore, we consider a power consumption model without the power consumption of the MBS. We denote the set of RRHs and UEs in a cell as \(S\) and \(U\), respectively, and define \(a_{s}\) as an indicator of the binary activation of an RRH \(s\), with values of 0 for a deactivated RRH and 1 for an activated RRH.  
The power consumption model includes the powers consumed by RRHs, fronthaul links, and backhaul links. The power consumption of an RRH can be expressed as:  
\[P_{RRH,s}=a_{s}\left(\Phi_{RRH}+\Delta s\sum_{u=1}^{U}b_{u}^{s}p_{t}^{s}\right) +(1-a_{s})\Phi_{Sleep},\quad s\in\{1,\cdots,S\},\;u\in\{1,\cdots,U\}, \tag{1}\]  
where \(\Phi_{RRH}\) is the circuit power of RRH s, which is activated, \(b_{u}^{s}\) is the binary association indicator of UE \(u\) in RRH s, with values of 1 for an associated UE and 0 for a nonassociated UE, \(p_{t}^{s}\) is the non-negative transmission power of RRH \(s\), \(\Phi_{Sleep}\) is the power consumption of the deactivated RRH \(s\), and \(\Delta s\) is the slope of the load-dependent power model of the RRH. The power consumption of a fronthaul link reported in [8] can be expressed as:  
\[P_{fh,s}=a_{s}\Big{(}\Phi_{fh}+\beta t_{s}\Big{)}, \tag{2}\]  
where \(\Phi_{fh}\) is the constant power consumption from the fronthaul transceiver and switch, \(\beta\) is the power consumption per bit/s at the fronthaul link, and \(t_{s}\) is the total traffic on the fronthaul link, which is associated with RRH s. The power consumption model of the backhaul link \(P_{bh}\) is also reported in [16], so that we set the power consumption of the backhaul link as a constant value. We express the total power consumption model of a single cell in the H-CRAN architecture as:  
\[P=\sum_{s=1}^{S}\Big{(}P_{RRH,s}+P_{fh,s}\Big{)}+P_{bh}. \tag{3}\]  
Figure 1: Architecture of H-CRAN.  
The SINR to UE \(u\) from RRH \(s\) can be expressed as:  
\[SINR_{s}^{u}=\frac{p_{t}^{s}g_{tu}^{s}}{\sum\limits_{i\
eq s}p_{t}^{i}g_{iu}^{i}+N _{0}}, \tag{4}\]  
where \(g_{tu}^{s}\) is the channel gain between RRH \(s\) and UE \(u\), and \(N_{0}\) is the noise power spectral density. The achievable data rate from RRH \(s\) can be expressed as:  
\[R_{s}=B\sum\limits_{u=1}^{U}\log_{2}(1+SINR_{u}^{s}), \tag{5}\]  
where \(B\) is the total channel bandwidth of the system. Therefore, the sum of achievable data rates from the RRHs is:  
\[R=\sum\limits_{s=1}^{S}a_{s}R_{s}. \tag{6}\]  
The total energy efficiency (EE) of the system can be defined as a ratio of the sum of achievable data rates to the total power consumption:  
\[EE=\frac{R}{P}. \tag{7}\]  
Our main goal is to maximize the total energy efficiency subject to a certain QoS constraint of UEs. The considered QoS constraint is outage probability, which is a ratio of the number of UEs that are not provided with a certain SINR threshold \(SINR_{th}\) to the total number of UEs. We denote the outage probability as \(P_{outage}\). The energy-efficiency maximization problem can be formulated as:  
\[\begin{split}\max\limits_{s,p_{t}^{s}}&\quad EE(p_{t }^{s})\\ \text{s.t.C1}:&\quad\sum\limits_{u=1}^{U}a_{s}b_{u}^{ s}=1\quad\quad\forall u,\\ \text{C2}:&\quad a_{s}\sum\limits_{u=1}^{U}b_{u}^{ s}p_{t}^{s}\leq p_{t}^{max}\quad\quad\forall s,\\ \text{C3}:&\quad P_{outage}\leq P_{outage}^{max} \end{split} \tag{8}\]  
where \(\forall u\in U,s\in S\), \(a=[a_{1},\ \cdots,a_{S}]\), \(b=[b_{1}^{1},\cdots,\ b_{U}^{S}]\), and \(p=[p_{t}^{1},\cdots,p_{t}^{S}]\). C1 implies that each user accesses only one RRH at a time, C2 ensures that the transmission power is not larger than the upper bound (the maximum transmission power is denoted as \(p_{t}^{max}\)), and C3 implies that the outage probability cannot exceed the maximum outage probability \(P_{outage}^{max}\)[17].