# Cell-Free Massive MIMO Power Consumption with Serial Front-Hauls
## 2 System Model and Problem Statement
### Problem Statement  
In canonical cell-free massive MIMO systems, the APs are coordinated to serve all users at the same time. Then, it is necessary for the exchange of the data of all users and the power coefficients between the CPU and all the APs. Consequently, for dedicated front-hauls, except for the power coefficients, the same information is sent through the \(N\) links, where \(N=M\). In order to describe the power consumption model for a dedicated front-haul deployment, we are going to focus on downlink transmission. During downlink transmission, the total power consumption depends on the power consumed in front-haul (\(P_{m}^{\text{FH}}\) for the \(m\)-th link) and the power consumed by each AP (\(P_{m}\) for the \(m\)-th AP), for both operation and  
Figure 1: Overview of different front-haul configurations in cell-free massive MIMO systems. transmission, as explained in the models proposed in [9, 10, 12]. Mathematically, the total power consumption can be expressed with the following set of equations:  
\[P_{\text{Total}}=\sum_{m=1}^{M}\left(P_{m}^{\text{FH}}+P_{m}^{\text{AP}}\right), \tag{1}\]  
\[P_{m}^{\text{FH}}=P_{m}^{\text{FH, fix}}+B_{Xm}^{\text{FH}}S_{\text{dl}}, \tag{2}\]  
\[P_{m}^{\text{AP}}=P_{m}^{\text{AP, fix}}+L_{\text{A}}P_{m}^{\text{AP, chain}}+\frac{\left(\tau_{\text{c}}-\tau_{\text{p}}\right)}{\tau_{\text{c}}} \frac{P_{m}^{\text{tr}}}{\alpha_{m}}. \tag{3}\]  
In (2), \(P_{m}^{\text{FH, fix}}\) is the traffic-independent power that is consumed in each front-haul link and depends on the system topology and the distance between each AP and the CPU [9]. The second term represents the traffic-dependent power, where \(B\) is the system bandwidth, \(\chi_{m}^{\text{FH}}\) is the traffic-dependent power consumption coefficient in each link, and \(S_{\text{dl}}\) is the sum spectral efficiency in the downlink taking into account all the users. For the sake of simplicity, in this work, \(P_{m}^{\text{AP}}\), \(P_{m}^{\text{FH, fix}}\), \(\chi_{m}^{\text{FH}}\), and in consequence, \(P_{m}^{\text{FH}}\), are assumed to be equal for all \(m\). In order to better understand (3), Figure 2 shows the antenna processing unit of each AP consisting of the digital signal processing module, the transceiver, and the interfaces to both the antennas and the front-haul. In (3), \(P_{m}^{\text{AP, fix}}\) is the traffic-independent power that is consumed by, e.g., the site cooling system, the local oscillator, and the baseband processors allocated in the digital processing unit [19], \(L_{\text{A}}\) is the number of Radio Frequency (RF) chains per AP, and \(P_{m}^{\text{AP, chain}}\) is the power consumption per RF chain, all of this contained in the transceiver module. Then, the transmitted power and the power used by the amplifier are also added and represented by \(P_{m}^{\text{R}}/\alpha_{m}\), where \(\alpha_{m}\) is the power amplifier coefficient. This last term is multiplied by a factor pointing out that the channel estimation is performed during the uplink (\(\tau_{\text{c}}\) is the coherence interval length, \(\tau_{\text{p}}\) is the training interval length). This estimation can be reused for the downlink if the Time Division Duplexing (TDD) mode is considered, which is a common assumption for cell-free massive MIMO systems [3]. It is worth mentioning that the \(P_{m}^{tx}\) is much smaller (100-200 mW) than the non-transmitted power (order of Watts), as is explained in [20]. Due to this issue, focusing on decreasing the consumption of non-transmitted power is important to reduce the total power consumed by the system. The model discussed above has been used in the context of cell-free massive MIMO in [9, 10, 12] and is based on the model presented for back-haul in [21, 22]. However, in [22], the authors understand the back-haul as a set of wireless microwave links. For obvious reasons, for serial deployments, this assumption is no longer valid since, for these cases, only a few or, in the most extreme case, only one link that interconnects all the APs and the CPU are considered.