needed (to sustain a fixed SNR per UE) as the network becomes denser.

Problem Statement and Energy Efficiency Optimization {#sec:problem}
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As described in Section \[sec:intro\], we concentrate on the UL EE defined as the benefit-cost ratio between the area spectral efficiency (ASE) \[bit/symbol/$\textrm{km}^2$\] and the area power consumption (APC) \[J/symbol/$\textrm{km}^2$\]. Using the novel and tight lower bound from Proposition \[prop:average-SE\], in this work the ASE is obtained as $$\mathrm{ASE} = \lambda K \, \underline{\mathrm{SE}}. \label{eq:def-ASE}$$ To specify the APC, we begin by observing that with the adopted power-control policy the average radiated power per UE is $$\frac{S-(\beta K -1)}{S} \mathbb{E} \{ p_{ji} \} 
= \left( 1- \frac{\beta K - 1}{S} \right)  \rho  \omega \frac{ \Gamma({\alpha}/{2} +1 ) }{(\pi \lambda)^{\alpha/2}}$$ where we have used and the fact that each user transmits one pilot symbol and $S-\beta K$ data symbols per block. Then, we observe that the APC must account not only for the radiated power, but also for the dissipation in analog hardware, digital signal processing, backhaul signaling, and other overhead costs (such as cooling and power supply losses). A detailed and generic model that takes all these factors into account was recently proposed in [@Bjornson2015a] and is such that 