transmit power at UE $i$ in cell $j$ will then be $$\label{p_{ji}_average}  
\mathbb{E}\{p_{ji}\} = \rho  \omega \mathbb{E}\{d_{jji}^{\alpha}\} = \rho  \omega \frac{ \Gamma({\alpha}/{2} +1 ) }{(\pi \lambda)^{\alpha/2}}$$ by using the distance distribution from Lemma \[lemma:distance-distribution\]. A key feature of the statistical power-control policy is that it provides the same average effective channel gain of $\mathbb{E}\left\{{p_{ji} \|{\mathbf{h}}_{jji}\|^2} \right\} = M {\rho}$ to all UEs (irrespective of their locations) without requiring any rapid feedback mechanism to provide UEs with instantaneous CSI. Note that this does not mean that the signal-to-interference-plus-noise ratio (SINR) is the same for all UEs, since the interference varies in the network.

Transceiver Hardware Impairments
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Any practical wireless transceiver is affected by hardware impairments; for example, unavoidable clock drifts in local oscillators, finite-precision digital-to-analog converters, amplifier non-linearities, non-ideal analog filters, etc. Such impairments can be partially mitigated by compensation algorithms [@wenk2010mimo], but not fully removed. Although negligible at low signal-to-noise ratios (SNRs), hardware impairments characterize the maximal achievable SE at high SNRs [@Bjornson2013c]. The vast majority of prior works on the design of energy-efficient networks assumes ideal transceivers, although hardware impairments might have a fundamental impact also on the EE. Motivated by this fact, the distortion noise arising from 