  We consider the downlink of Cell-Free Massive MIMO systems, where a very
large number of distributed access points (APs) simultaneously serve a much
smaller number of users. Each AP uses local channel estimates obtained from
received uplink pilots and applies conjugate beamforming to transmit data to
the users. We derive a closed-form expression for the achievable rate. This
expression enables us to design an optimal max-min power control scheme that
gives equal quality of service to all users.
  We further compare the performance of the Cell-Free Massive MIMO system to
that of a conventional small-cell network and show that the throughput of the
Cell-Free system is much more concentrated around its median compared to that
of the small-cell system. The Cell-Free Massive MIMO system can provide an
almost $20-$fold increase in 95%-likely per-user throughput, compared with the
small-cell system. Furthermore, Cell-Free systems are more robust to shadow
fading correlation than small-cell systems.
