  We consider a new approach to power control in decentralized wireless
networks, termed fractional power control (FPC). Transmission power is chosen
as the current channel quality raised to an exponent -s, where s is a constant
between 0 and 1. The choices s = 1 and s = 0 correspond to the familiar cases
of channel inversion and constant power transmission, respectively. Choosing s
in (0,1) allows all intermediate policies between these two extremes to be
evaluated, and we see that usually neither extreme is ideal. We derive
closed-form approximations for the outage probability relative to a target SINR
in a decentralized (ad hoc or unlicensed) network as well as for the resulting
transmission capacity, which is the number of users/m^2 that can achieve this
SINR on average. Using these approximations, which are quite accurate over
typical system parameter values, we prove that using an exponent of 1/2
minimizes the outage probability, meaning that the inverse square root of the
channel strength is a sensible transmit power scaling for networks with a
relatively low density of interferers. We also show numerically that this
choice of s is robust to a wide range of variations in the network parameters.
Intuitively, s=1/2 balances between helping disadvantaged users while making
sure they do not flood the network with interference.
