  In this paper, we consider uplink transmissions involving multiple users
communicating with a base station over a fading channel. We assume that the
base station does not coordinate the transmissions of the users and hence the
users employ random access communication. The situation is modeled as a
non-cooperative repeated game with incomplete information. Each user attempts
to minimize its long term power consumption subject to a minimum rate
requirement. We propose a two timescale stochastic gradient algorithm (TTSGA)
for tuning the users' transmission probabilities. The algorithm includes a
'waterfilling threshold update mechanism' that ensures that the rate
constraints are satisfied. We prove that under the algorithm, the users'
transmission probabilities converge to a Nash equilibrium. Moreover, we also
prove that the rate constraints are satisfied; this is also demonstrated using
simulation studies.
