Convergence conditions for iterative Transmission Power Control algorithms in wireless networks

Transmission Power Control (TPC) is required to ensure that certain quality requirements are met, to control interference and to improve energy efficiency. In general, an iterative algorithm is required to find a feasible power allocation for all the transmitters. We obtain simple sufficient conditions using existing bounds to test the convergence of an iterative power control algorithm in terms of the number of sub-carriers, channel gains and the target Signal to Interference and Noise Ratio (SINR) of the users. We consider two cases: 1) where the channel conditions are deterministic. 2) where the conditions change randomly based on some probability distribution. For the first case, we derive a condition which results in a Pareto optimal solution. For the random channel condition case, we determine a condition which leads to a solution that converges in distribution. We show through numerical examples that the powers of the nodes converge when sufficient conditions are satisfied, and diverge when they are not satisfied. We also show that our conditions can be applied in power based admission control.