On the optimality of the threshold-based scheduling in green communications: A new proof

Joint channel-aware and buffer-aware scheduling with power adaptation is an important and promising way to achieve delay minimum and energy efficiency for Green communications. Based on this idea, we exploited some various scenarios, and found that they can be modelled as linear programming problems with similar forms. Moreover, their optimal scheduling policies all have a threshold-based pattern. In this paper, we give a new and vigorous proof for the linear programming problems, using the system model from our former work, where probabilistic scheduling are proposed for random data arrival and time-varying channels. The proof is based on Lagrange duality theory, which can reveal the essence of the threshold-based scheduling.