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 commu-nications. 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.