Closed and open loop optimal control of buffer and energy of a wireless device

We study a decision problem faced by an energy limited wireless device that operates in discrete time. There is some external arrival to the device's transmit buffer. The possible decisions are: a) to serve some of the buffer content; b) to reorder a new battery after serving the maximum possible amount that it can; or c) to remain idle so that the battery charge can increase owing to diffusion process (which is possible in some commercially available batteries). We look at open and closed-loop controls of the system. The closed-loop control problem is a using the framework of Markov decision processes. We address both finite and infinite horizon discounted costs as well as average cost minimization problems. Without using any second order characteristics, we obtain results that include i) optimality of bang-bang control, ii) the optimality of threshold based policies, iii) parametric monotonicity of the threshold, and iv) uniqueness of the threshold. For the open-loop control setting we use recent advances in application of multimodular functions to establish optimality of bracket sequence based control.