Discrete Convexity and Stochastic Approximation for Cross-layer On– off Transmission Control

Abstract

This paper considers the discrete convexity of a cross-layer on – off transmission control problem in wireless communications. In this system, a scheduler decides whether or not to transmit in order to optimize the long-term quality of service (QoS) incurred by the queueing effects in the data link layer and the transmission power consumption in the physical (PHY) layer simultaneously. Using a Markov decision process (MDP) formulation, we show that the optimal policy can be determined by solving a minimization problem over a set of queue thresholds if the dynamic programming (DP) is submodular. We prove that this minimization problem is discrete convex. In order to search the minimizer, we consider two discrete stochastic approximation (DSA) algorithms: 1) discrete simultaneous perturbation stochastic approximation (DSPSA) and 2) ${\bm{L}}^{\natural}$ -convex stochastic approximation ( ${\bm{L}}^{\natural}$ -convex SA). Through numerical studies, we show that the two DSA algorithms converge significantly faster than the existing continuous simultaneous perturbation stochastic approximation (CSPSA) algorithm in multiuser systems. Finally, we compare the convergence results and complexity of two DSA and CSPSA algorithms where we show that DSPSA achieves the best tradeoff between complexity and accuracy in multiuser systems.