A Stochastic Primal-Dual Algorithm for Joint Flow Control and MAC Design in Multi-hop Wireless Networks

We study stochastic rate control for joint flow control and MAC design in multi-hop wireless networks with random access. Most existing studies along this avenue are based on deterministic convex optimization and the corresponding distributed algorithms developed therein involve deterministic feedback control. In a multi-hop wireless network, however, the feedback signal is obtained using error-prone measurement mechanisms and therefore noisy in nature. A fundamental open question is that under what conditions these algorithms would converge to the optimal solutions in the presence of noisy feedback signals, and this is the main subject of this paper. Specifically, we first formulate rate control in multi-hop random access networks as a network utility maximization problem where the link constraints are given in terms of the persistence probabilities. Using the Lagrangian dual decomposition method, we devise a distributed primal-dual algorithm for joint flow control and MAC design. Then, we focus on the convergence properties of this algorithm under noisy feedback information. We show that the proposed primal-dual algorithm converges (almost surely) to the optimal solutions only if the estimators of gradients are asymptotically unbiased. We also characterize the corresponding rate of convergence, and our findings reveal that in general the limit process of the interpolated process, corresponding to the normalized iterate sequence generated from the primal-dual algorithm, is a reflected linear diffusion process, not necessarily the Gaussian diffusion process.