Stochastic Throughput Optimization for Two-hop Systems with Finite Relay Buffers

Optimal queueing control of multi-hop networks remains a challenging problem even in the simplest scenarios. In this paper, we consider a two-hop half-duplex relaying system with random channel connectivity. The relay is equipped with a finite buffer. We focus on stochastic link selection and transmission rate control to maximize the average system throughput subject to a half-duplex constraint. We formulate this stochastic optimization problem as an infinite horizon average cost Markov decision process (MDP), which is well-known to be a difficult problem. By using sample-path analysis and exploiting the specific problem structure, we first obtain an equivalent Bellman equation with reduced state and action spaces. By using relative value iteration algorithm, we analyze the properties of the value function of the MDP. Then, we show that the optimal policy has a threshold-based structure by characterizing the supermodularity in the optimal control. Based the threshold-based structure and Markov chain theory, we further simplify the original complex stochastic optimization problem to a static optimization problem over a small discrete feasible set and propose a simple algorithm to solve the static optimization problem. Furthermore, we obtain the closed-form optimal threshold for the symmetric case. The analytical results obtained in this paper also provide design insights for two-hop relaying systems with multiple relays equipped with finite relay buffers.