This paper studies the joint optimization problem of energy and delay in a multi-hop wireless network. The optimization variables are the transmission rates, which are adjustable according to the packet queueing length in the buffer. The optimization goal is to minimize the energy consumption of energy-critical nodes and the packet transmission delay throughout the network. In this paper, we aim at understanding the well-known decentralized algorithms which are threshold based from a different research angle. By using a simplified network model, we show that we can adopt the semi-open Jackson network model and study this optimization problem in closed form. This simplified network model further allows us to establish some significant optimality properties. We prove that the system performance is monotonic with respect to (w.r.t.) the transmission rate. We also prove that the threshold-type policy is optimal, i.e., when the number of packets in the buffer is larger than a threshold, transmit with the maximal rate (power); otherwise, no transmission. With these optimality properties, we develop a heuristic algorithm to iteratively find the optimal threshold. Finally, we conduct some simulation experiments to demonstrate the main idea of this paper. 
A Jackson network model and threshold policy for joint optimization of energy and delay in multi-hop wireless networks