This paper presents optimal transmission policies for energy harvesting nodes in fading wireless channels. The goal is to maximize throughput by a deadline and minimize transmission completion time. The system model includes a rechargeable transmitter with a finite battery and a fading wireless channel. Energy is harvested over time and used for transmission, while the channel state fluctuates randomly. The paper considers both offline and online transmission policies. For the offline case, the authors propose a directional water-filling algorithm that optimally allocates transmit power based on energy arrivals and channel conditions. This algorithm ensures that energy is used efficiently and adapts to changes in energy and channel states. The algorithm is shown to be optimal for maximizing throughput. For the online case, the authors use stochastic dynamic programming to find an optimal policy that maximizes the average number of bits delivered by a deadline. They also propose near-optimal policies with reduced complexity. These policies are evaluated numerically under various configurations. The paper also introduces the concept of a maximum departure curve, which maps the transmission completion time minimization problem to the throughput maximization problem. This allows the authors to solve the transmission completion time minimization problem using the solution to the throughput maximization problem. The authors analyze the performance of different policies under various scenarios, including different energy arrival rates, channel fading conditions, and battery capacities. They show that the optimal online policy outperforms suboptimal policies in terms of throughput and transmission completion time. However, the performance of these policies can vary depending on the specific conditions of the system. The results demonstrate the effectiveness of the proposed algorithms in optimizing transmission policies for energy harvesting nodes in fading wireless channels.This paper presents optimal transmission policies for energy harvesting nodes in fading wireless channels. The goal is to maximize throughput by a deadline and minimize transmission completion time. The system model includes a rechargeable transmitter with a finite battery and a fading wireless channel. Energy is harvested over time and used for transmission, while the channel state fluctuates randomly. The paper considers both offline and online transmission policies. For the offline case, the authors propose a directional water-filling algorithm that optimally allocates transmit power based on energy arrivals and channel conditions. This algorithm ensures that energy is used efficiently and adapts to changes in energy and channel states. The algorithm is shown to be optimal for maximizing throughput. For the online case, the authors use stochastic dynamic programming to find an optimal policy that maximizes the average number of bits delivered by a deadline. They also propose near-optimal policies with reduced complexity. These policies are evaluated numerically under various configurations. The paper also introduces the concept of a maximum departure curve, which maps the transmission completion time minimization problem to the throughput maximization problem. This allows the authors to solve the transmission completion time minimization problem using the solution to the throughput maximization problem. The authors analyze the performance of different policies under various scenarios, including different energy arrival rates, channel fading conditions, and battery capacities. They show that the optimal online policy outperforms suboptimal policies in terms of throughput and transmission completion time. However, the performance of these policies can vary depending on the specific conditions of the system. The results demonstrate the effectiveness of the proposed algorithms in optimizing transmission policies for energy harvesting nodes in fading wireless channels.