In wireless networks, efficient energy storage and utilization plays a vital role, resulting in a prolonged lifetime and enhanced throughput. This factor becomes even more important in systems employing energy harvesting as compared to utility or battery powered networks, where a constant supply of energy is available. Therefore, it is crucial to design schemes that make the best use of available energy resources, keeping in view the practical constraints.

In this work, we consider data transmission with an energy harvesting transmitter which has hybrid energy storage with a perfect super-capacitor (SC) and an inefficient battery. The SC has finite storage space while the battery has unlimited storage space. The transmitter can choose to store the harvested energy in the SC or

in the battery, while draining energy from the SC and the battery simultaneously. Under this energy storage setup, we solve throughput optimal energy allocation problem over a point-to-point channel in an offline setting. The hybrid energy storage model with finite and unlimited storage capacities imposes a generalized set of

constraints on the transmission policy. We show that the solution is found by a sequential application of the directional water-filling algorithm.

Next, we consider offline throughput maximization in the presence of an additive time-linear processing cost in the transmitter's circuitry. In this case, the transmitter has to additionally decide on the portions of the processing cost to be drained from the SC and the battery. Despite this additional complexity, we show that the solution is obtained by a sequential application of a directional glue-pouring

algorithm, parallel to the cost-less processing case. Finally, we provide numerical illustrations for optimal policies and performance comparisons with some heuristic online policies.