Implantable bioelectronic devices with drug delivery capabilities have emerged as suitable candidates for biomedical applications focusing on localized drug delivery. These classes of miniaturized bioelectronics offer wireless operation and refillable designs that can be used for repeated animal behavioral studies without restricting their motion. The pumping mechanisms of these bioelectronic devices features soft materials, microfluidics, and electrochemical subsystems that can be scaled from behavioral studies in small animals to delivery of life-saving medication in humans. Here, we study the refillable aspect of these bioelectronic systems using an analytic model for the drug delivery time established from the ideal gas law when an initial gas volume is present in the device electrolyte reservoirs. The effect of the initial gas volume in delaying the drug delivery time is captured via a non-dimensional parameter identified as the normalized initial gas volume. An analytical solution is derived from the perturbation method, which agrees well with the numerical solution. These results have relevance in the reusability aspect of these bioelectronic systems since modifying the amount of initial gas in the device reservoirs for different experiments affects the total delivery time and can serve as a tunable parameter to ensure timely and successful delivery of the drug in the target region.