The Internet of Things (IoT) is transforming industries by enhancing productivity and efficiency; however, energy availability remains a significant challenge due to the limited capacity of batteries and supercapacitors powering IoT devices. The emergence of Green IoT (G-IoT) frameworks, which prioritize energy efficiency and renewable energy integration, offers a promising solution to address this challenge. Despite these advancements, energy storage systems (ESSs) face issues such as capacity degradation, leakage, and charge redistribution, which can lead to energy depletion and service disruptions. Traditional models often assume constant energy harvesting rates, overlooking the time-varying nature of environmental conditions that influence energy availability. In this paper, we propose a novel mathematical framework that incorporates time-dependent fluctuations in the energy harvesting rate to analyze the dynamic interactions between energy harvesting, leakage, and consumption in green IoT systems. Our approach includes an Energy Packet Network (EPN) model to represent transient energy dynamics and a Markov model to capture fluctuations in the energy harvesting rate. Through numerical simulations, we evaluate the impact of key design parameters, such as ESS capacity, mean energy consumption rate, energy leakage, and energy harvesting rate variations, on critical performance metrics including the probability of energy depletion and the transient mean number of stored energy packets. The results highlight the importance of considering time-varying energy harvesting in the design and optimization of IoT systems for long-term operation and sustainability.