# Analytical Model for the Duty Cycle in Solar-Based EH-WSN for Environmental Monitoring
## 1 Introduction
As stated in [1], the main research focus is again on the MAC layer, but according to a fundamental difference in the design principle of EH-WSN compared to their battery-powered counterpart: whereas the latter were developed with the objective of maximizing network lifetime, the energy-harvesting paradigm allows for paying more attention to other design objectives, namely performance metrics such as throughput and delay, in line with the high expectations from IoT. Note that QoS was already taken into account in the last generation of MAC protocols for battery-powered sensor networks (see, for instance, [6; 7]), but subject to rigid energy budget constraints. Contrarily, in EH-WSN the emphasis is put on network performance while ensuring a sustainable energy state (more formally, this is known as energy neutral operation, which means that, in a period of time, the energy drained from the environment is equal to or larger than the energy consumed by the node). Thus, it is of crucial importance to consider the energy-harvesting mechanism in the design of MAC protocols for EH-WSN; otherwise, it is not possible to provide performance guarantees under self-sustained operation [1]. Currently, such redesign process has already produced various MAC protocols for EH-WSN. These protocols again rely on duty-cycling the communication activity, but allowing for much larger duty cycles which, in turn, can be individually adapted to the energy harvesting and consumption rates of each node. The work presented in [1] proposes a taxonomy of duty-cycled MAC protocols for EH-WSN based on the element that triggers the communication process (sender, receiver or sink). This taxonomy and the most relevant protocols in each category are also examined in the next section. As it will be noticed, a common feature of current MAC protocols for EH-WSN is that they are based on heuristic approaches. Since the main design objective is network performance under self-sustained operation, these approaches focus on dynamically adjusting the duty cycle of nodes according to their energy level. So, nodes with more available energy increase their duty cycle in order to enhance throughput and/or delay, whereas those experiencing energy scarcity switch to sleep mode (to recharge batteries), thus diminishing their traffic activity and causing some performance degradation. However, the mathematical relationship between node duty cycle and other parameters such as energy harvesting rate and traffic load remains unknown. This is more or less successfully overcome by current protocols by imposing a tight control of the energy available at nodes via frequent measurements. Alternatively, in this paper, we follow an analytical approach, which allows for obtaining closed-form expressions for the duty cycle and initial energy storage that guarantee self-sustained operation at each node. This does not only contributes to a better understanding the dependence of node duty cycle on energy-related and traffic-related parameters, but we expect that it will result in helping the development of simpler and more effective adaptive schemes for adjusting the duty cycle of nodes. To fix ideas, we consider a widely deployed duty-cycled MAC protocol, which is the LPL (Low Power Listening) mechanism implemented in TinyOS sensor nodes, though the essential results can be extrapolated to other software (and corresponding hardware) platforms. We assume a typical environmental monitoring application, where nodes periodically sense and transmit towards a base station via multi-hop communication (time-driven paradigm). Based on these assumptions, we first derive an accurate model for the energy consumed by sensor nodes during each period of the communication process (communication round). Then, by using an experimentally tested theoretical model for the energy harvested by solar cells, we formulate the condition for energy neutral operation, from which we obtain the above mentioned closed-form expressions in terms of known and/or measurable parameters.