### Proposed Contribution and Related Works
All these papers present interesting solutions to increase energy efficiency in WSNs, however, few of them focus on the use of duty-cycling policies [15]. Anyway, in these works, no solution is adopted to adapt the activation and de-activation times of the nodes according to the synchronization errors and delays due to clock offsets and inaccuracies. We strongly believe that, in networks with very low sampling rates, the adoption of efficient duty-cycling policies and then the prolonged sleeping of the nodes, is the best way to reduce at minimum energy consumption. Hence, we consider a multi-hop thin Linear Sensor Network (LSN) architecture based on the LoRa technology for pervasive monitoring of the underground \"Bottini\" environment, where data delivery with very low sampling rate can be tolerated. In this scenario, the main goal is that of prolonging the network lifetime as much as possible. While some works discussing the application of multi-hop LoRa can be found in the literature [22; 23], to the best of our knowledge no solution has been proposed in the specific context of LSNs. More specifically, we leverage on the peculiarities of the considered scenario, to establish a duty cycling policy with long sleeping periods, alternated by relatively short transmitting periods. Several duty-cycle MAC protocols can be found in the literature, which can be roughly categorized into two categories: synchronous and asynchronous. In synchronous approaches, such as S-MAC [24], T-MAC [25], RMAC [26], and DW-MAC [27], neighbor nodes establish a common time scheduling. This goal is obtained by dividing the time into three periods, namely, _Synch_, _Data_ and _Sleep_. In particular, during the Synch period, nodes exchange the respective scheduling times with the goal of waking up at the same time in the subsequent Data periods, thus, minimizing the energy wasting due to idle listening and overhearing. In this paper we adopt the idea of dividing the time into Synch, Data and Sleep periods. In this setting, we propose a message exchange mechanism tailored to the LoRa based LSN scenario at hand, allowing each node to establish a common scheduling with adjacent nodes. In particular, the establishment of common scheduling is facilitated by the linear topology of the network, in which each node is simply requested to adopt the scheduling of the previous node in the chain. Moreover, we address the problem of possible node desynchronization after the sleeping periods. As a matter of fact, a fundamental assumption supporting synchronous approaches is that a separate protocol is used to synchronize the clocks during the Sync periods with required precision, e.g., by properly correcting the clock offsets, so that after a sleeping period adjacent nodes start the Synch operations at the same time. Several synchronization protocols could be considered to this aim that assume the presence of a deterministic trend characterizing the synchronization offsets among nodes (e.g., see [28; 29]). However, internal clocks of commercial micro-controllers present random clock offsets that are in many cases impossible to predict, and that can be only statistically characterized. In these situations, maintaining time synchronization after long sleeping periods remains a big challenge. To circumvent such difficulties, we propose an asynchronous mechanism for exchanging the scheduling times during Synch periods, where tx-rx synchronization is achieved employing low power listening (LPL) techniques [30]. In this setting, assuming a given statistics of the clock error, we propose an analytical framework based on density evolution that optimally evaluates the wake-up time of all nodes after a Sleep period with the aim of minimizing the average energy dissipation deriving from clock offsets. The rest of the paper is organized as follows. Section 2 presents the deployment scenario, i.e., the \"Bottini\", focusing on the problem of data transmission in this kind of environment.