Low-power wide-area network (LPWAN) technologies are gaining momentum for internet-of-things (IoT) applications since they promise wide coverage to a massive number of battery-operated devices using grant-free medium access. LoRaWAN, with its physical (PHY) layer design and regulatory efforts, has emerged as the widely adopted LPWAN solution. By using chirp spread spectrum modulation with qausi-orthogonal spreading factors (SFs), LoRa PHY offers coverage to wide-area applications while supporting high-density of devices. However, thus far its scalability performance has been inadequately modeled and the effect of interference resulting from the imperfect orthogonality of the SFs has not been considered. In this paper, we present an analytical model of a single-cell LoRa system that accounts for the impact of interference among transmissions over the same SF (co-SF) as well as different SFs (inter-SF). By modeling the interference field as Poisson point process under duty-cycled ALOHA, we derive the signal-to-interference ratio (SIR) distributions for several interference conditions. Results show that, for a duty cycle as low as 0.33%, the network performance under co-SF interference alone is considerably optimistic as the inclusion of inter-SF interference unveils a further drop in the success probability and the coverage probability of approximately 10% and 15%, respectively for 1500 devices in a LoRa channel. Finally, we illustrate how our analysis can characterize the critical device density with respect to cell size for a given reliability target.Index Terms-Low power wide area networks, LoRaWAN, interference analysis, coverage probability, fading channels A. Mahmood, L. Guntupalli, R. Rondón and M. Gidlund are with the

Duty cycling (DC) is a popular technique for energy conservation in wireless sensor networks that allows nodes to wake up and sleep periodically. Typically, a single packet transmission (SPT) occurs per cycle, leading to possibly long delay. With aggregated packet transmission (APT), nodes transmit a batch of packets in a single cycle. The potential benefits brought by an APT scheme include shorter delay, higher throughput and higher energy efficiency. In the literature, different analytical models have been proposed to evaluate the performance of SPT schemes. However, no analytical models for the APT mode on synchronous DC medium access control mechanisms exist. In this paper, we first develop a three-dimensional (3D) discrete time Markov chain (DTMC) model to evaluate the performance of an APT scheme with packet retransmission enabled. The proposed model captures the dynamics of the state of the queue of nodes and the retransmission status, as well as the evolution of the number of active nodes in the network, i.e., nodes with a non-empty queue. We then study the number of retransmissions needed to transmit a packet successfully. Based on the observations, we develop another less complex DTMC model with infinite retransmissions which embodies only two dimensions. Furthermore, we extend the 3D model into a four-dimensional model by considering error-prone channel conditions. The proposed models are adopted to determine packet delay, throughput, packet loss, energy consumption, and energy efficiency. Furthermore the analytical models are validated through discrete-event based simulations. Numerical results show that an APT scheme achieves substantially better performance than its SPT counterpart in terms of delay, throughput, packet loss and energy efficiency, and that the developed analytical models reveal precisely the behavior of the APT scheme.

Abstract-In this letter, we propose an analytical model to evaluate the performance of the S-MAC protocol. The proposed model improves the accuracy of previous models in two aspects. First, it incorporates the dependence among the nodes within a cluster by defining a DTMC that models the number of active nodes, whereas the previous models considered that nodes were mutually independent. Second, it proposes new methods for calculating packet delay and energy consumption. The analytical model is validated through discrete-event based simulations. Numerical results demonstrate that the proposed analytical model and methods yield accurate results under realistic assumptions.

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