  Ultra-narrowband (UNB) communications is an emerging paradigm that tackles
two challenges to realizing massive Internet-of-things (IoT) connectivity over
the unlicensed spectrum: the intra-network sharing, i.e., how the spectrum is
shared among IoT devices, and the inter-network sharing, i.e., the coexistence
of the IoT network with other incumbent networks. Specifically, intra-network
sharing is enabled by using extremely narrowband signals to connect a massive
number of IoT devices without any prior network synchronization. Further, to
enhance robustness to incumbent interference, each IoT packet is sent multiple
times, each at a different frequency within a single band. Nevertheless, the
interplay between intra-network sharing and inter-technology coexistence at a
large scale remains unclear. Thus, in this paper, we develop an analytical
framework to model and analyze UNB networks. We use stochastic geometry to
derive the probability of successful transmission, identifying the impact of
intra- and inter-network interference on the performance. In addition to
analyzing the existing single-band access protocols, we present two multiband
schemes, where each BS listens to a single band for practical implementation.
Different access protocols are further compared in terms of the transmission
capacity, i.e., the maximum number of IoT devices a UNB protocol can support in
the presence of incumbent networks. Several design insights are gleaned from
the derived closed-form expressions, and simulation results are further
provided to validate them.
